A Survey of Index Theory
and a Calculation of the
Truncated Equivariant
Witten Genus
Author:
Emanuele Dotto
Advisors:
Andrew Stacey, NTNU
Kathryn Hess Bellwald, EPFL
Work:
Master Project
Year:
Winter 2008/2009
1
Abstract. We explain the conjectures relating the Witten
Genus of a string manifold M defined as a formal power series
and the index of a Dirac operator 6∂ on the loop manifold LM .
Under some hypotheses, we generalize the Witten genus to
keep track of a given string action of a compact Lie group G
on M . When the action of G is not string, we use the FreedHopkins-Teleman theorem to see the index of 6∂ as an element
of the twisted equivariant K-theory τ +σ KG (G). Motivated by
a finite approximation of this result we define a twisted Ktheory for twistings in the second cohomology, and we explain
its role in index theory.
Contents
Introduction
1. Review of Mathematical Tools
1.1. G-Vector Bundles
1.2. Equivariant K-Theory
1.3. The Thom Isomorphism in Equivariant K-Theory
1.4. Application of the Thom Isomorphism
1.5. Characteristic Classes
1.6. The Thom Isomorphism in Cohomology and the Todd Class
2. Equivariant Index Theory
2.1. Elliptic Operators
2.2. The G-Index Theorem
2.3. The Atiyah-Singer Fixed Point Theorem
3. Spin Bundles and Dirac Operators
3.1. Clifford Algebras and Spin Representations in Finite Dimension
3.2. Spin Bundles
3.3. The Dirac Operator
3.4. The Index of the Dirac Operator
4. The Witten Genus
4.1. The Manifold of Smooth Loops
4.2. Clifford Algebras and Spinor Representations in Hilbert Spaces
4.3. Spin Structures and Dirac Operators on LM
4.4. The Witten Genus
5. The Witten Conjecture and the Truncated Equivariant Witten
Genus
5.1. The Truncated Witten Genus
5.2. The Truncated Equivariant Witten Genus
6. Index Theory and Twisted K-Theory
6.1. Cech Cohomology
6.2. K-Theory Twisted by Elements of the Second Cohomology
6.3. Projective Representations and Central Extensions
6.4. The Link with Index Theory
References
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2
Introduction
In his paper [18], Edward Witten defines the Witten genus of a finite
dimensional manifold M as a formal power series involving characteristic
classes of bundles over M . The definition of this formal power series comes
from a result of index theory, called the Atiyah-Singer fixed point theorem,
applied to the Dirac operator of the product manifold M n that is seen as
a finite dimensional approximation of the smooth loop manifold LM . Our
aim is to explain what motivates this definition, and to discuss the related
conjectures and problems.
The second section is mostly based on the lecture notes [15], and we explain the basis of index theory. This is the study of elliptic cochain complexes
of equivariant linear operators between sections of G-vector bundles. That
is, finite families of equivariant linear maps D = {Di : Γ(Ei ) −→ Γ(Ei+1 )},
where the Ei are complex G-vector bundles over a compact orientable manifold M , such that Di+1 Di = 0, and satisfying a certain additional condition.
This condition guarantees that the cohomologies of the complex are finite
dimensional representations of G, and then it is possible to define the index
of D as the element
X
IndexG D =
(−1)i ker Di / Im Di−1
of the representation ring RG . Using equivariant K-theory and the fact that
KG (∗) = RG it is possible to recover IndexG D "topologically". The AtiyahSinger fixed point theorem uses this topological treatment to compute the
traces of the operators g ∈ G on IndexG D using characteristic classes of
bundles over M .
In the third section we illustrate the most important example of elliptic
operator for this work: the Dirac operator 6∂ . We see which structure is
needed on the tangent bundle of M to define this operator. That is, we
discuss spin structures, Clifford algebras, spin groups and spinor representations. The principal reference used for this subject is [9]. Then we apply
the Atiyah-Singer theorem to obtain formulas for the index of 6∂ .
In the fourth and the fifth part we consider the Witten genus. On the
one hand, we illustrate Witten’s formal application of the Atiyah-Singer
fixed point theorem to a hypothetical Dirac operator which led to a first
construction of the Witten genus. The central idea of Witten’s definition is
to "truncate" the loop manifold LM to the compact manifolds M n , apply
the Atiyah-Singer theorem to its Dirac operator and take a limit when n goes
to the infinity to obtain a formal power series. This is based on the paper
[17]. Although much work has gone into making this limit rigorous, this
argument is purely formal and there is much yet to do to find a geometrical
interpretation of this. On the other hand, if the manifold M carries enough
structure (for instance if it is a string manifold) we illustrate how to define a
Dirac operator on LM , following the paper [16]. Under further assumptions
we define a formal power series that can be seen as some kind of equivariant
index of this Dirac operator. The main conjecture of this topic states that
this equivariant index is equal to the Witten genus. Here we explain in detail
this conjecture and the analytical problems preventing its proof. Then, still
using the "truncation technique" of [17], we generalize the Witten genus to
3
a formal power series that keeps track of a given action of a compact Lie
group G on M , supposing that this action "lifts to each structure" needed
to define the Dirac operator on LM .
This equivariant case leads to a further generalisation that is the subject
of section six. If the action of G on M does not lift to the string structure
of M we can define the index of the Dirac operator as a projective representation of the loop group LG of some level τ . In their paper [6], Freed,
Hopkins and Teleman define an object τ +σ KG (G), called twisted equivariant K-theory, and they show that τ +σ KG (G) is isomorphic to positive energy
representations of LG of level τ . Therefore if the index of the Dirac operator is of positive energy, it can be seen as an element of τ +σ KG (G). This
suggests that the twisted equivariant K-theory could play an important role
in the development of an index theory for loop manifolds. Motivated by a
finite dimensional version of this result we define a twisted K-theory for a
twisting in the second cohomology of a space, and we investigate its role in
index theory for finite dimensional manifolds.
1. Review of Mathematical Tools
1.1. G-Vector Bundles. In this part we recall the basic notions of vector
bundles. For most of this work we will use only finite dimensional vector
bundles. However, in section 4 we will need to deal with infinite dimensional manifolds and infinite dimensional vector bundles. For this reason the
definitions that we give here are completely general and do not require any
assumption on the finitude of the dimensions. Apart from this section and
section 4 all the bundles will be assumed to be of finite dimension.
Definition 1.1.1. A complex (resp. real) vector bundle over a connected
space B is
• a topological space E,
• a continuous map p : E −→ B,
• a complex (resp. real) vector space structure on the fibers Ex =
p−1 (x) for all x ∈ B compatible with the topology induced by E,
• a topological vector space V ,
satisfying the following conditions:
(1) there is an open covering U of B and for all U ∈ U a homeomorphism
φ : p−1 (U ) −→ U × V making the diagram
p
φ
/ U ×V
s
s
s
ss
s
ss pr1
sy ss
p−1 (U )
U
commute,
(2) the restrictions of the φ : p−1 (U ) −→ U × V to the fibers Ex are
linear maps φx : Ex −→ {x} × V ∼
= V for all x ∈ U and U ∈ U.
Such a set U together with a homeomorphism φ : p−1 (U ) −→ U × V is called
a local trivialisation of the vector bundle. The spaces E and B are called
respectively the total space and the base space, and V is the fibre of the
vector bundle.
4
A vector bundle over a general space B is a vector bundle on each connected component of B (in this case "the fibre" is not defined in general).
We will often use just the notation p : E −→ B or even E for all this data.
If the fibres are finite dimensional we say that E is a finite dimensional vector
bundle. If all the fibres have same finite dimension n the bundle is said to
be of dimension n or is called an n-vector bundle.
Convention 1.1.2. Most of the bundles that we will consider will be complex. Then by a vector bundle we generally mean a complex vector bundle.
However, if the context is clear we could refer to a real vector bundle just as
a vector bundle.
If the base space B is a manifold, it is natural to ask for an additional
property.
Definition 1.1.3. A smooth complex (real) vector bundle over a manifold B is a complex (real) vector bundle over B such that given two local trivialisations (U, φ) and (U 0 , φ0 ) with U ∩ U 0 6= ∅ the composition
φ ◦ φ−1 : (U ∩ U 0 ) × V −→ (U ∩ U 0 ) × V is smooth.
Note 1.1.4. If B is a manifold, we can suppose without loss of generality
that the covering U is an atlas of B. Suppose that B is connected so that all
the fibres are isomorphic. The family {p−1 (U )|U ∈ U} covers E and a local
trivialisation U of E induces a chart of E with domain p−1 (U ). Then E is
automatically a manifold and p : E −→ B is a smooth map. We say that
these charts are induced by the atlas U. If E is of finite real dimension n as a
vector bundle and B is a finite dimensional manifold then E is of dimension
n + dim(B) as a manifold.
Suppose now that a topological group G acts on B continuously.
Definition 1.1.5. A G-vector bundle over B is a vector bundle p : E −→
B together with a continuous action of G on E satisfying:
(1) the action on E covers the action on B, that is the map p : E −→ B
is G-equivariant,
(2) The fibrewise restriction of the action of an element g ∈ G is linear.
Note 1.1.6. A G-vector bundle over a point is just a vector space equipped
with a linear action of G. A finite dimensional G-vector bundle over a point
is thus a finite dimensional representation of G.
Definition 1.1.7. A morphism of G-vector bundles pE : E −→ B and
pF : F −→ B over B is an equivariant continuous map α : E −→ F such that
pF (α(e)) = pE (e) for all e ∈ E and that is fibrewise linear. A morphism of
G-vector bundles that is also a homeomorphism is called an isomorphism
of G-vector bundles. We denote by VectG (X) the set of isomorphism classes
of finite dimensional G-vector bundles over B.
Definition 1.1.8. Let p : E −→ B be a G-vector bundle, X a space on which
G acts continuously and f : X −→ B an equivariant map. The pull-back
of p : E −→ B by f is the map p : X ×f E −→ X, where
X ×f E = {(x, e) ∈ X × E|f (x) = p(e)}.
5
The fibre of a point x ∈ X by p is homeomorphic to Ef (x) . Then the vector
spaces structures on the fibres of E induce vector space structures on the
fibres of X ×f E. The preimages by f of the trivialisations of E define
trivialisations on X ×f E. Hence X ×f E is a vector bundle over X, which
is denoted by f ∗ E. The action of G on f ∗ E defined by g(x, e) = (gx, ge)
defines a structure of G-vector bundle on f ∗ E.
Let pE : E −→ B and pF : F −→ X be G-vector bundles. Then the map
pE × pF : E × F −→ B × X clearly defines a G-vector bundle over B × X.
Definition 1.1.9. Let E and F be two G-vector bundles over B and d : B −→
B × B the diagonal map. The Whitney sum of E and F is the G-vector
bundle E ⊕ F over B defined as
E ⊕ F = d∗ (E × F ).
1.2. Equivariant K-Theory. For this part we suppose that B is a locally
compact space and G a topological group. This section is purely topological,
and there is no hypothesis on the smoothness of the objects. Any G-vector
bundle will be of finite dimension.
Definition 1.2.1. Let pE : E −→ B and pF : F −→ B be two finite dimensional G-vector bundles over B. We define a the total space of a G-vector
bundle over B as the disjoint union of fibrewise tensor products
E ⊗ F = ∪x∈B Ex ⊗ Fx ,
and a projection mapping e ⊗ f ∈ Ex ⊗ Fx to x. The action of G on E ⊗ F
defined fibrewise as g(e⊗f ) = (ge)⊗(gf ) defines a G-vector bundle structure
on E ⊗ F . This bundle is called the tensor product of E and F .
Using the properties of the direct sum and the tensor product on vector
spaces it is easy to verify that VectG (B) together with the operations ⊕ and
⊗ is a semi-ring.
Definition 1.2.2. Suppose that B is compact. The G-equivariant KTheory of B is the ring completion of (VectG (B), ⊕, ⊗), and is denoted by
KG (B). That is, it is the quotient of VectG (B)×VectG (B) by the equivalence
relation (E, E 0 ) ∼ (F, F 0 ) if and only if there is a D ∈ VectG (B) such that
E ⊕ F 0 ⊕ D = F ⊕ E 0 ⊕ D. The equivalence class of (E, E 0 ) ∈ VectG (B) is
denoted E − E 0 . The operations
(E − E 0 ) + (F − F 0 ) = E ⊕ F − E 0 ⊕ F 0
and
(E − E 0 ) · (F − F 0 ) = (E ⊗ F ) ⊕ (E 0 ⊗ F 0 ) − (E ⊗ F 0 ) ⊕ (E 0 ⊗ F )
makes KG (B) into a ring. Any G-equivariant map f : X −→ B between
compact G-spaces induces a homomorphism f ∗ : KG (B) −→ KG (X) defined
by
f ∗ (E − F ) = f ∗ (E) − f ∗ (F ),
where f ∗ (E) denotes the pullback of a bundle.
Note 1.2.3. A finite dimensional G-vector bundle over a point is a representation of G. Then the G-equivariant K-Theory of a point is the representation ring of G. We will use the notation KG (∗) = RG .
6
Now we extend the definition of KG (B) to a space B that is locally compact, but not compact. For this definition we will not consider all the bundles
over B, but only a certain subclass.
Definition 1.2.4. Let X be a compact G-space with base point x0 ∈ X that
is fixed by the action, that is the inclusion i : {x0 } −→ X is G-equivariant.
e G (X)x as the kernal
We define the reduced G-equivariant K-theory K
0
∗
of i : KG (X) −→ RG .
The map i∗ associates to E − F ∈ KG (X) the difference of the fibres Ex0
and Fx0 in RG (the fibres over x0 are representations since x0 is fixed by
e G (X)x are then formal differences of vector
the action). The elements of K
0
bundles whose fibres over x0 are isomorphic representations of G. Note that
e G (X)x is the kernal of a homomorphism it is a ring without unit.
since K
0
We use this reduced equivariant K-theory to define the equivariant Ktheory of a locally compact, non-compact G-space B. Let B + be the onepoint compactification of B. The action of G on B extends canonically to
an action on B + that fixes (by definition) the point +.
Definition 1.2.5. If B is a locally compact but not compact G-space, we
define the G-equivariant K-Theory of B as
e G (B + )+ .
KG (B) = K
If f : X −→ B is a proper equivariant map, its extension f : X + −→ B +
define a homomorphism
f ∗ : KG (B) −→ KG (X)
by pullback.
Definition 1.2.6. The K-theory K(B) of a locally compact space B is the
equivariant K-theory of B with respect to the trivial group
K(B) = K{1} (B).
Note that, since the neighbourhoods of + are the complements of compact subspaces of B, a bundle over B + is a bundle over B which is trivial
outside some compact subspace of B. Then the equivariant K-theory of a
non compact (but locally compact) space B is formed by formal differences
of bundles trivial outside a compact and whose fibres over + are isomorphic
as representations of G. We will refer to KG (B) always as a ring, even if in
the non-compact case we mean "ring without unit".
The equivariant K-theory of a compact space B can be expressed as the
reduced equivariant K-theory of the disjoint union B ∪ {∗}, for some point
e G (B ∪ {∗})∗ is given
∗ outside B. The canonical isomorphism KG (B) −→ K
by extending a bundle over B to a bundle over B ∪ {∗} whose fibre over ∗ is
the zero representation. Then if we set B + = B ∪ {∗} for a compact space,
e G (B + )+ still holds.
the identity KG (B) = K
Note 1.2.7.
(1) There is an equivalent definition of the reduced equivariant K-theory of a compact space B. It is the set of equivalence
classes of G-vector bundles under the relation E ∼ E 0 if there are
representations V and V 0 of G such that E ⊗ V is isomorphic to
7
E 0 ⊗ V 0 . This is an abelian group under direct sum. If B has a
base point x0 fixed by the action then the function mapping a stable
class of bundles [E] to the class of E − Ex0 is an isomorphism of
abelian groups. The non-trivial step to show is the surjectivity. This
follows by a result stating that given a G-vector bundle E there is a
G-vector bundle E ⊥ such that E ⊕ E ⊥ is a trivial G-vector bundle
(cf. [14, Prop.2.4]).
(2) If B is not compact, a G-vector bundle does not define an element of
KG (B) in general. However, any G-vector bundle E over B defines a
multiplication E· : KG (B) −→ KG (B) (see 1.2.15). Then if f : X −→
Y is any map, we can see KG (Y ) as a module over KG (X).
There is another equivalent definition of equivariant K-theory that will
be very useful later on. Instead of considering formal differences G-vector
bundles over B we consider cochain complexes of G-vector bundles over B.
Definition 1.2.8. A cochain complex of G-vector bundles over B is
a sequence of G-vector bundles {Ei }i∈Z over B and morphisms
...
di−1
/ Ei
di
/ Ei+1
di+1
/ ...
such that di+1 di = 0 for all i and that Ei = X × {0} when i is large enough
in modulus. The morphisms di are called the differentials of the complex.
We denote a complex by (E∗ , d∗ ) or just as E∗ . A morphism of cochain
complexes (E∗ , d∗ ) and (F∗ , d∗ ) is a family of morphisms of G-vector bundles
fi : Ei −→ Fi commuting with the differentials, that is di fi = fi+1 di for all
i.
Definition 1.2.9. The direct sum of two cochain complexes E∗ and F∗
is the complex obtained by taking the levelwise Whitney sum of Ei and Fi
with differentials the obvious maps induced by the differentials of E∗ and
F∗ .
Definition 1.2.10. The tensor product of (E∗ , d∗ ) and (F∗ , d∗ ) is the
complex levelwise defined as
(E∗ ⊗ F∗ )i =
i
M
Ej ⊗ Fi−j ,
j=0
with differential defined at an element e ⊗ f ∈ Ej ⊗ Fi−j by
d(e ⊗ f ) = d(e) ⊗ f + (−1)j e ⊗ d(f ).
Definition 1.2.11. The support of a cochain complex E∗ over B is the set
of x ∈ B for which the sequence of vector spaces
...
di−1
/ (Ei )x
di
/ (Ei+1 )x
di+1
/ ...
is not exact. It is denoted Supp(E∗ ).
Definition 1.2.12. We define LG (B) as the isomorphism classes of cochain
complex of G-vector bundles over B which have compact support.
8
Since Supp(E∗ ⊗ F∗ ) = Supp(E∗ ) ∩ Supp(F∗ ) and Supp(E∗ ⊕ F∗ ) =
Supp(E∗ ) ∪ Supp(F∗ ), the set LG (B) is closed under direct sum and tensor product and it defines a semi-ring.
Definition 1.2.13. Two complexes E∗ and F∗ of LG (B) are homotopic
if there is a complex W∗ ∈ LG (B × [0, 1]) such that W∗ |B×{0} = E∗ and
W∗ |B×{1} = F∗ . We say that two complexes E∗ , F∗ ∈ LG (B) are equivalent
if there are complexes E∗0 and F∗0 on B with empty support such that E∗ ⊕E∗0
is homotopic to F∗ ⊕ F∗0 . The set of equivalent classes of this equivalence
relation is denoted LG (B)/∼ .
The proof of the following result can be found in [14, Prop.3.10]. An
explicit description of the isomorphism is in [1, §3].
Proposition 1.2.14. There is an isomorphism of rings
LG (B)/∼ ∼
= KG (B).
In the case where BPis compact this homomorphism maps a complex E∗ on
the alternating sum i (−1)i Ei . The tensor products of complexes induces
under this identification a multiplication
KG (B) ⊗ KG (B) −→ KG (B)
which gives the product of KG (B).
Note 1.2.15.
(1) Suppose B is non-compact. This theorem allows us
to define a product E· : KG (B) −→ KG (B) even if E does not define
an element of KG (B). An element a ∈ KG (B) defines a complex
A∗ over B with compact support. We can form the tensor product
E ⊗ A∗ which has compact support, and then map it back to an
element of KG (B), that we denote by Ea.
Let now f : B −→ X be any equivariant map, with B non-compact.
This map does not define a pullback KG (X) −→ KG (B) unless f is
proper. Even so, we can use the product above to see KG (B) as
a module over KG (X). Given a bundle E over X (or X + if X is
not compact) the pullback f ∗ E is a bundle over B and the product
f ∗ E· : KG (B) −→ KG (B) defines the module structure.
(2) It is shown in [1, §3] that we can associate to any complex E∗ over B
a bundle d(E∗ ) over B, and that for any map f : X −→ B naturality
is satisfied:
f ∗ (d(E∗ )) = d(f ∗ (E∗ )),
where f ∗ (E∗ ) is the pullback of the complex. The restriction of
this map to compactly supported complexes induces the isomorphism
above.
1.3. The Thom Isomorphism in Equivariant K-Theory. The description of the equivariant K-theory described in the previous section allows us
to define an isomorphism of abelian groups KG (B) ∼
= KG (E) for a G-vector
bundle E over a compact space B. It is called the Thom isomorphism and
it will be described in this section.
9
Definition 1.3.1. Let E be a G-vector bundle over a compact space B
and s : B −→ E a G-equivariant section. The Koszul complex of E with
respect to s is the cochain complex of G-vector bundles over B
0
d
/C
/ Λ1 E
d
/ Λ2 E
d
/ ... ,
where d : Λi E −→ Λi+1 E is defined at ξ ∈ Λi Ex by
d(ξ) = ξ ∧ s(x).
The support of the Koszul complex is the set of x ∈ X such that s(x) = 0.
For a G-vector bundle p : E −→ B the diagonal map δ : E −→ E ×p E is a
canonical section of the pull-back p∗ E. The Koszul complex associated to
p∗ E and δ : E −→ E ×p E is denoted by Λ•E . Note that the support of Λ•E
is the image of the zero-section of E, which is B embedded in E. Since B is
compact Λ•E has compact support.
Let F∗ ∈ LG (B) be a complex with compact support over B. Then
the pull-back by p : E −→ B at each level defines a complex p∗ F∗ over E
with support Supp(p∗ F∗ ) = p−1 (Supp(F∗ )). The complex Λ•E ⊗ p∗ F∗ over
E has compact support and so defines an element of LG (E). The map
φ : LG (B) −→ LG (E) defined by
φ(F∗ ) = Λ•E ⊗ p∗ F∗
induces a homomorphism of abelian groups
φ : KG (B) ∼
= LG (B)/∼ −→ LG (E)/∼ ∼
= KG (E).
Since B is compact, Λ•E is an element of LG (E), and we denote by
λE the element of KG (E) defined by Λ•E . By 1.2.15, the homomorphism
φ : KG (B) −→ KG (E) is given by
φ∗ (a) = λE · p∗ a,
where this product is the product defining the KG (B)-module structure on
KG (E).
Definition 1.3.2. This homomorphism φ∗ : KG (B) −→ KG (E) is called the
Thom homomorphism of p : E −→ B. The element λE ∈ KG (E) is called
the Thom class.
The following important result is in [14, Prop.3.2].
Proposition 1.3.3. The Thom homomorphism φ∗ : KG (B) −→ KG (E) is
an isomorphism of KG (B)-modules.
1.4. Application of the Thom Isomorphism. We use the Thom isomorphism to define a homomorphism
KG (T M ) −→ KG (∗) = RG
for any compact manifold M with a smooth action of a compact Lie group
G. This homomorphism will play a central role in index theory.
We start to built a homomorphism i! : KG (T M ) −→ KG (T Y ) from a given
G-equivariant embedding i : M −→ Y on a manifold Y . The embedding
i : M −→ Y induces an equivariant embedding T i : T M −→ T Y on the
tangent spaces (where the action on the tangent space is the action by the
differentials of the elements of G). Let N T M be the normal bundle of T M
10
with respect to this embedding and a G-invariant Riemannian metric on
T M . Then the action on T T Y restricts to an action on N T M . Choose
an equivariant embedding j : N T M −→ T Y making the following diagram
commutative
0
j
/ TY
:
vv
v
v
v
vv
vv T i
N TO M
T M.
The existence of this embedding in the non-equivariant case is shown in
[11, Th.11.1]. We can then see N T M as an open neighbourhood of T M in
T Y . Since N T M is not compact, the elements of KG (N T M ) are classes of
bundles which are trivial outside some compact subspace of N T M . Then
we can extend trivially these bundles to bundles over T Y . This defines
an "extension" homomorphism h : KG (N T M ) −→ KG (T Y ). Next, we
can give the real bundle T T Y over T Y a complex structure identifying
it with the underlying real bundle of π ∗ (T Y ) ⊗ C, where π : T Y −→ Y
is the canonical projection. This restricts to a complex structure on the
subbundle N T M of T T Y |T M . With respect to this complex structure we
have a Thom isomorphism φ∗ : KG (T M ) −→ KG (N T M ), that we compose
with the extension homomorphism h : KG (N T M ) −→ KG (T Y ) to define
i! : KG (T M ) −→ KG (T Y ) as the composition
KG (T M )
φ∗
/ KG (N T M )
h
/ KG (T Y ) .
7
i!
It can be showed that this construction is independent of the choice of the
embedding j : N T M −→ T Y .
For the compact manifold M choose a G-equivariant embedding i : M −→
Rn for some integer n large enough. The previous construction gives a homomorphism i! : KG (T M ) −→ KG (T Rn ). Furthermore, identifying T Rn = R2n
with Rn ⊗ C as G-spaces, we can see that total space T Rn is the total space
of a complex G-vector bundle T Rn −→ p over a point p. This gives a Thom
isomorphism φ : KG (p) ∼
= RG −→ KG (T Rn ).
Definition 1.4.1. The equivariant topological index of a compact Gmanifold M is the ring homomorphism BG : KG (T M ) −→ RG defined as the
composition
KG (T M )
i!
/ KG (T Rn )
φ−1
/
8 RG .
BG
1.5. Characteristic Classes. A characteristic class κ can be described as
a procedure associating to each (finite dimensional) vector bundle (real or
complex) E over B an element κ(E) ∈ H ∗ (B; R) for some ring (or eventually
field) R. We can require that κ "behaves well" with respect to some of
the operations defined on vector bundles, and that it only depends on the
isomorphism classes of the bundles. One of the more important properties
that is always required is naturality, that is for all maps f : B −→ B 0 we
11
have f ∗ (κ(E)) = κ(f ∗ E) for all vector bundles E over B 0 . The following
proposition shows that if κ is natural and constant on the isomorphism
classes it is completely determined on its value on sums of line bundles.
This is often called the "splitting principle", and a proof can be found in
[15, App.2].
Proposition 1.5.1. For all vector bundle E over B there is a space Y and
a map f : Y −→ B such that
(1) the pullback f ∗ E is isomorphic to a sum of line bundles,
(2) the induced homomorphism in cohomology f ∗ : H ∗ (B, Z) −→ H ∗ (Y, Z)
is injective.
By naturality of κ we obtain that
n
M
f (κ(E)) = κ(f E) = κ(
Li )
∗
∗
i=1
is determined by the value of κ on the sum of line bundles Li . Since f ∗ is
injective κ(p) is also determined.
In this section we will discribe the construction of some characteristic
classes. Most of these classes are defined from the Chern classes.
Proposition 1.5.2. For each topological space B there is a unique family of
maps ck : Vect(B) −→ H 2k (B, Z) for k ∈ N satisfying:
(1) for all k ∈ N and bundles E over B we have c0 (E) = 1 and ck (p) = 0
for all k > n if E is of dimension n,
(2) for all map f : X −→ B we have ck (f ∗ E) = f ∗ (ck (E)) for all vector
bundle E over B,
(3) if E and E 0 are bundles over the same space we have
ck (E ⊕ E 0 ) =
k
X
ci (E)ck−i (E 0 )
i=0
for all k ∈ N,
(4) for the canonical line bundle γn1 over CP n we have
c1 (γ1n ) = a,
where a is the generator of H ∗ (CP n , Z).
