DIFFERENTIAL GEOMETRY OF
GERBES AND TWO-VECTOR BUNDLES
The evolution of a classical physical system can often be expressed in terms of
differential equations, and these are often the Euler-Lagrange equations for critical
points of an action integral. Given two configurations x and y of the physical
system, at times t = r and t = s, and a possible time evolution t 7→ q(t) from
q(r) = x to q(s) = y, theRaction assigns a number S(q). In local theories, S(q)
s
is given as some integral r (−) dt over the time evolution, where the integrand
depends on q(t) locally. By Hamilton’s principle, the true evolution from x to y
will be a critical point for the action function q 7→ S(q). Here x and y are points
in a phase space X, q : [r, s] → X ranges over paths in X from x to y, and S is a
function on this path space Px,y X.
[In quantum mechanics x is replaced by a probability distribution, or more precisely, by a vector in a Hilbert space of sections in a bundle over the phase space,
whose square norm gives the probability distribution ...]
[In string theory the particles are viewed as closed loops, not points ...]
We think of a particle as a closed string, or free loop, γ : S 1 → X in some
base space X. Soon, X will be a smooth manifold, somewhat related to spacetime. To each string γ we wish to associate a state space H(γ). This is usually
a Hilbert space, but for us it will be a virtual vector space. As γ ranges through
the free loop space LX of X, the state spaces assemble to a virtual vector bundle
H → LX. In a quantized theory, it may be more appropriate to consider a ‘virtual’
Hilbert space of sections in this virtual bundle. For systems composed of multiple
particles, being strings γ1 , . . . , γp : S 1 → X, the state space is the tensor product
H(γ1 ) ⊗ · · · ⊗ H(γp ). When p = 0, this state space is C. A possible time evolution
is expressed by a compact (oriented) surface φ : Σ → X in the base space. The
boundary ∂Σ decomposes as p incoming circles, parametrizing free loops γ1 , . . . , γp ,
and q outgoing circles, parametrizing free loops γ1′ , . . . , γq′ . To this φ we wish to
associate an action functional S(φ) : H(γ1 ) ⊗ · · · ⊗ H(γp ) → H(γ1′ ) ⊗ · · · ⊗ H(γq′ ).
When Σ is closed, this is a linear isomorphism C → C, given by multiplication by
some nonzero scalar in C. The state spaces H(γ) and the action functional S(φ)
determine the evolution of the physical system, and are known as a quantum field
theory (QFT). Usually X is a Riemannian manifold, so Σ inherits a Riemannian
structure via φ. If the action functional S(φ) only depends on the topology of
Σ, we have a topological quantum field theory (TQFT). If it only depends on the
conformal structure on Σ, we have a conformal quantum fields theory (CQFT). The
latter has the most interesting relation to elliptic modular forms.
[Relate these points of view ...]
We propose to realize state spaces as action functionals over X in terms of parallel
transport in a differental-geometric object over X, called a two-vector bundle. This
object will be a categorification of the notion of a complex vector bundle over X,
replacing the ring (C, +, 0, ×, 1) of complex numbers with the semi-ring category (=
Typeset by AMS-TEX
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DIFFERENTIAL GEOMETRY OF GERBES AND TWO-VECTOR BUNDLES
rig category = bipermutative category) (V, ⊕, 0, ⊗, C) of finite-dimensional complex
vector spaces, and their linear isomorphisms. Maybe we should say that V is a semiring groupoid, or a bipermutative groupoid.
In a rank n complex vector bundle E over X, each fiber Ex is isomorphic to
the C-vector space Cn of n-tuples of complex numbers. Likewise, in a rank n twovector bundle E over X, each fiber Ex will be isomorphic to the category (actually
a groupoid) V n of n-tuples of vector spaces, via an isomorphism that (morally)
respects the V-action on V n induced by the tensor product. A rank 1 vector bundle
L is a line bundle, whereas a rank 1 two-vector bundle G is also known as a bundle
gerbe (always with band C∗ ).
A Cn -bundle is locally trivial, hence locally isomorphic to a product bundle
U × Cn → U , where U ⊂ X is open. A section in such a trivialization is an
n-tuple of complex functions on U . Two different trivializations over U differ by
a bundle isomorphism U × Cn → U × Cn , which corresponds to a map U →
GLn (C). Likewise, a rank n two-vector bundle will be locally isomorphic to the
product U × V n → U . A section is now an n-tuple of vector bundles over U .
More precisely, the sections over U form the groupoid V ect(U )n , with objects the
n-tuples of vector bundles (V1 , . . . , Vn ) over U , and isomorphisms the n-tuples of
vector bundle isomorphisms over U .
