Bundle Gerbes for Chern-Simons and Wess-Zumino-Witten Theories

TL;DR: The theory of Chern-Simons bundle 2-gerbes and multiplicative bundle gerbes associated to any principal G-bundle with connection and a class in H4(BG, ℤ) for a compact semi-simple Lie group was developed in this paper.

Abstract: We develop the theory of Chern-Simons bundle 2-gerbes and multiplicative bundle gerbes associated to any principal G-bundle with connection and a class in H4(BG, ℤ) for a compact semi-simple Lie group G. The Chern-Simons bundle 2-gerbe realises differential geometrically the Cheeger-Simons invariant. We apply these notions to refine the Dijkgraaf-Witten correspondence between three dimensional Chern-Simons functionals and Wess-Zumino-Witten models associated to the group G. We do this by introducing a lifting to the level of bundle gerbes of the natural map from H4(BG, ℤ) to H3(G, ℤ). The notion of a multiplicative bundle gerbe accounts geometrically for the subtleties in this correspondence for non-simply connected Lie groups. The implications for Wess-Zumino-Witten models are also discussed.

Summary

1. Introduction

• Quillen introduced the determinant line bundle of Cauchy-Riemann operators on a Hermitian vector bundle coupled to unitary connections over a Riemann surface.
• Briefly speaking, a degree p Deligne characteristic class for principal G-bundles with connection is an assignment to any principal G-bundle with connection over M of a class in the degree p Deligne cohomology group Hp(M,Dp) satisfying a certain functorial property.
• The authors will define three dimensional Chern-Simons gauge theories CS(G) as degree 3 Deligne cohomology valued characteristic classes for principal G-bundles with connection, but will later show that there is a global differential geometric structure, the Chern-Simons bundle 2-gerbe, associated to each Chern-Simons gauge theory.
• Next in the hierarchy, H2(M,D2) is the space of stable isomorphism classes of bundle gerbes with connection and curving, whose holonomy is defined for any 2-dimensional closed sub-manifold and whose characteristic class in H3(M,Z) is given by the Dixmier-Douady class of the underlying bundle gerbe.
• With the understanding of (2.3), the authors often identify a Deligne class with the corresponding Cheeger-Simons differential character.

3. From Chern-Simons to Wess-Zumino-Witten

• Dijkgraaf and Witten discuss a correspondence map between three dimensional Chern-Simons gauge theories and Wess-Zumino-Witten models associated to the compact Lie group G from the topological actions viewpoint, which naturally involves the transgression map τ : H4(BG,Z) → H3(G,Z).
• To classify the exponentiated Chern-Simons action in three dimensional ChernSimons gauge theories, the authors propose the following mathematical definitions of a three dimensional Chern-Simons gauge theory and a Wess-Zumino-Witten model.
• The next proposition shows that the map Ψ descends to the natural transgression map from H4(BG,Z) to H3(G,Z) and hence Ψ refines the Dijkgraaf-Witten correspondence.
• First it is now well understood how, given a WZW model, the authors can define an associated bundle gerbe over the group G, as WZW (G) = H2(G,D2) is the space of stable isomorphism classes of bundle gerbes with connection and curving [12, 38].

4. Bundle 2-gerbes

• A bundle 2-gerbe with connection and curving, defines a degree 3 Deligne class in H3(M,D3).
• For an exception, the authors denote by EG [] the associated simplicial manifold {EG[n]} for the universal bundle π : EG→ BG,.
• Changing to the trivialisation representing the bundle gerbe product for Q, the authors find that the extra terms involving δ(J) all cancel (in the sense of having canonical trivialisations), hence the associator line bundles for P and Q are the same, so the bundle 2-gerbe product for Q is well defined.
• Analogous to the fact that H2(M,D2) classifies stable equivalence classes of bundle gerbes with connection and curving, the authors have the following proposition, whose complete proof can be found in [34].