The existence of these classes are proved in [11, §14] and in [7, Th.3.2]
with two different approaches. The uniqueness is proved as well.
Definition 1.5.3. The collection of maps ck of the last proposition are called
the Chern classes. The k-th function ck is called the k-th Chern class.
Note 1.5.4. There is a family of characteristic classes for real vector bundles with value in H ∗ (B; Z/2) that satisfies analogous properties. They are
called the Stiefel-Whitney classes (cf. [11, §4]). In index theory all the
bundles considered are complex, and we will not use the Stiefel-Whitney
classes. However, studying the Dirac operator it will be useful to know a few
properties of these classes.
12
Definition 1.5.5. The total Chern class of a space B is the formal sequence
X
Y
c=
ck : Vect(B) −→
H i (B, Z).
i∈N
k∈N
Note that for each vector bundle E over B the value c(E) is a well defined
element of H ∗ (B, Z) since ck (E) is zero for k greater than
Q the dimension of
E. Since c0 (E) = 1 the elements c(E) are invertible in i∈N H i (B, Z). The
total Chern class is therefore a well defined map
Y
c : Vect(B) −→
H i (B, Z)× ,
i∈N
Q
H i (B, Z)×
where i∈N
is the group of invertible elements in
The third property of the Chern classes says that
Q
i∈N H
i (B, Z).
c(E ⊕ E 0 ) = c(E)c(E)0
for all bundles E and E 0 over the same space B. Then the total Chern class
is a (semi-group) homomorphism
Y
c : (Vect(B), ⊕) −→ ( H i (B, Z)× , ∪).
i∈N
If B is a finite CW-complex the groups H k (B; Z) are trivial for k bigger than the dimension of B. Therefore there is a canonical isomorphism
Q
i
∼ ∗
i∈N H (B, Z) = H (B, Z). In this case the total Chern class defines a
homomorphism
c : (Vect(B), ⊕) −→ H ∗ (B, Z)× .
There is a general construction that allows us to define new characteristic
classes from the Chern classes. Let Q[[x]]b be the multiplicative group of
formal power series in x with coefficients in Q and constant term 1. For an
element f ∈ Q[[x]]b consider the product f (x1 ) . . . f (xn ) for any n ∈ N. This
is symmetric in the xj ’s and therefore can be expressed as
f (x1 ) . . . f (xn ) = 1 + F1 (σ1 ) + F2 (σ1 , σ2 ) + . . . ,
for some family {Fi }∞
i=1 of polynomials Fi ∈ Q[t1 , . . . , ti ], where we denote
σk = σk (x1 , . . . , xn ) the k-th elementary symmetric function
X
σk =
xi1 . . . xik .
i1 <···<ik
{Fi }∞
i=1
The family
is unique, and it is independent of theQ
number of variables
xi . Using this family of polynomials we define a map F : i∈N H i (B, Z)b −→
Q
i
i∈N H (B, Z)b by
F (1 + a1 + a2 + . . . ) = 1 + F1 (a1 ) + F2 (a1 , a2 ) + . . .
Q
for any space B, where i∈N H i (B, Z)b are the sequences with constant term
1. This map is a group homomorphism (cf. [9, III.11.9]). We can recover
the formal series f ∈ Q[[x]]b from F since f (x) = F (1 + x). Note that
since the 0-th Chern class is constant 1 the total Chern class has image
13
Q
in i∈N H i (B, Z)b. Composing F with the total Chern class we obtain a
(semi-group) homomorphism
Y
F ◦ c : Vect(B) −→
H i (B, Z)b.
i∈N
Note that if a bundle E = ⊕ni=1 Li is a sum of line bundles then
F (c(E)) =
n
Y
F (1 + c1 (Li )) =
i=1
n
Y
f (c1 (Li ))
i=1
Definition 1.5.6. The Todd Class is the homomorphism obtained with
this procedure starting with the power series
x
f (x) =
∈ Q[[x]]b.
1 − e−x
It is denoted by td. At a sum of line bundles E = ⊕ni=1 Li it is given by
td(E) =
n
Y
c1 (Li )
.
−c1 (Li )
1
−
e
i=1
b
Definition 1.5.7. The total A-class
is obtained starting with the sequence
√
x/2
√
f (x) =
∈ Q[[x]]b
sinh( x/2)
b At a sum of line bundles E = ⊕n Li it is given by
and it is denoted A.
i=1
p
n
Y
c1 (Li )/2
b
p
A(E)
=
.
sinh( c1 (Li )/2)
i=1
Q Now, i we would like to define a semi-ring homomorphism Vect(B) −→
i∈N H (B, Q). Let σk = σk (x1 , . . . , xn ) be the k-th elementary symmetric
function. There is a unique polynomial sk (t1 , . . . , tk ) satisfying
n
X
sk (σ1 , . . . , σk ) =
xki ,
i=1
cf. [11, §16].
Definition 1.5.8. The Chern character is the function
Y
ch : Vect(B) −→
H i (B, Q)
i∈N
defined at a n-dimensional vector bundle E over B as
∞
X
sk (c1 (E), . . . , ck (E))
.
ch(E) = n +
k!
k=1
Note that if B is a finite CW-complex this sum is a well defined element
of H ∗ (B, Q). Suppose now that E = ⊕ni=1 Li is a sum of line bundles. Then
we have that
n
n
Y
Y
c(E) =
c(Li ) =
(1 + c1 (Li )).
i=1
i=1
Therefore ck is the k-th elementary symmetric function in the c1 (Li )
ck (E) = σk (c1 (L1 ), . . . , c1 (Ln )),
14
and by definition the Chern character is
P
sk (c1 (E),...,ck (E))
ch(E) = n + ∞
k!
Pk=1
sk (σ1 ,...,σk )
=n+ ∞
Pk=1 Pni=1k!c1 (Li )k
=n+ ∞
k=1
P
P k!c1 (Li )k
= ni=1 1 + ∞
k=1
k!
P
= ni=1 ec1 (Li ) .
0
Then if E ∼
= ⊕ni=1 Li and F ∼
= ⊕m
i=1 Li are sums of line bundles over the
same space
n
m
X
X
0
c1 (Li )
ch(E ⊕ F ) =
e
+
ec1 (Li ) = ch(E) + ch(F ).
i=1
i=1
Furthermore we can show that for line bundles L and L0 the equality c1 (L ⊗
L0 ) = c1 (L) + c1 (L0 ) holds (cf. [7, Prop.3.10]). Then
ch(E ⊗ F ) = ch(⊕ni=1 ⊕ij=1 (Lj ⊗ L0i−j ))
P P
0
= ni=1 ij=1 ec1 (Lj ⊗Li−j )
P P
0
= ni=1 ij=1 ec1 (Lj )+c1 (Li−j )
P P
0
= ni=1 ij=1 ec1 (Lj ) ec1 (Li−j )
= ch(E) ch(F ).
By the splitting principle and the naturality of the Chern classes this identities hold for any vector bundle. Therefore the Chern character defines a
semi-ring homomorphism
Y
ch : Vect(B) −→
H i (B, Q).
i∈N
From now on suppose that B is a finite CW-complex, so that the Chern
character defines a semi-ring homomorphism
ch : Vect(B) −→ H ∗ (B, Q).
Since ch preserves the sum and the multiplication, it induces a ring homomorphism in K-theory. If B is compact, the Chern character induces a ring
homomorphism
ch : K(B) −→ H ∗ (B; Z).
If B is only locally compact but not compact, recall that the elements of
K(B) are differences of bundles over B + . Therefore ch induces a ring homomorphism
ch : K(B) −→ H ∗ (B + ; Z).
But by naturality, the image by ch of an element of ker(i∗ : K(B) −→ K(+))
e ∗ (B + , Z). Then if B is
is included in ker(i∗ : H ∗ (B + ; Z) −→ H ∗ (+; Z)) = H
not compact the Chern character defines a homomorphism
e ∗ (B + , Z).
ch : K(B) −→ H
We want to define an "equivariant version" of the Chern character, that is
a ring homomorphism defined on KG (B). This is possible if G acts trivially
on B. Note that there is a homomorphism t : K(B) −→ KG (B) which gives
the trivial action to a vector bundle and a homomorphism pr∗ : RG −→
KG (B) induced by the projection to a point. For a trivial G-space B and
15
two G-vector bundles E and F over B we denote homG (Ex , Fx ) the set of
G-equivariant linear maps from Ex to Fx , for all x ∈ B. The union
[
homG (E, F ) =
homG (Ex , Fx )
x∈B
is itself a bundle over B (with trivial G-action). The following result is
proved in [14, Prop.2.2].
Proposition 1.5.9. If G acts trivially on B the homomorphism
t ⊗ pr∗ : K(B) ⊗ RG −→ KG (B)
is an isomorphism. Its inverse is induced by
X
µ(E) =
homG (B × Vi , E) ⊗ Vi ,
i
where the Vi run over the isomorphism classes of irreducible representations
of G.
We denote the inverse of t ⊗ pr∗ by µ : KG (B) −→ K(B) ⊗ RG . Applying the Chern character on the first factor we obtain a homomorphism
chG : KG (B) −→ H ∗ (B; Z) ⊗ RG . However, we want to associate to an element of KG (B) a cohomology class of B (or B + if B is not compact). We
can do this for each element g ∈ G taking the trace with respect to g.
Definition 1.5.10. Let B be a compact trivial G-space and g ∈ G. The
g-equivariant Chern character is the ring homomorphism defined as the
composition
ch ⊗ Trg
µ
chg : KG (B) −→ K(B) ⊗ RG −→ H ∗ (B; Z) ⊗ C = H ∗ (B; C).
If B is not compact, this same map defines a homomorphism
e ∗ (B + ; C).
chg : KG (B) −→ H
Note 1.5.11. The map µ gives a decomposition of each isomorphism class
E of G-vector bundles into a sum of G-vector bundles whose fibres are irreducible representations of G. Furthermore the g-equivariant Chern character
is given by
X
chg (E) =
ch(homG (B × Vi , E)) Trg (Vi ).
i
1.6. The Thom Isomorphism in Cohomology and the Todd Class.
Here we define the Thom isomorphism in cohomology and we compare it
with the Thom isomorphism in K-theory, using the Chern character. The
relation between these isomorphisms will involve the Todd class of a certain
bundle. This will be very useful in index theory, to obtain formulas for the
index of an operator.
Definition 1.6.1. An orientation of a real n-vector bundle p : E −→ B
is a choice of orientation of each fibre Eb such that at each point b0 ∈ B
there is a local trivialisation φ : p−1 (U ) −→ U × Rn for which the restriction
φ|Eb : Eb −→ Rn is orientation preserving for all b ∈ U . If an orientation
exists, we say that p : E −→ B is orientable. The bundle p : E −→ B is
oriented if we fix an orientation.
16
Note 1.6.2. The underlying real bundle of a complex vector bundle carries
a canonical orientation. By the orientation of a complex vector bundle we
will always mean the orientation on the underlying real bundle induced by
this complex structure.
For any real vector bundle E we denote by E0 = {v ∈ E|v 6= 0} the complement of the image of the zero section of the bundle. A choice of orientation
for a real n-vector bundle E is equivalent to a choice of a generator uF ∈
H n (F, F0 ; Z) for each fibre F of E, such that every point of the base space
admits a neighbourhood U and a cohomology class u ∈ H n (E|U , (E|U )0 ; Z)
whose restriction to (F, F0 ) gives uF for each fibre F (cf.[11, §9]). The following result is proved in [11, Th.9.1]
Proposition 1.6.3. Let E be a real oriented n-vector bundle. Then the
groups H i (E, E0 ; Z) are trivial for i < n and H n (E, E0 ; Z) contains a unique
cohomology class u such that
u|(F,F0 ) ∈ H n (F, F0 ; Z)
is the generator of F induced by the orientation for all fibre F of E. Furthermore the homomorphism H k (E; Z) −→ H k+n (E, E0 ; Z) mapping y to y ∪ u
is an isomorphism for all k ∈ N.
This result allows us to define a new characteristic class.
Definition 1.6.4. The Euler class of a real oriented n-vector bundle E is
the cohomology class e(E) ∈ H n (B; Z) defined as
e(B) = (π ∗ )−1 u|E .
Note that since the fibres of E are contractible the map π : E −→ B is a
homotopy equivalence, whose homotopical inverse is the zero section. Then
π ∗ : H ∗ (B; Z) −→ H ∗ (E; Z) is an isomorphism. The last result gives an
isomorphism H k (B; Z) −→ H k+n (E; E0 , Z).
Definition 1.6.5. The isomorphism ψ : H k (B; Z) −→ H k+n (E, E0 ; Z) given
by
ψ(x) = π ∗ (x) ∪ u
is called the Thom isomorphism.
Choose a Riemannian metric on E and denote by B(E) = {e ∈ E|kek ≤ 1}
the disk bundle of E and by S(E) = {e ∈ E|kek = 1} the sphere bundle
of E. Here the coefficients of the cohomology are integers. Note that by
excision H ∗ (E, E0 ) is isomorphic to H ∗ (B(E), B(E)\{0}). Since S(E) is a
deformation retract of B(E)\{0} we have another isomorphism
H ∗ (B(E), B(E)\{0}) ∼
= H ∗ (B(E), S(E)).
Now, (B(E), S(E)) is a good pair and so
e ∗ (B(E)/S(E)).
H ∗ (B(E), S(E)) ∼
=H
It’s not hard to see that B(E)/S(E) is homeomorphic to the one point
compactification E + . Putting all this together, we obtain an isomorphism
e ∗ (E + ).
j ∗ : H ∗ (E, E0 ) −→ H
17
Definition 1.6.6. The cohomology with compact supports of E is the
cohomology ring
e ∗ (E + ; Q).
Hc∗ (E; Q) = H
Note 1.6.7. The cohomology with compact supports of a k-dimensional
manifold X is usually defined as the cohomology h∗ (X) of the rational singular cochains with compact support on X. However, there is an isomorphism
DX : hi (X) −→ Hk−i (X; Q)
for all 0 ≤ i ≤ k called the Poincaré duality. We take rational coefficients. If
p : E −→ X is a n-dimensional real vector bundle consider the induced map
−1
DX
p∗ DE : hi+n (E) −→ hi (X).
−1
This is an isomorphism whose inverse is DE
i∗ DX , where i : X −→ E denotes
∗
the zero section. If X is compact h (X) = H ∗ (X). Furthermore there are
isomorphisms
hn+i (E) ∼
= Hk−i (E) ∼
= Hk−i (B(E)) ∼
= H n+i (B(E), S(E)) ∼
= H n+i (E, E0 ).
−1
i∗ DX corresponds to the Thom isomorphism
Under this identifications DE
(with rational coefficients). Furthermore h∗ (E) ∼
= Hc∗ (E).
Composing the Thom isomorphism with j ∗ we obtain an isomorphism
H k (B; Q) ∼
= Hck+n (E; Q). This isomorphism allows us to compare the
Thom isomorphism in K-theory and the Thom isomorphism in cohomology.
Given a complex vector bundle E of (complex) dimension n over a space
B consider the diagram
j∗ψ :
φ
K(B)
ch
H ∗ (B; Q)
ψ
/ H ∗ (E, E0 ; Q)
/ K(E)
j∗
ch
/ H ∗ (E; Q), ,
c
where φ and ψ are Thom isomorphisms in K-theory and in cohomology
respectively. Note that since E is non compact the Chern character is a map
with target Hc∗ (E; C). This diagram is in general not commutative. However,
the obstruction of the commutativity is measured by a certain characteristic
class. Recall that the isomorphism φ is given by
φ(a) = λE π ∗ a,
where λE is the element of KG (E) defined by the Koszul complex. Note
that even if π ∗ a is not an element of K(E), its equivariant Chern character
is a well defined element of H ∗ (E). We want to use the naturality of ch on
the product λE π ∗ a. But ch(λE ) is an element of Hc∗ (E) and ch(π ∗ a) is in
H ∗ (E). We can show that the following holds:
ch(λE π ∗ a) = j ∗ (ch(π ∗ a)(j ∗ )−1 (ch(λE ))).
Then we obtain
ψ −1 (j ∗ )−1 (ch(φ(a))) = ψ −1 (j ∗ )−1 (ch(λE π ∗ a))
= ψ −1 (ch(π ∗ a)(j ∗ )−1 (ch λE ))
= ψ −1 ((π ∗ ch(a)(j ∗ )−1 (ch λE )))
= ch(a)ψ −1 (j ∗ )−1 (ch λE ).
18
The last equality holds because
ψ(ch(a)ψ −1 (j ∗ )−1 (ch λE )) = π ∗ (ch(a)ψ −1 (j ∗ )−1 (ch λE ))u
= π ∗ ch(a)π ∗ (ψ −1 (j ∗ )−1 (ch λE ))u
= π ∗ ch(a)ψ(ψ −1 (j ∗ )−1 (ch λE )))
= π ∗ ch(a)(j ∗ )−1 (ch λE ).
Then the defect of commutativity is ψ −1 (j ∗ )−1 (ch λE ). It is shown in
[9, III.12.11] that this is given by
ψ −1 (j ∗ )−1 (ch λE ) = (−1)n td(E)−1 ,
where E is the conjugated bundle of E.
2. Equivariant Index Theory
In this section we explain the basis of index theory and in particular
the Atiyah-Singer fixed point theorem. This is the study of representations
associated to a certain class of linear equivariant operators on the vector
space of sections of G-vector bundles. The main idea is to use equivariant
K-theory to compute this representations.
2.1. Elliptic Operators. Let M be a compact manifold and G a compact
Lie group acting smoothly on M . Let {Ei }i∈Z be a family of complex Gvector bundles over M such that Ei = 0 for i large enough in modulus. We
denote by Γ(Ei ) the vector space of equivariant smooth sections of Ei . There
are canonical linear actions of G on the vector spaces Γ(Ei ) defined by
(g · s)(x) = gs(g −1 x)
for all g ∈ G and sections s of Ei .
Definition 2.1.1. A cochain complex of operators on {Ei }i∈Z is a family
of equivariant linear maps {Di : Γ(Ei ) −→ Γ(Ei+1 )}i∈Z such that
Di+1 ◦ Di = 0
for all i ∈ Z. We denote this family by (Γ(E∗ ), D∗ ).
We want to define a certain class of cochain complexes of operators so that
the quotients ker(Di )/ Im(Di−1 ) are finite dimensional representations of G.
Then we will use equivariant K-theory to compute these representations.
Definition 2.1.2. A linear operator D : Γ(E) −→ Γ(F ) is called a partial
differential operator if its local representations are partial differential operators. That is, for all sections s ∈ Γ(E), for all x0 ∈ M and charts
(UM , φM ) at x0 trivializing E and F we have
X
φF ◦ D(s) ◦ φ−1
=
aα ∂ α (φE ◦ s ◦ φ−1
M
M ),
0≤|α|≤k
for some k ∈ N, some multi-index α and some smooth complex matrix-valued
map aα defined on a neighbourhood of φM (x0 ). Here (UE , φE ) and (UF , φF )
denote charts on E and F induced by the local trivialisation. The expression
P
α
0≤|α|≤k aα ∂ is called a local representation of D at x0 .
A cochain complex of partial differential operators is a cochain
complex of operators (Γ(E∗ ), D∗ ) for which each Di is a partial differential
operator.
19
We now use local representations of a cochain complex of partial differential operators (Γ(E∗ ), D∗ ) to define a sequence
bundles and
P of G-vector
α
morphisms over the tangent bundle T M . Let 0≤|α|≤k aα ∂ be a local representation of Di associated to a chart (UM , φM ), and let n be the dimension
of M . For all vectors vx ∈ T UM over x and multi-indices α = (i1 , . . . , ik )
with ij ≤ n we define
v α = vi1 . . . vik ,
where the vij are the components of v with respect to the basis of Tx M induced by UM . We then define a matrix-valued map σUM : T UM −→ Md×h (C)
by
X
aα (x)v α .
σUM (vx ) =
|α|=k
Here d and h are the (complex) dimensions of Ei+1 and Ei respectively. Let
π : T M −→ M be the projection of the tangent bundle.
Definition 2.1.3. The symbol of the operator Di : Γ(Ei ) −→ Γ(Ei+1 ) is
the map σ
e(Di ) : π ∗ Ei −→ π ∗ Ei+1 defined on the fibre over vx ∈ Tx M by
σ
e(Di )(vx , e) = (v, LσU
M
(vx ) e),
where LσU (vx ) : (Ei )x −→ (Ei+1 )x is the linear map associated to the matrix
M
σUM (vx ) with respect to the real basis of (Ei )x and (Ei+1 )x induced by a
chart (UM , φM ) of M at x.
The symbol of a cochain complex (Γ(E∗ ), D∗ ) of partial differential operators is the sequence of maps of G-vector bundles
σ
e(D∗ ) = . . .
σ
e(Di−1 )
/ π ∗ Ei
σ
e(Di )
σ
e(Di+1 )
/ π ∗ Ei+1
/ ...
over T M .
It is easy to check that this definition is independent of the choice of
the chart (UM , φM ) of M trivializing Ei and Ei+1 . Futheremore the maps
σ
e(Di ) : π ∗ Ei −→ π ∗ Ei+1 are G-equivariant since Di is G-equivariant. Then
the symbol σ
e(D∗ ) is a sequence of morphisms of G-vector bundles.
Now we have all the notions that we need to define the cochain complexes
of operators that we want to study.
Definition 2.1.4. A cochain complex (Γ(E∗ ), D∗ ) of partial differential operators is elliptic if its symbol σ
e(D∗ ) defines an exact sequence when restricted to the fibres of non-zero vectors v ∈ T M , that is
σ
e(Di−1 )
...
/ (π ∗ Ei )v
σ
e(Di )
/ (π ∗ Ei+1 )v
σ
e(Di+1 )
/ ...
is exact for all v ∈ T M \{0}.
Note 2.1.5.
(1) If (Γ(E∗ ), D∗ ) is elliptic, its symbol σ
e(D∗ ) is a cochain
complex of G-vector bundles over T M (cf. definition 1.2.8). This is
because on the fibres of non-zero vectors σ
e(D∗ )i+1 ◦ σ
e(D∗ )i is zero
by exactness, and on the fibres over zero vectors all the σ
e(D∗ )i are
zero since σUM (0) = 0. Therefore the support of σ
e(D∗ ) is the set of
zero-vectors of T M .
20
(2) If (Γ(E∗ ), D∗ ) is composed just by a single partial differential operator D : Γ(E) −→ Γ(F ), it is elliptic if and only if
σ
e(D) : π ∗ E −→ π ∗ F
is an isomorphism on the fibres over non-zero vectors.
Example 2.1.6. Consider the (complexified) de Rham complex
d = ( ...
dk−1
dk
/ Γ(Λk (T ∗ M ⊗ C))
/ Γ(Λk+1 (T ∗ M ⊗ C))
dk+1
/ ... )
of a compact oriented manifold M .
If a compact Lie group G acts smoothly on M this action lifts to T M ∗ .
An element g ∈ G acts on T ∗ M by (dg −1 )∗ . This action on T ∗ M induces
an action on each Λk (T ∗ M ⊗ C) and thus on the sections Γ(Λk (T ∗ M ⊗ C)).
An element g ∈ G acts on a differential form ω ∈ Γ(Λk (T ∗ M ⊗ C)) by
the pull-back of the diffeomorphism g −1 : M −→ M . By naturality of the
exterior derivative with respect to the pull-back the de Rham complex is
G-equivariant.
Locally, dk is given by
X ∂ai ...i
1
k
dk (ai1 ...ik dxi1 ∧ · · · ∧ dxik ) =
dxj ∧ dxi1 ∧ · · · ∧ dxik ,
∂xj
j
that is
dk =
X
j
[dxj ∧ (−)]
∂
,
∂xj
where [dxj ∧ (−)] is the matrix of the left exterior product by dxj . Then dk
is a partial differential operator of order one, and for a vector vx ∈ Tx U we
have
X
X
σU (vx ) =
[vj dxj ∧ (−)] = [(
vj dxj ) ∧ (−)].
j
j
Now, choose a G-invariant Riemannian metric h−, −i on M so that we can
identify T M with T M ∗ as G-spaces.
P Then for a chart that induces an
orthonormal basis at x one has that j vj dxj is equal to hv, −ix . Then the
symbol of each dk is the morphism
σ
e(dk ) : π ∗ Λk (T ∗ M ⊗ C)) −→ π ∗ Λk+1 (T ∗ M ⊗ C)
given by
σ
e(dk )(v, e) = (v, hv, −i ∧ e).
By the algebraic properties of the exterior product it is easy to show that
σ
e(d) is elliptic.
2.2. The G-Index Theorem. Let (Γ(E∗ ), D∗ ) be an elliptic complex of
partial differential operators. It is a well-known fact about the theory of
elliptic operators that each of the quotients ker Di / Im Di−1 is a finite dimensional vector space. Since Di is equivariant the action of G on Γ(Ei )
restricts respectively to ker Di and Im Di−1 . Then ker Di / Im Di−1 is a finite
dimensional representation of G for each i.
21
Definition 2.2.1. The G-index of (Γ(E∗ ), D∗ ) is the sum of representations
X
IndexG (D∗ ) =
(−1)i ker Di / Im Di−1 ∈ RG ,
i
where RG denotes the ring of finite dimensional representations of G. For
each g ∈ G we define the g-index of D as the sum of the traces
X
Indexg (D∗ ) = Trg (IndexG (D∗ )) =
(−1)i Tr(g|ker Di / Im Di−1 ).
i
Note 2.2.2. Suppose that (Γ(E∗ ), D∗ ) is formed by a single elliptic operator
D : Γ(E0 ) −→ Γ(E1 ). In this case
IndexG (D∗ ) = ker D − E1 / Im D = ker D − coker D.
Then if G is the trivial group, RG ∼
= Z and IndexG (D) is just the index of
D as Fredholm operator under this isomorphism.
The aim of index theory is to use the symbol of (Γ(E∗ ), D∗ ) to compute
its index.
Example 2.2.3. Consider again the de Rham complex
d = ...
dk−1
dk
/ Γ(Λk (T ∗ M ⊗ C))
/ Γ(Λk+1 (T ∗ M ⊗ C))
dk+1
/ ...
of M . Then we have
IndexG (d) =
X
(−1)k H k (M ; C),
k
and for all g ∈ G
Indexg (d) =
X
(−1)k Tr(g|H k (M ;C) ) = L(g),
k
the Lefschetz number of g : M −→ M . For g = 1 the index Index1 (d) = L(1)
is the Euler characteristic χ(M ).
We saw in 2.1.5 that the support of the symbol σ
e(D∗ ) of an elliptic complex
(Γ(E∗ ), D∗ ) is the image of the zero section of the tangent bundle, which is
diffeomorphic to M . Since M is assumed to be compact, σ
e(D∗ ) has compact
support and therefore its homotopy class defines an element of KG (T M ) (cf.
proposition 1.2.14).
Definition 2.2.4. The element of KG (T M ) defined by the symbol of an
elliptic complex (Γ(E∗ ), D∗ ) of partial differential operators is denoted by
σ(D∗ ), and called the topological symbol of (Γ(E∗ ), D∗ ).
