Dirac family construction of K-classes
A.P.M. Kupers
May 6th 2010
Contents
1 Dirac family construction for tori
1.1 Tori and spectral flow . . . . . . . . . . . . . . . . . .
1.1.1 K-theory and spectral flow . . . . . . . . . . .
1.1.2 Heisenberg groups and Heisenberg-type groups
1.1.3 Twisted K-theory and spectral flow . . . . . .
1.1.4 Decomposition of loop groups of tori . . . . . .
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2
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2 Dirac family construction for a compact group
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Spinor fields and a Dirac operator . . . . . . . . . . . . . . . . . . .
2.2.1 The canonical Clifford extension of G . . . . . . . . . . . . .
2.2.2 The representation on spinor fields . . . . . . . . . . . . . . .
2.2.3 Konstant’s cubic Dirac operator . . . . . . . . . . . . . . . .
2.3 The family of cubic Dirac operators on an irreducible representation
2.3.1 The definition and first properties . . . . . . . . . . . . . . .
2.3.2 The kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Associated K-classes . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 The main theorem for compact groups . . . . . . . . . . . . .
2.4.2 Twisted K-theory of coadjoint orbits . . . . . . . . . . . . . .
2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Dirac family construction for loop groups
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Compact groups and loop groups: differences and similarities .
3.3 Sketch of construction . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 LP G and admissible central extensions . . . . . . . . . .
3.3.2 Finite energy and positive energy representations . . . .
3.3.3 The spin representations and canonical central extension
3.3.4 A family of Dirac operators on a loop group . . . . . . .
3.4 The twisted case and fractional loops . . . . . . . . . . . . . . .
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of LG
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A Prerequisites
A.1 Clifford algebras and spinor representations
A.1.1 Clifford algebras . . . . . . . . . . .
A.1.2 P inc . . . . . . . . . . . . . . . . . .
A.1.3 The spinor representation . . . . . .
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1
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1
1.1
Dirac family construction for tori
Tori and spectral flow
We will define K-theory classes by providing families of Dirac operators over the dual torus or the
torus, giving a Fredholm family. These will be simple cases of the construction of a Dirac family
for a compact group, where the fact that the torus is abelian simplifies the construction.
1.1.1
K-theory and spectral flow
Let T be a torus with Lie algebra t and choose a metric on t. Let ea denote a basis of t and let
Π denote the integer lattice in t, so that we can identify T with t/Π. Note that Π ∼
= π1 (T ). Let
t∗ denote the dual of the Lie algebra with dual metric and dual basis ea . Let Π∗ denote the dual
lattice Hom(Π, Z).
The metric on t allows us to define the Clifford algebra Clc (t∗ ). Let S be an irreducible spinor
representation, if dim T is odd with commuting Clc (1)-action. We will denote the Clifford action
by γ, and in particular Clifford multiplication by ea with γ a .
We will consider the spinor fields L = C ∞ (T ) ⊗ S. This has two actions: T acts by translation
on C ∞ (T ), hence t infinitesimally by differentiation Ra := −i ∂θ∂a , and t∗ by Clifford multiplication
on S.
Definition 1.1. Let D be the family of operators D : t∗ → End(L) given by
µ = µa ea 7→ Dµ = iγ a Ra + µa γ a
Note that for any µ ∈ g∗ , Dµ is an odd skew-adjoint operator. There is a special property of
Dµ with respect to multiplication by characters of T : they give a translation action of Π2 on the
family of Dirac operators. Every λ ∈ Π∗ gives us a character χλ : T → C∗ using the following
construction: λ extends to a map t → R, which maps the integral lattice to integers. Using
surjectivity of exp, we set χλ (exp(X)) = exp(2πiχλ (X)) and this is well-defined since the inegral
lattice is mapped to the integers.
Definition 1.2. For λ ∈ Π∗ , let Mλ : L → L be the operator given by f (t) 7→ χλ (t)f (t).
Proposition 1.3. Dµ Mλ = Mλ Dµ+λ .
Proof. You know how to differentiate, right?
This implies we can factor D has a family over T . Consider the quotient bundle t∗ ×Π∗ L where
Π acts on t∗ by translation and on L by the Mλ . This is a bundle over t∗ /Π∗ =: T ∗ , the dual
torus.
Since any Hilbert bundle is trivial by Kuiper’s theorem, we therefore get a family of skewadjoint Fredholm operators over T ∗ . If dim T is odd, we have a commuting Clc (1)-action. Therefore we get a class in K dim T (T ∗ ). To see this, we don’t even need the technology of [FHT07a].
Because the bundle of Fredholm operators is trivial, Atiyah-Singer’s results are enough.
The link with spectral flow, as defined in Atiyah-Patodi-Singer, is as follows. In the case of
the circle, i.e. T = S 1 , the operator Dµ : L → L has spectrum {i(n + µ)|n ∈ Z} and thus one
eigenvalues passes from the positive to the negative imaginary eigenspaces if we cross an integer.
On R/Z, the analog of T ∗ , this phenomena is known as spectral flow: walking in a circle makes
some eigenvalues pass 0. This number is independent of continuous deformations of the family of
Fredholm operators.
∗
1.1.2
Heisenberg groups and Heisenberg-type groups
To each locally compact topological abelian group T we can assign a Heisenberg group as follows:
let T̂ = Hom(T, T) by the Pontryagin dual of T . Then consider the group HT of elements T ×T× T̂
with multiplication (t1 , θ1 , ξ1 )(t2 , θ2 , ξ2 ) = (t1 + t2 , θ1 + θ2 + ξ1 (t2 ), ξ1 + ξ2 ).
2
Remember that L2 (T ) ∼
= L2 (T̂ ) using Fourier transform. Then HT acts on both: on the L2 (T ),
T acts by translation, T by scalar multiplication Mλ and T̂ by multiplication with the associated
character and on the L2 (T̂ ), T acts with the associated character, T by scalar multiplication
and T̂ by translation. The Fourier transform intertwines these two actions. Infinitesimally, we
get an action on L2 (T ) (or L2 (T̂ )) of multiplication and differentation satisfying the canonical
commutation relations (as in quantum mechanics).1
Now, let’s look at Heisenberg-type groups. These depend on a homomorphism V → T̂ , which
is used centrally extend T × V as the smallest subgroup of HT containing T , T and V as elements.
In the case of T our torus, T̂ ∼
= Π∗ , and we can get Heisenberg type groups from HTτ from a linear
∗
map τ : Π → Π . We can describe the group HTτ explicitly as the central extension of Π × T
defined using the following commutation rule, where p ∈ Π, t ∈ T :
ptp−1 = χτ (p) (t)t
1.1.3
Twisted K-theory and spectral flow
We will now prove a theorem which links twisted equivariant K-theory of the torus with irreducible
representations of a Heisenberg-type group derived from the torus.
Lemma 1.4. A map τ : Π → Π∗ represents a twist of KT∗ (T ), either as an element of HT3 (T ), or
by giving a central extension of a group.
If τ : Π → Π∗ has full rank, i.e. is an isomorphism of vector spaces after tensoring with Q, there
are finitely many unitary irreducible representations of HTτ , which are indexed by an equivalence
class [λ] in Π∗ /τ (Π). These are infinite dimensional and occur once in the L2 -completion of
C ∞ (T ) ⊗ S. We denote these by F[λ] and by definition F[λ] is exactly the completion of the direct
sum of the weight spaces of T in C ∞ (T ) ⊗ S for all weights in in the equivalence class [λ]. In fact,
this is quite easy to prove. The hard part is showing that these are all unitary irreducibles. But
we won’t need the latter statement, so we just prove the easy first statement.
L
Proposition 1.5. The subspace F[λ] = λ∈[λ] Fλ of weight spaces Fλ for the T -action on the L2 completion of C ∞ (T ) ⊗ S is an irreducible unitary HTτ -representation with respect to the L2 -inner
product.
Proof. The definition of the action of the T , T and T̂ easily shows that the action is unitary.
Note that the action of T and T preserves the weight spaces, and action of Mp sends the weight
space of weight λ isomorphically to the weight space of weight λ + τ (p). This proves that F[λ] is
HTτ -invariant. To see that it is irreducible, note that weight spaces Fλ are one-dimensional and
irreducible, using Peter-Weyl together with the fact that T is abelian.
The map τ extends to a linear map t → t∗ sending the integer lattice Π to the dual integer
lattice Π∗ . Thus there is an induced maps τ̃ : T → T ∗ . We can take the pullback of the Z/2Zgraded vector bundle t∗ ×Π∗ L over T ∗ using this map. This splits as vector bundles F[λ] ⊗ S. The
family Dξ of Dirac operators obtained by pullback of the family of operators Dµ preserves this
decomposition. In fact, it is invertible unless ξ ∈ exp(τ −1 ([−λ])).
To get more information from this construction, we lift the operators from a family over T to
a family over t. These are Γτ -equivariant.
Proposition 1.6. The family D : t → End(F[λ] ⊗ S) is a family of odd skew-adjoint Fredholm
operators and is Γτ -equivariant.
1 Take T = Rn , then T̂ ∼ Rn . In the case we can replace T by R to retain more information and we get the group
=
which is commonly known as the Heisenberg group. In this case the Stone-von Neumann theorem tells up for each
choice of Planck’s constant ~, there is a unique unitary irreducible representation up to unitary equivalence and
these are all unitary irreducible representations up to unitary equivalence. Note the contrast with the representation
theory of our Heisenberg groups.
3
Proof. The first property is obviously preserved by pullback. The Γτ -equivariance is not much
harder. Γτ acts on F[λ] ⊗ S as by the action of Γτ on F[λ] . On t its action factors through Π,
which acts by translation. The equivariance then follows from the relation Dµ Mλ = Mλ Dµ+λ for
the original Fredholm family.