5. Multiplicative bundle gerbes

• The simplicial manifold BG associated to the classifying space of G is constructed in [19], where the total space of the universal G-bundle EG also has a simplicial manifold structure.
• Xn which are compatible with the face and degeneracy operators for X. Brylinski and McLaughlin in [10] explain how one may inductively construct such a family of coverings by first starting with an arbitrary cover U0 of X0 and then choosing a common refinement U1 of the induced covers d−10 (U 0) and d−11 (U 0) of X1.
• As an immediate consequence of Proposition 5.3, the authors see that isomorphism classes of simplicial bundle gerbes on BG are classified by the simplicial Čech cohomology group H3(BG, U(1)).

6. The Chern-Simons bundle 2-gerbe

• The authors will construct a Chern-Simons bundle 2-gerbe over BG such that the proof of the main theorem (Theorem 5.9) follows.
• Alternatively, the authors could choose a smooth finite dimensional n-connected approximation Moreover the bundle gerbe G over G is equipped with a connection and curving with the bundle gerbe curvature given by ωφ.
• The theorem of Cheeger and Simons given above defines a unique differential character satisfying certain conditions, thus it uniquely defines a class in Deligne cohomology and an equivalence class of bundle 2-gerbes with connection and curving, so the authors must show that the corresponding Deligne class associated to the Chern-Simons bundle 2-gerbe satisfies the required conditions.

7. Multiplicative Wess-Zumino-Witten models

• The authors study the Wess-Zumino-Witten models in the image of the correspondence from CS(G) to WZW (G).
• Recall their correspondence map in Definition 3.3 and their integration map (2.4).
• The authors constructed Ψ from a canonical G-bundle over S1 ×G in Definition 3.3 such that the bundle gerbe holonomy for the multiplicative bundle gerbe G with connection and curving over G, being in the image of Ψ, corresponds to the bundle 2-gerbe holonomy for the Chern-Simons bundle 2-gerbe Q over S1 ×G as follows.
• In particular, for non-simply connected compact semi-simple Lie group G, the authors know that the Wess-Zumino-Witten model on G is only multiplicative at certain levels.

Full text

University of Zurich
Zurich Open Repository and Archive
Winterthurerstr. 190
CH-8057 Zurich
http://www.zora.uzh.ch
Year: 2005
Bundle gerbes for Chern-Simons and Wess-Zumino-Witten
theories
Carey, Alan; Johnson, Stuart; Murray, Michael; Stevenson, Danny; Wang, Bai-Ling
Carey, Alan; Johnson, Stuart; Murray, Michael; Stevenson, Danny; Wang, Bai-Ling (2005). Bundle gerbes for
Chern-Simons and Wess-Zumino-Witten theories. Comm. Math. Phys., 259(3):577-613.
Postprint available at:
http://www.zora.uzh.ch
Posted at the Zurich Open Repository and Archive, University of Zurich.
http://www.zora.uzh.ch
Originally published at:
Comm. Math. Phys. 2005, 259(3):577-613.
Carey, Alan; Johnson, Stuart; Murray, Michael; Stevenson, Danny; Wang, Bai-Ling (2005). Bundle gerbes for
Chern-Simons and Wess-Zumino-Witten theories. Comm. Math. Phys., 259(3):577-613.
Postprint available at:
http://www.zora.uzh.ch
Posted at the Zurich Open Repository and Archive, University of Zurich.
http://www.zora.uzh.ch
Originally published at:
Comm. Math. Phys. 2005, 259(3):577-613.