Note 2.2.5. By 1.2.15 the isomorphism α : LG (T M ) −→ KG (T M ) is natural for proper maps. Since the zero section i : M −→ T M is proper, we
obtain
σ(D∗ )|M = i∗ σ(D∗ ) = i∗ α(e
σ (D∗ )) = α(i∗ σ
e(D∗ )).
∗
But i σ
e(D∗ ) is an element of LG (M )/ ∼, and since M is compact α maps a
complex to the alternating sum of its bundles (cf. [14, §3]). Then
X
σ(D∗ )|M =
(−1)j i∗ (π ∗ Ej ),
j
22
but since π and i are mutually homotopic inverses
X
σ(D∗ )|M =
(−1)j Ej ∈ KG (M ).
j
We recall that the equivariant topological index of M is a ring homomorphism BG : KG (T M ) −→ RG (see Section 1.4).
Definition 2.2.6. The (equivariant) topological index of an elliptic
complex (Γ(E∗ ), D∗ ) is the representation BG (σ(D∗ )) ∈ RG obtained by
evaluating the equivariant topological index of M at the topological symbol
of (Γ(E∗ ), D∗ ).
Theorem 2.2.7. Let G be a compact Lie group acting smoothly on a compact manifold M . The G-index of an elliptic complex (Γ(E∗ ), D∗ ) of partial
differential operators on M is given by
IndexG (D∗ ) = BG (σ(D∗ )).
Idea of the proof. As a first step, we can show that the topological index
BG : KG (T M ) −→ RG is the unique homomorphism that satisfies
(1) if M = {p} is a point, then BG : KG (T M ) ∼
= RG −→ RG is the
identity,
(2) if i : N −→ M is an equivariant embedding of compact manifolds
then BG (i! a) = BG (a) for all a ∈ KG (T N ).
These two properties of BG are directly deduced from the properties of i! .
Suppose now that B 0 : KG (T M ) −→ RG is another homomorphism defined
for all compact manifolds M with this properties. Choose an equivariant
embedding M −→ Rk and an equivariant embedding Rk −→ S k (here S k
is seen as the one point compactification of Rk , and the action is extended
canonically). By the second property it is sufficient to show that BG = B 0
on the image of KG (T Rk ) in KG (S k ). Now, the homomorphism j! : RG =
KG (p) −→ KG (T Rk ) is an isomorphism, and then it is sufficient to show
that BG = B 0 on RG . This is true by the first property.
If the map σ from elliptic complexes of differential operators to KG (T M )
was surjective, one can try to define a homomorphism taking a ∈ KG (T M )
to IndexG (D∗ ) for some D∗ such that σ(D∗ ) = a. Then show that this map
satisfies the two properties above. However, σ is not surjective in general.
We can follow this line anyway defining a larger class of operators on Gvector bundles on M , namely pseudodifferential operators, and extend σ to
a surjective map on elliptic complexes of pseudodifferential operators. There
are some technical aspects in this procedure. For example to show that
an elliptic pseudodifferential operator is Fredholm, we have to extend it to
Sobolev sections. The details are in [15, §10].
After showing that σ extends to a surjective map over the class of elliptic
complexes of pseudodifferential operators, one can show that if σ(D∗ ) =
σ(D∗0 ) then D∗ and D∗0 have the same G-index. This allows us to define
a homomorphism B 0 : KG (T M ) −→ RG by B 0 (a) = IndexG (D∗ ) for some
elliptic complex of pseudodifferential operators D∗ such that σ(D∗ ) = a. It
remains to show that B 0 satisfies the two properties stated above. Then by
uniqueness BG = B 0 . This implies that for all differential operators D∗ on
23
M we have
BG (σ(D∗ )) = B 0 (σ(D∗ )) = IndexG (D∗ ).
The first property is easily proved for B 0 . If M = {p} is a point then an
elliptic complex of operators (Γ(E∗ ), D∗ ) is just a sequence of equivariant
linear maps Di : Ei −→ Ei+1 , and the symbol σ(D∗ ) is just the alternating
sum of the Ei , which is isomorphic as a representation to IndexG (D∗ ). For
the proof of the second property the reader is refered to [15, §11].
Suppose now that M is orientable. We need an orientation convention on
T M . Since locally T M is of the form U ×Rn for an open U ⊂ M , an orientation of Tvx T M is usually given by an oriented basis (w1 , . . . , wn , w10 , . . . , wn0 )
of Rn × Rn , where (w1 , . . . , wn ) induces an orientation of Tx M . However, we
will use the orientation induced by (w1 , w10 , . . . , wn , wn0 ), that will be more
convenient for our purposes. Note that a permutation of (w1 , . . . , wn ) induces an even permutation on the (w1 , w10 , . . . , wn , wn0 ). Thus if we invert
the orientation of M we do not change the orientation on T M . Thus we have
a well-defined orientation class [T M ] of T M independent of the orientation
of M . With this convention we have the following.
Corollary 2.2.8. Let G be a compact Lie group acting trivially on a compact
orientable manifold M . Then for all g ∈ G the g-index of (Γ(E∗ ), D∗ ) is
given by the formula
Indexg (D∗ ) = (−1)dim(M ) hchg (σ(D∗ )) td(TC M ), [T M ]i .
Proof. By the previous theorem we can show the formula for Trg (BG (σ(D∗ ))).
More generally, we show the formula for every element a ∈ KG (T M ). Consider first the non-equivariant case. The following diagram.
K(T M )
ch
φ
ch
Hc∗ (T M )
/ K(N T M )
ψ
h
ch
/ H ∗ (N T M )
c
/ K(T Rk )
k
φ0−1
/ K(∗) ∼
= R{1} ∼
=Z
ch=id
/ H ∗ (T Rk )
c
ψ 0−1
/ H ∗ (∗) ∼
=Z
c
is not commutative in general, and the obstruction to the commutativity of
each square is measured by the Todd classes (cf. 1.5). The index B of M is
by definition the composition of the homomorphisms of the upper row, that is
B(a) = φ0−1 hφ(a). Therefore φ0 (B(a)) = h(φ(a)). The Thom isomorphism
φ0 is given by φ0 (b) = (π ∗ b)λk , where π : T Rk −→ {∗} is the projection and
λk is the element in K(T Rk ) defined by the Koszul complex. Using this we
obtain
h(φ(a)) = (π ∗ B(a))λk .
Now, by section 1.5 we know that
(−1)k ψ 0 (td(T Rk )) ch((π ∗ b)λk ) = (−1)k ch((π ∗ b)λk ) = ψ 0 (ch b) = ψ 0 (b)
for any b ∈ Z. Therefore we obtain
ch(h(φ(a))) = ch((π ∗ B(a))λk ) = (−1)k ψ 0 (ch B(a)) = (−1)k ψ 0 (B(a)).
Now, ψ 0 is by definition ψ 0 (d) = dσk , where σk defines the orientation of
T Rk . Then
ch(h(φ(a))) = (−1)k B(a)σk .
24
Using the fact that the orientation classes agree with the extensions we have
B(a) = B(a)σk , [T Rk ]
= (−1)k ch(h(φ(a))), [T Rk ]
= (−1)k hch(φ(a)), [N T M ]i .
By the section 1.5 we know that ψ(td(N T M )) ch(φ(a)) = (−1)k−n ψ(ch(a)),
where k − n is the dimension of N T M . We compute td(N T M ). Since
N T M is the complexification of a real bundle N T M = N T M . Furthermore
T T M + N T M is trivial, and thus td(N T M ) is invertible and
td(N T M )−1 = td(T T M ) = td((π ∗ T M ) ⊗ C) = π ∗ (td(TC M )),
where π : T M −→ M is the projection. It follows that
ch(φ(a)) = (−1)k−n ψ(td(N T M )−1 )ψ(chg (a))
= (−1)k−n ψ(π ∗ (td(TC M )))ψ(ch(a)).
Substituting this in the expression above we obtain
B(a) = (−1)n hψ(π ∗ (td(TC M )))ψ(ch(a)), [N T M ]i
= (−1)n htd(TC M ) ch(a), [T M ]i .
For the equivariant case, just recall (cf.1.5.9) that every element a ∈
KG (T M ) decomposes as
X
a=
homG (Vi × M, a)(M × Vi ).
i
Since BG : KG (T M ) −→ RG is a ring homomorphism we have
P
Bg (a) = P
Trg (BG (a)) = Trg ( i BG (homG (Vi × M, a))BG ((M × Vi )))
= i Trg (BG (homG (Vi × M, a))) Trg (Vi ).
But the bundle homG (Vi × M, a) has no G-action, and so
Trg (BG (homG (Vi × M, a))) = B(homG (Vi × M, a)).
Using the formula for the non-equivariant index B we obtain
P
Bg (a) = Pi B(homG (Vi × M, a)) Trg (Vi )
= i (−1)n htd(TC M
P) ch(homG (Vi × M, a)), [T M ]i Trg (Vi )
n
= (−1) htd(TC M ) i ch(homG (Vi × M, a)) Trg (Vi ), [T M ]i
= (−1)n htd(TC M ) chg (a), [T M ]i .
There is a simplification of the formula when e(T M ) satisfies a certain
condition.
Proposition 2.2.9. Suppose that e(T M ) is not zero nor a zero divisor in
H ∗ (M ; C). Then the g-index of (Γ(E∗ ), D∗ ) is given by
P
i
n(n+1)/2 chg ( i (−1) Ei )
Indexg (D∗ ) = (−1)
td(TC M ), [M ] ,
e(T X)
where dim(M ) = n and
for each a ∈ H ∗ (M ; C).
a
e(T M )
denotes the unique solution x of a = xe(T M )
25
Proof. Applying the Thom isomorphism ψ : H n (M ) −→ Hc2n (T M ) we have
that for any b ∈ Hc2n (T M )
hb, [T M ]i = (−1)n(n+1)/2 ψ −1 (b), [M ] ,
the sign arising from the orientation convention on T M . We want to calculate ψ −1 (b) using the Euler class. By definition of the Thom isomorphism
b = ψ(ψ −1 (b)) = π ∗ (ψ −1 (b))u.
Let i : M −→ T M be the zero section. Note that this is a homotopic inverse
of the projection π. Then
i∗ (b) = ψ −1 (b)i∗ (u) = ψ −1 (b)e(T M ).
By the hypothesis on e(T M ), there is a unique solution to this equation,
that is
i∗ b
ψ −1 (b) =
.
e(T M )
Applying this to the formula of the previous proposition we obtain
∗
i chg (σ(D∗ ))
td(TC M ), [M ] .
Indexg (D∗ ) = (−1)n(n+1)/2
e(T X)
But now, i∗ chg (σ(D∗ )) = chg (i∗ σ(D∗ )) = chg (σ(D∗ )|M ). By 2.2.5 the restriction of σ(D∗ ) to M is the alternating sum of the Ei . This completes the
proof.
However, this condition on e(T M ) is quite strong. Further discussion on
the possible simplifications of the formula will be done after theorem 2.3.1.
When the action on M is not trivial the equivariant Chern character is
not defined. To avoid this problem we can "reduce" the computation of the
g-index to the fixed points of g in M . Here the action is trivial and we can
use the formula above to compute the g-index. This is the subject of the
next section.
2.3. The Atiyah-Singer Fixed Point Theorem. Let G be a compact
Lie group acting smoothly on a compact manifold M and let g ∈ G. The
Atiyah-Singer Fixed Point Theorem allows us to compute the g-index of an
elliptic complex (Γ(E∗ ), D∗ ) on M with respect to topological data of the
fixed points
M g = {x ∈ M |gx = x}.
Note that we can choose a G-invariant metric on M , and then G acts by
isometries. Then each connected component of M g is a compact submanifold
of M . Let {Mαg } be the (finite) collection of the connected components of
M g . Let jα : Mαg −→ M be the inclusion with associated
P normal bundle
N Mαg . For any vector bundle E we denote by Λ−1 E = k (−1)k Λk E.
Theorem 2.3.1 (Atiyah-Singer). The g-index of an elliptic complex of partial differential operators (Γ(E∗ ), D∗ ) on M is given by the formula
X
chg (jα∗ σ(D∗ ))
g
g
dim Mαg
td(TC Mα ), [T Mα ] .
Indexg (D∗ ) =
(−1)
chg (Λ−1 NC Mαg )
α
26
The problem with the connected components is that their dimensions
could be different. In this case M g is not properly a manifold, but just
the disjoint union of manifolds. Anyway, all the constructions with vector
bundles are valid also in this case. The only problem with the fixed point
formula is the sign. For ease of exposure we write the index formula
chg (j ∗ σ(D∗ ))
dim M g
g
g
Indexg (D∗ ) = (−1)
td(TC M ), [T M ] ,
chg (Λ−1 NC M g )
even if, in the case where the dimensions of the connected components of
M g are different, we mean by this the sum over the connected components,
with the right sign in front (j : M g −→ M denotes the inclusion).
To prove this theorem we need a result of Atiyah and Segal. The proof
can be found in [3, Prop.2.8]. Recall that the projection to a point induces
an RG -module structure on each ring KG (X). Furthermore the family of
representations V ∈ RG such that Trg (V ) 6= 0 forms a prime ideal of RG .
We denote by KG (X)g the localisation by this ideal.
Proposition 2.3.2. Let H be a topologically cyclic group generated by g ∈
H. Then the homomorphism j! : KH (T M g ) −→ KH (T M ) induces an isomorphism
(j! )g : KH (T M g )g −→ KH (T M )g .
g
Its inverse (j! )−1
g : KH (T M )g −→ KH (T M )g is given by
(j! )−1
g =
jg∗
,
Λ−1 (NC M g )
where jg∗ : KH (T M )g −→ KH (T M g )g is the map induced by the inclusion
after localisation, and N M g is the normal bundle of M g in M .
Proof of 2.3.1. We want to use the proposition above and 2.2.8 to prove the
theorem. First, we want to reduce the computation to a topologically cyclic
group. This is straightforward. Given a representation V of G we denote
V |H the restriction of V to a representation of a subgroup H ⊂ G. It is
sufficient to note that Trh (V ) = Trh (V |H ) for all h ∈ H, since the trace
only depends on the linear map h : V −→ V . Denoting by H = hgi the
topologically cyclic group generated by g we obtain that
Indexg (D∗ ) = Trg (IndexG (D∗ )) = Trg (IndexH (D∗ )).
Then it is sufficient to show the formula for H = hgi.
Let j : M g −→ M be the inclusion. Since the action of H restricts (trivially) to M g , the inclusion is H-equivariant. Then we can consider the
induced homomorphism j! : KH (T M g ) −→ KH (T M ). By definition of BH
the following diagram is commutative
j!
g
BH
/ R(H) ,
9
r
r
rr
r
r
r
rrr BH
KH (T M g )
KH (T M )
g
where we have denoted by BH
the sum of the indexes of the connected
g
components of M (note that the K-theory of a disjoint union is the direct
27
sum of the K-theories). Since the index homomorphism BH is actually a
homomorphism of RH -modules, the composition Trg ◦BH : KH (T M ) −→ C
induces a homomorphism BH,g : KH (T M )g −→ C on the localisation (and
g
g
similarly BH
induces BH,g
on KH (T M g )g ). Thus we have a commutative
diagram
g
BH,g
/
:C,
uu
u
uu
∼
= uuu
B
u
H,g
u
KH (T M g )
(j! )g
KH (T M )
where now (j! )g is an isomorphism by the proposition above. Then we have
Trg IndexH (D∗ ) = Trg (BH (σ(D∗ ))) = BH,g (σ(D∗ ))
j ∗ σ(D∗ )
= BH,g ((j! )−1
g (σ(D∗ ))) = BH,g ( Λ−1 (NC M g ) ).
Now, BH,g is the sum of the topological indexes on the connected components
of T M g . Since H acts trivially on T M g we can apply 2.2.8 and we obtain
∗
j σ(D∗ )
Trg IndexH (D∗ ) = BH,g ( Λ−1
(NC M gD
))
E
∗ σ(D ))
P
g
chg (jα
g
g
∗
dim
M
α
= α (−1)
g td(TC Mα ), [T Mα ] .
chg (Λ N M )
−1
C
α
We come back to the simplifications of this formula. What we need to
simplify the formula is a solution of the equation
i∗ (b) = ψ −1 (b)e(T M g ),
ch (j ∗ σ(D )) td(T M )
for b = gchg (Λ−1∗(NC M gC)) (cf. 2.2.9). Let’s study the equation i∗ (b) =
ψ −1 (b)e(T M g ) for a general b. The idea is to reformulate this equation
in the cohomology of some classifying space, solve it, and pull it back to
H ∗ (M g ; C).
Definition 2.3.3. Let H be a compact Lie group. We say that M g admits
an H-structure if there is a representation V of H and a commutative
diagram
TMg
/ EH ×H V
F
π
Mg
f
πH
/ BH,
such that T M g ∼
= f ∗ (EH ×H V ) as vector bundles, where EH is the universal
principal H-bundle.
We denote EH ×H V by EV . All the cohomologies here are with complex
coefficients. Consider the diagram
Hc∗ (T M g ) o
i∗
F∗
H ∗ (M g ) o
H ∗ (EV, EV0 )
f∗
i∗H
H ∗ (BH),
28
where iH : BH −→ EV is the zero section. Suppose furthermore that there
are elements eH ∈ H ∗ (BH) and bH ∈ H ∗ (EV, EV0 ) such that f ∗ (eH ) =
e(T M g ) and F ∗ (bH ) = b, and that eH is not zero nor a zero divisor in
H ∗ (BH). Let ψH : H ∗ (BH) −→ H ∗ (EV, EV0 ) be the universal Thom isomorphism. Therefore the equation
−1
i∗H (bH ) = ψH
(bH )eH
−1
has a unique solution in H ∗ (BH), that we denote by ψH
(bH ) =
Then
∗
−1
−1 ∗
∗ −1
∗ iH (bH )
ψ (b) = ψ F bH = f ψn (bH ) = f
eH
is the unique functorial solution of the equation
i∗H (bH )
eH .
i∗ (b) = ψ −1 (b)e(T M ).
To simplify the notation we write f ∗ (
i∗H (bH )
eH )
=
i∗ b
e(T M g ) .
Definition 2.3.4. We say that (Γ(E∗ ), D∗ ) is a H-universal elliptic complex with respect to g if
(1) there is a virtual hgi-vector bundle σH over EV such that
F ∗ σH = j ∗ σ(D∗ ) ∈ Khgi (T M g ),
(2) there is a virtual hgi-vector bundle NH over BH such that
f ∗ NH = Λ−1 (NC M g ) ∈ Khgi (M g ),
(3) there is a class tdH ∈ H ∗ (BH) such that f ∗ tdH = td(TC M g ),
If these conditions are satisfied, the element
b=
chg (j ∗ σ(D∗ )) td(TC M )
chg (Λ−1 (NC M g ))
admits a universal interpretation bH , and therefore the equation i∗ (b) =
ψ −1 (b)e(T M g ) admits a solution
ψ −1 (b) =
i∗ chg (j ∗ σ(D∗ )) td(TC M )
.
chg (Λ−1 (NC M g ))e(T M g )
But, by 2.2.5,
P
∗ (σ(D )|
k
i∗ chg (j ∗ σ(D∗ ) = chg (iP
∗ T M g )) = chg ( k (−1) Ek |M g )
k
∗
= chg ( k (−1) j Ek ).
Using the index formula with an argument completely analogous to the proof
of 2.2.9 we obtain the following.
Corollary 2.3.5. Suppose that (Γ(E∗ ), D∗ ) is a H-universal elliptic complex
with respect to g. Then its index is given by the formula
P
chg ( k (−1)k j ∗ Ek )
n(n+1)/2
g
g
Indexg (D∗ ) = (−1)
td(TC M ), [M ] ,
chg (Λ−1 NC M g )e(T M g )
H
where dim(M g ) = n and e(TaM ) denotes the pullback f ∗ ( aeH
) of the unique
aH
universal solution eH of aH = xeH .
29
Example 2.3.6. Consider the de Rham operator. Clearly any oriented
n-dimensional Riemannian manifold admits an SOn -structure. The Todd
class and the Euler class both have universal representatives tdn and en
in H ∗ (BSOn ). Suppose n = 2l is of even dimension. Then en is a nonzero element in the polynomial ring H ∗ (BSOn ), and therefore is not a zero
divisor. The bundle Λ−1 (NC M g ) clearly has a universal representative being
a vector bundle. Furthermore the sequence
...
n
/ Λk π ∗ (ESO ×
n SOn R ) ⊗ C
αk
n
/ Λk+1 π ∗ (ESO ×
n SOn R ) ⊗ C
/ ... ,
where π : ESOn ×SOn Rn −→ BSOn is the projection, defines a bundle
over ESOn . Here αk (ξ, v) = v ∧ ξ is the exterior multiplication. It can be
shown that this bundle is universal for σ(d). Then the index of the de Rham
operator is given by
chg (Λ−1 TC M |M g )
n(n+1)/2
g
g
Indexg (D∗ ) = (−1)
td(TC M ), [M ] .
chg (Λ−1 NC M g )e(T M g )
Using the equivariant decomposition T M |M g ∼
= T M g ⊕ N M g this formula
becomes
D
E
n(n+1)
ch(Λ−1 TC M g ) chg (Λ−1 NC M g )
g ), [M g ]
td(T
M
Indexg (d) = (−1) 2
g
g
C
C M )e(T M )
D chg (Λ−1 N
E
ch(Λ−1 TC M g )
l(n+1)
g
g
= (−1)
td(TC M ), [M ] .
e(T M g )
Note that by the Atiyah-Singer theorem this is the (non-equivariant) index
of the de Rham complex on M g .
By the splitting principle we can suppose that TC M g is a sum of line bundles Li . Furthermore, since TC M g is the complexification of a real bundle,
it is possible to choose this decomposition so that
TC M g = ⊕li=1 Li ⊕li=1 Li ,
see [15, §A3]. With respect to this decomposition
e(T M g ) =
l
Y
c1 (Li ),
i=1
cf. [15, §1,§A3]. Therefore, since c1 (Li ) = −c1 (Li ), the Todd class is given
by
l
l
Y
c1 (Li ) Y −c1 (Li )
g
td(TC M ) =
.
1 − e−c1 (Li ) i=1 1 − ec1 (Li )
i=1
The Chern character of Λ−1 TC M g is given by
ch(Λ−1 TC M g ) = ch(⊗li=1 Λ−1 (Li ) ⊗li=1 Λ−1 (Li ))
Q
Q
= li=1 ch(C − Li ) li=1 ch(C − Li )
Q
Q
= li=1 (1 − ec1 (Li ) ) li=1 (1 − e−c1 (Li ) ).
Then the product simplifies as
ch(Λ−1 TC M g )
td(TC M g ) = (−1)l e(T M g ).
e(T M g )
Therefore the index of d is
Indexg (d) = (−1)l(n+1) (−1)l he(T M g ), [M g ]i = he(T M g ), [M g ]i .
30
Since Indexg (d) is the index of the de Rham complex on M g by the formula
above, this is the Euler characteristic of M g . This also shows the general
result that he(T M g ), [M g ]i is the Euler characteristic of M g .
If the dimension is odd, the index of d is always zero, see [15, §4].
3. Spin Bundles and Dirac Operators
In this part we define a class of manifolds called spin manifolds. An even
dimensional spin manifold admits two particular complex bundles on it and
a canonical elliptic operator on these bundles called the Dirac operator. We
will study its index using the Atiyah-Singer fixed point theorem.
3.1. Clifford Algebras and Spin Representations in Finite Dimension. Let V be a K-vector space with K = R or C, and q a quadratic form
on V . Let T (V ) denote the tensor algebra
T (V ) =
∞ O
k
M
V,
k=1 i=0
and Iq the ideal of T (V ) generated by the elements of the form v ⊗ v + q(v)1
for v ∈ V .
Definition 3.1.1. The Clifford Algebra of V associated to q is the Kalgebra
Cl(V, q) = T (V )/Iq .
A map f : V −→ A into a K-algebra A is a Clifford map if
f (v)f (v) = −q(v)1
for all v ∈ V .
Note that there is a natural inclusion V ⊂ Cl(V, q) induced by the inclusion of V in T (V ) as 1-tensors. The Clifford algebra satisfies the following
universal property.
Proposition 3.1.2. Let A be a K-algebra and f : V −→ A a Clifford map.
Then there is a unique K-algebra homomorphism fe: Cl(V, q) −→ A making
the following diagram commute
V _
f
/A
;
∃!fe
Cl(V, q).
Proof. The map f extends uniquely to T (V ), and by hypothesis it descends
(uniquely) to the quotient Cl(V, q).
This property allows us to see the orthogonal group
O(V, q) = {f ∈ GL(V )|f ∗ q = q}
as a subgroup of the group of automorphisms of Cl(V, q).
31
Consider now the automomorphism α : Cl(V, q) −→ Cl(V, q) induced by
the reflection isometry v 7−→ −v on V . Since α is of order two, the Clifford
algebra splits as the sum
Cl(V, q) = Cl(V, q)0 ⊕ Cl(V, q)1 ,
where Cl(V, q)i is the eigenspace of α of eigenvalue (−1)i . Since α is a homomorphism of algebras, the product Cl(V, q)i Cl(V, q)j is included in Cl(V, q)ij .
Thus this splitting defines a Z/2-grading of Cl(V, q).
Definition 3.1.3. The subalgebra Cl(V, q)0 is called the even part of
Cl(V, q), and Cl(V, q)1 is its odd part.
Let Cl(V, q)× be the group of invertible elements in Cl(V, q). Note that if
v ∈ V with q(v) 6= 0 then −v/q(v) is an inverse for v, and so v ∈ Cl(V, q)× .
Definition 3.1.4. The Pin group of (V, q) is the subgroup Pin(V, q) of
Cl(V, q)× generated by those elements v ∈ V with q(v) = ±1. The Spin
group of (V, q) is defined as the intersection
Spin(V, q) = Pin(V, q) ∩ Cl(V, q)0 .
Definition 3.1.5. The twisted adjoint representation of Cl(V, q) is the
f : Cl(V, q)× −→ GL(Cl(V, q)) defined by
homomorphism Ad
f φ (ψ) = α(φ)ψφ−1 .
Ad
f v is well
If v ∈ V with q(v) 6= 0 then v is invertible and so in this case Ad
defined.
f v (V ) ⊂ V .
Proposition 3.1.6. For all v ∈ V with q(v) 6= 0 we have Ad
f v |V : V −→ V is the reflection associated to
Furthermore the restriction Ad
the orthogonal complement of v in V .
Proof. Let w ∈ V be any vector. By definition of Cl(V, q) we have that
q(v + w)1 = −(v + w)(v + w) = q(v)1 + q(w)1 − vw − wv,
that is vw + wv = q(v)1 + q(w)1 − q(v + w)1 = −2q(v, w). Then
f v (w) = −vwv −1 = 1 vwv = 1 (vwv + vvw − v 2 w)
Ad
q(v)
q(v)
1
= q(v)
(v(wv + vw) + q(v)w)
=
1
q(v) (−2q(v, w)v
+ q(v)w) = −2 q(v,w)
q(v) v + w,
which is an element of V . Furthermore this is the formula of the reflection
described above.
As an immediate consequence we have the following.
f : Cl(V, q)× −→
Corollary 3.1.7. The twisted adjoint representation Ad
GL(Cl(V, q)) restricted to Pin(V, q) and Spin(V, q) gives two representations
ξ : Pin(V, q) −→ O(V, q) and ξ0 : Spin(V, q) −→ O(V, q) of V .
Now, suppose that V is finite dimensional. Then we can define the subgroup SO(V, q) of O(V, q) consisting of isometries with determinant 1.