Note that since t is a Π-principal bundle over T , the quotient map by Π gives a map t//Γ →
T //T which is a local equivalence of groupoids. Furthermore Γτ is a central extension of Γ,
therefore (t → T, Γτ , ) gives a twist of equivariant twisted K-theory of T . This means that the
Fredholm operators define a class in KTτ +dim T (T ). The twisted K-theory obtained in this manner
form a basis of KTτ +dim T (T ), which means that the following theorem holds:
Theorem 1.7. The construction of a twisted K-theory class from a representation F[λ] induces
an isomorphism of abelian groups:
Ψ : R(Γτ ) → KTτ +dim T (T )
1.1.4
Decomposition of loop groups of tori
This section is a small preview, but not irrelevant since it appears in the proof of the theorem
of section 4. We will describe how the theorem of the last section in fact describes isomorphism
classes of positive energy representations of LG at the level τ .
We define two special subgroups of LG. The Lie algebra g has a center z. Let Lz denote the
Lie subalgebra of Lg of loops with value in the center. Note that any invariant inner product g
induces an inner product of Lg which restricts to an inner product on Lz. This gives rise to a
notion of projection and orthogonal complement.
Definition 1.8. Let Γ be the subgroup of LG of loops such that the velocity (dγ)γ −1 has constant
projection to z.
Definition 1.9. Let V be the orthogonal complement to z in Lz with respect to the L2 inner
product. Then exp(V ) is an abelian subgroup of LG.
In this definition, any subspace in V complementary to z would suffice.
Proposition 1.10. LG is the semidirect product of Γ and exp(V ). If G is abelian, this is a direct
product.
Proof. The map Γ×exp(V ) → LG of sets given by multiplication of loops is bijective. This follows
from the fact that the map of Lie algebras LΓ × V → Lg is a bijection, where LΓ denotes the Lie
algebra of Γ.
The multiplication of Γ × exp(V ) such that the map is homomorphism of groups is given by
(γ1 , exp(v1 ))(γ2 , exp(v2 )) = (γ1 γ2 , exp(v1 ) exp(Ad(γ1 )v2 )). If G is abelian, then Ad is always the
identity, to the semidirect product becomes a product.
If G = T , a torus, then the above proposition describes it as a product. Since z = t, we can
identify Γ explicitly.
Lemma 1.11. For a torus T , the space Γ of loops with velocity such that the projection z is
constant is isomorphic to Π × T .
Proof. LT can be described as the space of paths γ̃ : [0, 1] → t such that γ̃(1) − γ̃(0) ∈ Π, modulo
translation by Π. Then dγγ −1 is equal to the path γ̃ 0 . The projection to z is constant if and
only if γ̃ 0 is constant. So Γ can be identified with the paths in t modulo translation by Π which
have constant velocity. The starting point of such a map is given by an element of t/Π = T . The
velocity must be an element of Π otherwise γ̃(1) − γ̃(0) ∈
/ Π.
4
Now we use that for admissible representations of loops, exp(V ) is a product of central extensions. Hence (LT )τ ∼
= Γτ × (exp(V ))τ . The (exp(V ))τ has a unique irreducible unitary representation F , the Fock representation. So the positive energy representations of (LT )τ are in bijective
correspondence with the representations of Γτ . Hence we can apply the theory of the last section
to obtain the following theorem:
Theorem 1.12. There is an isomorphism of abelian groups
Ψ : Rτ (LG) → KTτ +dim T (T )
It might seem weird that there is no contribution σ to the twist. But this twist is trivial over
T and LT , since the adjoint representation maps everything to the identity in the case of abelian
groups.
2
2.1
Dirac family construction for a compact group
Overview
The main source for this construction is [FHT07b, section 1], but there is also a summary in
[FHT05, section 4]. An overview of the construction can be found in figure 2.1.
2.2
2.2.1
Spinor fields and a Dirac operator
The canonical Clifford extension of G
Let G be a compact Lie group. Fix a G-invariant inner product on g , i.e. (−, −) on g such that
(gX, gY ) = (X, Y ) for all g ∈ G. For a general compact Lie group averaging any inner product
with respect to invariant measure gives us such an inner product. For semi-simple Lie groups, the
Killing form suffices. Then the adjoint representation gives us a map Ad : G → O(g).
Definition 2.1. The graded central extension Gσ is obtained as G ×O(g) P inc (g), with grading
induced by the grading of P inc (g). We call it the canonical Clifford extension.
In the case of simple group, σ can be identified with the dual Coxeter number ȟ, after identifying
the twists with Z.
Lemma 2.2. 1 → T → Gσ → G → 1 is indeed a graded central extension and induces a split
extension of Lie algebra 0 → iR → gσ → g → 0 which has a splitting g → gσ .
Proof. For the first statement, it suffices to prove that the kernel of the map Gσ → G is exactly T.
First note that the following diagram commutes, where the map T → G is constant the identity
and T → P inc (g) is the earlier inclusion, hence there is an induced map T → Gσ :
T 4TTTTT
TTTT
44
TTTT
44
TTTT
44
TTTT
#
44
σ
/* G
44 G
44
44
/ O(g)
P inc (g)
The kernel of Gσ → G is easy to describe if we use an explicit description of Gσ . Gσ is
the subgroup of G × P inc (g) of elements (g, x) such that Ad(g) = Tx . The maps Gσ → G and
G → P inc (g) are then simply the projections. This means that the kernel of Gσ → G consists of
elements (e, x) such that Tx = id, where Tx is the element of O(V ) determined by a x ∈ P inc (it
is explicitly described in the appendix as v 7→ xvα(x)−1 ). But Tx = id if and only if x lies in the
image of T → P inc (g). So we see that the kernel is exactly the image of T → Gσ .
The exact sequence of Lie algebra is a direct consequence of the first exact sequence. The
splitting is induced by the splitting of pinc → o.
5
2.2.2
The representation on spinor fields
Let S be the spinor representation of Clc (g∗ ) with a compatible metric. This means that P inc (g)
acts unitarily and therefore that Gσ acts unitarily on S through the map Gσ → P inc (g). If dim G
is odd there is a commuting Clc (1) action. Furthermore, taking the infinitesimal representation
of this unitary representation or equivalently restricting to pinc (g) ⊂ Clc (g∗ ), we get a Lie algebra
representation of gσ on S.
Let {ea } be a basis of g and {ea } be the dual basis of g∗ . As in [FHT07b, 1.2], we define the
following tensors using the inner product and Lie bracket on g:
(ea , eb ) = gab
(ea , eb ) = g ab
c
[ea , eb ] = fab
ec
([ea , eb ], ec ) = fabc
c
c
c
Lemma 2.3. The tensor fabc is skew in all indices and satisfies fab
fcde + fbd
fcae + fda
fcbe = 0.
Proof. The first statement is a consequence of the antisymmetry of the bracket: [ea , eb ] = −[eb , ea ]
and the invariance of the inner product ([ea , eb ], ec ) = −(ea , [eb , ec ]).
The second statement is a consequence of inserting the Jacobi identity in (−, ee ).
We will define spinor fields. There are three action on these spinor fields: the Clifford action γ
of g∗ , the spinor action σ of g and the infinitesimal translation action R of g. We summarize the
commutation relations for convenience and give details below:
[γ a , γ b ] = −2g ab
c
[σa , σb ] = fab
σc
b c
[σa , γ b ] = −fac
γ
c
[Ra , Rb ] = fab
Rc
[Ra , γ b ] = [Ra , σa ] = 0
Clifford action γ a . We can let elements of g∗ act on S by Clifford multiplication. Denote the
action of ea by γ a . Since S is a graded module, γ a will necessarily be odd and since the metric
is compatible, it will be a skew-Hermitian transformation. Being a Clifford representation,
the graded commutator will be:
[γ a , γ b ] = −2g ab
Spinor action σa . We can let elements of g act on S using the splitting g → gσ and then mapping
gσ into pinc to get a Lie-algebra reprsentation. By the following commutative diagram, it
suffices to calculate the image in pinc of ad(X) ∈ so(g).
gσ o
/g
pinc o
/ so(g)
The action of ea can be calculated as follows. First calculate the components of ad(ea ) with
respectP
to the basis: these are given by ([ea , eb ], ec ) = fabc . Hence in pinc we get the element
σa := a<b 41 fabc (γ b γ c − γ c γ b ) = 41 fabc γ b γ c using skewness of fabc . Note that σa will be
even and since fabc is skew and the γ a are skew-Hermitian, it will be skew-Hermitian as well.
6
The fact that we are dealing with a Lie algebra representation gives us the graded commutator of σa and σb . However, one could also do this calculation directly using nothing but
c
c
c
the formula for the commutator of γ a ’s and the relation fab
fcde + fbd
fcae + fda
fcbe = 0. This
gives us exactly the same result:
c
[σa , σb ] = fab
σc
So we have two actions of S. How do these two action interact?
Lemma 2.4. The following formula holds:
b c
[σa , γ b ] = −fac
γ
Proof. The proof uses the commutation relations for γ a .
[σa , γ b ] =
=
=
=
=
1
(facd γ c γ d γ b − facd γ b γ c γ d )
4
1
(facd γ c γ d γ b + facd γ c γ b γ d − 2facd g bc γ d )
4
1
(facd γ c γ d γ b − facd γ c γ d γ b − 2facd g bc γ d + 2facd g db γ c )
4
1
b c
(−2fac
γ − 2fadc g db γ c )
4
1
b c
b c
b c
(−2fac
γ − 2fac
γ ) = −fac
γ
4
Alternatively, for a semisimple Lie algebra, one can simply define the action of g using the
b c
c
σ commutation relations [σa , γ b ] = −fac
γ and [σa , σb ] = fab
σc hold. These relations define
σa , uniquely, for if σa0 is a second such action, then σa − σa0 commutes with γ b and since S is
c
c
irreducible, must be a scalar λa . Then fab
σc = [σa , σb ] = [σa + λa , σb + λb ] = fab
(σc + λc ) implies
c
that fab
λc = 0. If g is semisimple [g, g] = g implies that there exist complex numbers µa , υ b such
c
that ([µa ea , υ b eb ], ed ) = δdc and therefore µa υ b fab
= δdc and we get λc = 0.