arXiv:math/0410013v2 [math.DG] 12 Sep 2005
BUNDLE GERBES FOR CHERN-SIMONS AND
WESS-ZUMINO-WITTEN THEORIES
ALAN L. CAREY, STUART JOHNSON, MICHAEL K. MURRAY, DANNY STEVENSON,
AND BAI-LING WANG
Abstract. We develop the theory of Chern-Simons bundle 2-gerbes and mul-
tiplicative bundle gerbes associated to any principal G-bundle with connection
and a class in H
4
(BG, Z) f or a compact semi-simple Lie group G. The Chern-
Simons bundle 2- gerbe realises differential geometrically the Cheeger-Simons
invariant. We apply these notions to refine the Dijkgraaf-Witten corresp on-
dence between three dimensional Chern-Simons functionals and Wess-Zumino-
Witten models associated to the group G. We do this by introducing a lifting
to the level of bundle gerbes of the natural map from H
4
(BG, Z) to H
3
(G, Z).
The notion of a multiplicative bundle gerbe accounts geometrically f or the
subtleties in this correspondence for non-simply connected Lie groups. The
implications for Wess-Zumino-Witten models are also discussed.
Contents
1. Introduction 1
2. Deligne characteristic classes for principal G-bundles 5
3. From Chern-Simons to Wess-Zumino-Witten 10
4. Bundle 2-gerbes 15
5. Multiplicative bundle gerbes 20
6. The Chern-Simons bundle 2-gerbe 26
7. Multiplicative Wess-Zumino-Witten models 32
References 35
1. Introduction
In [42] Quillen introduced the determinant line bundle of Cauchy-Riemann oper-
ators on a Hermitian vector bundle coupled to unitary connections over a Riemann
surface. This work influenced the development of many lines of investigation in-
cluding the study of Wess-Zumino-Witten actions on Riemann surfaces. Note that
Quillen’s determinant line bundle also plays an essential role in the construction of
the universal bundle gerbe in [15], see also [8].
The relevance of Chern-Simons gauge theory has been noted by many a uthors,
starting with Ramadas-Singer-Weitsman [43] and recently Dupont-Johansen [20],
who used gauge covariance of the Chern-Simons functional to give a ge ometric
2000 Mathematics Subject Classification. 55R65, 53C29, 57R20, 81T13.
The authors acknowledge the support of the Australian Research Council. ALC thanks MPI f¨ur
Mathematik in Bonn and ESI in Vienna and BLW thanks CMA of Australian National Uni versity
for their hospitality during part of the writing of this paper.
1

2 A.L. CAREY, S. JOHNSON, M.K. MURRAY, D. STEVENSON, AND BAI-LING WANG
construction of Quillen line bundles. The curvatures of these line bundles in an
analytical set-up were studied extensively by Bismut-Freed [3] and in dimension
two, went back to the Atiyah-Bott work on the Yang-Mills equations over Riemann
surfaces. [2].
A new element was introduced into this picture by Fr e e d [25] and [24] (a re-
lated line of thinking was started by some of the present authors [14]) through the
introduction of higher algebr aic structures (2-categories) to study Chern-Simo ns
functionals on 3-ma nifolds with boundary and corners. For closed 3-manifolds one
needs to study the behaviour of the Chern- Simons ac tion under gluing formulae
(that is topological quantum field theories) generalising the corresponding picture
for Wess -Zumino-Witten. Heuristically, there is a Cher n-Simons line bundle as in
[43], such that for a 3-manifold with boundary, the Chern-Simons action is a sec-
tion o f the Chern-Simons line bundle assoc iated to the boundary Riemann surface.
For a codimension two submanifold, a closed circle, the Chern-Simons action takes
values in a U(1)-gerbe or an abelian group-like 2-category.
Gerbes first began to enter the picture with J -L Brylinski [6] and Breen [5]. The
latter developed the notion of a 2-gerbe as a sheaf of bicategories extending Giraud’s
[29] definition of a gerbe as a sheaf of groupoids. J-L Brylinski used Giraud’s gerbes
to study the central extensions of loop groups, string structures and the relation
to Deligne cohomology. With McLaughlin, Brylinski developed a 2- gerbe over a
manifold M to realise degre e 4 integral cohomology on M in [10] and introduced
an expression of the 2-gerbe holonomy as a Cheeger-Simons differential character
on any manifold with a tr iangulation. This is the starting point for Gomi [31],[32]
who developed a local theory of the Chern-Simons functional along the lines of
Freed’s suggestion. A different approach to some of these matters using simplicial
manifolds has been found by Dupont and Kamber [21 ].
Our contribution is to develop a global differential geometric realization of Chern-
Simons functionals using a Chern-Simons bundle 2-gerbe and to apply this to the
question raised by Dijkgraaf and Witten about the relation between Chern-Simons
and Wess-Zumino-Witten models. Our approach provides a unifying perspective
on all of this previous work in a fashion that can be directly related to the physics
literature on Chern-Simons field theory (thought of as a path integral defined in
terms of the Chern-Simons functional).
In [23] it is shown that three dimensional Chern-Simons gauge theories with
gauge gro up G can be classified by the integer cohomology group H
4
(BG, Z), and
conformally invariant sigma models in two dimension with targe t space a compact
Lie gr oup (Wess-Zumino-Witten models) can be classified by H
3
(G, Z). It is also
established that the co rrespondence between three dimensional Chern-Simons gauge
theories and Wess-Zumino-Witten models is related to the tra nsgression map
τ : H
4
(BG, Z) → H
3
(G, Z),
which explains the subtleties in this correspondence for compact, semi-simple non-
simply connected Lie groups ([36]).
In the present work we introduce Cher n-Simons bundle 2-gerbe s and the notion
of multiplicative bundle gerbes, and apply them to explore the geometry of the
Dijkgraaf-Witten correspondence. To this end, we will assume throughout that G
is a compact semi-simple Lie group.
The role of Deligne cohomology as an ingredient in topological field theories goes
back to [27] and we add a new feature in se ction 2 by using Deligne cohomology