[May alternatively want to work with the associated principal two-bundle, with
fibers isomorphic to GLn (V).]
Let ∇ be a connection on E. For each curve γ : [0, 1] → X from x to y, and
each vector v ∈ Ex , there is a unique horizontal lift γ̃ : [0, 1] → E covering γ,
with γ̃(0) = v. The end-point γ̃(1) ∈ Ey depends linearly on v, so there is a linear
isomorphism Tγ : Ex → Ey , given by parallel transport of vectors in (E, ∇). When
γ : S 1 → X is a closed loop based at x, Tγ : Ex → Ex is a linear automorphism, with
a well-defined determinant. For each fixed trivialization Cn ∼
= Ex , Tγ corresponds to
a matrix in GLn (C), but different trivializations give conjugate matrices. Different
base points for γ also give conjugate matrices, but the same determinant.
There is also a notion of connection in a two-vector bundle E, called a connective
structure Co. For each curve γ : [0, 1] → X we would ideally like to specify a notion
of parallel transport of n-tuples of vector spaces W = (W1 , . . . , Wn ) ∈ E, giving a Vlinear functor Tγ : Ex → Ey . When γ : S 1 → X is a closed loop, Tγ : Ex → Ex should
ideally be a V-linear endofunctor. For each fixed trivialization V n ∼
= Ex , Tγ should
ideally translate to multiplication by a matrix in GLn (V), whose determinant is a
virtual vector space. The idea is that this virtual vector space, the holonomy of
(E, Co) around γ, shall define the state space component H(γ) of a QFT. As γ
varies in the free loop space LX, the virtual vector spaces H(γ) should assemble
to a virtual vector bundle over LX, the holonomy bundle.
For n = 1, when we are considering bundle gerbes G over X, a construction
of this ideal notion of parallel transport is described in Section 6.6 of Brylinski’s
book. The holonomy around a closed loop γ is given by the tensor product with a
line H(γ), and these assemble to a line bundle H over LX, known as the anomaly
bundle. See Brylinski’s Section 6.2. A first goal in this seminar is to extend this
construction to higher rank two-vector bundles.
Given a bundle gerbe with connective structure (G, Co) over X, there is a secondary notion of connection, called a curving K. It should have a generalization
to rank n two-vector bundles with connective structure (E, Co). When n = 1, the
curving K determines a connection D on the anomaly bundle H over LX, and
DIFFERENTIAL GEOMETRY OF GERBES AND TWO-VECTOR BUNDLES
3
hence a notion of parallel transport in (H, D) over paths in LX. A path in LX
from γ to γ ′ is the same as a map φ : Σ = S 1 × [0, 1] → X, where γ and γ ′ map
from the incoming and outgoing boundary circles of Σ, respectively. The parallel
transport Tφ : H(γ) → H(γ ′ ) then provides the action functional component S(φ)
of the QFT, at least for cylinders. More generally, the curving K specifies an action
functional S(φ) : H(γ1 ) ⊗ · · · ⊗ H(γp ) → H(γ1′ ) ⊗ · · · ⊗ H(γq′ ), with suitable gluing
and invariance properties, defining a full QFT. A second goal is to also extend this
construction to higher rank two-vector bundles.
Line bundles L over X are classified, up to isomorphism or concordance, by
homotopy classes of maps from X to BC∗ ≃ K(Z, 2), hence correspond bijectively
to elements c1 (L) of H 2 (X; Z). Similarly, bundle gerbes with band C∗ over X
are classified, up to a suitable notion of equivalence, by homotopy classes of maps
from X to BBC∗ ≃ K(Z, 3), hence correspond bijectively to elements of H 3 (X; Z).
There is a transgression map H 3 (X; Z) → H 2 (LX; Z), taking the class of a gerbe G
to the first Chern class c1 (H) of the associated anomaly line bundle (for any choice
of connective structure Co on G).
Line bundles with connection (L, ∇) are instead classified by elements of a
smooth Deligne cohomology group H 2 (X; Z( 2)D ). This is the hypercohomology
of X with coefficients in the complex of sheaves
dlog
C∗X −−→ Ω1 (X).
Similarly, bundle gerbes with connective structure and curving (G, Co, K) are classified by elements of the Deligne cohomology group H 3 (X; Z(3)D ), which is the
hypercohomology of X with coefficients in the complex of sheaves
dlog
d
C∗X −−→ Ω1 (X) −
→ Ω2 (X).