Differentation action Ra . Until now we have just considered spinors, not spinor fields. Because
g can be identified with left-invariant vector fields of G, we get a Lie algebra action of g on
C ∞ (G) by applying the corresponding vector fields to functions. Denote the action of ea by
Ra . This satisfies:
c
[Ra , Rb ] = fab
Rc
Now consider C ∞ (G) ⊗ S, which we can identify by left translation with C ∞ (G, S), the spinor
fields. In this situation Ra are even self-adjoint operators. Because we trivialized using left
translation and the Ra are left-invariant vector fields, we get the following interaction between the
Ra and γ a and σa :
[Ra , γ b ] = [Ra , σa ] = 0
The space C ∞ (G) ⊗ S embeds into L2 (G) ⊗ S, where the L2 norm is with respect to the
invariant measure. L2 (G) with left translation by G is a very useful representation, playing the
role of fundamental representation for compact Lie groups. This means we have the following
decomposition theorem.
Theorem 2.5 (Corollary of Peter-Weyl). L2 (G) ⊗ S decomposes as follows:
M
L2 (G) ⊗ S ∼
V∗⊗V ⊗S
=
V ∈Ĝ
L
where
denotes the completed tensor product, Ĝ the set of irreducible representations of G, G
acts on V ∗ by right translation, on V by left translation and projective on S through Gσ .
7
The identification of V with a subspace of L2 (G) is done using matrix coefficients, which are
smooth functions. Therefore each summand is a finite dimensional space is smooth spinor fields
and the actions of g and g∗ described earlier restrict to actions on the summand. Furthermore,
which these actions only work on S or on V by left translation, the action on V ∗ is trivial and we
can forget about it. From now on, we will be interested in V ⊗ S only.
2.2.3
Konstant’s cubic Dirac operator
The skewness of fabc allows us to define the following element of Λ3 (g∗ ):
Ω=
1
fabc ea ∧ eb ∧ ec
6
It is G-invariant for the action induced by the coadjoint action on g∗ as a consequence of the
invariance of (−, −). This allows us to introduce the following Dirac operator on C ∞ (G) ⊗ S:
Definition 2.6 (The Dirac operator D0 ). Define D0 : C ∞ (G) ⊗ S → C ∞ (G) ⊗ S as follows:
i
i
i
D0 = iγ a Ra + γ a σa = iγ a Ra + fabc γ a γ b γ c = iγ a Ra + γ(Ω)
3
12
2
where γ(Ω) is given by the analogue Λ3 (g∗ ) → Clc (g∗ ) of the canonical map Λ2 (g) → Clc (g∗ )
defined earlier.
One can took a look at Landweber’s article for applications of this Dirac operator to an analogue
of the Borel-Bott-Weil theorem for loop groups.
Proposition 2.7. D0 is an odd skew-adjoint operator. It is Gσ -invariant. The square is given as
follows:
1
D02 = g ab (Ra Rb + σa σb )
3
so we can conclude that D0 is indeed a Dirac operator (its principal symbol is that of a generalized
Laplacian).
Proof. The first statement is trivial consequence of the definition. For Gσ -invariance, remember
that Gσ acts on C ∞ (G) through G by left translation and on S through P inc (g). Let (g, x) ∈ Gσ ,
Aba given by Ad(g)(ea ) = Aba eb . Note that the the coadjoint action on ea = (ea , −) is given
by the contragredient action g(ea , −) = (ea , g −1 −) and hence CoAd(g)(ea ) = (A−1 )ab eb . From
this we conclude that (g, x)Ra (f )(h) = Aba Rb (f )(gh), (g, x)σa (ψ) = Aba σb (xψ) for ψ ∈ S and
(g, x)γ a (ψ) = (A−1 )ab γ b (xψ). Using this, we establish Gσ -invariance:
i
(g, x)D0 (f ⊗ ψ(h)) = (g, x)iRa (f ) ⊗ γ a ψ(h) + (g, x)f ⊗ γ a σa ψ(h)
3
i
b
−1 a c
= iAa Rb (f ) ⊗ (A )c γ (xψ)(gh) + f ⊗ (A−1 )ac γ c Aba σb (xψ)(h)
3
= D0 (f ⊗ (xψ)(gh)) = D0 ((x, g)f ⊗ ψ)(h)
Note that we could have used the invariance of γΩ to replace a part of the above calculation,
but two variations of a proof are better than one. To compute the square, we note that since D0
is odd, D02 = 21 [D0 , D0 ]. Now it is just a matter of applying the commutation relations.
Note that the invariance and the fact that only left translation appears, imply D0 restricts to
V ⊗ S.
8
2.3
2.3.1
The family of cubic Dirac operators on an irreducible representation
The definition and first properties
Our next goal is to define a family of Dirac operators in End(V ⊗ S) depending on a paramter in
g∗ . We do this in the simplest way possible, using g∗ to define an additional Clifford multiplication.
Definition 2.8. Define D(V ) : g∗ → End(V ⊗ S) as follows:
µ = µa ea 7→ Dµ = D0 + µa γ a = D0 + γ(µ)
Proposition 2.9. For each µ ∈ g∗ , Dµ is an odd skew-adjoint operator, i.e. Dµ restricts as
Dµ : V ⊗ S + → V ⊗ S − and Dµ : V ⊗ S − → V ⊗ S + , and Gσ -equivariant, i.e. (g, x)Dµ =
DCoAd(g)(µ) (g, x). Its square is given as follows:
Dµ2 = D02 − |µ|2 − 2iµb g ba (Ra + σa )
so we can conclude that D0 is indeed a Dirac operator (its principal symbol is that of a generalized
Laplacian).
Proof. The first statement is trivial. The second follows from Gσ -invariance of D0 combined with
the fact that (g, x)µa γ a = µa (A−1 )ab γ b = CoAd(µ)b γ b .
To calculate the square, we note that since Dµ is odd, we have Dµ2 = 21 [Dµ , Dµ ]. This can
be expanded as Dµ2 = D02 + 12 [µa γ a , µa γ a ] + [D0 , µa γ a ], where we use the fact that the graded
commutator of two odd elements in symmetric in this entries. Using the commutating relations,
we derive [µa γ a , µb γ b ] = −2µa µb g ab = −2|µ|2 and
i
[D0 , µa γ a ] = [iγ a , µb γ b ]Ra + [γ a µd γ d ]
3
= −2iµb g ab Ra − 2iµb g ab σa
This suffices to prove the proposition.
2.3.2
The kernel
We want to know over which points of g∗ the family D(V ) induces an isomorphism of the fiber
and over which it doesn’t. Since the fibers are finite dimensional, it suffices to find to kernel of
Dµ . We start by doing the calculation in the case that G is connected. After that we’ll formulate
a proposition for general G.
To do the calculation, we decompose our irreducible representation V in a convenient way. Fix
a µ ∈ g∗ and fix a maximal torus Tµ in Zµ ⊂ G. We have chosen this for the following reason:
it holds that for t ∈ Tµ , CoAd(t)(µ)(X) = µ(Ad(t−1 )(X)) = µ(X). Infinitesimally, this implies
µ([H, X]) = 0 for H ∈ tµ . Since µ([H, X]) = iα(H)µ(X) for X ∈ gα , we conclude that µ now
annihilates all non-zero root spaces. This is equivalent to µ ∈ t∗µ .
If µ is regular, then we can choose a Weyl chamber such that µ is antidominant. If µ is not
regular, we can chosen a Weyl chamber such that µ is a negative
wall of the dual Weyl chamber.
L
After fixing a Weyl chamber, we P
can decompose gC as tC ⊕ α∈∆+ (g−α ⊕ gα ), where ∆+ ⊂ t∗ are
the positive roots. Define ρ = 12 α∈∆+ α ∈ t∗µ . Note that it depends on µ.
We give an alternative proof to the one in [FHT07b, proposition 1.19]. This is the one suggested
in [FHT07b, footnote 7] and [FHT05, section 4.1].
Proposition 2.10. Suppose V is irreducible with lowest weight −λ. Then Dµ ∈ End(V ⊗ S) is
nonsingular unless µ is regular and µ = −λ − ρ. In the latter case ker Dµ = K−λ ⊗ S−ρ , where
K−λ ⊂ V is the one-dimensional root space of lowest weight −λ and S−ρ ⊂ S is the root space of
lowest weight −ρ.
9
Proof. The proof proceeds in the following steps:
(a) Show D02 is multiplication by a real constant and D0 is Clifford multiplication on K−λ ⊗S−ρ .
(b) Link Dµ2 to D02 .
(c) Show that Dµ2 has value 0 on K−λ ⊗ S−ρ and a lower value on all other weight spaces.
(d) Link Dµ to D0 on K−λ ⊗ S−ρ and show that ker Dµ coincides with ker Dµ2 .
Let’s proceed with this program.
(a) Let π̇ : g → End(V ⊗ S) be the total infinitesimal action on V ⊗ S. It is given by
π̇(ξ) = ξ a (Ra + σa )
if ξ = ξ a ea ∈ g. Then it is easy to check that the following two commutation relations hold:
[D02 , π̇(ξ)] = 0
[D02 , γ(ξ a gab eb )] = 0
Because V is irreducible V is generated from the lowest weight vector by applying γ(ξ a gab eb )
and π̇(ξ). This means that D02 is determined by its value on the lowest weight space of V ,
which is K−λ ⊗ S−ρ .
2
We will show
L that D0 is multiplication by a constant on K−λ ⊗ S−ρ . To do this, write
gC = tC ⊕ α∈∆+ (g−α ⊕ gα ) and choose a compatible basis etj , eα , e−α (it is well-known
that the gα are one-dimensional).