Proposition 3.1.8. If V is of finite dimension and q is non-degenerate,
ξ0 (φ) is in SO(V, q) for all φ ∈ Spin(V, q). Furthermore the representation
ξ0 : Spin(V, q) −→ SO(V, q) is surjective.
32
Proof. For v ∈ V a non-zero vector we denote by ρv the reflection with respect to the q-orthogonal complement in V of the vector subspace generated
by v. We assume the result claiming that each element ρ of O(V, q) can be
written as a composition of reflections
ρ = ρv 1 . . . ρ v r ,
with r ≤ dim(V ). Choosing a basis e1 , . . . , en such that e1 = v and q(v, ej ) =
0 for 2 ≤ j ≤ dim(V ) we see that det(ρv ) = −1. Then
SO(V, q) = {ρv1 . . . ρvr |r is even}.
Since by definition Spin(V, q) = {v1 . . . vr |q(vi ) = ±1, r is even}, it is clear
that ξ0 (φ) is in SO(V, q) for all φ ∈ Spin(V, q). Now, we prove the surjectivity. It is clear that ρv = ρtv for all non-zero t ∈ K. Let ρv1 . . . ρvr ∈ SO(V, q)
be a general element. Then
ρv 1 . . . ρ v r = ρw 1 . . . ρ w r ,
with wi = √ vi
|q(vi )|
. Note that q(vi ) is non-zero since q is non-degenerate.
Then q(wi ) = ±1, that is w1 . . . wr is in Spin(V, q). Furthermore we have
that
ξ0 (w1 . . . wr ) = ξ0 (w1 ) . . . ξ0 (wr )
= ρw 1 . . . ρ w r = ρv 1 . . . ρ v r .
We now consider theP
case where V = Rn and q is the canonical quadratic
n
2
form q(x1 , . . . , xn ) =
i=1 xi . In this case we use the notations Cln =
n
n
Cl(R , q), Spinn = Spin(R , q), On = O(Rn , q) and SOn = SO(Rn , q). The
following result is proved in [9, Th.I.2.10].
Proposition 3.1.9. The sequence
ξ0
0 −→ Z/2 −→ Spinn −→ SOn −→ 1
ξ0
is exact and Spinn −→ SOn represents the universal double covering of SOn .
Note that there is a C-algebra isomorphism Cln ⊗C ∼
= Cl(Cn , q ⊗ C).
n
Furthermore all non-degenerate quadratic forms on C induce isomorphic
Clifford
algebras. Then denoting Cln = Cl(Cn , qC ) with qC (z1 , . . . , z n ) =
Pn
2
i=1 |zi | we have
Cln ∼
= Cln ⊗C.
Under this isomorphism we can see Spinn as embedded in Cln , and we can
restrict representations of Cln and Cln to Spinn .
Definition 3.1.10. A real spinor representation of Spinn is the restriction to Spinn of an irreducible real representation of Cln .
A complex spinor representation of Spinn is the restriction to Spinn
of an irreducible complex representation of Cln .
The following results give a complete classification of all the complex irreducible representations of Cln and of all complex spinor representations.
All the proofs can be found in [9, §4,§5]. Let Mn (C) be the algebra of n × n
matrices with complex coefficients.
33
Proposition 3.1.11. For n = 2l there is an isomorphism Cln ∼
= M2l (C).
M
⊕
M
.
For n = 2l + 1 we have Cln ∼
= 2l
2l
Proposition 3.1.12. The standard representation ρ of Mn (C) on Cn is the
unique irreducible complex representation of Mn (C) up to equivalence. There
are exactly two equivalence classes of irreducible complex representations of
M2l ⊕ M2l . They are given by
ρ1 (α, β) = ρ(α) and ρ2 (α, β) = ρ(β).
Corollary 3.1.13. If n is even, there is only one irreducible complex representation of Cln up to equivalence. If n is odd, there are exactly two inequivalent representations of Cln .
Proposition 3.1.14. When n is even, the unique complex spinor representation of Spinn splits into a sum of two inequivalent irreducible complex
representations of Spinn . When n is odd, the two inequivalent irreducible
representations of Cln induce the same complex spinor representation. Furthermore, this is an irreducible representation of Spinn .
Definition 3.1.15. For any n, the unique complex spinor representation is
denoted
∆C
n : Spinn −→ GL(V ).
If n is even, we denote the decomposition of ∆C
n in irreducible inequivalent
complex representations of Spinn by
+
−
C
C
∆C
n = ∆n ⊕ ∆n .
3.2. Spin Bundles. Here we use the representations of Spinn defined before
to construct some bundles from a boundle admitting a particular structure.
Let G be a topological group. Recall that a principal G-bundle is a bundle
p : E −→ B with fibre G and endowed with a continuous right G-action on
E such that p is constant on the orbits and the local trivialisations are Gequivariant.
Also recall that an orientation of a real n-vector bundle p : E −→ B is
a choice of orientation of each fibre Eb with an additional condition on the
local trivialisations (see def.1.6.1). We can associate to an oriented Riemannian real n-vector bundle p : E −→ B the bundle PSO (E) of oriented
orthonormal frames. This is the bundle over B whose fibre at b ∈ B is the
space of orientation-preserving isometries from Rn to Eb . It is a principal
SOn -bundle, whose action on the fibres is given by precomposition. Using
ξ0
the adjoint representation Spinn −→ SOn we define a right Spinn -action on
PSO (E).
Definition 3.2.1. Suppose n ≥ 3. A spin structure on p : E −→ B is a
principal Spinn -bundle PSpin (E) over B and a Spinn -equivariant morphism
of bundles
ξ : PSpin (E) −→ PSO (E)
which is a double covering.
If n = 2, a spin structure is a principal SO2 -bundle PSpin (E) with a double
covering PSpin (E) −→ PSO (E) defining a morphism of principal bundles.
If n = 1 we have that PSO (E) = B and we define a spin structure just as
a double covering of B.
34
Note 3.2.2.
(1) If the base space B is a manifold, it can be shown that a
spin structure on an oriented Riemannian vector bundle over B exists
if and only if its second Stiefel-Whitney class is zero. Furthermore
the distinct spin structures are in one to one correspondence with
the elements of H 1 (B, Z/2) (cf. [9, Th.II.1.7]).
(2) Supposing that E is a smooth bundle, one can show that PSO (E) is
also smooth. Since a spin structure is a double covering of PSO (E),
it is a smooth bundle too.
Now, we want to "glue" the complex spinor representation of Spinn to a
spin structure. Before doing that we recall the construction of the associated
bundle. Let E be a principal G-bundle and ρ : G −→ Homeo(F ) a continuous
group action. The associated bundle is the orbit space of E × F under the
action defined by
g(e, f ) = (eg −1 , gf )
with projection to the base space of E. The class of a couple (e, f ) ∈ E × F
is denoted by [e, f ]. This bundle has fibre F , and it is denoted by E ×ρ F .
In the case where E is a smooth bundle and G is a Lie group acting on F
by diffeomorphisms by a continuous ρ : G −→ Diff(F ) the associated bundle
is a smooth bundle.
Definition 3.2.3. Let PSpin (E) be a spin structure. The complex spinor
bundle of E associated to the spin structure is the complex vector bundle
SC (E) = PSpin (E) ×∆Cn V,
where ∆C
n : Spinn −→ GL(V ) is the unique complex spinor representation
of Spinn (cf 3.1.14).
We recall that if the dimension n = 2l of E is even, there is a decomposition of ∆C
n in irreducible representations
+
−
C
C
∆C
n = ∆n ⊕ ∆n .
This decomposition induces a decomposition of SC (E) as the sum of two
complex vector bundles
SC (E) = SC+ (E) ⊕ SC− (E),
where
l
SC± (E) = PSpin (E) ×∆C ± C2 .
n
Note 3.2.4. The spinor representation is smooth. So if E is a smooth bundle
the complex spinor bundles SC (E) and SC± (E) are also smooth bundles.
The case we are interested in is when the base space B is a manifold.
We now reformulate all the definitions above for the tangent bundle of a
manifold.
Definition 3.2.5. Let M be a finite dimensional manifold. We say that M
is orientable (oriented) if its tangent bundle is. A spin structure on an
oriented Riemannian manifold is a spin structure on its tangent bundle. A
manifold with a fixed spin structure is called a spin manifold. We define
the spinor bundle of a spin manifold M by
SC (M ) = SC (T M ).
35
If M is of even dimension we can also define
SC± (M ) = SC± (T M ),
and there is a splitting
SC (M ) = SC (M )+ ⊕ SC (M )− .
Suppose now that a compact Lie group G acts smoothly on a compact
manifold M by orientation-preserving isometries. Then this action lifts to
one orthonormal oriented frame bundle PSO (T M ) and it commutes with the
SOn -action. Suppose now that we have a fixed spin structure PSpin (T M ) on
M.
Definition 3.2.6. The action of G on M is a spin action (with respect
to the spin structure) if it lifts to an action on PSpin (T M ) which commutes
with the Spinn -action.
Note 3.2.7. A spin structure on M corresponds to a unique element a ∈
H 1 (M, Z/2). If the group G is finite, the action of G on M is spin if and
only if the maps g ∗ : H 1 (M, Z/2) −→ H 1 (M, Z/2) satisfy g ∗ a = a for all
g ∈ G. However, if G is not finite, this condition does not assure that the
lifting of the action on the spin structure is continuous (cf. [15, §19]).
A spin action of G on M induces an action on SC (M ), SC+ (M ) and SC− (M )
defined by g[e, f ] = [ge, f ]. This actions are well-defined since the action on
the spin structure commutes with the action of Spinn . Then SC (M ), SC+ (M )
and SC− (M ) are complex G-vector bundles over M .
Recall that if V is a representation of H and E is a principal H-bundle,
an H-invariant inner product on V induces a Riemannian metric on the
associated bundle. The inner product of [e, v] and [e0 , v 0 ] in the same fibre is
defined by
[e, v], [e0 , v 0 ] = φ2 (e)v, φ2 (e0 )v 0 ,
for a trivialisation φ = (φ1 , φ2 ) : p−1 (U ) −→ U × H of E. This definition
is independent of the choice of the trivialisation by the H-invariance of the
inner product on V . Then the spin bundles SC (M ), SC+ (M ) and SC− (M ) are
also Riemannian vector bundles.
3.3. The Dirac Operator. Let M be a spin manifold of even dimension
n = 2l.
Definition 3.3.1. The space of smooth sections Γ(SC (M )) of the spinor
bundle of M is called the space of spinor of M . We call the spaces of
smooth sections Γ(SC (M )+ ) and Γ(SC (M )− ) the 21 -spinor and − 21 -spinor
of M respectively.
We want to define a linear elliptic operator 6∂ : Γ(SC (M )+ ) −→ Γ(SC (M )− ),
that is also equivariant if M carries a spin action.
Definition 3.3.2. Let E be a smooth real vector bundle over M . A covariant derivative on E is a linear map
∇ : Γ(E) −→ Γ(T ∗ M ⊗ E)
such that
∇(f s) = df ⊗ s + f ∇s
36
for all f ∈ C ∞ (M ) and all section s ∈ Γ(E).
Given a covariant derivative ∇ on E and a vector field V ∈ Γ(T M ), we
define a linear map ∇V : Γ(E) −→ Γ(E) by
∇V (s) = ∇(s)(V ).
Definition 3.3.3. Suppose that E carries a Riemannian metric. A covariant
derivative ∇ on E is called Riemannian if
V [ s, s0 ] = ∇V s, s0 + s, ∇V s0
for all V ∈ Γ(T M ) and s, s0 ∈ Γ(E), where V [hs, s0 i] is the value of the
derivation V on the function hs, s0 i ∈ C ∞ (M ).
We want to define a covariant derivative on SC (M ) and use it to define
the Dirac operator. To define this covariant derivative we use some general
results about connections and associated bundles.
Definition 3.3.4. Let π : P −→ M be a smooth principal G-bundle, and
g be the Lie algebra of G. A connection on P is a g-valued 1-form ω ∈
Γ(T ∗ P ⊗ g) such that
d
ωp ( p · exp(tv)|t=0 ) = v
dt
for all v ∈ g, and
g ∗ ωp = Adg−1 ◦ωp : Tp P −→ g
for all p ∈ P and g ∈ G.
The following result shows the relation between connections and covariant
derivatives (cf.[9, Prop.II.4.4]). Consider a real oriented Riemannian vector
bundle E over M and let PSO (E) be the principal SOn -bundle of oriented
orthonormal frames on E. Let (e1 , . . . , en ) be an oriented orthonormal family
of local sections of E, i.e. a local section of PSO (E).
Proposition 3.3.5. A connection ω on PSO (E) induces a Riemannian covariant derivative on E uniquely defined by
n
X
∇ei =
w
eij ⊗ ej ,
j=1
where w
e is the local 1-form on M defined by w
e = (e1 , . . . , en )∗ ω. Conversely,
given a Riemannian covariant derivative ∇ on E, there is a unique connection ω on PSO (E) satisfying the relation
n
X
∇ei =
w
eij ⊗ ej .
j=1
Now consider the situation where we have a smooth principal G-bundle
P −→ M and a representation ρ : G −→ SOm of G. Then the associated
bundle P ×ρ Rm is a Riemannian bundle. We can show that PSO (P ×ρ Rm )
is actually the associated bundle
PSO (P ×ρ Rm ) = P ×ρ0 SOm ,
where ρ0 : G −→ Homeo(SOm ) is the action defined at a g ∈ G as left
multiplication by ρ(g). Given a connection ω on P , we can extend it trivially
37
on P × SOm and push it forward to P ×ρ0 SOm . This gives a connection
on PSO (P ×ρ Rm ) = P ×ρ0 SOm . Applying the previous result we obtain a
Riemannian covariant derivative on P ×ρ Rm .
We apply this general construction in order to obtain a Riemannian covariant derivative on SC (M ) = PSpin (M ) ×∆n V . What we need is a connection
on the principal Spinn -bundle PSpin (M ). Since M carries a Riemannian metric, there is a unique Riemannian covariant derivative on T M . Using this
last proposition we obtain a connection on PSO (T M ). Since the spin structure defines a double covering PSpinn (M ) −→ PSO (T M ), we can extend this
connection to a connection on PSpinn (M ). The previous construction gives
a covariant derivative
∇ : Γ(SC (M )) −→ Γ(T ∗ M ⊗ SC (M ))
on SC (M ). Note that the metric on T M induces a diffeomorphism T M ∗ ∼
=
T M of vector bundles. This induces a linear isomorphism Γ(T ∗ M ⊗SC (M )) ∼
=
Γ(T M ⊗ SC (M )). Furthermore there are canonical isomorphisms T M ⊗
SC (M ) = T M ⊗ C ⊗C SC (M ) = TC M ⊗C SC (M ). Thus we can see the
covariant derivative as a C-linear map
∇ : Γ(SC (M )) −→ Γ(TC M ⊗C SC (M )).
In order to use this covariant derivative to define the Dirac operator we
need another map, that we define using Clifford multiplication. Since the
representation ∆n of Spinn is the restriction of an irreducible complex representation V of Cln , we have a C-linear map ca : V −→ V for all a ∈ Cln .
A local chart for M at x ∈ M induces isomorphisms (TC M )x ∼
= Cn and
n
∼
SC (M )x = V . Under the inclusion C ⊂ Cln we obtain a C-linear map
cw : SC (M )x −→ SC (M )x for all wx ∈ (TC M )x which does not depend on
the choice of the local chart. This defines a global morphism of complex
vector bundles
c : TC M ⊗C SC (M ) −→ SC (M ).
This can be pointwise extended to a C-linear map on sections
c : Γ(TC M ⊗C SC (M )) −→ Γ(SC (M )).
Composing this with the covariant derivative we obtain a C-linear map
∇
c
D : Γ(SC (M )) −→ Γ(TC M ⊗C SC (M )) −→ Γ(SC (M )).
Since the dimension of M is even, there is a splitting SC (M ) = SC+ (M ) ⊕
This splitting is induced by a splitting of representations, and
then the restriction of the covariant derivative on SC+ (M ) defines a covariant
derivative
∇ : Γ(SC+ (M )) −→ Γ(TC M ⊗C SC+ (M )).
SC− (M )).
The map c : TC M ⊗C SC (M ) −→ SC (M ) is defined using Clifford multiplication by elements of Cn . Since these elements are of odd degree, the map c
shifts the degrees up by one, and its restriction to TC M ⊗C SC+ (M ) induces
a C-linear map on sections
c : Γ(TC M ⊗C SC+ (M )) −→ Γ(SC− (M )).
38
Definition 3.3.6. The Dirac operator of an even dimensional spin manifold M is the C-linear map defined as the composition
∇
c
6∂ : Γ(SC+ (M )) −→ Γ(TC M ⊗C SC+ (M )) −→ Γ(SC− (M )).
We now see what happends when a Lie group acts on the manifold M .
Recall that a spin action on the spin manifold M is an action by orientation
preserving isometries that lifts to the spin structure commuting with the
Spinn -action. In this case there are induced actions on SC+ (M ) and SC− (M )
and therefore on Γ(SC+ (M )) and Γ(SC+ (M )).
Proposition 3.3.7. If M is an even dimensional spin manifold with a spin
action of a compact Lie group G, the Dirac operator is G-equivariant.
Proof. Since G acts on M by isometries, the covariant derivative on T M is
G-equivariant. By definition, the induced connection on PSO (T M ) is also
invariant by the action. Since the lift of the action to the spin structure commutes with the Spinn -action, the induced connection on the spin structure
is also G-invariant. Then the induced covariant derivative
∇ : Γ(SC+ (M )) −→ Γ(TC M ⊗C SC+ (M ))
is G-equivariant, where the action of G on TC M ⊗C SC+ (M ) is g(v ⊗ a) =
(gv)⊗(ga). Furthermore Clifford multiplication satisfies (gv)·(ga) = g(v ·a).
Therefore
c : Γ(TC M ⊗C SC+ (M )) −→ Γ(SC+ (M ))
is also G-equivariant. This shows that 6∂ is G-equivariant.
Note 3.3.8. The construction of the operator D : Γ(SC (M )) −→ Γ(SC (M ))
is valid even without the assumption that the dimension of M is even. Then
we could define D as the Dirac operator of any spin manifold M . If there is
a spin action of G on M , the operator D is G-equivariant by a proof totally
analogous to the last proposition. Furthermore it is elliptic. However, this
operator is not interesting from the point of view of the index theory. To see
why, define an inner product on Γ(SC (M )) by
Z
0
s, s =
s(x), s0 (x) x ,
M
hs(x), s0 (x)ix
where
are the fibrewise inner products of SC (M ). With respect
to this inner product we have
Ds, s0
x
= s, Ds0
x
+ div(X)x
s, s0
for all
∈ Γ(SC (M )) and for some vector field X on M depending on
0
s and s (cf.[9, Prop.II.5.3]). We integrate this equality and, since M is
closed, Stokes Theorem implies that D is self-adjoint. Furthermore this
inner product is G-invariant. This two facts mean that ker(D) ∼
= coker(D)
as representations of G. The G-index of D is therefore zero.
3.4. The Index of the Dirac Operator. Let M be an even dimensional
spin manifold with a spin action of a compact Lie group G. In order to apply
the Atiyah-Singer fixed point theorem to the Dirac operator of M , we have
to prove that 6∂ is elliptic and characterize its topological symbol. We denote
SC± (M ) just as S ± (M ).
39
Proposition 3.4.1. The Dirac operator is elliptic. In particular its symbol
σ
e(6∂ ) : π ∗ S + (M ) −→ π ∗ S − (M ) is given by
σ
e(6∂ )(v, e) = (v, v · e),
where · is Clifford multiplication.
Proof. First, we need a local expression for the Dirac operator. Note that
a covariant derivative is a local operator, that is the value at a section is
determined by the local expression of the section. Let U be the domain
of a chart of M trivializing S + (M ). Take an orthonormal oriented family
of sections (e1 , . . . , em ) of S + (M )|U . Then any section s of S + (M ) can be
uniquely written as s = f1 e1 + · · · + fm em over U , where the fi are smooth
functions on U . Over U we have
∇(s) = ∇(f1 e1 + · · · + fm em ) =
m
X
(dfi ⊗ ei + fi ∇ei ).
i=1
Since the covariant derivative is induced by a connection on the oriented
orthonormal frames, the second summand is given by
X
X
fi ∇ei =
fi ω
eij ⊗ ej ,
i
i,j
which is a derivation P
of order zero. Identifying the tangent and the cotangent
∂fi ∂
. Then the Dirac operator is given locally
bundle dfi becomes nj=1 ∂x
j ∂xj
by
P
P
6∂ (s) = c( m
dfi ⊗ ei + i,j ω
eij ⊗ ej )
i=1
Pm Pn ∂fi ∂
P
eij · ej
= i=1 ( j=1 ∂xj ∂xj ) · ei + i,j ω
Pn
Pm ∂fi
P
∂
= j=1 ∂xj · ( i=1 ∂xj ei ) + i,j ω
eij · ej
P
Pn
∂
j
eij · ej ,
= j=1 ∂xj · (∂ s) + i,j ω
where · is Clifford multiplication. To characterize the symbol of 6∂ we are
only interested in the terms of highest degree of derivation, that is the first
sum. Then the matrix-coefficients of the partial operators ∂ j with multiindex j of norm one are the matrix-coefficients of Clifford multiplication by
∂
∂xi . The matrix-valued map σU over T U is then
σU (v) =
n
n
X
X
∂
∂
[
·]vj =
[vj
·] = [v·],
∂xj
∂xj
j=1
j=1
where [a·] denotes the matrix of the left Clifford multiplication by an element
a ∈ T U . The symbol σ
e(6∂ ) : π ∗ S + (M ) −→ π ∗ S − (M ) of the Dirac operator
is therefore
σ
e(6∂ )(v, e) = (v, v · e),
that is Clifford multiplication by elements of T M . We already saw that an
v
element v ∈ T M with non-zero norm has inverse − kvk
in the Clifford algebra
of T M . This shows that σ
e(6∂ ) restricted to the fibres of non-zero vectors of
v
T M is an isomorphism (whose inverse is (v, f ) 7→ (v, − kvk
f )). Hence the
Dirac operator is elliptic.
40
We can now apply the Atiyah-Singer theorem, to obtain
chg (j ∗ σ(6∂ ))
g
g
g
Indexg (6∂ ) = (−1)dim M
td(T
M
),
[T
M
]
.
C
chg (Λ−1 NC M g )
However, we would like a simplified formula. Note that since G acts by
isometries the dimension of each connected component M g is of even dimension n. Suppose furthermore that T M g is a spin manifold. A more detailed
proof of the following result is in [15, §A5].
Proposition 3.4.2. The Dirac operator is a Spinn -universal elliptic operator.
Sketch of the proof. A spin manifold of dimension n always has a Spinn structure. The projection Spinn −→ SOn induces an isomorphism
∼ H ∗ (B Spin ; C).
H ∗ (BSOn ; C) =
n
∗
Thus the images in the ring H (B Spinn ; C) of the universal interpretations in
H ∗ (BSOn ; C) of td(TC M g ) and e(T M g ) (cf. Ex.2.3.6) satisfy all the required
properties. By the same reason Λ−1 NC M g has a universal interpretation.
The universal bundle for σ(6∂ ) can be defined as the bundle induced by
α
π ∗ S + (E Spinn ×Spinn V ) −→ π ∗ S − (E Spinn ×Spinn V ),
where π : E Spinn ×Spinn V −→ B Spinn is the bundle associated to the universal Spinn -bundle, V is the fundamental representation of Spinn , and
α(ξ, a) = ξ · a is Clifford multiplication. A description of the hgi-action
on this bundle is in [15, §A5].
By the Corollary 2.3.5 the formula of the index is
∗
+
−
dim M g /2 chg j (S (M ) − S (M ))
g
g
Indexg (6∂ ) = (−1)
td(TC M ), [M ] .
e(T M g ) chg (Λ−1 NC M g )
We want to simplify this formula a little more. Suppose that j ∗ T M decomposes as j ∗ T M = T M g ⊕ N M g as spin bundles. Then there is the following
result.
Corollary 3.4.3. The g-index of the Dirac operator on a spin manifold M
is
+
g
−
g
l
g chg (S (N M ) − S (N M ))
g
b
, [M ] ,
Indexg (6∂ ) = (−1) A(TC M )
chg (Λ−1 NC M g )
where M g is of dimension n = 2l.
Proof. The decomposition j ∗ T M = T M g ⊕ N M g induces the decomposition
j ∗ (S + (M ) − S − (M )) ∼
= (S + (M g ) − S − (M g )) ⊗ (S + (N M g ) − S − (N M g ))
on the spin bundles. Note that the action of g on M g is trivial, and so it is
trivial also on S ± (M g ). Then
chg j ∗ (S + (M ) − S − (M )) = ch(S + (M g ) − S − (M g )) chg (S + (N M g ) − S − (N M g )).
The g-index of 6∂ is therefore
D
E
dim M g
ch(S + (M g )−S − (M g )) td(TC M g ) chg (S + (N M g )−S − (N M g ))
g] .
,
[M
(−1) 2
e(T M g )
chg (Λ−1 NC M g )
41
b C M g ) agrees with the
It remains to show that the characteristic class A(T
+
g
−
g
g
(M )) td(TC M )
class ch(S (M )−S
when the manifold M g is spin. By the splite(T M g )
ting principle it is sufficient to show that this is true when TC M g decomposes
as sum of line bundles
TC M g = ⊕li=1 Li ⊕li=1 Li .
Furthermore as real bundles we have
T M g = ⊕li=1 (Li )R ,
where (Li )R is the underlying real bundle of Li . If this last decomposition
is a decomposition of spin bundles we can use the following argument. We
have that
S + (M g ) − S − (M g ) =
l
O
(S + ((Li )R ) − S − ((Li )R )),
i=1
and one can show (cf.[15, §7]) that for complex line bundles
S + ((Li )R ) ⊗ S + ((Li )R ) ∼
= Li
and
S − ((Li )R ) ⊗ S − ((Li )R ) ∼
= Li
as complex vector bundles. Therefore we have
Q
ch(S + (M g ) − S − (M g )) = li=1 ch(S + ((Li )R ) − S − ((Li )R ))
Q
= li=1 ch(S + ((Li )R )) − ch(S − ((Li )R ))
Q
= li=1 (ec1 (Li )/2 − e−c1 (Li )/2 ).
We saw that the splitting can be chosen so that
g
e(T M ) =
l
Y
c1 (Li ).
i=1
Finally, the Todd class is
td(TC M g ) =
=
=
Ql
c1 (Li )
−c1 (Li )
i=1 1−e−c1 (Li )
i=1 1−ec1 (Li )
Ql
Ql
c1 (Li )
−c1 (Li )
i=1 (1−e−c1 (Li ) )ec1 (Li )/2
i=1 (1−ec1 (Li ) )e−c1 (Li )/2
2
Ql c1 (Li )
.
i=1 ec1 (Li )/2 −e−c1 (Li )/2
Ql
Therefore the product
ch(S + (M g )−S − (M g )) td(TC M g )
e(T M g )
√
c1 (Li )
i=1 ec1 (Li )/2 −e−c1 (Li )/2
Ql
=
=
=
=
is
√
Ql
c1 (Li )
−c1 (Li )
√
√
√
√
i=1 e c1 (Li )/2 −e −c1 (Li )/2
i=1 e −c1 (Li )/2 −e c1 (Li )/2
√
√
Ql
Ql
√c1 (Li )
√−c1 (Li )
i=1 sinh( −c (L )/2)
i=1 sinh( c (L )/2)
1
1
i
i
l
l
b
b
A(⊕i=1 Li )A(⊕i=1 Li )
b C M g ).
