Topological gauge theories and group cohomology

Dijkgraaf, Robbert; Witten, Edward

doi:10.1007/bf02096988

Cited by 901 publications

(1,375 citation statements)

References 19 publications

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“…It is shown in [23] that three dimensional Chern-Simons gauge theories with gauge group G can be classified by the integer cohomology group H 4 (BG, Z), and conformally invariant sigma models in two dimension with target space a compact Lie group (Wess-Zumino-Witten models) can be classified by H 3 (G, Z). Recall the commutative diagram…”

Section: From Chern-simons To Wess-zumino-wittenmentioning

confidence: 99%

“…In [23] it is shown that three dimensional Chern-Simons gauge theories with gauge group G can be classified by the integer cohomology group H 4 (BG, Z), and conformally invariant sigma models in two dimension with target space a compact Lie group (Wess-Zumino-Witten models) can be classified by H 3 (G, Z). It is also established that the correspondence between three dimensional Chern-Simons gauge theories and Wess-Zumino-Witten models is related to the transgression map τ : H 4 (BG, Z) → H 3 (G, Z), which explains the subtleties in this correspondence for compact, semi-simple nonsimply connected Lie groups ( [36]).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Carey

Johnson

Murray

et al. 2005

Commun. Math. Phys.

146

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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 H 4 (BG, Z) for a compact semi-simple Lie group G. The ChernSimons 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-ZuminoWitten 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 for the subtleties in this correspondence for non-simply connected Lie groups. The implications for Wess-Zumino-Witten models are also discussed.

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Section: From Chern-simons To Wess-zumino-wittenmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Carey

Johnson

Murray

et al. 2005

Commun. Math. Phys.

146

View full text Add to dashboard Cite

show abstract

“…There are not that many such theories, and one can try to guess what it is. In the simplest situations (unitary symmetry action, no fermions) the result must be a Dijkgraaf-Witten theory [15], which are labeled by certain group cohomology classes of the symmetry group. For example, in three spacetime dimensions and G = Z/n, these classes are very explicit, and the actions resemble that of Chern-Simons theory.…”

Section: Introductionmentioning

confidence: 99%

Framed Wilson operators, fermionic strings, and gravitational anomaly in 4d

Thorngren

2015

J. High Energ. Phys.

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Abstract:We study gapped systems with anomalous time-reversal symmetry and global gravitational anomaly in three and four spacetime dimensions. These systems describe topological order on the boundary of bosonic Symmetry Protected Topological (SPT) Phases. Our description of these phases is via the recent cobordism proposal for their classification. In particular, the behavior of these systems is determined by the geometry of Stiefel-Whitney classes. We discuss electric and magnetic operators defined by these classes, and new types of Wilson lines and surfaces that sit on their boundary. The lines describe fermionic particles, while the surfaces describe a sort of fermionic string. We show that QED with a fermionic monopole exhibits the 4d global gravitational anomaly and has a fermionic π-flux.

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“…of all finite groups up to isomorphism. According to [10] there is a unique untwisted TQFT over R associated to each F k .…”

Section: 2mentioning

confidence: 99%

“…For more details, see [10], [13], [36], [12] and [47]. Here we consider only the (2+1)-dimensional untwisted theories, and we ignore gluing with corners, higher codimension gluing, etc.…”

Section: Appendix a Finite Group Tqftsmentioning

confidence: 99%

Positivity of the universal pairing in 3 dimensions

Calegari

Freedman

Walker

2009

J. Amer. Math. Soc.

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Associated to a closed, oriented surface S S is the complex vector space with basis the set of all compact, oriented 3 3 -manifolds which it bounds. Gluing along S S defines a Hermitian pairing on this space with values in the complex vector space with basis all closed, oriented 3 3 -manifolds. The main result in this paper is that this pairing is positive, i.e. that the result of pairing a nonzero vector with itself is nonzero. This has bearing on the question of what kinds of topological information can be extracted in principle from unitary ( 2 + 1 ) (2+1) -dimensional TQFTs. The proof involves the construction of a suitable complexity function c c on all closed 3 3 -manifolds, satisfying a gluing axiom which we call the topological Cauchy-Schwarz inequality, namely that c ( A B ) ≤ max ( c ( A A ) , c ( B B ) ) c(AB) \le \max (c(AA),c(BB)) for all A , B A,B which bound S S , with equality if and only if A = B A=B . The complexity function c c involves input from many aspects of 3 3 -manifold topology, and in the process of establishing its key properties we obtain a number of results of independent interest. For example, we show that when two finite-volume hyperbolic 3 3 -manifolds are glued along an incompressible acylindrical surface, the resulting hyperbolic 3 3 -manifold has minimal volume only when the gluing can be done along a totally geodesic surface; this generalizes a similar theorem for closed hyperbolic 3 3 -manifolds due to Agol-Storm-Thurston.

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Topological gauge theories and group cohomology

Cited by 901 publications

References 19 publications

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

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

Framed Wilson operators, fermionic strings, and gravitational anomaly in 4d

Positivity of the universal pairing in 3 dimensions

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