α c
γ using the commutation relations.
For α ∈ ∆+ , we rewrite the term 3i γ α σα as 3i σα γ α + 3i fαc
α
α c
Note that fαc is non-zero only if c = tj and then has value −iα(etj ), hence 3i fαc
γ = 31 γ(α).
2
+
Summing over α ∈ ∆ , this will contribute a term 3 γ(ρ). Now we can write D0 as follows:
i
i
i
2
D0 = i(γ α Rα + γ −α R−α + γ tj Rtj ) + ( γ −α σ−α + σα γ α + γ tj σtj ) + γ(ρ)
3
3
3
3
Note that since K−λ ⊗ S−ρ is of lowest weight, R−α and σ−α for α ∈ ∆+ will vanish on it.
α c
Furthermore, since [σt , γ α ] = −ftc
γ and ftαj c is non-zero only if c = α and then has value
iα(etj ). This shows that γ α maps the weight space of S of weight ω to that of weight ω − α.
Therefore γ α will vanish on the lowest weight space as well.
These considerations imply that the terms γ α Rα and γ −α R−α vanish and only the terms
coming from tC survive. Similarly, the terms γ −α σ−α and σα γ α vanish and only the terms
coming from tC survive. So on K−λ ⊗ S−ρ we can write D0 as:
i
2
D0 = iγ tj Rtj + γ tj σtj + γ(ρ)
3
3
Now note that Rtj acts as multiplication by −iλ(etj ) and σtj acts as multiplication by
−iρ(etj ). There we obtain that
1
2
D0 = γ(λ) + γ(ρ) + γ(ρ) = γ(λ + ρ)
3
3
It is now a trivial consequence of the relations in the Clifford algebra that
D02 = −|λ + ρ|2
10
(b) We introduce two operators adapted to µ. The first is the µ-shifted g-action π̇µ : g →
End(V ⊗ S). It is given by:
π̇µ (ξ) = ξ a (Ra + σa − iµa )
if ξ = ξ a ea ∈ g. It is skew-adjoint.
The second is the µ-shifted energy Eµ ∈ End(V ⊗ S). It is given by:
Eµ = iπ̇µ (g ab µa eb ) −
|µ|2
|µ|2
= iµa g ab (Rb + σb ) +
2
2
It is self-adjoint. We can claim that Eµ has constant value ( µ2 − ω, µ) on the weight space
of weight ω. This is a simple consequence of the definition of the µ-shifted action. On the
weight space of weight ω, π̇µ equals multiplication by i(µ, ω). Then Eµ is easily seen to be
2
multiplication by −(µ, ω) + |µ|2 = ( µ2 − ω, µ).
Our earlier calculations in proposition 2.9 show that Dµ2 + 2Eµ = D02 . This implies that
Dµ2 + 2Eµ is multiplication by the constant −|λ + ρ|2 , which is therefore independent of µ.
(c) We look a bit closer at Eµ . Note that the antidominancy of µ implies that −(µ, ω) and
therefore ( µ2 − ω, ω) is minimal on the lowest weight. Note that if µ is regular, it has strictly
higher value on other weights. Otherwise µ might have the same value on some other weights.
Anyway, since Eµ and D02 are multiplication by a constant on weight spaces, Dµ2 is as well
and we can conclude that the value Dµ2 is maximal on on the weight space K−λ ⊗ S−ρ . To
get this maximum, we calculate the value Dµ2 on K−λ ⊗ S−ρ using the value of D02 and Eµ
there.
Note that on K−λ ⊗S−ρ the operator 2Eµ is multiplication by |µ|2 +2(ρ+λ, µ). This implies
that Dµ2 is multiplication by
−|λ + ρ|2 − |µ|2 − 2(ρ + λ, µ) = −|λ + ρ + µ|2
From this we get Dµ2 = 0 on K−λ ⊗ S−ρ if and only if −|µ + λ + ρ|2 = 0 if and only if
µ = −λ − ρ. Furthermore, because Dµ2 is nonpositive, if Dµ2 is not zero on K−λ ⊗ S−ρ , it is
not nonsingular on the entire space V ⊗ S.
Note that if µ is not regular, then the fact that µ lies on the wall of some dual Weyl chamber
implies that µ + ρ ∈
/ Z∆+ . But λ ∈ Z∆+ , being a root, hence −|µ + λ + ρ|2 6= 0.
(d) Finally, to check that ker Dµ2 = ker Dµ we note that our calculation that D0 is γ(λ + ρ) on
K−λ ⊗S−ρ implies that Dµ is equal to γ(µ+λ+ρ) there. We conclude that for ψ ∈ K−λ ⊗S−ρ ,
|γ(µ + λ + ρ)ψ|2 = |µ + λ + ρ|2 |ψ|2 , using the skew-adjointness of the γ a , and therefore
γ(µ + λ + ρ)ψ = 0 if and only if |µ + λ + ρ|2 = 0.
Alternatively, one can use that Dµ is a skew-adjoint operator and therefore if ψ ∈ ker Dµ2 ,
(Dµ ψ, Dµ ψ) = (Dµ2 ψ, ψ) implies that ψ ∈ ker Dµ as well.
We made the assumption that G is connected. If this is not the case, there are some minor
changes. K−λ doesn’t have to be one-dimensional any more, but Zµ acts irreducibly on K−λ .
This means that D02 must still act as a constant on K−λ and the theorem continues to hold.
2.4
2.4.1
Associated K-classes
The main theorem for compact groups
Note that each Dµ is a Fredholm operator, since it acts on a finite-dimensional space. Therefore
we can consider D(V ) as a family of odd skew-adjoint Z/2Z-graded Fredholm operator. This
11
family is compactly supported, i.e. invertible outside a compact subset, and Gσ -equivariant. If
dim G is odd, there is a commuting Clc (1)-action. Therefore it corresponds to a class as follows
σ+dim G ∗
[D(V )] ∈ KG
(g )cpt
where σ is an abbreviation for the twisting (g∗ → g∗ , Gσ , σ ). These maps assemble into a map
σ+dim G ∗
Ψ : R(G) → KG
(g )cpt . Then the following theorem holds:
Theorem 2.11. Let G be a compact Lie group. Then there is an isomorphism of abelian groups:
σ+dim G ∗
Ψ : R(G) → KG
(g )cpt
0
To prove this we will require the standard isomorphism KG
(pt) ∼
= R(G).
Proof. The Thom isomorphism for the G-bundle g∗ over a point in twisted equivariant K-theory
is usually given by a pushforward of j : {0} ,→ g∗ (see [FHT07a, section 3.6]). It assigns to a
representation the family of Clifford multiplication operators, supported at the origin.
σ+dim G ∗
0
j : KG
→ KG
(g )cpt
V 7→ (V ⊗ S, 1 ⊗ γ) =: [γ]
Let π : g∗ → {0} be the projection. Consider the following diagram:
/
j∗
0 o
KG
π∗
σ+dim G ∗
KG
(g )cpt
By naturality j∗ and π∗ are inverse. It suffices to show that j∗ ([V ]) = [D(V )]. For this note
that γ and D(V ) through compactly supported families of operators D0 + µa γ a , with ∈ [0, 1],
hence D(V ) is a compact perturbation of γ. This implies [γ] = [D(V )].
2.4.2
Twisted K-theory of coadjoint orbits
The kernels patch together to a Gσ -equivariant bundle K ⊗ S 0 over the coadjoint orbit O of
µ = −λ − ρ. The subbundle K is G-equivariant and S 0 is a twisted bundle of twist σ. The
spinor bundle of the normal bundle to O has twists σ(N ). As a consequence of the bundle
L = HomClc (N ) (S(N ), S 0 ) has twist σ − σ(N ). But T O ⊕ N is trivial, hence the corresponding
spinor bundles S(O) gives a twisting σ(O) = σ − σ(N ). So K ⊗ L is a σ(O)-twisted bundle. If G is
connected and simply-connected, σ(O) = σ(N ) = σ = 0, since G admits no nontrivial twistings.
We then obtain the following commutative diagram:
σ(O)+dim O
KG
pp
p
pp
p
p
p
p ∗
w pp
p
0
KG (pt) o
j∗
π∗
(O)
QQQ
QQQk∗
QQQ
QQQ
Q(
/ σ+dim G ∗
KG
(g )cpt
We already know that π∗ = (j∗ )−1 and by construction k∗ ([K ⊗L]) = [D(V )] and p∗ ([K ⊗L]) =
V as a consequence.
2.5
2.5.1
Examples
Finite groups
In the case the Lie algebra, Dirac operator, spinors and the associated twisted σ are zero. The
theorem then becomes trivial, the isomorphism going between two equal groups.
12
2.5.2
Tori
We compare the case of tori to our earlier construction of K-classes for tori. Note that fabc = 0,
since T is abelian, and therefore the Dirac operator reduces to the one we considered early on
Lt. Alternatively, one can construct that Dirac operators used there by pullback of the one in the
compact group case.
On the one hand we have the representation theory, which says R(G) = Z(Π∗ ), where Π∗ =
Hom(T, T) = Hom(Π, Z) are the characters. All these representations are one-dimensional, so
Dµ : End(Vλ ⊗ S) for λ ∈ Π∗ reduces to v ⊗ ξ 7→ iλa γ a + iµa γ a , where we identify λ with a linear
map of t which sends Π to the integers. Therefore the kernel of Dµ is exactly in supported in
{−λ} ⊂ t∗ . Since T is abelian, the coadjoint orbits consists of points.