BUNDLE GERBES FOR CHERN-SIMONS AND WESS-ZUMINO-WITTEN THEORIES 3
valued characteristic classes for pr incipal G-bundles with connection. Briefly speak-
ing, a degree p Deligne characteristic class for principal G-bundles with connection
is an as signment to any principal G-bundle with connection over M of a class in
the deg ree p Deligne cohomology group H
p
(M, D
p
) satisfying a certain functor ial
property. Deligne cohomolog y valued characteristic classes refine the characteristic
classes for principal G-bundles.
We will define three dimensional Chern-Simons gauge theories CS(G) as degree
3 Deligne cohomology valued characteristic classe s for principal G-bundles with
connection, but will later show that there is a global differential geometric structure,
the Chern-Simons bundle 2-gerbe, asso c iated to each Chern-Simons gauge theory.
We will interpret a Wess-Zumino-Witten model as arising from the curving of a
bundle ge rbe associated to a degree 2 Deligne cohomology c lass on the Lie group
G as in [12] and [28]. We then use a certain canonical G bundle defined on S
1
× G
to construct a transgression map between class ic al Chern-Simons gauge theories
CS(G) and classical Wess-Zumino-Witten models W ZW (G) in section 3, which is a
lift of the transgression map H
4
(BG, Z) → H
3
(G, Z). The resulting correspondence
Ψ : CS(G) −→ W ZW (G)
refines the Dijkgraaf-Witten correspondence between three dimensional Chern-
Simons g auge theories and Wess-Zumino-Witten models associated to a compact
Lie group G. On Deligne cohomology groups, our correspondence Ψ induces a
transgres sion map
H
3
(BG, D
3
) −→ H
2
(G, D
2
),
and refines the natural tr ansgression map τ : H
4
(BG, Z) → H
3
(G, Z) (Cf. Propo-
sition 3.4). See [9] for a related transgres sion of Deligne cohomology in a different
set-up.
For any integral cohomology class in H
3
(G, Z), there is a unique stable equiv-
alence class of bundle gerbe ([37, 38]) whose Dixmier-Do uady clas s is the given
degree 3 integral cohomology class. Geometrically H
4
(BG, Z) can be regarded as
stable equiva le nce classes of bundle 2-gerbes over BG, whose induced bundle gerbe
ove r G ha s a c e rtain multiplicative structure.
To study the geometry of the correspondence Ψ, we revisit the bundle 2-gerbe
theory developed in [44] and [34] in Section 4. Note that transformatio ns be tween
stable isomorphisms provide 2-morphisms making the category BGrb
M
of bundle
gerbes over M and stable isomorphisms between bundle gerbes into a bi-category
(Cf. [44]).
For a smooth surjective submersion π : X → M , consider the face operators
π
i
: X
[n]
→ X
[n−1]
on the simplicial manifold X
•
= {X
n
= X
[n+1]
}. Then a bundle
2-gerbe on M consists of the data of a smooth surjective submersion π : X → M
together with
(1) An object (Q, Y, X
[2]
) in BGrb
X
[2]
.
(2) A stable isomorphism m: π
∗
1
Q⊗π
∗
3
Q → π
∗
2
Q in BGrb
X
[3]
defining the bun-
dle 2- gerbe product w hich is asso ciative up to a 2- morphism φ in BGrb
X
[4]
.
(3) The 2 -morphism φ satisfies a natural coherency condition in BGrb
X
[5]
.
We then develop a multiplicative bundle gerbe theory over G in section 5 as
a simplicial bundle gerbe on the simplicial manifold associated to BG. We say a
bundle gerbe G over G is transgressive if the Deligne class of G, written d(G) is
in the image of the correspondence map Ψ : CS(G) → W ZW (G) = H
2
(G, G
2
).