There is a transgression map H 3 (X; Z(3)D ) → H 2 (LX; Z( 2)D ), taking the class of
a bundle gerbe with connective structure and curving (G, Co, K) to the class of the
anomaly bundle with its induced connection, (H, D).
[Discuss generalized cohomological classification of two-vector bundles, possibly
with connective structure or curving. The latter should be the true elliptic objects.]
Definition: Let X be a topological space. A totally ordered open cover of X is a
collection
S U of open subsets Uα ⊂ X indexed by α in a totally ordered set I, such
that α Uα = X. We write Uαβ = Uα ∩ Uβ , Uαβγ = Uα ∩ Uβ ∩ Uγ , etc.
More general partial orderings can be allowed. May also wish to allow local
homeomorphisms Uα → X, submersions Uα → X, or fibrations Uα → X.
The sheaf version of the following definition is due to Giraud, and explained by
Brylinski. The bundle version is due to Murray, Chatterjee and Hitchin.
Definition: Let X be a topological space with an ordered open cover U. A
charted bundle gerbe G consists of
(i) a “gluing” line bundle Lαβ over Uαβ for each pair α < β in I,
and
(ii) a “coherence” isomorphism
∼
=
φαβγ : Lαβ ⊗ Lβγ −→ Lαγ
between line bundles over Uαβγ , for each triple α < β < γ in I,
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DIFFERENTIAL GEOMETRY OF GERBES AND TWO-VECTOR BUNDLES
such that
(iii) the two resulting composite isomorphisms
∼
=
Lαβ ⊗ Lβγ ⊗ Lγδ −→ Lαδ
of line bundles over Uαβγδ are equal.
We think of G as being a locally trivial V-bundle, so that G|Uα is isomorphic to
Uα × V over Uα . For α < β there are two trivializations over Uαβ , which can be
compared by a V-linear isomorphism Uαβ × V → Uαβ × V over Uαβ . At the level of
sections, this isomorphism should take vector bundles over Uαβ to vector bundles
over Uαβ , in a way that is compatible with the V-module structure. In other words,
the image of the trivial line bundle Uαβ × C → Uαβ must be some vector bundle
Lαβ over Uαβ , and the functor of sections is given by the tensor product with Lαβ .
To give an isomorphism of categories, this tensor product must admit an inverse
up to isomorphism, so the vector bundle Lαβ must in fact be a line bundle.
Alternatively, we may think of G in terms of a principal GL1 (V)-bundle, where
GL1 (V) = V1 ⊂ V is the groupoid of lines. It is equivalent to the groupoid with
one object and C∗ as morphisms. Then G|Uα is isomorphic to Uα × GL1 (V) → Uα ,
and the two trivializations over Uαβ are still compared by an isomorphism given by
tensor product with a line bundle Lαβ .
[Discuss equivalence of charted gerbes, generated by elementary changes of trivializations, and by refinement of the ordered cover.]
[Discuss the Dirac magnetic monopole G over S 3 , with Uα = S 3 − {S}, Uβ =
S 3 − {N }, Uαβ = S 3 − {S, N } ∼
= R3 − {0}, Lαβ the Hopf line bundle over S 3 −
2
{S, N } ≃ S . Note that the principal GL1 (V)-bundle (of groupoids) has no global
object, since this would restrict to a line bundle V α over Uα and a line bundle V β
over Uβ , subject to the relation V α = Lαβ ⊗ V β over Uαβ . Both Uα and Uβ are
contractible, so V α and V β are trivial, which can only happen if Lαβ is also trivial.
By construction, it is not.]
[Another model for the magnetic monopole over S 3 views the three-sphere as
the join S 1 ∗ S 1 , where S 1 is covered by three open intervals with empty triple
intersection. This gives a cover of S 3 by three contractible open subsets Uα , Uβ and
Uγ , such that each double intersection is contractible, but Uαβγ ∼
= S 1 × R2 ≃ S 1 .
The gluing bundles Lαβ , Lβγ and Lαγ can all be taken to be trivial, but the
coherence isomorphism φαβγ : Uαβγ × C → Uαβγ × C is given by multiplication by a
function Uαβγ → C∗ of degree 1. In this case the gerbe admits global objects over
each covering set, for example given by a trivial line bundle V α over Uα , etc. As
this example shows, these can be compatible over the double intersections, since
V α = Lαβ ⊗ V β over Uαβ , etc. However, the gerbe does still not admit a global
object, since the two identifications V α = Lαβ ⊗ Lβγ ⊗ V γ and V α = Lαγ ⊗ V γ are
compared over Uαβγ by the non-identity coherence isomorphism φαβγ .]