In the special case T = S 1 we can be even more explicit. We have t = iR, which is generated by
i of norm 1 if we take the inner product (ia, ia0 ) = aa0 . Then we identify t∗ with iR through this,
and the dual vector to i is i. Under these choices, if we fix iR≥0 asL
the positive Weyl chamber, we
have that Π∗ = iZ and ρ = 0, since no roots appear in gC = tC ⊕ α∈∆+ (gα ⊕ g−α ). The spinor
representation S is given by C ⊕ C, where γ t is given by:
0 i
γi =
i 0
Each irreducible is labelled by a single weight in. Then Dia on Vin ⊗ S is given by:
0
i(n + a)
Dµ =
i(n + a)
0
2
Indeed, Dia = γ(in + ia) on the lowest weight space and Dia
= −(n + a)2 Id. Furthermore Dia
is singular only if ia = −in, so everything works out fine.
On the other hand, we have the twisted K-theory. First note that σ will be trivial over T ,
because Ad : T → O(t) is the constant map with value identity. Therefore, we are actually dealing
with the usual equivariant K-theory. Each [D(V )] is a class in KTdim T (t∗ )cpt supported above its
lowest weight character, which is easily seen to be isomorphic to KT0 (pt) = R(T ). Alternatively,
one can see this as an instance of the usual Thom isomorphism in equivariant K-theory.
3
Dirac family construction for loop groups
3.1
Overview
The main source for this construction is [FHT07b, sections 2-5], but there is also a summary in
[FHT05, section 8-13] for more general cases. An overview of the construction can be found in
figure 3.1. There is a proof for the case that G is a finite group in [Wil05].
3.2
Compact groups and loop groups: differences and similarities
Next we turn to the study of loop groups. Our goal will be to generalize the results of the
last paragraphs to loop groups. That is, we want to find a map from some variation on the
representation ring to some twisted K-theory group and show that this is an isomorphism.
Of course, since the authors of the papers [FHT07a], [FHT07b] and [FHT05] thought about the
way they set up their articles, the general idea of the procedure remains the same. Again we will
define a spinor representation, take a look at spinor fields, define families of Dirac operators for
representations and use these to give a map. However, technically things become more involved:
• We will be interested in representations of central extensions of loop groups, because all
interesting representations are projective. Additional complications will come from the fact
that not every central extension support enough structure to make the theory work. Therefore we restrict ourselves to admissible extensions (don’t worry, for semisimple compact Lie
groups, every central extension is admissible [FHT07b, proposition 2.15]).
13
More generally, we can extend our theory to gauge transformations of principal G-bundles
over the circle. These are called twisted loop groups by Pressley-Segal [Pre86, section 3.7].
• The only representations for which a theory similar to that of compact Lie groups works are
the so-called positive energy representations. However, these representations are in general
infinite-dimensional. Furthermore, the Lie algebra is infinite-dimensional. All this makes
the definitions harder and requires us to use functional analysis in some places.
• The Dirac family will no longer be indexed by g∗ , but by the affine space AP of connections
of the principal G-bundle over the circle. These will be isomorphic with the affine space of
linear splittings of (LP g)τ → LP g, as in the case of projective representations of compact
Lie groups [FHT07b, section 1.5].
• There will be some convergence problems. To deal with these we need to work on the dense
subspace of finite energy loops. For this we need a correct notion of energy with respect to
a connection.
The result will be the following theorem. in its most general form:
Theorem 3.1. Let (LP G)τ be a positive definite admissible graded central extension of LP G.
Then there is an isomorphism of graded free abelian groups
τ +dim G
Ψ : Rτ −σ (LP G) → KG
(G[P ])
where Rτ −σ (LP G) denotes the free abelian group on the irreducible positive energy representations
τ +dim G
at level τ − σ and KG
(G[P ]) denotes the twisted equivariant K-theory of the image of the
holonomy of P in G.
3.3
Sketch of construction
We will give a sketch in the untwisted case, i.e. P = G × S 1 , and provide some additional details
in each section, mainly for my own benefit.
3.3.1
LP G and admissible central extensions
Sketch. A loop group LG is the space of smooth loops in a Lie group. This has a Lie algebra Lg.
Because we will study projective representations, we want to look at central extensions of LG by T.
To get a good definition of energy, we need to able to rotate loops and therefore look at the slightly
larger group L̂G. Admissible extensions are those central extensions with the correct properties
to define the energy with respect to a connection and the Dirac operator in later sections: there
is a compatibility with the central extension and L̂G. Admissible extensions occur often and have
nice properties.
Details. We start with the definition of a loop groups and its extension to free rotation loops.
Definition 3.2. For G a compact Lie group and LG denote the space of smooth loops in G.
Note that this is the same as the space of G-equivariant diffeomorphisms of G × S 1 convering the
identity of S 1 . Furthermore, let L̂G denote the group of G-equivariant diffeomorphisms of G × S 1
convering a rigid rotation S 1 .
The topology and manifold structure on LG are considered in [Pre86, section 3.2]. We note
that the group L̂G can be described as a semidirect product of LG with S 1 . The product of
(γ, ϕ)(η, φ) is given by (rφ (γ)η, ϕ + φ), where (rφ (γ)η(ϕ))(θ) = γ(θ + φ)η(θ). The group L̂G is
very useful to consider, because it gives us elements gives rotate loops. For this reason, it also
occurs naturally in the theory of the root system and Kac-Moody algebras, see e.g. [Pre86, chapter
5] and in particular [Pre86, page 71].
14
The definition of L̂G gives short exact sequences of groups and Lie algebras:
1 → LG → L̂G → T̂rot → 1
1 → Lg → L̂g → iR̂rot → 1
Because we want to deal with projective representations, we need to look at central extensions
of LG by T. However, we need to restrict to a special class of central extensions, the admissible
ones. They roughly have the properties that the central extension can be extended to L̂G and
that we have a nice inner product on the Lie algebra of this extension (L̂G)τ .
Definition 3.3. A central extension (LG)τ is admissible if:
(a) There is a compatible extension (L̂G)τ of L̂G, which means that the following diagram is
commutative:
1
1
1
/T
/ (LG)τ
/ LG
/1
1
/T
/ (L̂G)τ
/ L̂G
/1
Trot
Trot
1
1
In this case, there is a commutative diagram of Lie algebras as follows:
1
1
1
/ iR
/ (Lg)τ
/ Lg
/1
1
/ iR
/ (L̂g)τ
/ L̂g
/1
iRrot
iRrot
1
1
Let K denote the generator i ∈ iR.
(b) On (L̂g)τ there exists a (L̂G)τ -invariant inner product −, − such that K, d = −1
for all d ∈ (L̂g)τ which are mapped to i ∈ iRrot .
This extension and inner product are considered part of the data of an admissible central
extension.
The class of admissible extensions can be enlarged to that of topologically and analytically
regular extensions [FHT05, section 2]. Note that the d’s in the definition of admissible extension
are exactly G-invariant vector fields tranverse to the fibers, thus generating a rotation-like flow.
We first note some trivial and non-trivial properties of the class of admissible extensions:
15
Proposition 3.4. (a) Let G be a semisimple Lie group, then every central extension of LG is
admissible and for each (L̂G)τ the inner product is unique.
(b) If G is simply connected, then the isomorphism classes of central extensions are in bijection
with elements of H 2 (LG; Z).
(c) Let (LG)τ1 and (LG)τ2 be two admissible central extensions. Then (LG)τ1 +τ2 := (LG)τ1 ×LG
(LG)τ2 is also admissible.
(d) For every admissible central extension (LG)τ there exists an inverse admissible central extension (LG)−τ .
Sketch of proof of part (a). This proof is an elaborate version of Pressley-Segal’s proof that every
central extension of a simply connected G is determined by a G-invariant bilinear form on g, using
the density of polynomial loops.
The importance of the inner product is the following proposition, linking it to connections
and splittings of the map (L̂g)τ → (Lg)τ (compare with the case of projective representations
[FHT07b, section 1.5]). First we need a lemma as preparation:
Lemma 3.5. The inner product −, − has the following properties:
K, (LG)τ = 0
K, K = 0
Proof. Note that the second statement is a consequence of the first. The first follows by noting
that if we fix a d ∈ (L̂g)τ which is mapped to i ∈ iRrot , then for all ξ ∈ (LG)τ there exists a
unique d0 ∈ (L̂P g)τ which is mapped to i ∈ iRrot such that d − d0 = ξ, where we identify (LG)τ
with its image in (L̂g)τ . This implies that K, ξ = K, d − K, d0 = −1 + 1 = 0.
Let’s remind ourselves a bit about connections in principal G-bundles. There are many equivalent definitions, most importantly as horizontal distributions, collections of lifts of vector fields
of the base space or simply as an affine space A (a Ω1 (S 1 ; g) ∼
= Lg-torsor). For the untwisted case,
there is a distinguished connection, corresponding to the canonical horizontal distribution A0 of
d
the product G × S 1 . The correspoding G-invariant horizontal vector field dA0 is dθ
. Using this
1
1
canonical connection, the Lg-torsor Ap can be identified with Ω (S , g), the g valued one-forms.
Proposition 3.6. Let (LG)τ be an admissible extension. If −, − is nondegenerate, then
the following sets of data are equivalent:
(a) A connection A ∈ A.
(b) A lift of the horizontal vertical field dA ∈ L̂g to an element dA of (L̂g)τ such that dA , dA =
0.
(c) A (Lg)τ -equivariant splitting of the map (L̂g)τ → (Lg)τ .
If −, − is not nondegenerate, then (a) and (b) are equivalent, and a datum of type (b)
gives one of type (c). However, the latter map might not be a bijection.
Proof. (a) ⇔ (b) Any connection is completely determined by its horizontal distribution. The
d
on the circle, thus determines
vector field dA , the unique horizontal lift of the vector field dθ
the connection uniquely. Now we must show that for each inner product and each dA ,
there exists a unique element of (L̂P g)τ such that dA , dA = 0. Any two lifts differ
by a multiple of K. Thus let d0A be a second lift satisfying d0A , d0A = 0. But we now
d0A = dA + cK. Then we get
d0A , d0A = dA + cK, dA + cK = 2c K, dA = −2c
So we get c = 0 and dA = d0A .