A(T
Ql
42
If the decomposition above is not of spin bundles, an argument involving
classifying spaces shows that the identity
ch(S + (M g ) − S − (M g )) =
l
Y
(ec1 (Li )/2 − e−c1 (Li )/2 )
i=1
still holds (cf. [15, §A4]). Therefore the same argument applies to a decomposition in line bundles that is not necessarily spin.
There is a further simplification of this formula in the case where N M g is
equivariantly the underlying real bundle of a complex bundle U of dimension
r with c1 (U ) = 0. Since Λr (U ) is of dimension one, g acts on Λr (U ) by
multiplication for some ζ ∈ C. Suppose furthermore that ζ is a p-th root of
unity.
Proposition 3.4.4. Under these assumptions suppose in addition that p is
odd. Then the index formula of 6∂ is
D
E
b C M g ) chg (Λ−1 U )−1 , [M g ] ,
Indexg (6∂ ) = (−1)r+l ζ 1/2 A(T
where ζ 1/2 is the unique primitive root of unity that squares to ζ.
Proof. Since p is odd ζ 1/2 is unique. Since c1 (U ) = 0, the top exterior
power Λr (U ) is non-equivariantly a trivial line bundle. Therefore the line
bundle L which is non-equivariantly trivial and on which g acts by ζ 1/2 is an
equivariant square root of Λr (U ). It is a general fact (cf. [15]) that in this
case we have the identity
S + (UR ) − S − (UR ) = (−1)r L ⊗ Λ−1 U .
Therefore we obtain
chg (S + (N M g )−S − (N M g )) = (−1)r chg (L⊗Λ−1 U ) = (−1)r ζ 1/2 chg (Λ−1 U ).
Since N M g = UR we have that NC M g = (UR )C = U ⊕ U . Then we obtain
chg (S + (N M g )−S − (N M g ))
chg (Λ−1 NC M g )
=
=
(−1)r ζ 1/2 chg (Λ−1 )U
chg (Λ−1 U ⊗Λ−1 U )
(−1)r ζ 1/2 (chg (Λ−1 U ))−1 .
The result follows substituting this in the formula of 3.4.3.
4. The Witten Genus
The Witten genus of a "string manifold" M was first defined by Witten
in [18] as a formal power series. The motivation of his definition comes from
the study of the index of a formal Dirac operator on the loop manifold LM
and from the Atiyah-Singer fixed point theorem. The relations between the
Witten genus of M and the index of an actual Dirac operator on LM remain
an open problem today.
In this section we explain the geometry of the loop space LM and we
discuss the construction of a Dirac operator on it. Then we explain the main
conjecture relating the Witten genus and the Dirac operator.
43
4.1. The Manifold of Smooth Loops.
Definition 4.1.1. The loop space of a finite dimensional orientable manifold M is the set of smooth maps
LM = C ∞ (S 1 , M ).
We want to define a manifold structure on LM . For us, a manifold is the
following.
Definition 4.1.2. Let W be a locally convex vector space and X a set. A
local chart on X is a pair (U, φ), where U ⊂ X is a non-empty subset and
φ : U −→ W is a bijection onto an open subset φ(U ) ⊂ W . We say that two
charts (U, φ), (V, ψ) satisfying U ∩ V 6= ∅ are compatible if the restriction
φ ◦ ψ −1 |ψ(U ∩V ) : ψ(U ∩ V ) −→ φ(U ∩ V )
is a diffeomorphism between open subsets of W . A (smooth) atlas on X
is a collection of charts {(Ui , φi )|i ∈ I} satisfying
S
(1) i∈I Ui = X,
(2) all charts (Ui , φi ), (Uj , φj ) are compatible.
A maximal atlas of X is an atlas A that is maximal in the sense that
if (V, ψ) is any chart compatible with all charts in A then (V, ψ) ∈ A. A
manifold (modelled on W ) is a set X together with a maximal atlas A.
A chart (U, φ) ∈ A is called admissible. We say that (U, φ) is a chart at
x ∈ X if x ∈ U . The vector space W is called the model space of X. If
the model space is a Fréchet space (resp. Hilbert space) we say that X is a
Fréchet (resp. Hilbert) manifold.
A manifold structure on a set X always induces a topology on X. A subset
A of X is open if and only if φ(A ∩ U ) is open in the model space for all
chart (U, φ) of X. A manifold X will always carry this topology.
Proposition 4.1.3. The set of smooth loops LM of an orientable finite
dimensional manifold M admits a Fréchet manifold structure.
Sketch of the proof. We define the charts on LM , but we will skip the proof
that these charts form an atlas. As model space we choose the smooth loops
LRn , where n is the dimension of M . The vector space structure on LRn is
the pointwise structure. The topology on LRn is defined as follows. For each
positive integer m the space of continuous functions C(S 1 , Rm ) is endowed
with the (locally convex) topology induced by the ∞-norm
kf k∞ = sup kf (x)k.
z∈S 1
The topology on LRn is the initial topology for the maps jk : LRn −→
C(S 1 , Rnk ) defined as
jk (α)(z) = (α(z), α0 (z), . . . , αk−1 (z)).
Since each of the C(S 1 , Rnk ) is a locally convex vector space LRn is locally
convex. Furthermore it is complete since each of the C(S 1 , Rnk ) is a Banach
space.
44
We now define the local charts. Given a map f : M −→ N we denote
Lf : LM −→ LN the map defined as
Lf (α) = f ◦ α.
Let π : T M −→ M be the projection and s0 : M −→ T M be its zero section.
A local addition on M is a map η : T M −→ M such that η ◦ s0 = idM
and there is a neighbourhood V of the diagonal in M × M such that π ×
η : T M −→ M × M is a diffeomorphism onto V . It can be shown that any
finite dimensional manifold admits a local addition. We use this to define a
chart at a loop α ∈ LM as follows. Let Uα be the subset of LM defined as
Uα = {β ∈ LM |(α, β) ∈ LV }.
We want to identify Uα with the space of sections Γ(α∗ T M ). First note that
a section s ∈ Γ(α∗ T M ) is a map s = (s1 , s2 ) : S 1 −→ S 1 × T M such that
s1 = idS 1 and
α ◦ s1 = α = π ◦ s2 = Lπ(s2 ).
Furthermore s is smooth if and only if s1 and s2 are smooth. Then the
correspondence s = (s1 , s2 ) 7−→ s2 defines a bijection between Γ(α∗ T M )
and
(Lπ)−1 (α) = {γ ∈ LT M |Lπ(β) = α}.
Note that for a γ ∈ (Lπ)−1 (α) we have
L(π × η)(γ) = (α, η ◦ γ).
Therefore (Lπ)−1 (α) is the preimage by L(π × η) of the set of pairs of the
form (α, β) in LV , which is by definition {α} × Uα . Then the restriction of
L(π × η) to (Lπ)−1 (α) defines a bijection
L(π×η)
Γ(α∗ T M ) ∼
= (Lπ)−1 (α) −→ {α} × Uα ∼
= Uα .
The vector spaces Γ(α∗ T M ) are manifolds modelled on LRn (where n =
dim(M )). This follows from a general result since α∗ T M is a smooth orientable vector bundle over S 1 . The idea is to choose a diffeomorphism from
α∗ T M to S 1 × Rn that allows us to identify Γ(α∗ T M ) to Γ(S 1 × Rn ), that
is naturally identified to LRn . This construction does not depend on the
choice of the diffeomorphism α∗ T M ∼
= S 1 × Rn . We still should show that
the charts do not depend of the choices of local additions, and that the
transition functions are smooth. For this, we refer to [8].
We now state some properties of the manifold structure on LM . All the
proofs of what follows are in [8].
Proposition 4.1.4. Let M and N be orientable finite dimensional manifolds. The manifold structures on the loop space have the following properties.
(1) for any smooth map f : M −→ N , the map Lf : LM −→ LN defined
as
Lf (α) = f ◦ α
is smooth.
45
(2) let X be a (possibly infinite dimensional) manifold. For any map
f : X −→ LM the adjoint map f ∨ : S 1 × X −→ M defined as
f ∨ (z, x) = f (x)(z)
is smooth if and only if f is smooth. Therefore this defines a bijection
C ∞ (X, LM ) ∼
= C ∞ (S 1 × X, M ),
(3) the evaluation map e : S 1 × LM −→ M defined as
e(z, α) = α(z)
is smooth,
(4) the inclusion of constant maps i : M −→ LM defined as
i(x)(z) = x
for all x ∈ M and z ∈
S1
is an embedding.
Now consider an oriented smooth finite dimensional (real) vector bundle
p : E −→ M . Note that the fibre of a loop α ∈ LM by the map Lp : LE −→
LM is
(Lp)−1 (α) = {β ∈ LE|p(β(z)) = α(z) ∀z ∈ S 1 }.
Then given two loops β, γ ∈ (Lp)−1 (α), the points β(z) and γ(z) are in
the fibre Eα(z) for all z ∈ S 1 . We can then define the loops β + γ and λβ for
all scalars λ using the vector space structure of the fibres of E:
(β + γ)(z) = β(z) + γ(z), (λβ)(z) = λβ(z),
for all z ∈
S1.
Proposition 4.1.5. This fibrewise vector space structure defines a (real)
vector bundle structure on Lp : LE −→ LM .
For a vector bundle E over M we denote by LE this vector bundle over
LM .
Note 4.1.6. This construction is also valid for principal bundles. Let P be
a principal G-bundle over M . Then the pointwise action induces an LGaction on LP . It can be showed that the local trivialisations of P induces a
principal LG-bundle structure on LP over LM .
One can ask what is the relation between the tangent bundle T LM of LM
and the bundle LT M .
Proposition 4.1.7. The unique map f : T LM −→ LT M whose adjoint is
s ×id
de
0
f ∨ : S 1 × T LM −→
T S 1 × T LM −→ T M
is an isomorphism of vector bundles over LM .
Note that this isomorphism induces a LR-module structure on T LM . This
will be important for the definition of a spin structure on LM (cf. section
4.3).
Given a Riemannian metric h−, −i on M we can define a (weak) Riemannian metric on LM using this isomorphism. Let α ∈ LM be a loop. We define
an inner product on LT Mα as follows. Let v, w ∈ LT Mα . This means that
46
for any z ∈ S 1 the values v(z) and w(z) are in the same fibre T Mα(z) . Then
we can define
Z
hv, wiα =
hv(−), w(−)iα(−)
S1
(we take this integral with respect to the Haar measure on S 1 ).
The manifold LM also carries a canonical S 1 -action by rotation. It is
defined as
(zα)(z 0 ) = α(zz 0 )
for all z, z 0 ∈ S 1 .
Proposition 4.1.8. The action of S 1 on LM is smooth, and the Riemannian
metric on LM induced by any Riemannian metric on M is S 1 -invariant.
Proof. The adjoint of the map S 1 × LM −→ LM defining the action is the
composition
δ
·×id
e
S 1 × (S 1 × LM ) ∼
= S 1 × S 1 × LM −→ S 1 × LM −→ M,
where δ flips the two S 1 components and · is the product in S 1 . This composition is smooth by 4.1.4, and always by 4.1.4 the action is therefore smooth.
Given two smooth curves αt , βt in LM such that α0 = β0 ∈ LM the inner
product of zα00 and zβ00 is
R
hzα00 , zβ00 i = RS 1 hde(0z 0 , (zαt )00 , de(0z 0 , (zβt )00 i
= S 1 hα00 (zz 0 )), β00 (zz 0 )i .
Since the multiplication by z is an orientation preserving diffeomorphism of
S 1 the change of variable shows that this is equal to
Z
α00 (−), β00 (−) = α00 , β00 .
S1
Now, suppose that G is a compact Lie group acting on M . Then LG acts
on LM by pointwise action:
(γα)(z) = γ(z)α(z)
for all z ∈ S 1 , for γ ∈ LG and α ∈ LM . Furthermore LG is a Lie group,
since the group law has adjoint
e
·
S 1 × (LG × LG) −→ G × G −→ G,
which is smooth.
Proposition 4.1.9. The action of LG on LM is smooth. Furthermore the
Riemannian metric on LM induced by a G-invariant Riemannian metric on
M is LG-invariant.
Proof. The adjoint of the map LG × LM −→ LM defining the action is
e
·
S 1 × (LG × LM ) −→ G × M −→ M,
which is smooth. Therefore the action is smooth.
Let αt , βt be two smooth curves in LM such that α0 = β0 ∈ LM and
γ ∈ LG. The action of γ on the tangent vector defined by αt is
γ · α00 = dγ(α00 ) = (γαt )00 ,
47
and similarly for βt . Then the inner product of γ · α00 and γ · β00 is
R
hγ · α00 , γ · β00 i = RS 1 hde(0z , (γαt )00 ), de(0z , (γβt )00 )i
= RS 1 h(γ(z)αt (z))00 , (γ(z)βt (z))00 i
= RS 1 hdγ(z)α00 (z), dγ(z)β00 (z)i
= RS 1 hγ(z) · α00 (z), γ(z) · β00 (z)i
= S 1 hα00 (z), β00 (z)i
= hα00 , β00 i .
Note 4.1.10. Given
principal G-bundle P
By 4.1.6 LP carries a
smooth LG-manifold.
a manifold F with smooth G action and a smooth
over M we can form the associated bundle P ×G F .
canonical principal LG-bundle structure and LF is a
Furthermore we have the identity
∼ LP ×LG LF.
L(P ×G F ) =
We are interested in the action of a certain Lie group on LM that "contains" both the actions of S 1 and LG.
Definition 4.1.11. The semidirect product of LG and S 1 is the manifold
LG × S 1 with the group law defined as
(γ, z) · (γ 0 , z 0 ) = (γ · (z · γ 0 ), zz 0 ).
We denote the resulting Lie group LG o S 1 .
The semidirect product LG o S 1 acts on LM by
(γ, z) · α = γ · (z · α).
Proposition 4.1.12. The action of LG o S 1 on LM is smooth and a Ginvariant Riemannian metric on M induces a LG o S 1 -invariant metric on
LM .
Proof. This action is the composition of two smooth actions and it is therefore smooth. Since a G-invariant metric on M induces an LG-invariant metric on LM which is also S 1 -invariant, this metric is LG o S 1 -invariant. 4.2. Clifford Algebras and Spinor Representations in Hilbert Spaces.
We want to generalize the constructions of the spin bundles and of the Dirac
operator done in section 3 for the infinite dimensional manifold LM . Then
study the index of the Dirac operator on LM with the action of LG o S 1
described in the last section.
We start with the definition of the spin group in infinite dimension and
the choice of its fundamental representation. Note that the definition of the
Clifford algebra of a complex vector space of section 3 did not require that the
vector space in of finite dimension. Suppose that V is an infinite dimensional
real vector space with an inner product h−, −i. This inner product extends
canonically to an hermitian product on the complexification VC = V ⊗ C.
We denote by C(V ) the complex Clifford algebra Cl(VC , q), where q is the
quadratic form induced by this hermitian product on VC . Recall that there is
an involution α : C(V ) −→ C(V ) induced by − idV inducing a Z/2-grading
on C(V ).
48
Definition 4.2.1. Let B be an algebra with involution ∗ . A Clifford map
f : V −→ B is called skew-adjoint if
f (v)∗ = −f (v).
The skew-adjoint Clifford maps satisfy the following universal property.
Proposition 4.2.2. A skew-adjoint Clifford map f : V −→ B induces a
unique involution preserving algebra homomorphism F : C(V ) −→ B such
that F |V = f .
We want to work with a bigger object than C(V ), which will be a Banach
algebra with involution.
Definition 4.2.3. A C ∗ -algebra is a Banach algebra A together with an
involution ∗ : A −→ A satisfying
kaa∗ k = kak2 .
Let A and A0 be two C ∗ -algebras. A C ∗ -morphism is a continuous algebra
homomorphism f : A −→ A0 that commutes with the involutions.
Let MV be the set of all the algebra homomorphisms from V to all C ∗ algebras which commute with the involution. The following is proved in
[12, §1.2].
Proposition 4.2.4. For all a ∈ C(V ) the supremum
kak∞ = sup{kπ(a)k|π ∈ MV }
is finite, and it defines a norm on C(V ).
Definition 4.2.5. The C ∗ -Clifford algebra of V is the C ∗ -algebra C[V ]
defined as the completion of C(V ) with respect to the norm k.k∞ . The
involution is the map α : C[V ] −→ C[V ] induced by − idV .
The C ∗ -Clifford algebra satisfies the following universal property (cf. [12,
§1.2]).
Proposition 4.2.6. Let B be a C ∗ -algebra and f : V −→ B a skew-adjoint
Clifford map. Then there is a unique isometric C ∗ -morphism F : C[V ] −→ B
such that F |V = f .
We give a list of the basic properties satisfied by the C ∗ -Clifford algebra.
The proofs are in [12, §1.2].
Proposition 4.2.7.
(1) The canonical embedding V −→ C[V ] is an
isometry,
(2) each linear isometry f : V −→ V 0 extends to a unique isometric C ∗ morphism
fe: C[V ] −→ C[V 0 ]
such that fe|V = f ,
(3) the involution defines a Z/2-grading on C[V ]
C[V ] = C[V ]+ ⊕ C[V ]− ,
and C[V ]± is the closure of C(V )± in C[V ],
49
(4) if V is the Hilbert space completion of V then the inclusion V −→ V
induces an isomorphism
C[V ] ∼
= C[V ]
As in section 3 we want to define a fundamental representation of C[V ].
From now on, suppose that V is also complete with respect to the inner
product topology, then it is an Hilbert space. This representation will depend
on the choice of a unitary structure on V .
Recall that a unitary structure on V is a linear isometry J ∈ O(V ) satisfying J 2 = − idV . Since V is infinite-dimensional such a structure always
exists. Let VJ be the resulting complex vector space, whose underlying real
vector space is V and the scalar multiplication by i ∈ C is defined as
iv = J(v)
for all v ∈ V . The inner product h−, −i on V extends to an hermitian
product on VJ defined by
hv, wiJ = hv, wi + i hv, J(w)i .
Now, for all n ∈ N consider the n-th complex exterior power Λn (VJ ) with
the inner product defined by bilinear extension of
hv1 ∧ · · · ∧ vn , w1 ∧ · · · ∧ wn i = det[hvi , vj iJ ].
This inner product induces canonically an inner product on the (algebraic)
direct sum
M
Λ(VJ ) =
Λn (VJ ).
n∈N
The space Λ(VJ ) is not complete with respect to this inner product.
Definition 4.2.8. The Fock space of VJ is the completion HJ of Λ(VJ ).
The Fock space is the vector space that will carry the fundamental representation of C[V ]. Now we need to define an action of C[V ] on HJ . We start
by defining a map πJ : V −→ B(HJ ), where B(HJ ) is the space of bounded
endomorphisms of HJ .
Definition 4.2.9. For all v ∈ V , the creator of v is the complex linear
operator cJ (v) : HJ −→ HJ defined by
cJ (v)(ζ) = v ∧ ζ
on Λ(VJ ) and extended to HJ .
The annihilator of v is the complex linear operator aJ (v) : HJ −→ HJ
defined on elements of the form w1 ∧ · · · ∧ wn by
n
X
aJ (v)(w1 ∧ · · · ∧ wn ) =
(−1)j+1 hwj , vi w1 ∧ · · · ∧ w
bj ∧ · · · ∧ wn
j=0
and extended by linearity and completion (where w
bj means omitted).
The following property is proved in [12, §2.3].
Proposition 4.2.10. For v ∈ V and for all ξ, η, ∈ Λ(VJ ) we have
hcJ (v)(ξ), ηi = hξ, aJ (v)(η)i .
50
Consider for each v ∈ V the map πJ (v) : HJ −→ HJ defined as
πJ (v) = cJ (v) − aJ (v).
Recall that the Banach algebra B(HJ ) is a C ∗ -algebra with respect to the
involution defined as the adjunction.
Proposition 4.2.11. For all v ∈ V the operator πJ (v) is bounded, and the
function πJ : V −→ B(HJ ) associating πJ (v) to v extends to a C ∗ -morphism
πJ : C[V ] −→ B(HJ ).
Proof. The properties of the creators and annihilators imply that
πJ (v)2 = kvk2 id
and that πJ is R-linear, that is πJ is a Clifford map. Furthermore by the
last proposition πJ (v) is a skew-adjoint operator. Therefore
kπJ (v)(η)k2 = hπJ (v)(η), πJ (v)(η)i
= η, πJ (v)2 (η)
,
= η, kvk2 (η)
= kvk2 kηk2
that is πJ (v) is bounded for all v ∈ V .
Since each πJ (v) is skew-adjoint, the Clifford map πJ : V −→ B(HJ ) is
skew-adjoint and therefore induces a C ∗ -morphism
C[V ] −→ B(HJ )
by proposition 4.2.6.
Definition 4.2.12. The Fock representation of C[V ] is the representation defined by the C ∗ -morphism
πJ : C[V ] −→ B(HJ )
The following is proved in [12, §2.4.2].
Proposition 4.2.13. The Fock representation of C[V ] is irreducible.
The restriction of the Fock representation on the sub-algebra C[V ]+ is no
longer irreducible.
Definition 4.2.14. The even Fock space H+
J is the closure in HJ of the
subspace
M
Λ2n (VJ ).
n∈N
Analogously the odd Fock space H−
J is the closure of
M
Λ2(n+1) (VJ )
n∈N
in the same space.
Let ΓJ be the unitary operator on HJ defined on the subspace Λn (VJ ) by
ΓJ |Λn (VJ ) = (− id)n .
Note that the even and the odd Fock spaces can be written as
H±
J = (id ±ΓJ )(HJ ).
51
It is shown in [12, §2.5.1] that ΓJ satisfies the condition
ΓJ πJ (a)ΓJ = πJ (α(a))
for all a ∈ C[V ] (recall that α is the involution of C[V ]). Then if a ∈
C[V ]+ the operator πJ (a) commutes with ΓJ , and therefore it preserves the
−
subspaces H+
J and HJ . By restriction we obtain two representations
πJ± : C[V ]+ −→ H±
J
that satisfy the following (cf. [12, §2.5.3,§2.5.5]).
Proposition 4.2.15. The representations πJ+ and πJ− are irreducible inequivalent representations of C[V ]+ .
We return now to the Fock representation of C[V ]. One can ask when
two Fock representations associated to two different unitary structures J
and K on V are unitary equivalent (that is there is an equivariant unitary
transformation HJ −→ HK ).
Definition 4.2.16. An operator L ∈ B(V ) is a Hilbert-Schmidt operator if there is an orthonormal Hilbert basis {ei , i ∈ I} of V such that the
real sequence
X
kLei k2
i∈I
converges. In this case we denote
!1/2
kLkHS =
X
kLei k2
.
i∈I
The set of Hilbert-Schmidt operators on V is denoted HS(V ).
If there is such a basis, then the convergence is satisfied for any other
orthonormal Hilbert basis. Furthermore it can be shown that the set HS(V )
is an ideal of B(V ) and kLkHS defines a norm on HS(V ). The topology
induced by the norm k.kHS is independent of the choice of the basis. We
always endows HS(V ) with this topology.
The following is proved in [12, §3.4.1].
Proposition 4.2.17. Two Fock representations πJ and πK are equivalent if
and only if
K − J ∈ HS(V ).
As in section 3 we want to define the spin group of V , and its spin representation induced by the Fock representation. The group playing the role of
On in this context will not be the orthogonal group O(V ), but the restricted
orthogonal group of V . This group depends on the unitary structure J on
V.
Definition 4.2.18. The restricted general group of V is defined as
GLJ (V ) = {A ∈ GL(V )|[J, A] ∈ HS(V )},
where [J, A] = JA − AJ is the commutator.
52
The set of the operators A ∈ B(V ) such that [J, V ] is in HS(V ) form a
Banach algebra BJ (V ) with respect to the norm
kAkJ = kAk + k[J, A]kHS ,
where kAk is the operator norm. Then GLJ (V ) is the group of unit of
BJ (V ). Therefore it is open in BJ (V ), and thus it is a Banach Lie group.
Definition 4.2.19. The restricted orthogonal group of V is
OJ (V ) = O(V ) ∩ GLJ (V ) = {A ∈ O(V )|[J, A] ∈ HS(V )}.
Note that an operator A ∈ O(V ) is a unitary operator in U (VJ ) if and
only if AJ = JA. Therefore U (VJ ) is a subgroup of OJ (V ).
The principal motivation for the use of OJ (V ) is the resolution of "implementation problems".
Definition 4.2.20. Let g ∈ O(V ) inducing ge : C[V ] −→ C[V ] on the C ∗ Clifford algebra. We say that ge can be implemented in the Fock representation πJ if there is a unitary operator U ∈ U (HJ ) such that
πJ (e
g (a)) = U πJ (a)U ∗
for all a ∈ C[V ]. We say that an operator U ∈ U (HJ ) satisfying this property
implements ge.
The following result shown in [12, §3.3.5] characterizes the elements of
O(V ) whose extension to C[V ] can be implemented in πJ .
Proposition 4.2.21. An element ge : C[V ] −→ C[V ] can be implemented in
πJ if and only if g ∈ OJ (V ).
This new characterisation of OJ (V ) allows us to construct the central
extension analogous to Z/2 −→ Pinn −→ On .
Definition 4.2.22. The Pin group of V with respect to J is the set PinJ (V )
of unitary operators U ∈ U (HJ ) that implements some ge : C[V ] −→ C[V ]
induced by a g ∈ OJ (V ).
Proposition 4.2.23. The set PinJ (V ) is a subgroup of U (HJ ) and an element U ∈ PinJ (V ) implements a unique ge. The resulting map PinJ (V ) −→
OJ (V ) induces a central extension
S 1 −→ PinJ (V ) −→ OJ (V ).
Proof. Suppose that U ∈ PinJ (V ) implements two ge, e
h : C[V ] −→ C[V ].
Then
πJ (e
g (a)) = U πJ (a)U ∗ = πJ (e
h(a))
for all a ∈ C[V ]. In particular for v ∈ V we have
πJ (g(v)) = πJ (h(v)).
Since πJ (w)2 = kwk2 id for all w ∈ V , the map πJ is injective on V . Therefore g = h. Since the extensions ge, e
h are unique we obtain ge = e
h.
0
e
Given U implementing ge and U implementing h we show that U U 0 implements the composition ge ◦ e
h. This is the map induced by gh. Then
53
this will show that PinJ (V ) is a group and that PinJ (V ) −→ OJ (V ) is a
homomorphism. A computation shows this result:
U U 0 πJ (a)(U U 0 )∗ = U U 0 πJ (a)(U 0 )∗ U ∗
= U πJ (e
h(a))U ∗
= πJ (e
g (e
h(a))),
for all a ∈ C[V ].
By irreducibility of πJ it follows that the kernel of this homomorphism is
S1.
Now we want to find the infinite dimensional analogs of SOn and Spinn .
For this, we define a group homomorphism µ : PinJ (V ) −→ {−1, 1} that
allows us to define another homomorphism : OJ (V ) −→ {−1, 1} that plays
the role of a determinant.
Recall that ΓJ is the unitary operator on HJ defined on the subspace
Λn (VJ ) by
ΓJ |Λn (VJ ) = (− id)n .