Here is the higher rank version, as defined by Baas, Dundas and Rognes.
Definition: Let X be a topological space with an ordered open cover U. A
charted rank n two-vector bundle E consists of
αβ n
(i) an n × n matrix of “gluing” bundles E αβ = (Eij
)i,j=1 over Uαβ for each pair
αβ
α < β in I, with det(dim E ) = +1 or −1,
and
(ii) an n × n matrix of “coherence” isomorphisms
∼
=
αβ
n
· E βγ −→ E αγ
φαβγ = (φαβγ
ij )i,j=1 : E
DIFFERENTIAL GEOMETRY OF GERBES AND TWO-VECTOR BUNDLES
5
between n × n matrices of vector bundles over Uαβγ , for each triple α < β < γ in I,
such that
(iii) the two resulting composite isomorphisms
∼
=
E αβ · E βγ · E γδ −→ E αδ
of n × n matrices of vector bundles over Uαβγδ are equal.
Here E αβ · E βγ is the matrix product, with (i, k)-th entry
n
M
αβ
βγ
Eij
⊗ Ejk
.
j=1
More precisely, we have defined a bundle gerbe with band C∗ . The role of
C is that the gluing data in such a gerbe, over the double intersections Uαβ ,
is given by a map Uαβ → BC∗ to the classifying space of the group C∗ . Over
the triple intersections Uαβγ the coherence data makes use of the tensor product
represented by a pairing BC∗ × BC∗ → BC∗ . This is the map induced by the
group multiplication C∗ × C∗ → C∗ , which is a group homomorphism because C∗
is abelian.
This is fine for the classical discussion of photons and electromagnetism, since
these have state spaces given by spaces of sections in U (1)-bundles. However, to
model the weak and strong nuclear forces, sections in SU (2)- and SU (3)-bundles
are used, and a theory of non-abelian gerbes may be required.
The gluing data in a rank n two-vector bundle, over a double intersection, is given
by a map to the
space |GLn (V)| of a groupoid, rather than of a group.
Qclassifying
n
Here Mn (V) = i,j=1 V is the groupoid of n×n matrices of vector spaces, and their
isomorphisms. We let GLn (V) be the full subgroupoid with objects W = (Wij )ni,j=1
where the dimension matrix dim(W ) = (dim(Wij )ni,j=1 has determinant +1 or
−1. Let GLn (N) = GLn (Z) ∩ Mn (N) be the multiplicative monoid of invertible
integer matrices D = (dij )ni,j=1 with non-negative entries. The groupoid GLn (V)
decomposes as
∗
a
GLn (V) ≃
n
Y
GLdij (C)
D∈GLn (N) i,j=1
a
≃
n
Y
U (dij )
D∈GLn (N) i,j=1
and its classifying space satifies
|GLn (V)| ≃
a
n
Y
BU (dij ).
D∈GLn (N) i,j=1
The matrix product required for the coherence data over the triple intersections
Uαβγ is now induced by a functor
·
GLn (V) × GLn (V) −
→ GLn (V)
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DIFFERENTIAL GEOMETRY OF GERBES AND TWO-VECTOR BUNDLES
with the induced map
·
|GLn (V)| × |GLn (V)| −
→ |GLn (V)|
of classifying spaces. Note that GL1 (V) ≃ C∗ as groupoids, and hence |GL1 (V)| ≃
BC∗ .
In other words, the extension from C∗ -gerbes to two-vector bundles is based on
extending the product C∗ × C∗ → C∗ to the tensor product GLd (C) × GLe (C) →
GLde (C), which is a group homomorphism, rather than trying to work with the
group multiplication GLd (C) × GLd (C) → GLd (C), which is not a group homomorphism for d ≥ 2.
The groupoid GLn (V), with the monoidal pairing given by matrix product, is a
monoidal groupoid. Monoidal groupoids where each object admits an inverse up to
isomorphism have been studied by Sinh, Ulbrich and Breen. In Brylinski’s Section
7.3 they are called groupoids with tensor product. Two-vector bundles are more
general than gerbes with gluing data in such, groupoids with tensor product, since
most objects in GLn (V) do not admit inverses up to isomorphism.
[Define connective structure for gerbes and two-vector bundles in terms of connections on the gluing bundles that are compatible via the coherence isomorphisms.
Explain the construction of the holonomy (= anomaly) line bundle for a gerbe with
connective structure, as well as its parallel transport. Seek to generalize to twovector bundles with connective structure. Compare two choices of connections on
the same local sections and integrate along a path to get a matrix of vector space
endomorphisms. What part of this should be extracted as the parallel transport?]