16
τ
τ
τ
(b) ⇒ (c) The splitting is given by β 7→ βA
where βA
is the unique lift such that βA
, dA = 0.
τ
A similar argument as above shows that indeed βA is unique. The equivariance follows from
the fact that gd(φ−1 )∗ A = φ∗ (dA ) and the invariance of inner product allows us to move φ∗
τ
to βA
.
(b) ⇐ (c) if the inner product is nondegenerate. If −, − is nondegenerate and s :
(LG)τ → (L̂G)τ is a splitting, then the property that K, dA = −1 and s((LP G)τ ), dA =
0 determines dA uniquely.
Proposition 3.7. Fix a connection A0 , then we have already noted the Lg-torsor A can be
identified with Lg through the map ξ 7→ A0 + ξdθ. The dependence of dA on ξ is as follows:
dA0 +ξdθ = dA0 − ξ.
τ
τ
In this case, βA
is given by βA
− β, ξ K.
0
0 +ξdθ
Proof. dA0 +ξdθ is defined by the properties π∗ (dA0 +ξdθ ) = i where π : P × S 1 → S 1 is projection
to the base and (A0 + ξdθ)(dA0 +ξθ ) = 0, the pairing of vector fields with a 1-form.
But dA0 − ξ clearly satifies the first, since ξ is vertical. For the second property, note that
(A0 + ξdθ)(dA0 − ξ) = A(dA0 ) + ξ − ξ = 0 where we’ve used that dθ annihilates ξ and that
dθ(dA0 ) = 1 and that A0 (ξ) = ξ, since ξ is vertical.
τ
τ
τ
βA
is characterised by βA
, dA0 − ξ = 0. We claim that βA
is given by
0 +ξdθ
0 +ξdθ
0 +ξdθ
τ
τ
βA0 − β, ξ K. To see this, note that βA0 , dA0 = 0, − β, ξ K, dA = β, ξ ,
τ
βA
, −ξ = − β, ξ and − β, ξ K, ξ = 0.
0
3.3.2
Finite energy and positive energy representations
Sketch. For a connection A, [dA , −] behaves like an energy operator. The direct sum of its
eigenspaces is the space of finite energy loops. A positive energy representation ρ has the property
that dA act as operators with discrete spectrum bounded below. For a positive energy representation, we can define a dense subspace of finite energy vectors on which finite energy loops act
nicely. These will be the natural domain of the Dirac operator.
Details. Our analysis of loop groups will use the concept of energy of a loop, roughly the angular
momentum of the wave packet the loop represents. The energy depends on the choice of a connection, because a connection determines which wave packets wrap around straight or what exactly
the component of the angular momentum in the transversal direction to the rotation direction of
the circle is.
Fix a connection A, then the space (LgC )f in (A) of finite energy loops with respect to A will
be constructed as a direct sum of eigenspaces of dA . The details of this construction are of lesser
importance to the ideas of the proof, but more important is the fact that the finite energy loops
are a dense subspace of (LgC )f in (A) in which energy element can be decomposed as a finite sum
of eigenvectors of dA . Secondly, the notion of finite energy lifts to (LG)τ .
Parallel transport with respect to A defines a map of the fiber above 1 ∈ S 1 , given by multiplication with an element of
R G1 . This is called the holonomy hol : A → G1 . In our case, hol assigns
to βdθ the element exp( S 1 βdθ) ∈ G. Infinitesimally, we get a unitary map exp(−2πiSA ) of the
complexification gC of the fiber above 1 ∈ S 1 of the adjoint bundle, where SA is a self-adjoint
linear operator with eigenvalues strictly between −1 and 1 (if these occur, we modify SA to replace
them with 0).
Proposition 3.8. For ξ ∈ g, let ξA ∈ Lg be given by applying exp(iθSA ) to the parallel transport
of ξ with respect to A. (Note that the parallel transport of ξ will not give a loop unless ξ is in
the 0-eigenspace of SA ; ξA is always a loop.) Define z n ξA (θ) := einθ ξA (θ), then [dA , z n ξA ] =
iz n (SA + n)(ξ)A .
17
Proof. The parallel transport of ξ, which we also denote ξ, is uniquely determined by [dA , ξ] = 0.
d
, we hage [dA , z n ξA ] = inz n ξA +z n [dA , ξA ] = z n (in +iSa )ξA +
Now note that since dA is a lift of dθ
[dA , ξ]. The last term is zero, hence we’re done.
L
Definition 3.9. (LgC )f in (A) is the direct sum n∈Z z n gC .
We now continue by defining the only notion of representation for which much is known [Pre86,
chapter 9] and which is interesting [Pre86, remark (ii) after theorem 9.3.5].
Definition 3.10. Let (LG)τ be an admissible graded central extension of LG and let ρ : (LG)τ →
U (V ) be a unitary representation on a Z/2Z-graded Hilbert space such that T acts by scalars.
(V, ρ) is called a positive energy representation of level τ if the following conditions holds:
(a) The representation ρ extends to a unitary representation ρ : (L̂G)τ → U (V ).
(b) For all A ∈ A, the operator ρ̇(dA ) is a skew-adjoint operator iEA such that EA has discrete
spectrum which is bounded below.
We denote the eigenspaces of EA of energy e by Ve (A). We now note some properties of the
energy operator EA for later use.
Proposition 3.11. Suppose that ξ is an eigenvector of SA of eigenvalue , then [EA , ρ̇(z n ξA )] =
(n + )ρ̇(z n ξ). This implies that ρ̇(z n ξA ) maps Ve (A) to Ve+n+ (A).
Positive energy representations have great properties; they behave exactly like the representations of a compact Lie group. Most of these properties can be found in [Pre86, chapter 5, 9,
11].
Proposition
L3.12. (a) Positive energy representations are unitarizable and decomposable as
V =
e V (e) where each V (e) is finite-dimensional. Here V (e) are the eigenspaces of
ρ̇(dA0 ).
(b) If property (b) of definition 3.10 hold for one connection A, it holds for all elements of A.
(c) Positive energy representations are completely reducible.
(d) Positive energy representations admit an (projective) intertwining action of Dif f + (S 1 ).
(e) For each level τ there is a finite set of irreducible positive energy representations.
(f ) The isomorphism classes of irreducible positive energy representations of level τ are parametrized
by the set of antidominant weights.
Fix a connection A, then we define Vf in (A) as the direct sum of the eigenspaces Ve (A) of ρ̇(dA ).
L
Proposition 3.13. The decomposition e Ve (A) has the following properties:
(a) If ξA is a eigenvector of SA with eigenvalue , then ρ̇((z n ξA )τA )(Ve (A)) ⊂ Ve+n+ (A).
(b) If V is irreducible and Ω is a vector of lowest energy, then the vectors for ρ̇((z nj (ξj )A )τA ) · · · ρ̇((z n1 (ξ1 )A )τA )Ω
span a dense subspace of V .
L
(c) If V is finitely reducible, each
e<s Ve (A) is finite dimensional for all s ∈ R and their
dimension grows approximantely linearly with respect to s.
τ
(d) If A0 = A + βdθ, then EA0 = EA + iρ̇(βA
)+
18
β,β
.
2
3.3.3
The spin representations and canonical central extension of LG
Sketch. The construction of an adjoint representation, using this to create a canonical Clifford
central extension of LG and the construction of a spin representation proceed analogously to the
finite dimensional case. The main difference is the inclusion of a polarization to keep track of
positive and negative energy.
Details. The construction of the infinite dimensional analogous of P inc and S depend on the
choice of a polarization of the Hilbert space. In our case the polarization will keep track of the
positive and negative energy loops.
Definition 3.14. A complex structure on H is an orthogonal skew-adjoint operator J : H → H
such that J 2 = −1. A polarizing structure is a Fredholm operator such that its restriction to the
orthogonal complement ia complex structure and the extension J : H ⊗ C → H ⊗ C has non-zero
eigenvalues {±i} of infinite multiplicity.
A polarizing structure J0 determines a polarization J consisting of compatible complex structures:
If dim ker J0 is even. Then J consists of all complex structures J such that J − J0 is HilbertSchmidt.
If dim ker J0 is odd. Then J consists of all real skew-adjoint Fredholm operators which dim ker J =
1 and J a complex structure on (ker J)⊥ such that J − J0 is Hilbert-Schmidt.
Lemma 3.15. J is independent on the choice of J0 ; any other J ∈ J will generate the same
polarization.
Proof. This follows trivially from the definition.
Definition 3.16. The restricted orthogonal group OJ (H) is given by all operators T ∈ O(H)
such that T JT −1 ∈ J for all J ∈ J .
Using the constructions described in [Pre86, chapter 12], we can define the group P incJ (H),
which acts irreducibly unitary of a Z/2Z-graded Hilbert space S of spinors. Analogously to the
finite dimensional case, this action comes from an Clifford multiplication action H ∗ → End(S). A
summary of the properties of these constructions is the following theorem, which we state without
proof:
Theorem 3.17. Given a Hilbert space H and a polarization J there exists a group P incJ (H)
which fits in the graded central extension:
1 → T → P incJ (H) → OJ (H) → 1
There is a unique irreducible unitary graded representation S of P incJ , whose action we denote by χ. S admits a Clifford multiplication γ : H ∗ → End(S) such that γ(CoAd(g)(µ)) =
χ(g)γ(µ)χ(g)−1 and γ(µ) is skew-hermitian for all µ ∈ H ∗ and g ∈ P incJ .
We turn again to loop groups. For each connection on G × S 1 → S 1 , there is a canonical
polarizing structure JA on the L2 -completion H of Lg. It is given on the dense subspace of finite
energy loops by:
(
0
on H0 (A)
JA = dA
on He (A)
|e|
To check that this is independent of the A, we prove the following lemma.
Lemma 3.18. JA − JA0 is Hilbert-Schmidt.