The property of ΓJ stated above says exactly that ΓJ implements the automorphism α of C[V ] inducing the grading. Then given U ∈ U (HJ ) implementing ge we have that ΓJ U ΓJ also implements ge:
ΓJ U ΓJ πJ (v)Γ∗J U ∗ Γ∗J
= ΓJ U πJ (−v)U ∗ Γ∗J
= −ΓJ πJ (e
g (v))Γ∗J
= πJ (e
g (v)).
Therefore ΓJ U ΓJ = µ(U )U for some µ(U ) ∈ S 1 . Since Γ2J = id the coefficient µ(U ) is 1 or −1. This defines a group homomorphism
µ : PinJ (V ) −→ {−1, 1}.
Definition 4.2.24. The Spin group of V with respect to J is the subgroup
of PinJ defined as
SpinJ (V ) = µ−1 (1).
The value µ(U ) only depends on the implemented endomorphism ge. Suppose that U and U 0 implement ge. Then U 0 = zU for some z ∈ S 1 , and
so
µ(U 0 )U 0 = ΓJ U 0 ΓJ = zΓJ U ΓJ = zµ(U )U = µ(U )U 0 .
Therefore µ(U ) = µ(U 0 ).
This defines a group homomorphism : OJ (V ) −→ {−1, 1} by (g) = µ(U )
for some U implementing ge.
Definition 4.2.25. The reduced special orthogonal group of V with
respect to J is the subgroup of OJ (V ) defined as
SOJ (V ) = −1 (1).
By definition the central extension S 1 −→ PinJ (V ) −→ OJ (V ) restricts
to a central extension
S 1 −→ SpinJ (V ) −→ SOJ (V ).
We want to define representations of SpinJ (V ). It is possible to show that
each element U ∈ SpinJ (V ) defines an inner automorphism of HJ , associated
54
to an element g ∈ C[V ]+ . Furthermore the automorphisms π ± (g) ∈ B(H±
J)
±
depend only on U . This defines maps SpinJ (V ) −→ B(HJ ).
Definition 4.2.26. The maps SpinJ (V ) −→ B(H±
J ) are called spinor representations.
4.3. Spin Structures and Dirac Operators on LM . Let M be an oriented Riemannian manifold of even dimension n. The basic idea for the
definition of a spin structure on LM is the same as in finite dimension. However, in infinite dimensions we have to deal with Hilbert space completions
in order to consider the spin groups.
The Hilbert space we consider here is the space of square-integrable loops
L2 Rn = L2 (S 1 , Rn ). Since n is assumed to be even, the space L2 Rn has a
standard unitary structure. It is defined as follows. Let J0 : Rn −→ Rn be
the standard complex structure defined as
J0 (e2k ) = e2k−1 , J0 (e2k−1 ) = −e2k ,
where {ei } is the standard basis. The standard unitary structure of L2 Rn is
the unique linear operator J : L2 Rn −→ L2 Rn that satisfies
J(v cos(kθ)) = v sin(kθ),
J(v sin(kθ)) = −v cos(kθ),
J(v) = J0 (v),
for all k ∈ Z\{0}, all θ ∈ R and v ∈ Rn , where we inject Rn in L2 Rn as
constant loops.
We always consider L2 Rn with this unitary structure J. It is shown in
[13, §12.5] that the action of LSOn on LRn extends to an action on L2 Rn
that defines an inclusion LSOn ,→ SOJ (L2 Rn ).
Definition 4.3.1. The frame bundle PSOJ (LM ) of LM is the principal
SOJ (L2 Rn )-bundle over LM defined as
PSOJ (LM ) = LPSO (T M ) ×LSOn SOJ (L2 Rn ),
where PSO (T M ) is the frame bundle of M .
This bundle allows us to define a spin structure on LM in a similar way
as in finite dimensions.
Definition 4.3.2. A spin structure on LM is a principal SpinJ (L2 Rn )bundle PSpin (LM ) over LM together with a SpinJ (L2 Rn )-equivariant morphism PSpin (LM ) −→ PSOJ (LM ) whose restriction on the fibres gives the
central extension
S 1 −→ SpinJ (L2 Rn ) −→ SOJ (L2 Rn ).
It is possible to find conditions on M that guarantee the existence of a
spin structure on LM . We discuss that briefly. For a deeper discussion the
reader is invited to look in [10]. Suppose that M is a spin manifold. This
means that there is a lift
B Spin
n
;
vv
v
vv
vv
vv
/ BSOn
M
fe
f
55
classifying the spin structure, where f : M −→ BSOn classifies T M . This
"loops" to a diagram
BL Spinn .
s9
Lfe sss
LM
The inclusion LSOn ,→ SOJ
sss
sss
Lf
/ BLSOn
(L2 Rn )
defines a map
L Spinn −→ LSOn ,→ SOJ (L2 Rn ).
The pullback of the central extension SpinJ (L2 Rn ) −→ SOJ (L2 Rn ) by this
e Spinn of L Spinn .
map defines an S 1 -extension L
e
Therefore if the map Lf lifts to a map
e Spinn
BL
C
e fe
L
/ B Spin (L2 Rn ) ,
J
BL Spinn
t:
Lfe ttt
LM
tt
tt
tt
Lf
/ BLSOn 
/ BSOJ (L2 Rn )
e fe
L
e Spinn −→ B SpinJ (L2 Rn ) classifying a spin
we obtain a map LM −→ B L
structure on LM .
Definition 4.3.3. A string structure on a spin manifold M is a choice of
e fe. A string manifold is a spin manifold with a chosen string structure.
lift L
e fe exists it classifies a line bundle over the "looped" spin
If this lifting L
structure LPSpin (T M ) of M that pulls back to the central extension
e Spinn −→ L Spinn .
L
Line bundles over a loop manifold are related to "gerbes" over the original manifold. Such a line bundle exists if there is a gerbe over PSpin (T M )
that pulls back to the gerbe of Spinn associated to the canonical generator of H 3 (Spinn ; Z). This is the same as the existence of an element of
H 3 (PSpin (T M ); Z) that restricts to the canonical generator of H 3 (Spinn ; Z)
on each fibre. Using the Serre spectral sequence it is possible to show that a
string structure exists if and only if the image λ of the canonical generator by
the fourth differential d4 : H 3 (Spinn ; Z) −→ H 4 (M ; Z) is zero. Furthermore
λ satisfies the relation
2λ = p1 (M ),
where p1 (M ) = −c2 (T M ⊗ C) is the first Pontryagin class. Therefore if the
cohomology group H 4 (M ; Z) is "good enough", the Pontryagin class p1 (M )
can carry important information about the existence of a spin structure on
LM . This concludes our overview on the existence of a spin structure on
LM .
Now suppose that LM carries a fixed spin structure PSpin (LM ). The spin
bundles of LM are defined as follows.
56
Definition 4.3.4. The (complex) spinor bundles of LM are the complex
vector bundles over LM defined as
S + (LM ) = PSpin (LM ) ×SpinJ H+
J
and
S − (LM ) = PSpin (LM ) ×SpinJ H−
J.
Suppose now that M is a string manifold carrying a spin action of G that
lifts to the string structure of M . Then the smooth action of LG on LM
lifts to the spin structure. The action of S 1 on LM is always a spin action.
Therefore S + (LM ) and S − (LM ) are LG o S 1 -vector bundles.
We now define a Dirac operator 6∂ : Γ(S + (LM )) −→ Γ(S − (LM )). The
result 3.3.5 is still valid, in the sense that a connection on PSpin (LM ) induces
a covariant derivative
∇ : Γ(S + (LM )) −→ Γ(hom(T LM, S + (LM ))),
where hom(T LM, S + (LM )) is the bundle of morphisms of vector bundles
from T LM to S + (LM ).
As in the finite dimensional situation we would like to compose ∇ with
Clifford multiplication. However in infinite dimension it is not true that
hom(T LM, S + (LM )) is isomorphic to the tensor product T LM ∗ ⊗S + (LM ).
Furthermore, since the Fock representation is defined from L2 Rn , if we want
to extend Clifford multiplication to a bundle we need a bundle which has
L2 Rn as model space. For this reason we define the bundle L2 T M of squareintegrable loops.
Definition 4.3.5. The bundle of square-integrable loops of T M is the
Hilbert vector bundle L2 T M over LM defined as
L2 T M = LPSO (T M ) ×LSOn L2 Rn .
As in finite dimensions, we define Clifford multiplication as a map
c : L2 T M −→ B(HJ ).
Since L2 Rn is a Hilbert space there is a vector bundle isomorphism L2 T M ∼
=
(L2 T M )∗ . Therefore Clifford multiplication can be seen as a map
c : (L2 T M )∗ −→ B(HJ ).
It was shown in [16] that there is an inner product on (LRn )∗ defining a
topology so that one could view (L2 T M )∗ as a Hilbert completion of LT M ∗ .
Thus by modifying the set-up Clifford multiplication restricts to a map
c : (LT M )∗ −→ B(HJ ).
It is proved in [16, §5.10] that this map induces a vector bundle morphism
c : hom(T LM, S + (LM )) −→ S − (LM ).
Therefore there is an induced map on sections
c : Γ(hom(T LM, S + (LM ))) −→ Γ(S − (LM )).
Definition 4.3.6. The Dirac operator on the spin manifold LM is the
composition
∇
c
6∂ : Γ(S + (LM )) −→ Γ(hom(T LM, S + (LM ))) −→ Γ(S − (LM )).
57
It is shown in [16] that the S 1 -action on LM lifts to actions on S + (LM )
and S − (LM ) and that the Dirac operator is S 1 -equivariant. It is natural to
ask if 6∂ : Γ(S + (LM )) −→ Γ(S − (LM )) can be defined to be LG-equivariant,
for the action of a general Lie group G that lifts to the spin structure and to
the string structure of M . This subject is still fairly new and unstudied and
this is one of the many open questions about the Dirac opertors on LM . It
seems however that this is true at least for a homotopy equivalent subgroup
of LG (cf. [16]).
4.4. The Witten Genus. In the last section we constructed an operator
6 : Γ(S + (LM )) −→ Γ(S − (LM )) on the loop manifold LM of a string mani∂
fold M , which is S 1 -equivariant. Furthermore for a string action of a compact
Lie group G it should be possible to define this operator to be LG o S 1 equivariant (or at least, a group equivalent to LG o S 1 ). The idea of the
Witten genus is to take the ξ-index of 6∂ for a topological generator ξ ∈ S 1 .
For the equivariant Witten genus we would like to consider its (γ, ξ)-index
for some γ ∈ LG and a topological generator ξ of S 1 . However, since the
dimensions are infinite many deep analytical problems arise, and some concepts still have to be understood. Our aim here is to illustrate these problems
and explain some of the related conjectures.
The first difficulty concerns the choices of the topologies on Γ(S ± (LM )),
and what a representation means in this context. So far we always meant by
"representation" a finite dimensional vector space with a continuous linear
action. Since we are dealing with infinite dimensional manifolds we cannot
expect ker 6∂ and coker 6∂ to be finite dimensional spaces, and so we have to
drop the finite dimensional condition. By representation we mean a locally
convex complete topological vector space with a continuous linear action.
Definition 4.4.1. The representation ring of an (infinite dimensional) Lie
group K is the group completion of the semi-group of isomorphism classes of
representations of K with respect to the direct sum. This group is denoted
RK .
Another analytical problem in infinite dimensions is that there is no canonical locally convex topology on the completion of the tensor product of two
locally convex complete vector spaces. We avoid this problem considering
only sums of representations. We would like to define the LG o S 1 -index of
6∂ . Here the choice of the topology on Γ(S ± (LM )) is clearly a problem. To
consider ker 6∂ and coker 6∂ as representations of LG o S 1 we need at least locally convex complete topologies on Γ(S ± (LM )) for which ker 6∂ and coker 6∂
are closed subspaces. It is possible to define a topology on C ∞ (M, S ± (LM ))
so that a curve c : R −→ C ∞ (M, S ± (LM )) is continuous if its adjoint map
č : R × LM −→ S ± (LM ) is smooth. However, the induced topology on
Γ(S ± (LM )) has not been studied but is not likely to be even locally convex.
But it is possible to find a finer locally convex topology so that the associated
smooth curves of Γ(S ± (LM )) are the same as the smooth curves induced
by those of C ∞ (M, S ± (LM )). It is also possible to try to put a Hilbert
space structure on Γ(S ± (LM )). Picking a Riemannian metric on S ± (LM )
58
we would like to define an inner product on Γ(S ± (LM )) by
Z
0
s(α), s0 (α) α ”
” s, s =
LM
as in finite dimensions. To use this approach we clearly need to know what
an integral over an infinite dimensional manifold is. Since we are dealing
with loop spaces, it is possible to define such an integral, embedding LM
in a certain class of Sobolev loops. However, this point of view contains
considerably hard analytical issues.
Assuming that this topology can be defined we can define the index of 6∂
as follows.
Definition 4.4.2. The LG o S 1 -index of the Dirac operator on LM is the
formal difference of representations
IndexLGoS 1 6∂ = ker 6∂ − coker 6∂ ∈ RLGoS 1 .
It is called the G-equivariant Witten genus of M .
The second important problem to discuss is the index theory for operators
over LM . The key result of index theory for finite dimensional manifold is
the commutativity of the diagram
{elliptic G-equivariant operators on M }
IndexG
RG ∼
= KG (∗) o
φ−1
σ
/ KG (T M )
i!
KG (T Rm )
that allows us to compute the index of an elliptic operator topologically. It
is natural to ask if it is possible to construct an analogous diagram for LM .
The first problem in doing this is that in infinite dimension the definition
of ellipticity does not make sense. The second obstacle is the definition of
K-theory. It is possible to define it for non-locally compact spaces using
classifying spaces and spectra, but a description in terms of vector bundles
is no longer available. Another approach could be to consider a completely
analogous construction as in finite dimension, but considering a certain class
of infinite dimensional vector bundles over LM . In this case there will be
a problem with the completeness and the tensor products. It is however
possible to construct a "symbol" σ(6∂ ) of the Dirac operator on LM as a
virtual bundle over the cotangent bundle T ∗ LM . Even supposing that this
virtual bundle admits a LG o S 1 -vector bundle structure we cannot go so
far as we would like to. Since a well-defined K-theory is not available we
cannot use maps defined on the whole K-theory as in finite dimensions (as
for instance Thom isomorphisms or the map i! ) to map this symbol down
to a bundle over a point. This would be a formal difference of vector spaces
with LG o S 1 -action, that with a lot of luck could be in the representation
ring of LG o S 1 and be the same as the G-equivariant Witten genus. We
can however try to built a formal difference of representations from σ(6∂ ), as
the composition φ−1 ◦ i! does in finite dimensions. Even if this is possible,
we cannot expect to show that this is the G-equivariant Witten genus using
the same methods as in finite dimensions. This because in the proof of 2.2.7
we really use properties of maps defined on the whole K-theory. The piece
59
of theory that is really missing here is a framework, a universe where our
bundles live and maps moving these objects.
Third, we can still try to consider the traces on the G-equivariant Witten
genus. We forget for a moment the G-action on M . Even if the previous
construction works, we do not have any of the algebraic topology tools such
as characteristic classes since there is no K-theory involving vector bundles
in infinite dimensions. However, we try to give some sense of the ξ-index
of 6∂ , for a topological generator ξ of S 1 . Since the spaces ker 6∂ and coker 6∂
are infinite dimensional the trace does not make sense for now. The AtiyahSinger fixed point theorem suggests that in order to define some kind of index
it should be sufficient to look over the fixed points of the action. The fixed
point manifold of the S 1 -action on LM is M embedded as constant loops.
The idea is to cut the bundles S ± (LM )|M as a direct sum with respect to the
S 1 -action, and then take the (hopefully defined) ξ-index of the restrictions
of 6∂ on the sections of these sub-bundles. We can form the direct sum of
sub-bundles Ek± of S ± (LM )|M defined as
Ek± = {a ∈ S ± (LM )|M |z · a = z k a for all z ∈ S 1 }.
The direct sum of this bundles is dense in S ± (LM )|M , but we do not know
if they are of finite dimension. However, by equivariancy of 6∂ we can restrict
it to operators
6∂k : Γ(Ek+ ) −→ Γ(Ek− ).
If the 6∂k are elliptic, it is possible to define the following.
Definition 4.4.3. The ξ-index of 6∂ is the formal power series W M (ξ) on ξ
defined by
X
W M (ξ) =
(Indexξ 6∂k )ξ k .
It is called the Witten genus of M .
The following conjecture has been suggested first by Edward Witten in
his famous paper [18]. It remains an open problem.
Conjecture 4.4.4. Let M be a compact string manifold of even dimension
d, and suppose that the Witten genus on M can be defined. Then it is given
by the formula
*
!
+
∞
O
b C M ) ch
W M (ξ) = (−1)d/2 ξ −d/24 A(T
(Λ−ξk TC M )−1 , [M ] .
k=1
Note 4.4.5. In the ring K(M )[[t]] there is the identity
(Λ−t E)−1 = St (E),
P k k
where St (E) =
t S (E) is the sum of the tensor powers of E. Then the
Witten genus formula is sometimes written as
*
!
+
∞
O
M
d/2 −d/24
b C M ) ch
W (ξ) = (−1) ξ
A(T
(Sξk TC M ) , [M ] .
k=1
In the following section we will explain why it is plausible to think that this
conjecture could be true. The key is the Atiyah-Singer fixed point theorem.
60
It is natural to ask if it is possible to associate to the G-equivariant Witten
genus a formal power series, keeping track of the LG-action. This procedure
of "cutting" in order to take the trace does not work in the equivariant case,
since the LG-action does not restrict to the spaces Ek± . In section 5.2 we will
consider these traces in a "truncated case", and define a formal power series
for "good loops" in LG. In this case there is no known relation with the
actual Dirac operator on LM , but this formal power series will be a formal
limit of a finite dimensional formula.
5. The Witten Conjecture and the Truncated Equivariant
Witten Genus
In section 5.1 we explain how to arrive at the formula of the Witten
conjecture 4.4.4, using the "limit of a truncated formula". In his original
paper [18], Witten works directly with the infinite dimensional normal bundle
of M in LM . He shows that the fixed points of the S 1 -action on LM is the
manifold M embedded as constant loops. To state his conjecture he uses
a hypothetical infinite dimensional version of formula 3.4.4. Here, we will
use a slightly different approach. We "cut" LM to a compact manifold,
approximating S 1 with the n-cyclic group Cn . This allows us to use formula
3.4.4 on the compact manifolds Map(Cn , M ) with Cn -action, and then taking
the limit over n to obtain (up to a sign) the Witten genus formula. The
two methods are very similar, and the reader is invited to follow Witten’s
construction in [18] while reading the proof of 5.1.2.
In section 5.2 we follow the same idea to generalize the "truncated formula" to the equivariant case. Then we see under which assumption it is
possible to pass to the limit to define an equivariant analogous of the Witten
formula.
5.1. The Truncated Witten Genus. This is a summary of the paper [17],
and all the detailed proofs can be found there.
Let M be an 2l-dimensional spin manifold, and Map(Cn , M ) the manifold
of maps from the n-cyclic group Cn to M . As a manifold, this is just the
product M ×n . We use the notation Map(Cn , M ) to look at this group as
a "truncated" approximation of the manifold LM . This also allows us to
define a smooth action of Cn on Map(Cn , M ) by permutation. We choose a
generator σ of Cn and we define an action on Map(Cn , M ) by
σ(x1 , . . . , xn ) = (xn , x1 , x2 , . . . , xn−1 ).
This action is also a truncated version of the action of S 1 on LM by rotation.
First note that the product of spin manifolds is also spin. This is easily
seen using Stiefel-Whitney classes. By 3.2.2, M × N is spin if and only if
w2 (M × N ) = 0. Now, this is equal to
w2 (M × N ) = w2 (M ) × 0 + 0 × w2 (N ) + w1 (M ) × w2 (N ).
Then if M and N are spin w2 (M ) = w2 (N ) = 0, and w1 (M ) = w1 (N ) = 0
since M and N are orientable. This shows w2 (M × N ) = 0. Thus the
manifold Map(Cn , M ) is spin. Furthermore the choice of spin structure on
M induces a choice of spin structure on Map(Cn , M ).
61
In [17, §3.3] it is showed that the action of Cn on Map(Cn , M ) is a spin
action. Then we can consider the Dirac operator
6∂n : S + (Map(Cn , M )) −→ S − (Map(Cn , M )),
and calculate its σ-index for a generator σ ∈ Cn .
Definition 5.1.1. The (n)-truncated Witten genus of M is the σ-index
of 6∂n .
Note that in general the g-index of an operator only depends on the conjugacy class of g. Then the truncated Witten genus does not depend of
the choice of the generator σ of Cn . The aim of this section is to show the
following.
Proposition 5.1.2. Suppose that n = 2p + 1 is odd. Then the n-truncated
Witten genus of M is
*
!
+
p
O
b C M ) ch
Indexσ 6∂n = (−1)(p+1)l σ lp(p+1)/2 A(T
(Λ−σk TC M )−1 , [M ] .
k=1
Before proving this result note the similarity with the Witten genus formula. In the conjecture, the terms between boundaries are the limit when n
goes to the infinity of those for Indexσ 6∂n . Up to the sign, there is only the
power of ξ that considerably differs from the truncated formula. This is also
obtained taking the limit when p goes to the infinity:
lim σ lp(p+1)/2 = σ l lim σ
p→∞
p→∞
Pp
i=0
i
= σl σ
P∞
i=0
i
= σ l σ ζ(−1) = σ −l/12 ,
P
where ∞
i=0 i is interpreted as the value of the Riemann zeta function at −1,
which is −1/12. Then, up to a sign, the Witten’s formula is the limit when
n goes to the infinity of Indexσ 6∂n .
Proof of 5.1.2. To compute this index we need to characterize the fixed
points of σ. Note that
σ(x1 , . . . , xn ) = (xn , x1 , x2 , . . . , xn−1 ) = (x1 , . . . , xn )
if and only if
x1 = xn = xn−1 = · · · = x2 ,
that is the fixed points of σ are the constant maps of Map(Cn , M ), which is
a manifold diffeomorphic to M .
So far, the Atiyah-Singer theorem for the Dirac operator (cf. 3.4.3) gives
the formula
chσ (S + (N M ) − S − (N M ))
l
b
, [M ] ,
Indexσ (6∂n ) = (−1) A(TC M )
chσ (Λ−1 NC M )
where N M is the normal bundle of M embedded in Map(Cn , M ) as constant
maps.
We would like to use formula 3.4.4 to simplify this more. First, we need
to find a complex vector bundle whose underlying real bundle is N M . It is
shown in [17, §3.4] that N M is equivariantly isomorphic to
NM ∼
= T M ⊗ Const(Cn , R)⊥ ,
62
where Const(Cn , R)⊥ is the Cn -orthogonal complement of constant maps
in Map(Cn , R), and Cn acts on Map(Cn , R) by permutation (just as on
Map(Cn , M )). We denote by C[k] the representation of Cn on C where σ
acts by multiplication by σ k . Since n = 2p + 1 is odd, Const(Cn , R)⊥ is the
underlying real representation
p
Const(Cn , R)⊥ ∼
= (⊕ C[k])R ,
k=1
and therefore
p
p
NM ∼
= (T M ⊗ (⊕k=1 C[k]))R .
= T M ⊗ (⊕k=1 C[k])R ∼
Note that the second isomorphism does not always preserve the orientations,
but there is a correction sign of (−1)lp . Now, T M ⊗ (⊕pk=1 C[k]) is the complexification of a real bundle, and thus its first Chern class is zero. Furthermore its top exterior power is the trivial line bundle with fibre C[lp(p + 1)].
Hence 3.4.4 applies. Also note that
chσ (Λ−1 (T M ⊗ ⊕pk=1 C[k])) = ch(Λ−σk (⊕pk=1 TC M )).
Substituting this in the formula 3.4.4 we obtain the result.
5.2. The Truncated Equivariant Witten Genus. Let G be a compact
Lie group acting on a compact string manifold M , and suppose that action
lifts to the string structure. We follow the same ideas of the last section: we
truncate everything using the cyclic group Cn in order to find a formula for
the trace on the equivariant index of the Dirac operator on Map(Cn , M ).
Let Map(Cn , G) be the group of maps from Cn to G with group structure
given by pointwise multiplication. As a Lie group, this is just the compact
product G×n . It is isomorphic as a Lie group to the group of periodic maps
Ln G = {γ : Z −→ G|γ(k + n) = γ(k) for all k ∈ Z},
which is our "truncated approximation" of the smooth loop group LG. We
denote an element (g1 , . . . , gn ) ∈ Map(Cn , G) as g, and a (x1 , . . . , xn ) ∈
Map(Cn , M ) as x. The group Map(Cn , G) acts smoothly on Map(Cn , M )
by pointwise action. Furthermore, Cn acts smoothly on Map(Cn , M ) and
Map(Cn , G) as defined in section 5.1.
We define the semidirect product Map(Cn , G) o Cn as the product manifold Map(Cn , G) × Cn endowed with the operation
(g, α)(h, β) = (g(α · h), αβ),
for all g, h ∈ Map(Cn , G) and α, β ∈ Cn .
Note that this semidirect product acts smoothly on Map(Cn , M ) by
(g, α) · x = g(αx).
This action is our truncated model for the action of LG o S 1 on LM . Since
the action of G on M is spin the action of Map(Cn , G) o Cn on Map(Cn , M )
is also spin, and the Dirac operator is Map(Cn , G) o Cn -equivariant.
Definition 5.2.1. The (n)-truncated g-equivariant Witten genus of
M is the (g, σ)-index of the Dirac operator 6∂n .
We now apply the Atiyah-Singer fixed point theorem to obtain formulas
for the g-equivariant Witten genus of M , for a generator σ ∈ Cn . Before
stating a formula for this index we compute the fixed points of (g, σ).
63
Lemma 5.2.2. Let σ ∈ Cn be a generator, and g = (g1 , . . . , gn ) be an
element of Map(Cn , G). Then the following diagram commutes
Map(Cn , M )
((gn gn−1 ...g1 ,1,...,1),σ)
(g1 ,g2 g1 ,...,gn ...g1 )·
Map(Cn , M )
/ Map(Cn , M )
(g1 ,g2 g1 ,...,gn ...g1 )·
(g,σ)
/ Map(Cn , M ).
Proof. Let (x1 , . . . , xn ) be an element of Map(Cn , M ). Then
(g, σ) · ((g1 , g2 g1 , . . . , gn . . . g1 ) · (x1 , . . . , xn )) =
(g, σ) · (g1 x1 , g2 g1 x2 , . . . , gn . . . g1 xn ) =
(g1 gn . . . g1 xn , g2 g1 x1 , g3 g2 g1 x2 , . . . , gn gn−1 . . . g1 xn−1 ) =
(g1 , g2 g1 , . . . , gn gn−1 . . . g1 ) · (gn . . . g1 xn , x1 , x2 , . . . , xn−1 ) =
(g1 , g2 g1 , . . . , gn gn−1 . . . g1 ) · ((gn . . . g1 , 1, . . . , 1) · (xn , x1 , x2 , . . . , xn−1 )) =
(g1 , g2 g1 , . . . , gn gn−1 . . . g1 ) · (((gn . . . g1 , 1, . . . , 1), σ) · (x1 , x2 , . . . , xn )).
