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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).