4 A.L. CAREY, S. JOHNSON, M.K. MURRAY, D. STEVENSON, AND BAI-LING WANG
The main results of this pap e r are the following two theorems (Theorem 5.8 and
Theorem 5.9)
(1) The Dixmier-Douady class of a bundle gerbe G over G lies in the image
of the transgression map τ : H
4
(BG, Z) → H
3
(G, Z) if and only if G is
multiplicative.
(2) Let G be a bundle ge rbe over G with connection and curving, whose Deligne
class d(G) is in H
2
(G, D
2
). Then G is transgressive if and only if G is
multiplicative.
Let φ be an element in H
4
(BG, Z). The corresponding G-invariant polynomial
on the Lie alg e bra under the universal Chern-Weil homomorphism is denoted by
Φ. For any connection A on the universal bundle EG → BG with the curvature
form F
A
,
(φ, Φ(
i
2π
F
A
)) ∈ H
4
(BG, Z) ×
H
4
(BG,R)
Ω
4
cl,0
(BG)
(where Ω
4
cl,0
(BG) is the space of closed 4-forms on BG with periods in Z), defines
a unique degree 3 Deligne class in H
3
(BG, D
3
). Here we fix a smooth infinite
dimensional model of EG → BG by embedding G into U (N) and letting EG be
the Stiefel manifold of N orthonormal vectors in a separable complex Hilbert space.
We will show that H
3
(BG, D
3
) clas sifies the stable equivalence classes of bundle
2-gerbes with curving on BG, (we already know that the second Deligne cohomology
classifies the stable equivalence classes of bundle gerbes with curving). These are the
universal Chern-Simons bundle 2-gerb e s Q
φ
in section 6 (cf. Proposition 6.4) giving
a geometric realisation of the degr ee 3 Deligne class determined by (φ, Φ(
i
2π
F
A
)).
We show that for any principal G-bundle P with connection A over M, the
associated Chern-Simons bundle 2-gerbe Q
φ
(P, A) over M is obtained by the pull-
back of the universal Chern-Simons bundle 2-gerbe Q
φ
via a classifying map. The
bundle 2-gerbe curvature of Q
φ
(P, A) is given by Φ(
i
2π
F
A
), and the bundle 2-gerbe
curving is g iven by the Chern-Simons form associated to (P, A) and φ.
Under the canonical isomorphism between Deligne cohomology and Cheeger-
Simons cohomology, there is a canonical holonomy map for any degree p Deligne
class from the group of smooth p-cocycles to U(1 ). This holonomy is known as the
Cheeger-Simons differential character associated to the Deligne class.
The bundle 2-gerbe holonomy for this C hern-Simons bundle 2-gerbe Q
φ
(P, A)
ove r M as given by the Cheeger-Simons differential character is used in the in-
tegrand for the path integral for the Chern-Simons quantum field theor y. In the
SU (N) Chern-Simons theory, Φ is chosen to be the second Cher n polynomial. For
a smooth ma p σ : Y → M , under a fixed trivialisation of σ
∗
(P, A) over Y , the cor-
responding holonomy of σ is given by e
2πiCS(σ,A)
, where CS(σ, A) ca n be written
as the following well-known Chern-Simons form:
k
8π
2
Z
Y
T rσ
∗
(A ∧ dA +
1
3
A ∧ A ∧ A),
Here k ∈ Z is the level determined by φ ∈ H
4
(BSU(N), Z)
∼
=
Z.
We will establish in Theorem 6.7 that the Chern-Simons bundle 2-gerbe Q
φ
(P, A)
ove r M is eq uivalent in Deligne cohomolo gy to the Cheeger-Simons invariant asso-
ciated to the principal G-bundle P with a connection A and a class φ ∈ H
4
(BG, Z).