This Lemma shows that under the diffeomorphism
(g1 , g2 g1 , . . . , gn . . . g1 ) : Map(Cn , M ) −→ Map(Cn , M )
the action of (g, σ) corresponds to the action of ((gn . . . g1 , 1, . . . , 1), σ). It
will be often convenient to consider the (equivalent) action of Map(Cn , G) o
Cn on Map(Cn , M ) obtained by transporting the usual action by the diffeomorphism (g1 , g2 g1 , . . . , gn . . . g1 ). For instance, it is easier to describe the
fixed points of ((gn . . . g1 , 1, . . . , 1), σ) instead of the fixed points of (g, σ).
Proposition 5.2.3. The fixed point manifold Map(Cn , M )((gn ...g1 ,1,...,1),σ) is
the image of the fixed points M gn ...g1 embedded in Map(Cn , M ) as constant
maps.
Proof. A point (x1 , . . . , xn ) ∈ Map(Cn , M ) is fixed by ((gn . . . g1 , 1, . . . , 1), σ)
if and only if (gn . . . g1 xn , x1 , . . . , xn−1 ) = (x1 , . . . , xn ). This means that
xn = xn−1 = · · · = x1 = gn . . . g1 xn ,
that is the components of (x1 , . . . , xn ) are all equal and fixed by gn . . . g1 . This Proposition already gives a first result.
Corollary 5.2.4. If gn . . . g1 = 1 ∈ G, the (g, σ)-equivariant index of the
Dirac operator 6∂n on Map(Cn , M ) is
Index(g,σ) 6∂n = Indexσ 6∂n .
This means that in this case the index does not depend on the action of
G on M . The explicit formula for this index is formula 5.1.2. We will return
to this particular case at the end of this section, when we will consider a
"passage to the limit" of the truncated formula.
Without any assumption on the product gn . . . g1 , we find an expression for
the normal bundle N Map(Cn , M )(g,σ) of Map(Cn , M )(g,σ) in Map(Cn , M )
depending on the normal bundle N M gn ...g1 of M gn ...g1 in M . Choose a
G-invariant metric on M . This metric induces a Map(Cn , G)-invariant metric on Map(Cn , M ), which is also Map(Cn , G) o Cn -invariant. Then the
64
action of (g, σ) on T Map(Cn , M ) = Map(Cn , T M ) can be restricted to
Map(Cn , N M gn ...g1 ). We denote by Const(Cn , R)⊥ the Cn -invariant complement of the constant maps in Map(Cn , R), on which σ acts by permutation
of the components. With respect to this metric we have the following result.
Proposition 5.2.5. The bundle N Map(Cn , M )(g,σ) is (g, σ)-equivariantly
isomorphic to the sum
N Map(Cn , M )(g,σ) ∼
= (T M gn ...g1 ⊗ Const(Cn , R)⊥ ) ⊕ Map(Cn , N M gn ...g1 ),
where (g, σ) acts on T M gn ...g1 ⊗ Const(Cn , R)⊥ by 1 ⊗ σ.
Proof. The equivariant decomposition T M |M gn ...g1 = T M gn ...g1 ⊕ N M gn ...g1
induces the decomposition
gn ...g1
∼
) ⊕ Map(Cn , N M gn ...g1 ).
Map(Cn , T M )|
(g,σ) = Map(Cn , T M
Map(Cn ,M )
Under this isomorphism, the action of (g, σ) splits. Since the action of (g, σ)
on T M gn ...g1 is equivalent to the action of ((gn . . . g1 , 1, . . . , 1), σ) and gn . . . g1
acts trivially on T M gn ...g1 , the element (g, σ) acts on Map(Cn , T M gn ...g1 ) by
σ.
Now, T Map(Cn , M )(g,σ) is diffeomorphic to the image of T M gn ...g1 embedded in Map(Cn , T M ) as the constant maps. Then T Map(Cn , M )(g,σ) is included in the summand Map(Cn , T M gn ...g1 ) under the previous isomorphism.
Thus N Map(Cn , M )(g,σ) is equivariantly isomorphic to the sum of the complement of the constant maps in Map(Cn , T M gn ...g1 ) and Map(Cn , N M gn ...g1 ).
Since for any vector space V we have Map(Cn , V ) ∼
= V ⊗ Map(Cn , R) there
is an isomorphism of vector bundles
Map(Cn , T M gn ...g1 ) ∼
= T M gn ...g1 ⊗ Map(Cn , R),
under which the action of σ on Map(Cn , T M gn ...g1 ) becomes the permutation of the components of Map(Cn , R). The T M gn ...g1 -valued constant functions correspond to the tensors product with R-valued constant functions.
The Cn -invariant complement of the constant maps in Map(Cn , T M gn ...g1 )
is T M gn ...g1 ⊗Const(Cn , R)⊥ under this isomorphism. Then the complement
of T Map(Cn , M )(g,σ) in T Map(Cn , M ) is given by
N Map(Cn , M )(g,σ) ∼
= (T M gn ...g1 ⊗ Const(Cn , R)⊥ ) ⊕ Map(Cn , N M gn ...g1 ).
The first summand of this decomposition has already been treated in
the last section and we have explicit formulas for the equivariant Chern
characters of its spin bundles, in particular when n is odd. We now focus
on the second summand. In order to apply formula 3.4.4 we need to find a
complex structure on Map(Cn , N M gn ...g1 ).
Since gn . . . g1 acts by isometries on N M gn ...g1 we can decompose it as an
equivariant sum
m
M
gn ...g1 ∼
NM
N θk ,
=
k=1
N θk
where gn . . . g1 acts on
by rotation by θk . Actually, N θk is the bundle of
gn . . . g1 -equivariant morphisms from Vθk to N M gn ...g1 , where Vθk is the twodimensional real representation of the topologically cyclic group generated
65
by gn . . . g1 on which gn . . . g1 acts by rotation by θk (cf. [15, §18]). This
rotation action gives a complex structure on N θk provided that θk 6= π, 0.
With respect to this complex structure gn . . . g1 acts on N θk by multiplication
by eiθk . We denote by U θk the bundle N θk endowed with this complex
structure.
For the rest of the work we suppose that none of the θk of this decomposition is 0 nor π (note that this is satisfied if gn . . . g1 is an element of finite
odd order). Thus we have shown the following.
Proposition 5.2.6. There is a gn . . . g1 -equivariant decomposition of bundles
m
M
N M gn ...g1 ∼
N θk ,
=
k=1
N θk
where each
is equivariantly the underlying real bundle of a complex bundle U θk on which gn . . . g1 acts by multiplication by eiθk .
Note 5.2.7. The previous result gives a complex structure on the normal
bundle N M gn ...g1 . However, the orientation induced by this complex structure may not agree with the original orientation of N M gn ...g1 . For this reason
we write
m
M
gn ...g1 ∼
c
NM
N θk ,
= (−1)
k=1
where c = 0 if this two orientations agree and c = 1 if they do not.
We define (N M gn ...g1 )j to be the vector bundle N M gn ...g1 endowed with
Lm
the action of gn . . . g1 defined under the isomorphism N M gn ...g1 ∼
= k=1 N θk
by
j
vm ),
gn . . . g1 (v1 , v2 , . . . , vm ) = (ζ1j v1 , ζ2j v2 , . . . , ζm
where ζk is a n-th primitive root of eiθk . Then we have the following decomposition.
Proposition 5.2.8. There is an equivariant isomorphism
n
M
Map(Cn , N M gn ...g1 ) ∼
(N M gn ...g1 )j ,
= (−1)nc
j=1
where
(−1)c
is the sign of 5.2.7.
Proof.L
By the previous Proposition there is a decomposition N M gn ...g1 ∼
=
m
c
θk , and then
(−1)
N
k=1
Map(Cn , N M
gn ...g1
)∼
= Map(Cn , (−1)c
m
M
N )∼
= (−1)nc
θk
k=1
m
M
Map(Cn , N θk ).
k=1
, N M gn ...g1 )
is equivalent to the
Recall that the action of (g, σ) on Map(Cn
action of ((gn . . . g1 , 1, . . . , 1), σ). Then, under the isomorphism above, the
action of (g, σ) splits. On a summand Map(Cn , N θk ) it is given by
(g, σ)(v1 , . . . , vn ) = (eiθk vn , v1 , . . . , vn−1 ),
that is the action of ((gn . . . g1 , 1, . . . , 1), σ) on Map(Cn , N θk ). Recall that
N θk is the underlying real bundle of a complex bundle U θk . Therefore
66
Map(Cn , N θk ) is the underlying real bundle of Map(Cn , U θk ). The complex matrix of (g, σ) on a fibre of Map(Cn , U θk ) with respect to the basis
induced by the basis of U θk defining the complex structure is
0 eiθk
.
id 0
The eigenvalues of this matrix are the roots of the polynomial

−t 0 . . .
1 −t . . .
0
1
..
..
.
.
0
..
.
0
0 ...




det 





eiθk
0 



.. 
= (−t)n + (−1)n+1 eiθk .
. 



−t
0 1 −t
0
Thus the eigenvalues are the n-th roots of eiθk . Let ζk be a primitive n-th
root of eiθk . Then the eigenvalues are ζkj for 1 ≤ j ≤ n. Since the vector
bundle formed by the eigenspaces fibrewise associated to a ζkj is isomorphic
to N θk as vector bundles, we have that
Map(Cn , N ) ∼
=
θk
n
M
(N θk )j ,
j=1
where the action of (g, σ) on (N θk )j is by multiplication by ζkj . By definition
of (N M gn ...g1 )j , there is an equivariant isomorphism
(N M gn ...g1 )j ∼
=
m
M
(N θk )j .
k=1
Reordering the summands we obtain
Lm
θk
Map(Cn , N M gn ...g1 ) ∼
= (−1)nc Lk=1 Map(C
n, N )
m Ln
θ
nc
∼
k
= (−1)
j=1 (N )j
Lk=1
n Lm
nc
∼
= (−1) Lj=1 k=1 (N θk )j
n
∼
= (−1)nc j=1 (N M gn ...g1 )j .
Note that since each of the N θk is even dimensional the sign of the orientation
does not change when we commute the sum over k with the sum over j. Theorem 5.2.9. Suppose that n = 2p + 1 and that there is no θk equal 0 or
π. Let ζk be a primitive n-th root of eiθk , and dk be the complex dimension
of U θk . Then the n-truncated g-equivariant Witten genus of M is given by
the product of
D
E
b M gn ...g1 ) ch Np (Λ−σk TC M gn ...g1 )−1 ch Nn Nm (Λ j U θk )−1 , [M gn ...g1 ]
A(T
j=1
k=1
k=1
−ζ
k
with the factor
Pm
(−1)
k=1
dk +l(p+1)+c
(σ
lp(p+1)/2
m
Y
k=1
n(n+1)/2 1/2
ζk
)
.
67
Proof. By 5.2.5, 5.1.2 and 5.2.8 the normal bundle N Map(Cn , M )(g,σ) is
equivariantly isomorphic to the underlying real bundle of
m M
n
M
(−1)nc+lp (T M ⊗ (⊕pk=1 C[k]))
(U θk )j ,
k=1 j=1
where (−1)nc+lp = (−1)c+lp is the sign correcting the orientations. The first
Chern class of this complex bundle is the cup product of the first Chern
class of the two summands. We saw that c1 (T M ⊗ (⊕pk=1 C[k])) = 0. Thus
proposition 3.4.4 applies. The sign in front of the formula is (−1)r+l , where
r
r is the
Pm dimension of this big complex bundle. The sign (−1) is given by
(−1) k=1 dk , since the first summand has even dimension and n is odd. Note
that (g, σ) acts on the top exterior power of this bundle by
σ
lp(p+1)/2
m
Y
n(n+1)/2
ζk
.
k=1
Then the formula 3.4.4 gives the expected result.
We would like to use this formula to define a power series analogous to
the one appearing in the Witten conjecture. The problem here is that we
cannot pass to the limit as in the non-equivariant case, since the fixed point
manifold depends on n.
We saw that when the product gn . . . g1 is the identity the fixed point
manifold is M and the (g, σ)-index of the Dirac operator is equal to its σindex. We would like to find an analogous condition on loops of LG. Consider
the action of G on itself by left multiplication. If the induced action of (g, σ)
on Map(Cn , G) admits a fixed point h, then g is of the form
g = h(σh)−1 .
A simple computation shows that in this case the product gn . . . g1 is equal
to one. Therefore the (g, σ)-index is equal to the σ-index. Suppose then that
a (γ, ξ) ∈ LG o S 1 admits a fixed point δ in LG. Then γ is of the form
γ = δδ(ξ−)−1 .
In this case left multiplication by δ gives a commutative diagram
LM
ξ
δ
LM
/ LM
δ
(γ,ξ)
/ LM
and then the action of (γ, ξ) is equivalent to the action of ξ. We can define in
this case a power series for (γ, ξ), it will be the same as the Witten formula
for ξ. This suggests that for such a γ the action should restrict to the Γ(Ek ),
and the (γ, ξ)-index of 6∂k should be the same as its ξ-index.
In the general case, we can still say something on the fixed point manifold. For an element (γ, ξ) ∈ LG o S 1 , recall that we denote by (LM )(γ,ξ)
the submanifold of the loops on M fixed by (γ, ξ). The evaluation map
e1 : LM −→ M is the smooth map associating to a loop α ∈ LM its value
α(1).
68
Proposition 5.2.10. The restriction of the evaluation map e1 : LM −→ M
to (LM )(γ,ξ) is an injective immersion.
Proof. Let α ∈ LM be a loop fixed by (γ, ξ). Then for all z ∈ S 1
γ(z)α(ξz) = α(z).
Taking z = ξ k for some k ∈ Z we obtain γ(ξ k )α(ξ k+1 ) = α(z k ), that is
α(ξ k+1 ) = γ(ξ k )−1 α(ξ k ).
This recursive relation shows that for all k ∈ Z
α(ξ k ) = γ(ξ k−1 )−1 γ(ξ k−2 )−1 . . . γ(1)−1 α(1).
If β ∈ LM is another loop fixed by (γ, ξ) with β(1) = α(1) then
β(ξ k ) = γ(ξ k−1 )−1 . . . γ(1)−1 β(1) = γ(ξ k−1 )−1 . . . γ(1)−1 α(1) = α(ξ k ).
Since ξ is a topological generator of S 1 the set {ξ k |k ∈ Z} is dense in S 1 . By
continuity of α and β we obtain α = β.
We now proof that the differential de1 : T (LM )(γ,ξ) −→ T M is injective.
Let α00 be a tangent vector in T (LM )(γ,ξ) . Suppose that
de1 (α00 ) = (αt (1))00 = 0.
This means that the curve αt (1) of e1 ((LM )(γ,ξ) ) ⊂ M is constant in t. Then
all the loops αt (−) of (LM )(γ,ξ) are based at the same point. But we just
saw that this implies that αt = αt0 for all t, t0 ∈ R, that is (αt (z))00 = 0 for
all z ∈ S 1 . This means that
α00 = 0,
therefore de1 is injective.
Note that if the G-action is trivial the manifold (LM )(γ,ξ) = LM ξ consists
of constant loops. Therefore the map e1 provides a diffeomorphism between
LM ξ and M . This suggests that the submanifold e1 ((LM )(γ,ξ) ) ⊂ M should
play the role of M in the Witten genus formula for the equivariant case. This
means that the manifold M gn ...g1 should be substituted with e1 ((LM )(γ,ξ) )
in the formula 5.2.9. To continue with this idea we need to decompose the
normal bundle of e1 ((LM )(γ,ξ) ) in M with respect to some action of an
element analogous to gn . . . g1 . It is not clear how to do this in a completely
satisfactory way. We should try to associate to γ an element g ∈ G such
that
M g = e1 ((LM )(γ,ξ) ).
We need this to obtain an element acting on M , that allows us to decompose
the normal bundle of M g in M with respect to its action.
Following the lines of the truncated truncated case, this element should
be something of the form of an infinite product
g=
∞
Y
γ(ξ k ).
k=0
We know that a loop α ∈ LM is fixed by (γ, ξ) if and only if it satisfies
α(ξ k ) = γ(ξ k−1 )−1 . . . γ(ξ)−1 γ(1)−1 α(1)
69
for all k. Take a non constant sequence {kn }n∈N such that ξ kn converges to
1. Then
α(1) = lim α(ξ kn ) = lim (γ(ξ kn −1 )−1 . . . γ(ξ)−1 γ(1)−1 α(1)).
n→∞
n→∞
γ(ξ kn −1 )−1 . . . γ(ξ)−1 γ(1)−1
Therefore if the product
converges to some g ∈
(γ,ξ)
g
G, the manifold e1 ((LM )
) will be M . Unfortunately, we have no clue
about this convergence. In the case where γ is of the form γ = δδ(ξ−)−1 ,
the product γ(ξ kn −1 )−1 . . . γ(ξ)−1 γ(1)−1 is equal to δ(ξ kn )δ(1)−1 , which converges to 1 when n goes to the infinity. Therefore this is a generalization of
the case above.
In the case where this product converges, it is possible to pass to the limit
in the formula 5.2.9, to define the following.
Definition 5.2.11. Under this assumption, the (γ, ξ)-equivariant Witten
genus of M is the formal power series in ξ defined as the product of

*
! ∞ m
+
∞
O
OO
b M g ) ch
A(T
(Λ−ξk TC M g )−1 ch 
(Λ j U θk )−1  , [M g ] ,
j=1 k=1
k=1
and the factor
Pm
(−1)
k=1
dk +l+c
(ξ −l/12
m
Y
−ζk
−1/12 1/2
ζk
)
.
k=1
Where U θk , dk and ζk ∈ C are defined as in the truncated case, but with
respect to the element g.
Even if this is definable, there is no obviuos geometrical interpretation
involving the Dirac operator on LM .
6. Index Theory and Twisted K-Theory
The motivation for this section is the Freed-Hopkins-Teleman (FHT) theorem. Consider the following situation. A compact Lie group G acts on a
string manifold M by an action that lifts to the spin structure, but not to
the string structure. Then the loop group LG acts on LM , but this is not
a spin action. Therefore LG acts only projectivly on the spin structure of
e of LG by S 1 that acts on the spin
LM , and it has a central extension LG
structure. Therefore the index of the Dirac operator is a projective representation of LG o S 1 of some level τ . We suppose that this is a positive energy
representation, that is the representation splits following the S 1 -action as a
sum
M
Ek
S1,
k>0
∈ S1
of representations of
where z
acts on Ek by multiplication by z k .
In their paper [6] FHT define an object τ +σ KG (X) for any G-space X,
called "twisted equivariant K-theory". In the case where G acts on itself by
conjugation, they built an isomorphism between τ +σ KG (G) and the ring of
positive energy representations of LG o S 1 of level τ . This theorem suggests
that if the index of the Dirac operator is of positive energy, we can see it
as an element of some twisted K-theory. Maybe, this could be the missing
70
framework to develop an index theory for LM . However, this is an extremely
complicated result, and we will not go into more details here.
In what follows, we consider the following finite dimensional version of
the FHT theorem. Writing G as BLG the FHT theorem says that there is
"some twisted K-theory" τ +σ KG (BLG) that is isomorphic to some projective
representations of LG. Then for G, we would like a twisted K-theory that
applied to BG gives twisted representations of G. For the non-twisted case,
this is a result of Atiyah and Segal (cf. [2]). They shows that the K-theory
K(BG) is isomorphic to a completion of the representation ring RG . Here we
define a twisted version of this. We define a twisted K-theory for elements in
the second cohomology of a space. Applying this to BG we obtain an abelian
group τ K(BG) that contains projective representations of G of level τ . We
do not know yet if we can see τ K(BG) as some completion of projective
representations of G of level τ .
The first part of this section concerns the definition of this object. In the
last part we will see how to fit this in the finite dimensional index theory.
6.1. Cech Cohomology. Let U be an open cover of a topological space X,
and A an abelian group.
Definition 6.1.1. The Cech complex of U = {Ui }i∈I is the cochain complex defined at degree k ∈ N as
C k (U; A) = {c : I k+1 −→ A}.
The differential dk : C k (U; A) −→ C k+1 (U; A) is defined as
dk (c)(i0 , . . . , ik+1 ) =
k+1
X
c(i0 , . . . , ibj , . . . , ik+1 ).
j=0
The Cech cohomology of U is the cohomology of the Čech complex. It is
denoted by Ȟ ∗ (U; A).
This cohomology is functorial, in the following sense. Given a continuous map f : X −→ Y and an open covering U = {Ui }i∈I of Y , the family
f −1 (U) = {f −1 (Ui )}i∈I covers X. There are induced maps fk : C k (U; A) −→
C k (f −1 (U); A) defined by
fk (c)(i0 , . . . , ik ) = c(i0 , . . . , ik ).
These maps induce a pullback f ∗ : Ȟ ∗ (U; A) −→ Ȟ ∗ (f −1 (U); A).
Definition 6.1.2. Let U = {Ui }i∈I and V = {Vj }j∈J be two open covers of
X. We say that V is finer than U if there is a map s : J −→ I such that
Vj ⊂ Us(j)
for all j ∈ J. In this case we write U ≺ V.
A refinement of a covering U is a finer covering U ≺ V together with a
choice of map s : J −→ I. The map s : J −→ I is called a refinement of U
over V.
71
The relation ≺ defines a partial order on the set of open covers of X.
Furthermore if U ≺ V and s : J −→ I is an associated map, there is a map
s∗ : C ∗ (U; A) −→ C ∗ (V; A) of cochain complexes defined as
sk (c)(j0 , . . . , jk ) = c(s(j0 ), . . . , s(jk )).
It is possible to show (cf. [4, §1.3]) that two maps s, s0 : J −→ I defining two
refinements of U ≺ V induce chain homotopic maps s∗ and s0∗ . Therefore
for two covers U ≺ V of X there is a well defined map in cohomology
Ȟ ∗ (U; A) −→ Ȟ ∗ (V; A).
Definition 6.1.3. The Cech cohomology of X is defined as the limit of
groups
Ȟ ∗ (X; A) = lim Ȟ ∗ (U; A),
U
with respect to ≺ and the maps defined above.
The following result is well known.
Proposition 6.1.4. For any CW-complex X there is a canonical natural
isomorphism
Ȟ ∗ (X; A) ∼
= H ∗ (X; A).
For manifolds, the Cech cohomology behaves even better. The following
is in [4, §1.3].
Proposition 6.1.5. Let U be a covering such that any finite non empty
intersections of elements of U is contractible. Then there is a canonical
isomorphism
Ȟ ∗ (U; A) ∼
= H ∗ (X; A).
Given a finite dimensional Riemannian manifold M , it is always possible
to construct such a covering U using geodesics. Therefore we have a canonical
isomorphism
Ȟ ∗ (U; A) ∼
= H ∗ (M ; A).
6.2. K-Theory Twisted by Elements of the Second Cohomology.
The idea here is to define "local bundles" over a space X. These local
bundles cannot be glued together to define a global bundle over X, and the
obstruction to this is measured by an element in H 2 (X; Z/2). The use of
Z/2 coefficients will be justified later.
In order to define this we recall the relation between vector bundles and
transition functions. Given a finite family Ui1 , . . . , Uin of subspaces we denote
by Ui1 ,...,in the intersection Ui1 ∩ · · · ∩ Uin . We also denote by GLn the group
of linear isomorphisms GL(Cn ).
Definition 6.2.1. A set of transition functions (of dimension n) over
a connected space X is a covering {Ui }i∈I together with continuous maps
h = {hij : Uij −→ GLn } satisfying the cocycle condition
hij (x) = hik (x)hkj (x)
for all i, j, k ∈ I and x ∈ Uijk .
Given a continuous map f : Y −→ X we define the pullback of h as the
family
f ∗ h = {hij ◦ f : f −1 (Uij ) −→ Cn }.
72
Given two sets of transition functions on a space X it is possible to define
a sum and a product.
Definition 6.2.2. Let {hij : Uij −→ GLn } and {gαβ : Vαβ −→ GLm } be
two set of transition functions over X.
Their Whitney sum is the transition functions h ⊕ g = {hij ⊕ gαβ : Uij ∩
Vαβ −→ GLn+m } defined by
hij (x)
0
(hij ⊕ gαβ )(x) =
.
0
gαβ (x)
Their tensor product is the transition functions h⊗g = {hij ⊗gαβ : Uij ∩
Vαβ −→ GL(Cn ⊗ Cm ) = GLnm } defined by
(hij ⊗ gαβ )(x) = hij (x) ⊗ gαβ (x).
Definition 6.2.3. Let h and g be sets of transition functions over X of the
same dimension n. An isomorphism between h and g is a collection of
maps {rαi : Ui ∩ Vα −→ GLn } satisfying
rαi (x)hij (x) = gαβ (x)rβj (x)
for all x ∈ Uij ∩ Vαβ and all i, j, α, β. The set of isomorphism classes of
transition functions over X is denoted T F (X).
It is not hard to check that the Whitney sum and the tensor product
descend to T F (X).
Proposition 6.2.4. There is natural semi-ring isomorphism between isomorphism classes of transition functions over X and isomorphism classes of
finite dimensional complex vector bundles over X.
Proof. Given a vector bundle p : E −→ X we choose an atlas {Ui }i∈I of E
with trivialisations φi = (φ1i , φ2i ) : p−1 (Ui ) −→ Ui × Cn . For any non-empty
intersection Uij the composition
φ−1
φj
i
Uij × Cn −→
p−1 (Uij ) −→ Uij × Cn
is of the form
2 −1
φj (φ−1
i (x, v)) = (x, φj (φi (x, v))).
n
For a fixed x, the map φ2j (φ−1
i (x, −)) is a linear automorphism of C , and
therefore
φ2j (φ−1
i (x, v)) = hij (x) · v
for a unique continuous function hij : Uij −→ GLn . It is not hard to see that
the set h = {hij : Uij −→ GLn } satisfies the cocycle condition, and therefore
defines a set of transition functions over X. Furthermore the definition of
isomorphism of transition functions says exactly that the isomorphism class
of h does not depend on the choice of the atlas {Ui }i∈I of E. Therefore this
defines a map from isomorphism classes of vector bundles to isomorphism
classes of transition functions. It is easy to see that this is a semi-ring
homomorphism.
Given a set of transition functions h = {hij : Uij −→ GLn } we define a
vector bundle over X by
a
E=
Ui × Cn /∼ ,
i
73
where ∼ is the equivalence relation identifying (x, v) ∈ Ui ∩ Cn with (y, w) ∈
Uj ∩ Cn if and only if x = y and v = hij (x)w. The cocycle condition
guarantees that ∼ is an equivalence relation. The map p : E −→ X of the
bundle is the projection on the first component. It is not hard to prove that
this is a vector bundle, and a trivializing atlas is given by the covering {Ui }
with trivializations p−1 (Ui ) −→ Ui × Cn defined as [(x, v)] = (x, v). Once
again the isomorphism class of E depends only on the isomorphism class of
h.
It is also clear that this isomorphism commutes with the pullback.
This shows that for a compact space B the K-theory K(B) is naturally
isomorphic to the ring completion of T F (B). For a non-compact locally
compact space B the K-teory K(B) is isomorphic to the kernel of the pullback by the map induced by the projection to the basepoint on the ring
completion of T F (B + ).
The twisted K-theory considered here is defined using transition functions
that do not satisfy the cocycle condition. Therefore they are not able to
define a global bundle. We use the multiplicative notation for the group
Z/2, that is Z/2 = {−1, 1}.
Definition 6.2.5. Let V = {V }i∈I be a covering of X and c ∈ C 2 (V; Z/2).