19
Proof. Let SA denote the map
(
SA =
0
on H0 (A)
on He (A)
1
|e|
which is Hilbert-Schmidt by the growth of the eigenvalues of dA . Then JA = SA dA and we can
write JA − JA0 = SA dA − SA0 dA + SA0 dA − SA0 dA0 . The second term SA0 (dA − dA0 ) is HilbertSchmidt since the difference of dA and dA0 is bounded, being multiplication by an element of
Lg. Then first term is Hilbert-Schmidt, even though dA is unbounded, since the difference of
eigenvalues goes asymptotically as |e|1 2 .
Thus we get a canonical polarization J0 . For loop groups, there is a analogue of the adjoint
representation of a Lie group on its Lie algebra:
Proposition 3.19. There is a continuous homomorphism LG → OJ0 (H).
Proof. See [Pre86, section 6.3] for two different proofs.
We can now proceed as before, taking the pullback of the central extension of OJ0 (H) to obtain
a central extension of LG.
Definition 3.20. The graded central extension (LG)σ is obtained as LG ×OJ0 (H) P incJ0 (H).
R
This graded central extension is admissible with the basic bilinear form S 1 (β1 (s), β2 (s))ds.
This gives a splitting Lg → (Lg)σ . Through P incJ0 , (LG)σ acts on S.
Proposition 3.21. S is a positive energy representation of LG. The minimal energy with respect
to a fixed connection A can be made to be zero and then S0 (A) is an irreducible graded Clc (zA )module. Secondly Clifford multiplication by an element β ∗ ∈ Lg of energy e with respect to A
lowers energy by e, i.e. [EA , γ(β ∗ )] = −eγ(β ∗ ).
σ
→ ZA is isomorphic to the central extension constructed in the finite
The restriction ZA
dimensional case. If A = A0 , ZA0 = G and the induced splitting g → gσ is the canonical splitting.
3.3.4
A family of Dirac operators on a loop group
Sketch. For each irreducible admissible positive energy representation, we can define a family of
Dirac operators indexed by the connections A. This family is analogous to the family in the finite
2 21
dimensional case. However, it is general unbounded, which we control by dividing by (1 − DA
) .
This gives us a class in twisted K-theory.
Details. We have gathered sufficient tools to define a family of Dirac operators on a loop group.
There are some problems with defining an Clifford action of the 3-form Ω given Ω(β1 , β2 , β3 ) =
[β1 , β2 ], β3 , but these are solved in [FHT07b, section 3.3], giving an operator γ(Ω)A depending
on a connection A ∈ A. This operator preserves energy and is (LG)σ -equivariant.
Let (LG)τ be a central extension satisfying a nondegeneracy condition: −, − is positive
definite. Then let V be a finite reducible positive energy representation of (LG)τ −σ . Then V ⊗ S
is a finite energy representation of (LG)τ . For each connection A, choose a Hilbert basis {ep } of
H such that each basis vector has finite energy E(ep ) with respect to energy EA . By density of
the finite energy vectors this is possible. Let ep by the corresponding dual basis. Denote by (Rp )A
the action of eτp −σ lifted using the splitting (Lg)τ −σ → Lg given by A.
Definition 3.22. We define a Dirac family D(V ) parametrized by A as linear operator on (V ⊗
S)f in (A).
i
A 7→ DA := iγ p (Rp )A + γ(Ω)A
2
This is well-defined because (Rp )A can be defined using the splitting of induced by A and
because only a finite number of terms are nonzero on each finite energy vector (Rp raises energy,
γ p lowers it). The family of Dirac operators has formal properties similar to the famil yof Dirac
operators in the compact Lie group case.
20
Proposition 3.23. For each A ∈ A, DA is an odd skew-adjoint operator on (V ⊗ S)f in (A). DA
preserves energy, i.e. [EA , DA ] = 0 and the dependence of DA on A is as follows: if A = A0 +βdθ,
then DA = DA0 + γ(β ∗ ).
Proof. That DA is odd and skew-adjoint is trivial. To prove that it preserves energy, we note that
we have already shown that (Rp )A raises energy by E(ep ), i.e. [EA , (Rp )A ] = E(ep )(Rp )A . The
properties of the spin representation give us that Clifford multiplication γ p decreases energy by
p
p
E(e
symmetry and (LG)σ -invariance of the bilinear form
R p ), i.e. [EA , γ ] = −E(ep )γ . Finally, the
σ
(β1 (s), β2 (s))ds for the extension (LG) tells us that γ(Ω)A preserves enegy, i.e. [EA , γ(Ω)A ] =
S1
0. This means that DA preserves energy, i.e. [EA , DA ] = 0.
For the dependence on the connection, note that the lift eτp −σ changes to eτp −σ − β, ξ K.
But K acts centrally as multiplication by i, so iγ p (Rp )A+ξdθ = iγ p (Rp )A + γ p ep , ξ and the
latter is equal to γ(β ∗ ). The term γ(Ω)A doesn’t change.
However, DA will in general be unbounded. To control this, note that DA is odd skew-adjoint
2
and DA
is nonpositive.
Definition 3.24. We define a controlled Fredholm family F (V ) : A → B(V ⊗ S).
1
2 2
)
A 7→ FA := DA /(1 − DA
Proposition 3.25. The operator FA is bounded and the map F (V ) is (LG)τ -equivariant.
Proof. This first can be easily seen from the boundedness of FA on the dense subspace of finite
energy vectors. The second is a direct consequence of (LG)τ -equivariance of DA , which follows
from (LG)τ -equivariance of γ(Ω)A , the compatibility χ(g)γ(µ) = γ(CoAd(g)(µ))χ(g) and the fact
that the splitting which determines (Rp )A is equivariant with respect to the action.
Now comes the direct analogue of the large calculation we did for the Dirac operator in the
finite dimensional case:
Proposition 3.26. If V is an irreducible positive energy representation then FA is an odd skewadjoint Fredholm operator and ker FA ⊂ (V ⊗ S)emin (A).
Proof. That FA is bounded, odd and skew-adjoint is trivial. The last statement is completely
analogous to the case of a compact group G. The steps are as follows:
2
(a) Show DA
+ 2EA is multiplication by a real constant.
2
(b) Show that DA
is maximal on (V ⊗ S)emin (A) = Vemin ⊗ S0 (A).
2
on Vemin ⊗ S0 (A) reduces to the operator of a compact group G [FHT07b,
(c) Show that DA
proposition 4.12].
That it is Fredholm follows from the final statement, since by skew-adjointness the cokernel of
FA has the same dimension as the kernel of FA .
τ +dim G
Hence we obtain a map Ψ0 : Rτ −σ (LP G) → K(LG)
(A). However, note that the twisting
τ
τ
τ = (A → G1 , (LG) , ) where the local equivalence A//LG → G1 //G, where LG action on A by
pullback of a connection under the induced diffeomorphism of G × S 1 of an element of LG and
G acts on G1 by conjugation, is given by the holonomy hol : A → G1 . Hence, we can consider
τ +dim G
τ +dim G
(G1 ). The map Ψ : Rτ −σ (G) → KG
(G1 ) assigns to an
Ψ0 instead as a map Ψ into KG
irreducible positive energy representation V the class [F (V )]. Then the following theorem holds:
Theorem 3.27. Let G be a compact Lie group. Then there is an isomorphism of abelian groups:
τ +dim G
Ψ : Rτ −σ (G) → KG
(G1 )
21
3.4
The twisted case and fractional loops
In the twisted case, we start with a non-trivial principal G-bundle P . We define the gauge groups
or twisted loop groups as follows:
Definition 3.28. For G a compact Lie group and LG denote the space of G-equivariant diffeomorphisms of P convering the identity of S 1 . Furthermore, let L̂G denote the group of G-equivariant
diffeomorphisms of P convering a rigid rotation S 1 .
Most of the stuff of the untwisted case works similarly, although the holonomy needs some
modification. To a principal bundle we can assign a union of connected components of G, which
is a G-space under conjugation. This union will be denoted by G[P ]. It can be given in two ways:
• A principal G-bundle is classified up to isomorphism by a homotopy class in [S 1 , BG]. These
are exactly conjugacy classes in π1 (BG), which can be identifed with π0 (G).
• Let AP be the affine space of all connections on P . Fix a fiber and a p ∈ P in this fiber.
Then parallel transport over the circle gives us the holonomy map hol : AP → G, which
assigns to each connection its holonomy with respect to p. Its image is G[P ].
A consequence of the first remark is that all principal G-bundles over the circle are trivial
if G is connected and intuitively all principal G-bundles over the circle come from permuting
components.
The rest of the theory was set up in such a way that the theory remains almost the same. This
would have been different if we singled out the standard connection A0 to identify the Lg-torsor
A with Lg.
A second generalization is to replace the Trot appearing in the definition of the extended loop
group L̂G with a finite cover T̃rot of Trot of finite degree n ∈ Z: the new definition of L̂n G is
that it is the pullback L̂GTrot T̃rot . The results require no modifications, but this generalization is
useful in the case of tori, because a double cover occurs in the proof.
A
A.1
A.1.1
Prerequisites
Clifford algebras and spinor representations
Clifford algebras
Let F denote R or C and let V be a F-vector space . We denote by T (V ) , Λ(V ) and S(V ) are
the tensor algebra, exterior algebra and symmetric algebra on a F-vector space V respectively.
Usually Λ(V ) and S(V ) are constructed as quotients of T (V ). Alternatively, these algebras can
be defined using universal properties. For example, if U : FAlg → FVect is the forgetful functor,
then T (V ) is the universal F-algebra with embedding iV : V → U (T (V )) such that every map of
F-vector spaces V → U (A) where A is a F-algebra factors as U (f ) ◦ i for a map f : T (V ) → A of
F-algebras:
i /
U (T (V ))
T (V )
V GG
GG
GG
U (f )
f
GG
G# U (A)
A
Λ(V ) and S(V ) satisfy similar universal properties for antisymmetric and symmetric F-algebras
respectively.