Frequently asked questions

Q1. What is the definition of a bundle gerbe?

A multiplicative bundle gerbe over a compact Lie group G is defined to be a simplicial bundle gerbe on the simplicial manifold BG• associated to the classifying space of G. 

Q2. What is the choice of a finite dimensional principal G-bundle?

Given a principal G-bundle P with a connection A over M and an integer n > max{5, dimM}, there is a choice of nconnected finite dimensional principal G-bundle En with a connection 

Q3. What is the corresponding left-invariant closed 3-form on BG?

The corresponding left-invariant closed 3-form ωφ on G is an integer multiple of the standard 3-form < θ, [θ, θ] > where θ is the left-invariant MaurerCartan form on G and < , > is the symmetric bilinear form on the Lie algebra of G defined by Φ ∈ I2(G). 

Q4. What is the holonomy of the bundle gerbe G over 0?

The authors can construct a flat G-bundle over Σ0,3 ×Σ with boundary orientation given in such a way that the usual holonomies for flat G-bundle are σ1, σ2 and σ1 · σ2 respectively. 

Q5. What is the proof of the definition of a Deligne characteristic class?

Note that the characteristic classes only depend on the underlying topological principal G-bundle, in order to define a Deligne cohomology valued characteristic class, the authors will restrict ourselves to differentiable principal G-bundles. 

Q6. What is the space of 2-morphisms between two stable isomorphisms?

Note that the space of 2-morphisms between two stable isomorphisms is one-to-one corresponding to the space of line bundles over M .Consider the face operators πi : X [n] → X [n−1] on the simplicial manifold X• ={Xn = X [n+1]}. 

Q7. What is the holonomy of the so-called B field?

It implies that for the transgressive Wess-Zumino-Witten models, the so-called B field satisfies a certain integrality condition. 

Q8. What is the definition of a three dimensional Chern-Simons gauge theory?

The authors make the following definitions:(1) A three dimensional Chern-Simons gauge theory with gauge group G is defined to be a Deligne characteristic class of degree 3 for a principal Gbundle with connection. 

Q9. what is the proof of a class of principal G-bundles?

The proof is: given a characteristic class c, the authors have, of course, c(EG) ∈ H∗(BG,Z) and conversely if ξ ∈ H∗(BG,Z) is given, then defining cξ(P ) = f∗(ξ) for any classifying map f : M → BG gives rise to a characteristic class for the isomorphism class of principal G-bundles defined by the classifying map f . 

Q10. What is the definition of a bundle gerbe on a simplicial manifold?

The authors begin with the definition of a simplicial bundle gerbe as in [44] on a simplicial manifold X• = {Xn}n≥0 with face operators di : Xn+1 → Xn (i = 0, 1, · · · , n+ 1). 

Q11. What is the equivalence class of a bundle 2-gerbe with connection and?

Recall that for a connection on a line bundle over M , a gauge transformation is given by a smooth function M → U(1), and an extended gauge transformation for a bundle gerbe with connection and curving is given by a line bundle with connection over M . 

Q12. What is the relevance of the Chern-Simons gauge theory?

The relevance of Chern-Simons gauge theory has been noted by many authors, starting with Ramadas-Singer-Weitsman [43] and recently Dupont-Johansen [20], who used gauge covariance of the Chern-Simons functional to give a geometric2000 Mathematics Subject Classification. 

Q13. What is the commutative diagram of a homotopy class?

This commutative diagram gives rise to a homotopy class of maps ψ̂ : G→ K(Z, 3) which determines a classcψ ∈ H 3(G,Z) ∼= [G,K(Z, 3)]. 

Q14. What is the role of Deligne cohomology in topological field theories?

The role of Deligne cohomology as an ingredient in topological field theories goes back to [27] and the authors add a new feature in section 2 by using Deligne cohomologyvalued characteristic classes for principal G-bundles with connection.