A c-twisted vector bundle of dimension n over V is a set of continuous
functions h = {h : Vij −→ GLn } satisfying the (c-)twisted cocycle condition
hij (x) = c(i, k, j)hik (x)hkj (x),
for all x ∈ Vijk . The set of c-twisted vector bundles over V is denoted
Vect(V, c).
Note 6.2.6. We take Z/2-coefficients because we want to apply this theory
to index theory for the Dirac operator. Since Spinn is a Z/2-extension of
On , we need only Z/2-coefficients. However, we never really use properties
of Z/2, unless the fact that each element is the inverse of itself. Being careful
with the inverses, it is possible to generalize all that for coefficients in any
multiplicative subgroup of C∗ .
Note 6.2.7. Suppose that h is a c-twisted vector bundle. Therefore c satisfies
c(i, j, l)hij hjl = hil
= c(i, k, j)hik hkl
= c(i, k, j)(c(i, j, k)hij hjk )hkl
= c(i, k, j)c(i, j, k)hij (hjk hkl )
= c(i, k, j)c(i, j, k)hij c(j, k, l)hjl
= c(i, k, j)c(i, j, k)c(j, k, l)hij hjl ,
that is 1 = c(i, j, l)c(i, k, j)c(i, j, k)c(j, k, l) = d2 (c)(i, j, k, l). This means
that if a c-twisted vector bundle exists c has to be a cocycle. From now on
we suppose that all the twistings c are cocycles.
Given two c-twisted vector bundles h, h0 ∈ Vect(V, c) we can define their
direct sum by
hij 0
0
(h ⊕ h )ij =
.
0 h0ij
74
This clearly satisfies the c-twisted cocycle condition. However a tensor product is not well-defined as an operation on Vect(V, c). This because it changes
the twistings.
Now we define isomorphisms of twisted bundles in a similar way as we did
for transition functions.
Definition 6.2.8. An isomorphism between two elements h and h0 of
Vect(V, c) is a family of continuous map r = {ri : Vi −→ GLn } such that
ri hij = h0ij rj .
The set of isomorphism classes of c-twisted vector bundles over V is denoted
by Vect(V, c).
Note how this definition differs from the non-twisted case. Here we compare transition functions only over the same covering. However, two nontwisted vector bundles over two coverings V and W are isomorphic as vector
bundles if and only if their restrictions on V ∩W are isomorphic in this sense.
Proposition 6.2.9. The sum descends to isomorphism classes over of twisted
vector bundles over V.
Proof. Take an isomorphism r between h and g, and an isomorphism r0
between h0 and g 0 . Then we show that the family defined by
ri 0
R = {Ri =
}
0 ri0
is an isomorphism between h ⊕ h0 and g ⊕ g 0 :
ri hij 0
0
Ri (h ⊕ h )ij =
r0 ih0ij 0
gij rj 0
=
0 r0
0
gij
j
0
= (g ⊕ g )ij Rj .
We want to compare the groups Vect(V, c) and Vect(V, c0 ) for different
twistings c and c0 .
Lemma 6.2.10. If c, c0 ∈ C 2 (V; Z/2) are equivalent cocycles the semi-groups
Vect(V, c) and Vect(V, c0 ) are isomorphic.
Proof. Note that for any h ∈ Vect(V, c) and any b ∈ C 1 (V; Z/2) we have
b(i, j)hij
= b(i, j)c(i, k, j)hik hkj
= b(i, j)b(i, k)b(k, j)c(i, j, k)b(i, k)hik b(k, j)hkj
= d1 (b)(i, k, j)c(i, j, k)(b(i, k)hik )(b(k, j)hkj ),
that is the family bh = {b(i, j)hij } is d1 (b)c-twisted. Since c and c0 are
equivalent, there is a b ∈ C 1 (V; Z/2) such that c0 = d1 (b)c. Therefore this
element b defines a map ψb : Vect(V, c) −→ Vect(V, c0 ) sending h to bh. We
show that this map induces an isomorphism
ψb : Vect(V, c) −→ Vect(V, c0 )
on isomorphism classes. Since the coefficients are Z/2 we have that d1 (b)2 =
1. Therefore c = d1 (b)c0 and b also defines a map Vect(V, c0 ) −→ Vect(V, c)
75
which is an inverse of ψb . Then it is sufficient to show that ψb descends
to the isomorphism classes. Let r = {ri : Vi −→ GLn } be an isomorphism
between h and h0 . Then r also defines an isomorphism between bh and bh0 :
ri b(i, j)hij
= b(i, j)ri hij
= b(i, j)h0ij rj .
We would like to find out when this isomorphism Vect(V, c) ∼
= Vect(V, c0 )
∗
is canonical. Note that each element x ∈ C (V; Z/2) induces an element of
xS 1 ∈ C ∗ (V; S 1 ), just by composition with the obvious injection.
Lemma 6.2.11. Suppose that for all b, b0 ∈ C 1 (V; Z/2) such that c0 =
d1 (b)c = d1 (b0 )c there is some a ∈ C 0 (V; S 1 ) such that d0 (a) = (bb0 )S 1 .
Then there is a canonical isomorphism Vect(V, c) ∼
= Vect(V, c0 ).
Proof. Recall that we have isomorphisms ψb : Vect(V, cV ) −→ Vect(V, c0V ).
We prove that under this hypothesis ψb does not depend on b. We need to
show that for any h ∈ Vect(V, c) there is an isomorphism r between bh and
b0 h. By hypothesis there is an a ∈ C 0 (V; S 1 ) such that d0 (a) = (bb0 )S 1 . This
means
d0 (a)(i, j) = a(i)a(j)−1 = b(i, j)b0 (i, j),
that is a(i)b(i, j) = a(j)b0 (i, j). We define an isomorphism r by ri =
a(i) idGLn . Therefore we obtain
ri b(i, j)hij
= a(i)b(i, j)hij
= a(j)b0 (i, j)hij
= b0 (i, j)hij rj .
Note 6.2.12. For any b, b0 so that c0 = d1 (b)c = d1 (b0 )c the product bb0
is a cocycle, since d1 (bb0 ) = cc = 1. Therefore if H 1 (V; Z/2) is zero, the
element bb0 is the image of an a ∈ C 0 (V; Z/2). In this case a actually has
Z/2-coefficients. Then each pair of such b, b0 defines the same isomorphism
ψb = ψb0 . We obtain in this case a canonical isomorphism Vect(V, c) ∼
=
Vect(V, c0 ).
We want to define a semi-group of τ -twisted vector bundles, for an element
τ ∈ H 2 (X; Z/2). That is, we want a definition independent on the atlas and
on the choice of a representative c ∈ τ . Note that a refinement s of a covering
V over a finer covering W induces a map s∗ : Vect(V, c) −→ Vect(W, s∗ (c))
defined at a c-twisted bundle h = {hij : Vij −→ GLn } by
s∗ h = {s∗ hij : Wαβ ,→ Vs(α)s(β)
hs(α)s(β)
−→ GLn }.
This map clearly descends to a map
s∗ : Vect(V, c) −→ Vect(W, s∗ (c)).
First suppose that X admits a good cover U, that is a covering whose finite
intersections are contractible (this is the case if X is a finite dimensional
manifold). Recall that in this case there is a canonical isomorphism
H 2 (X; Z/2) ∼
= Ȟ 2 (U; Z/2).
76
Proposition 6.2.13. Given two good covers U and U 0 and two cocycles
c ∈ C 2 (U; Z/2) and c0 ∈ C 2 (U 0 ; Z/2) defining the same cohomology class
τ ∈ H 2 (X; Z/2), there is an isomorphism
Vect(U, c) ∼
= Vect(U 0 , c0 ).
Proof. Take a cover V finer than U and U 0 which is also good and choose
refinements s of U over V and s0 of U 0 over V. Since c and c0 both define τ , the
cocycles s∗ (c) and s0∗ (c0 ) are equivalent. Therefore by 6.2.10 the semi-groups
Vect(V, s∗ (c)) and Vect(V, s0∗ (c0 )) are isomorphic. Therefore it is sufficient
to show that the map
s∗ : Vect(U, c) −→ Vect(V, s∗ (c))
is an isomorphism (in the same way Vect(U 0 , c0 ) and Vect(V, s0∗ (c0 )) are isomorphic). This result follows from contractibility. The idea is the following.
For any h ∈ Vect(V, s∗ (c)) we built a g ∈ Vect(U, c). For any Ui ∈ U the
inclusion of Ui in X defines a twisted bundle hUi ∈ Vect(V ∩ Ui , s∗ (c)Ui ) over
Ui . Since Ui is contractible, the cocycle s∗ (c)Ui is a coboundary and therefore Vect(V ∩ Ui , s∗ (c)Ui ) is isomorphic to Vect(V ∩ Ui , 0). Then hUi defines
a vector bundle gi over Ui . Since Ui is contractible gi is trivial. Doing this
for any i we obtain a family {gi }i∈I , and choosing trivialisations for each gi
we obtain transition functions g = {gij }. This family g defines an element
of Vect(U, c). This procedure defines a map
Vect(V, s∗ (c)) −→ Vect(U, c)
which is an inverse of s∗ .
Definition 6.2.14. Let τ be an element of H 2 (X; Z/2). The semi-group of
τ -twisted vector bundles over X is defined as
τ
Vect(X) = Vect(U, c)
for a good cover U and a representative c ∈ τ . The τ -twisted K-theory of
X is the group completion τ K(X) of τ Vect(X).
Note 6.2.15. For different choices of representative c ∈ τ and good coverings
U of X the semi-groups Vect(U, c) are isomorphic. Furthermore by 6.2.12,
this isomorphisms are canonical if the cohomology H 1 (X; Z/2) is trivial.
This is the case for example when X is simply-connected.
Suppose now that X does not admit a good cover. This case is considerably more complicated. We want to define τ Vect(X) as a semi-group
containing isomorphism classes of twisted bundles on several coverings, for
a cohomology class τ ∈ H 2 (X; Z/2) ∼
= Ȟ 2 (X; Z/2).
The ideal situation will be when we a cocycle c ∈ τ defines a uniform choice
of cocycles cV ∈ C 2 (V; Z/2) for any covering V of X. Then we could consider
isomorphism classes of cV -twisted vector bundles. This would be the case
if we could define a limit C ∗ (X; Z/2) of the complexes C ∗ (V; Z/2) whose
cohomology is Ȟ ∗ (X; Z/2). Since there are several maps C ∗ (V; Z/2) −→
C ∗ (W; Z/2) for W finer than V, it is not possible to define this limit in
general, at least not as a complex of groups (the limits C k (X; Z/2) should
be definable as sets).
77
To avoid this we proceed as follows. Given a cohomology class τ ∈
H 2 (X; Z/2) ∼
= Ȟ 2 (X; Z/2) we choose a covering U = {Ui }i∈I of X and
a cocycle c ∈ C 2 (U; Z/2) whose cohomology class is mapped to τ to the
limit Ȟ 2 (X; Z/2).
We define a partial order over the family of pairs (V, cV ), where V =
{Vi }i∈I is a covering and cV is a cocycle of C 2 (V; Z/2). A pair (W, cW ) is
greater than (V, cV ) if W = {Wα }α∈A is finer than V and there is a refinement
s : A −→ I satisfying
s∗ (cV ) = cW .
This is denoted (V, cV ) ≺ (W, cW ).
We only consider the family of pairs (V, cV ) that are greater than (U, c).
This assures that all the cocycles considered induce the same cohomology
class τ . Note that for (V, cV ) ≺ (W, cW ) there is a well defined map
Vect(V, cV ) −→ Vect(W, cW )
induced by a refinement s : A −→ I such that s∗ (cV ) = cW . Another such
refinement induces the same map on isomorphism classes. We would like to
define Vect(X, c) as the limit of Vect(V, cV ) for (U, c) ≺ (V, cV ). In order to
do this, we need this order to be inductive, that is for each (U, c) ≺ (V, cV )
and (U, c) ≺ (V 0 , cV 0 ) there is a (W, cW ) grater than (V, cV ) and (V 0 , cV 0 ).
This is true if and only if there are refinements s of V over W and s0 of V 0
over W such that
s∗ (cV ) = s0∗ (cV 0 ).
We do not know if this is true in general. The only thing we can say is that
s∗ (cV ) and s0∗ (cV 0 ) are equivalent. For the rest of the work we assume that
this condition is satisfied.
Note 6.2.16. We find a condition that guarantees that this order is inductive. Suppose that for each covering W and each b ∈ C 1 (W; Z/2) there is a
covering W 0 finer than W and a refinement r such that r∗ (b) is a cocycle in
C 1 (W 0 ; Z/2).
Given two pairs (U, c) ≺ (V, cV ) and (U, c) ≺ (V 0 , cV 0 ), there is a covering
W and refinements s of V over W and s0 of V 0 over W such that s∗ (cV ) is
equivalent to s0∗ (cV 0 ). This because the cohomology classes of cV and of cV 0
define the same class τ to the limit (W can be chosen to be the intersection
V ∩ V 0 ). Therefore there is a b ∈ C 1 (W; Z/2) such that
s0∗ (cV 0 ) = d1 (b)s∗ (cV ).
By hypothesis there is W 0 finer than W and a refinement r such that r∗ (b)
is a cocycle. Therefore we obtain
(s0 ◦ r)∗ (cV 0 ) = r∗ (d1 (b)s∗ (cV ))
= d1 (r∗ (b))r∗ (s∗ (cV ))
= (s ◦ r)∗ (cV ).
This means that the pair (W 0 , (s◦r)∗ (cV )) is grater than (V, cV ) and (V 0 , cV 0 ).
Definition 6.2.17. The set of (isomorphism classes of ) c-twisted vector bundles over X is defined as the limit
Vect(X, c) =
lim
(U,c)≺(V,cV )
Vect(V, cV ).
78
Next, we show that Vect(X, c) does not depend on c, but only on the
cohomology class τ .
Proposition 6.2.18. Let (U, c) and (U 0 , c0 ) be two pairs such that the classes
of c and of c0 defines the same element τ ∈ Ȟ 2 (X; Z/2). Then there is an
isomorphism
Vect(X, c) ∼
= Vect(X, c0 ).
Proof. Since (U, c) and (U 0 , c0 ) both define τ , there is a covering U 00 =
{Uα00 }α∈A finer than U = {Ui }i∈I and than U 0 = {Uj0 }j∈J and refinements
s : A −→ I and s0 : A −→ J such that the class of s∗ (c) is equivalent to the
class of s0∗ (c0 ) in C 2 (U 00 ; Z/2). Furthermore there are isomorphisms
Vect(X, c) =
lim
Vect(V, cV )
lim
Vect(V, c0V ).
(U 00 ,s∗ (c))≺(V,cV )
and
Vect(X, c0 ) =
(U 00 ,s0∗ (c0 ))≺(V,c0V )
Note that for any (U 00 , s∗ (c)) ≺ (V, cV ) and (U 00 , s0∗ (c0 )) ≺ (V, c0V ) the cocycles cV and c0V are equivalent. Therefore to show that Vect(X, c) and
Vect(X, c0 ) are isomorphic it is sufficient to show that Vect(V, cV ) is isomorphic to Vect(V, c0V ) if cV and c0V are equivalent cocycles over the same
covering V. We already showed this in 6.2.10.
This result allows us to define the following.
Definition 6.2.19. The semi-group of τ -twisted vector bundles over X
is defined as
τ
Vect(X) = Vect(X, c)
2
for a c ∈ C (U; Z/2) whose cohomology class is mapped to τ to the limit
Ȟ 2 (X; Z/2). The τ -twisted K-theory of X is the group completion τ K(X)
of τ Vect(X).
Note 6.2.20. If X admits a good cover U this definition agrees with the
definition above. This because the semi-groups Vect(V, cV ) are all isomorphic to Vect(U, cU ) for good covers (U, cU ) ≺ (V, cV ) (cf. proof of 6.2.13).
Furthermore note that for τ = 0 ∈ H 2 (X; Z/2), this definition agrees with
the usual K-theory K(X).
Example 6.2.21. An example of twisted vector bundles is the "spin bundle"
for a n-dimensional oriented Riemannian vector bundle E over X that does
not admit a spin structure. The structure group GLn of E can be reduced to
SOn , that is we can find transition functions gij : Vij −→ SOn for E. Choose
the atlas V fine enough so that each gij (Vij ) is included in an open subset of
SOn belonging to an atlas trivializing the Z/2-bundle Spinn −→ SOn . Then
for each gij we can choose a lift geij : Vij −→ Spinn . If this choice of liftings
is good, the family {e
gij } satisfies the cocycle condition, and so defines a
principal Spinn -bundle, which is a spin structure. However, this good choice
does not always exist. A spin-structure on E exists if and only if there is
a choice of liftings {e
gij } that satisfies the cocycle condition. Furthermore,
inequivalent choices of such liftings define inequivalent spin structures. We
saw that such a structure exists if and only if w2 (E) = 0 ∈ H 2 (X; Z/2).
79
This means that there is a choice of liftings {e
gij } so that the failure of the
cocycle condition is measured by a Čech cocycle that defines w2 (E) to the
limit. Therefore if w2 (E) is zero this family {e
gij } defines an actual bundle.
However, even if E does not admit a spin structure, there is a choice of
geij that satisfies a twisted cocycle condition, for a cocycle defining w2 (E).
Therefore we can compose this geij with the complex spinor representation to
obtain a w2 (E)-twisted vector bundle. If w2 (E) = 0, this defines the bundle
SC (E) associated to a spin structure.
As a conclusion of this section we define pullbacks. They are defined as is
the non-twisted case.
Definition 6.2.22. Let f : Y −→ X be a continuous map, and h ∈ Vect(V, c)
for a covering V of X. The pullback of h by f is the family
hij
f
f ∗ h = {f ∗ hij : f −1 (Vij ) −→ Vij −→ GLn }
in Vect(f −1 (V); f ∗ c).
It is clear that f ∗ h is a (f ∗ c)-twisted vector bundle over f −1 (V). This
defines a map
f ∗ : Vect(V, c) −→ Vect(f −1 (V), f ∗ c).
Proposition 6.2.23. The map f ∗ descends to a semi-group homomorphism
f ∗ : Vect(V, c) −→ Vect(f −1 (V), f ∗ c)
on the isomorphism classes.
Proof. Take h ∈ Vect(V, c) isomorphic to g ∈ Vect(V, c). Then there is a
family r = {ri : Vi −→ GLn } such that
ri hij = gij rj .
We define f ∗ r as the family
f
r
i
f ∗ r = {f ∗ ri : f −1 (Vi ) −→ Vi −→
GLn }.
This is an isomorphism between f ∗ h and f ∗ g:
(f ∗ r)i (y)(f ∗ hij )(y) = ri (f (y))hij (f (y))
= gij (f (y))rj (f (y))
= (f ∗ g)ij (y)(f ∗ r)j (y).
It remains to show that the map f ∗ : Vect(V, c) −→ Vect(f −1 (V), f ∗ c) is
additive. Let h, h0 ∈ Vect(V, c), and we compute
hij (f (y)) 0
∗
∗
0
(f h ⊕ f h )ij (y) =
0
0
hij (f (y))
hij 0
=
(f (y))
0
h0ij
= (f ∗ (h ⊕ h0 ))ij (y).
80
Therefore this induces a pullback on the limit
∗
f ∗ : τ Vect(X) −→ f τ Vect(Y )
and on the twisted K-theory
∗
f ∗ : τ K(X) −→ f τ K(Y ).
6.3. Projective Representations and Central Extensions. We briefly
recall some definition and general facts about projective representations. Let
A be an abelian Lie group.
Definition 6.3.1. A (τ -)projective representation (by A) of a Lie group
K is a locally convex vector space V together with a continuous map ρ : K −→
GL(V ) such that ρ(1) = idV and
ρ(g)ρ(g 0 ) = τ (g, g 0 )ρ(gg 0 ),
for a smooth map τ : K × K −→ A. If the Lie group K is of finite dimension, we consider only finite dimensional projective representations. The
group completion of the semi-group of isomorphism classes of τ -projective
representations of K under the direct sum is denoted τ RK . We call τ the
level of the representation.
Given a τ -projective representation ρ : K −→ GL(V ) of K we can built
an A-central extension τ K of K and a representation of τ K. The extension
τ K is define as the manifold K × A with group law
(g, z) · (g 0 , z 0 ) = (gg 0 , τ (g, g 0 )zz 0 ).
The map ρe: τ K −→ GL(V ) defined as
ρe(g, z) = zρ(g)
defines an effective representation of τ K. This procedure defines an injection
τ R ,→ Rτ .
K
K
Here we are interested in two cases. First when K = LG o S 1 is is the
semidirect product of a loop group with S 1 , and A = S 1 is the circle. In this
case we saw that if G acts on M , but the action is not string, it is possible
under sum assumptions to view the index of the Dirac operator on LM as a
projective representation of LG o S 1 by S 1 .
The second case is when K = G acts by a non-spin action on M , and
A = Z/2. In this case the index of the Dirac operator on M is a projective
representation of G by Z/2. The study of this representation is the subject
of the next section.
6.4. The Link with Index Theory. In order to fit the twisted K-theory in
the context of index theory we need a "universal interpretation" of the topological symbol BG : KG (T M ) −→ KG (∗). Consider the universal principal
G-bundle EG −→ BG. For each G-space X there is a homomorphism
EG ×G − : KG (X) −→ K(EG ×G X)
mapping a G-vector bundle F over X to the bundle EG×G F over EG×G X.
Furthermore this homomorphism EG ×G − is "almost" an isomorphism, in
b G (X) of KG (X) to which EG ×G −
the sense that there is a completion K
81
b G (X) −→ K(EG ×G X). This is
extends to an isomorphism EG ×G − : K
proved in [2].
Applying this to the tangent bundle T M of a compact manifold M we
obtain a map
KG (T M ) −→ K(EG ×G T M ).
Applying this to a point we obtain
RG ∼
= KG (∗) −→ K(EG ×G ∗) ∼
= K(BG).
This map associates to a representation V the associated bundle EG ×G V
over BG. It is possible to show that there is a homomorphism δ : K(EG ×G
T M ) −→ K(BG) for which BG is the pullback
/ K(EG ×G T M )
KG (T M )
BG
δ
/ K(BG)
RG
In this sense δ is a "universal index". Then there is a commutative diagram
{elliptic G-equivariant operators on M }
BG σ
VVVV
VVVδ(EG×
VVVV G −)σ
VVVV
VVVV
V*
/ K(BG)
RG
where the horizontal map becomes an isomorphism after completion, and σ
is the symbol. Then up to completion we can see the index of an operator as
an element of K(BG). For this reason we will denote also the composition
δEG ×G − by BG .
It is in this diagram that we want to fit the twisted K-theory. Consider
an elliptic differential operator D on a manifold M , but suppose that G acts
only Z/2-projectivly on the vector bundles over M and that D is projectivly
equivariant. In this case its index ker D − coker D is a projective representation of G of some level τ , and it is a Z/2-extension τ G of G that acts on
ker D − coker D. Denoting by τ D the set of elliptic τ -projectivly equivariant
operators on M , there is a commutative diagram
τD
GG
GGIndexτ G
GG
IndexG
GG
#
τR
/ Rτ G
G
For example, this is the case when M is an even dimensional spin manifold
with the action of a compact Lie group G, but the action does not lift to the
spin structure. Therefore G acts only projectivly on the spin structure of M
and the Dirac operator is projectivly equivariant.
82
Considering also the universal interpretations of the topological symbols
we obtain a commutative diagram
τD
IndexG
τR

G
BQBQQQ
BB QQQBτ G σ
BB QQQQ
BB
QQ(
BB
BB
K(B τ G)
O
Indexτ G BBB
BB
BB E τ G×τ G −
B!
/ Rτ G
Note that something is missing on the left. On the right part of the
diagram we have the classical index theory with respect the action of the
central extension τ G. We would like to have some "twisted index theory"
on the left part, working directly with projective representations.
Recall that the level τ of the projective representation induces the central
extension
φ
Z/2 −→ τ G −→ G.
This is a Z/2-bundle over G that is classified by a map α : G −→ BZ/2 =
K(Z/2, 1). Therefore the induced map
Bα : BG −→ BK(Z/2, 1) = K(Z/2, 2)
classifies an element in H 2 (BG; Z/2), that we still denote τ . That is, τ is a
twisting for BG. We suppose that a condition needed to define the twisted
K-theory τ K(BG) is satisfied (that is BG admits a good cover or the order
on the couples (V, cV ) is inductive). We still need to find maps making the
following commute
τ K(BG)
O
/ K(B τ G)
O
E τ G×τ G −
τR
/ Rτ G
G
The horizontal map is just the pull back by the map Bφ : B τ G−→BG defining the central extension. By definition, the twisting τ ∈ H 2 (BG; Z/2)
satisfies φ∗ τ = 0 ∈ H 2 (B τ G; Z/2), and therefore the pullback by Bφ is a
well defined map
Bφ∗ : τ K(BG) −→ 0 K(B τ G) = K(B τ G).
The vertical map is trickier. We want to define a twisted version of the
map EG ×G −. We saw that vector bundles are the same thing as transition
functions. Actually, this is true in general for principal G-bundles P over X.
Taking transition functions g = {gij : Uij −→ G} for P and a representation
ρ : G −→ GL(V ), the vector bundle P ×G V has transition functions
gij
ρ
Uij −→ G −→ GL(V ).
We want to do the same thing in the twisted case. We fix a covering V of
BG trivializing the universal principal G-bundle EG −→ BG. This gives a
set of functions
u = {uij : Vij −→ G}.
83
Those are actual transition functions, satisfying the (non-twisted) cocycle
condition. We define a map EG ×τG − : τ RG −→ τ K(BG) sending a projective representation ρ : G −→ GL(V ) of G to the element of τ Vect(BG)
defined by the isomorphism class of the family
uij
ρ
u ×τG V = {Uij −→ G −→ GL(V )}.
The twisted cocycle condition satisfied by this family is the following:
ρ(uij (x))ρ(ujk (x)) = τ (uij (x), ujk (x))ρ(uij (x)ujk (x))
= τ (uij (x), ujk (x))ρ(uik (x)).
Since the class τ ∈ H 2 (BG; Z/2) is defined from the central extension, it is
possible to choose the covering V so that there is a cocycle c ∈ τV satisfying
τ (uij (x), ujk (x)) = c(i, j, k).
Therefore u ×τG V defines a τ -twisted vector bundle. The diagram so defined
τ K(BG)
O
EG×τG −
τR
φ∗
/ K(B τ G)
O
E τ G×τ G −
G
/ Rτ G
is commutative by construction. The complete diagram becomes
τD
BQBQQQ
|
BB QQQBτ G σ
|
|
BB QQQQ
|
|
BB
QQ(
||
∗
BB
|
φ
|
τ K(BG)
B
|
B / K(B τ G)
|
O
O
Indexτ G BBB
|| IndexG
|
BB
|
|
EG×τG −
BB E τ G×τ G −
||
B!
~||
τR
/
Rτ G
G
Note that a map τ D −→ τ K(BG) is still missing. We could just define
it has the composition EG×G τ IndexG , but it can be interesting to built
it directly from τ D, defining a symbol σ(D) as an element of some object
τ K (T M ), and then take the image of σ(D) by some twisted topological
G
index τ BG : τ K G (T M ) −→ τ K(BG). To define a twisted topological index
we need pushforwards and Thom isomorphisms in twisted K-theory. This
subject is studied in [5], but with a different notion of twisted K-theory.
84
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[15]
[16]
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