Alternatively, one can define these algebras in terms of a basis e1 , . . . , en of V . For example,
Λ(V ) is generated by e1 , . . . , en subject to the relation ei ej + ej ei = 0.
Definition A.1 (Clifford algebra). Let V be a finite dimensional vector space with bilinear
form (−, −). Then Cl(V ) is the R-algebra T (V )/I where I is the ideal generated by {v ⊗ w −
(v, w)1|v, w ∈ V }.
22
From now on, we’ll assume (−, −) is non-degenerate. There are several alternative constructions, all of which give the same F-algebra. Furthermore, Cl(V ) can be described as the solution
to an universal problem. In terms of a basis e1 , . . . , en , it is generated by e1 , . . . , en subject to the
relation ei ej + ej ei = 2(ei , ej ). Clifford algebras have several important features:
Relation with Λ(V ). Note that Λ(V ) is Cl(V ) for V with zero bilinear form. It is then easy to
see that Λ(V ) and Cl(V ) are isomorphic as F-vector spaces and we can conclude dim Cl(V ) =
2dim V .
Z/2Z-grading. The ideal generated by {v ⊗w −(v, w)1|v, w ∈ V } is concentrated in even degrees.
Therefore, the Z≥0 -grading induces a Z/2Z-grading on Cl(V ).
(Anti-)automorphisms. The Clifford algebra has several (anti)automorphisms, which are induced from (anti)automorphisms of T (V ). The transpose (−)t is induced by the map of
T (V ) which reverses ordering, i.e. maps an element of the form v1 ⊗ v2 ⊗ . . . ⊗ vk to
vk ⊗ . . . ⊗ v2 ⊗ v1 .
The automorphism of T (V ) given by v 7→ −v on generators, induces an automorphism α
of Cl(V ). This is called the grading involution. The composition of the transpose and the
grading involution x 7→ x̄ := α(xt ) is again an antiautomorphism of Cl(V ).
Functoriality The construction of Clifford algebras is functorial in V . Any linear map f : V → W
preserving the inner product induces a map f ∗ : Cl(V ) → Cl(W ).
There are several varieties of Clifford algebra associated to a real vector space. There are
two R-algebras: Applying the Clifford algebra construction to V with bilinear form (−, −) gives
Cl+ (V ) , applying the Clifford algebra construction to V with bilinear form −(−, −) gives Cl− (V )
. However, the important case for us will the complexication of a Clifford precise. To be precise,
we will define Clc (V ) as Cl− (V )⊗C . It is isomorphic to Cl(V ⊗C) if we use the bilinear extension
of (−, −), not the hermitian one.
A.1.2
P inc
We can make the earlier linear isomorphism between Λ(V ) and Cl(V ∗ ) ∼
= Cl(V ) (where V ∗ has
∗
the dual inner product) more explicit. There is a filtration of Cl(V ) induced by multiplication
with generators. The associated graded algebra of Cl(V ∗ ) is then Λ(V ).
Let {ea } be a basis of V and {ea } be the dual basis. Let gab = (ea , eb ) denote the metric
tensor and g ab = (ea , eb ) the dual metric tensor. Using the filtration Λ2 (V ) can be identified with
the linear span of 41 (ea eb − eb ea ) for 1 ≤ i < j ≤ n. To be precise, we use the somewhat unusual
identification ea ∧ eb 7→ 41 (ea eb − eb ea ). In fact Λ2 (V ) is a Lie algebra and the reason for the factor
1
4 is that it is necessary to get a basis of a Lie algebra which is closed under the Lie bracket.
Proposition A.2. Λ2 (V ) ⊂ Cl(V ∗ ) is a Lie algebra under the graded commutator.
Proof. It suffices to calculate the commutator of two basis elements of the form 12 ea eb :
1
1
[ (ea eb − eb ea ), (ec ed − ed ec )] =
4
4
1
1
1
1
g ac (eb ed − ed eb ) + g bc (ea ed − ed ea ) + g ad (eb ec − ec eb ) + g bd (ea ec − ec ea )
4
4
4
4
This also shows that the image of the bracket is contained in Λ2 (V ). Antisymmetry and the
Jacobi identity are clear from the definition of the bracket as coming from the commutator.
In terms of a basis of V the Lie algebra so(V ) of skew-symmetric matrices is generated by
matrices Eab for 1 ≤ a < b ≤ dim V , given by gcb (ea ⊗ ec − ec ⊗ ea ). If the basis is orthonormal,
then this matrix has components which are zero except for the (a, b)’th entry, which is 1, and the
23
(b, a)’th entry, which is -1, and we get the usual generators of so(V ). These satisfy the commutation
relations
[Eab , Ecd ] = g ac Ebd + g bc Ead + g ad Ebc + g bd Eac
There is a isomorphism of vector spaces between so(V ) and Λ2 (V ), mapping Eab to ea ∧ eb ,
which gets identified with 41 (ea eb − eb ea ) ∈ Λ2 (V ) ⊂ Cl(V ∗ ). Our earlier calculations show this is
an isomorphism of Lie algebras.
If we want to work coordinate-invariantly, like we should, then the isomorphism of Lie algebras
is implemented by the composition of the following two maps:
A 3 so(V ) 7→ (A−, −) ∈ Λ2 (V )
α 3 Λ2 (V ) 7→ µ(i ⊗ i)(α) ∈ Cl(V ∗ )
where in the last line µ is Clifford multiplication and i in the inclusion V ∗ ,→ Cl(V ∗ ) and we
consider Λ2 (V ) ,→ V ∗ ⊗ V ∗ with balanced representative, i.e. a ∧ b 7→ 21 (a ∧ b − b ∧ a).
Then we define the Lie groups Spin± (V ) as the groups given by exponentiating Λ2 (V ) in
±
Cl (V ∗ ). These form a double cover of SO(V ), which can extended to a double cover of O(V ),
giving the Lie groups P in± (V ):
1
/ Z/2Z
/ Spin± (V )
/ SO(V )
/1
1
/ Z/2Z
/ P in± (V )
/ O(V )
/1
Similarly, we obtain P inc (V ) ⊂ Clc (V ∗ ). This is a central extension of P in− (V ) by T:
1
/ Z/2Z
/ P in− (V )
/ O(V )
/1
1
/T
/ P inc (V )
/ O(V )
/1
Alternatively definitions of the P in groups abound. Two well-known alternatives are the
following: one can take P in to be the invertible elements in the appropiate Clifford algebra such
that xvα(x)−1 ∈ V for all v ∈ V and xt x = 1. Using the Pfaffian of a skew-symmetric map, one
can define the spin group as a element A of O(V ⊗ C) together with a square root of a polynomial
defined on a chart of the space of complex structures.
A.1.3
The spinor representation
We want to a construct an irreducible Z/2Z-graded Clc (V )-module for later use. To do this we
first note that there is a Clc (V )-module structure on Λ(V ) ⊗ C. There are two main constructions:
As a subrepresentation of Λ(V ) ⊗ C The idea is that the action of Clc (V ) on V extends to an
action Λ(V ), which is reducible. By choosing any complex structure on V , we can construct
a representation of Clc (V ) on Λ(W ), where W the +i eigenspace of J in V ⊗ C. Generators
of Clc (V ) coming from W will act as creation operators, generators coming from W̄ will act
as annihilation operators. This representation is irreducible. For more information, look at
[Woi08, section 2.3] or [Pre86].
As holomorphic sections of a line bundle We can define a holomorphic line bundles P f over
J (V ), the space of complex structures on V , which can be identified with a submanifold
over Gr(V ) by representing a complex structure by its isotropic i-eigenspace. We can define
the spinor representation as the holomorphic section Γ(P f ∗ ) of the dual bundle to P f .
If dim V is odd, there is a single irreducible Z/2Z-graded Clc (V ) representation. It has a
commuting action of Clc (1), which we consider part of the structure. If dim V is even, there are
two irreducible Z/2Z-graded Clc (V ) representations. These are distinguished by the action of the
volume form. We fix one of them.
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References
[FHT05]
D.S. Freed, M.J. Hopkins, and C. Teleman. Loop groups and twisted k-theory iii. 2005.
arxiv:math/0312155v3.
[FHT07a] D.S. Freed, M.J. Hopkins, and C. Teleman. Loop groups and twisted k-theory i. 2007.
arxiv:0711.1906v1.
[FHT07b] D.S. Freed, M.J. Hopkins, and C. Teleman. Loop groups and twisted k-theory ii. 2007.
arxiv:math/0511232v2.
[Pre86]
S. Pressley, A. & Segal. Loop Groups. Oxford University Press, 1986.
[Wil05]
S. Willerton. The twisted drinfeld double of a finite group via gerbes and finite
groupoids. 2005. arXiv:math/0503266v1.
[Woi08]
P. Woit. Quantum field theory and representation theory: A sketch. 2008. arxiv:hepth:0206135v1.
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Compact
Lie group G
Spinor representation
S of Clc (g∗ )
Spinor fields
C ∞ (G) ⊗ S
Peter-Weyl
theorem
Subrepresentation
V ⊗ S for V
irreducible
Konstant’s
cubic Dirac
operator D0
Family of Dirac
operators D(V )
Class [D(V )] in
σ+dim G ∗
KG
(g )cpt
Figure 1: An figure of the construction of K-classes for irreducible representations of a compact
Lie group.
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Loop group LG
with admissible central
extension τ
Spinor representation
S of P incJ
Vf in ⊗ S
Cubic Dirac
operator D0
Family of Dirac
operators D(V )
Positive energy
representation V
Class [D(V )] in
σ+τ +dim G
KG
(G)
Figure 2: An figure of the construction of K-classes for irreducible positive energy representations
at leavel τ of a loop group with admissible central extesnsion τ .
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