On the Brauer Groups of Symmetries of Abelian Dijkgraaf–Witten Theories

Fuchs, Jürgen; Priel, Jan; Schweigert, Christoph; Valentino, Alessandro

doi:10.1007/s00220-015-2420-y

Cited by 33 publications

(35 citation statements)

References 26 publications

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“…For details we refer to [FPSV14]. We repeat their very interesting question linking this natural functor from the TFT construction to the ENOM functor, which they solve in the case Vect G for G abelian by explicit calculation using the bundle construction:…”

Section: Defects In 3d Topological Field Theoriesmentioning

confidence: 99%

“…The main motivation for our initial work [LP15b] was the case H = C[G] for G abelian, as treated in the second authors joint paper [FPSV14]. In particular let G ∼ = Z n p with p a prime number.…”

Section: Motivationmentioning

confidence: 99%

“…and Z(M ) by so-called pull-push-construction, that sums over all possible continuations of bundles on Σ to M , see e.g. [FPSV14].…”

Section: Defects In 3d Topological Field Theoriesmentioning

confidence: 99%

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Three natural subgroups of the Brauer-Picard group of a Hopf algebra with applications

Lentner¹,

Priel²

2017

Bull. Belg. Math. Soc. Simon Stevin

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In this article we construct three explicit natural subgroups of the Brauer-Picard group of the category of representations of a finite-dimensional Hopf algebra. In examples the Brauer Picard group decomposes into an ordered product of these subgroups, somewhat similar to a Bruhat decomposition.Our construction returns for any Hopf algebra three types of braided autoequivalences and correspondingly three families of invertible bimodule categories. This gives examples of so-called (2-)Morita equivalences and defects in topological field theories. We have a closer look at the case of quantum groups and Nichols algebras and give interesting applications. Finally, we briefly discuss the three families of group-theoretic extensions. 1 We choose these names EV, BV for compatibilities with previous conventions. Be advised that V does not necessarily have complement subgroups B, E in BV, EV in the most general cases. arXiv:1702.05133v1 [math.QA] 16 Feb 2017 2Example (Sec. 4.1.5). Let G ∼ = Z n p with p a prime number. Our decomposition reduces to the Bruhat decomposition of BrPic(Vect G ), which is the Lie group O 2n (F p ) over the finite field F p . In this case BV, EV are lower and upper triangular matrices, intersecting in the subgroup V = GL n (F p ). The partial dualizations are Weyl group elements. More precisely, our result reduces to the Bruhat decomposition of the Lie groups D n relative to the parabolic subsystem A n−1 , so reflections are actually equivalence classes corresponding to n + 1 cosets of the parabolic Weyl group.The present article is devoted to start the discussion of the more general case C = H-mod. We shall not try to prove a decomposition theorem, but focus our attention on establishing and discussing the expected natural subgroups V, BV, EV, R of the Brauer Picard group. We will also briefly discuss several interesting applications of our results, in particular when H is the Borel part of a quantum group resp. a Nichols algebra.In Section 2 we briefly recall the induction functor and the ENOM-functor [ENOM09]In view of interesting examples and the applications to defects in mathematical physics and Nichols algebras we state the obvious generalization of these concepts to the groupoid setting, so that arbitrary monoidal equivalences C ∼ → D give rise to invertible C-Dbimodule categories, and these are in bijection to braided equivalences Z(C)In Section 3 we define and derive for each subgroup V, BV, EV and the subset R explicit expressions for the braided autoequivalence as well as the invertible bimodule categories.On one hand BV resp. EV are obtained using induction functors from H-mod resp. H * -mod. So the bimodule categories in BrPic(H-mod) resp. BrPic(H * -mod) are given by definition. We then calculate explicitly the images under the ENOM functor using Bigalois objects and finally we describe again the preimage of EV now in BrPic(H-mod). As linear categories, the bimodule categories in BV are all equal to C, while the bimodule categories in EV are representation categories of Bigalois o...

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Section: Defects In 3d Topological Field Theoriesmentioning

confidence: 99%

Section: Motivationmentioning

confidence: 99%

See 1 more Smart Citation

Three natural subgroups of the Brauer-Picard group of a Hopf algebra with applications

Lentner¹,

Priel²

2017

Bull. Belg. Math. Soc. Simon Stevin

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“…In this section we study the action of a finite symmetry group G on a classical Dijkgraaf-Witten theory L ω : D-Cob n −→ Vect with gauge group D and topological action ω ∈ Z n (BD; U (1)). General symmetries of abelian quantum Dijkgraaf-Witten theories are discussed in [FPSV15]. We only consider symmetries arising from an action of G on D-gauge fields which preserve ω ∈ Z n (BD; U (1)).…”

Section: Discrete Symmetries and 'T Hooft Anomaliesmentioning

confidence: 99%

“…Following [FPSV15] we call these kinematical symmetries. We show that they extend to the quantum theory and study their gauging.…”

Section: Discrete Symmetries and 'T Hooft Anomaliesmentioning

confidence: 99%

’t Hooft Anomalies of Discrete Gauge Theories and Non-abelian Group Cohomology

Müller

Szabo

2019

Commun. Math. Phys.

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We study discrete symmetries of Dijkgraaf-Witten theories and their gauging in the framework of (extended) functorial quantum field theory. Non-abelian group cohomology is used to describe discrete symmetries and we derive concrete conditions for such a symmetry to admit 't Hooft anomalies in terms of the Lyndon-Hochschild-Serre spectral sequence. We give an explicit realization of a discrete gauge theory with 't Hooft anomaly as a state on the boundary of a higher-dimensional Dijkgraaf-Witten theory. This allows us to calculate the 2-cocycle twisting the projective representation of physical symmetries via transgression. We present a general discussion of the bulkboundary correspondence at the level of partition functions and state spaces, which we make explicit for discrete gauge theories. Dijkgraaf-Witten theoryDijkgraaf-Witten theories are topological gauge theories with finite gauge group D. In this section we introduce the framework of (extended) functorial quantum field theory, and subsequently construct classical and quantum Dijkgraaf-Witten theories.

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Two and Three Dimensional Calculus

Dyke

2018

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Quantum field theory has various projective characteristics which are captured by what are called anomalies. This paper explores this idea in the context of fully-extended three-dimensional topological quantum field theories (TQFTs).Given a three-dimensional TQFT (valued in the Morita 3-category of fusion categories), the anomaly identified herein is an obstruction to gauging a naturally occurring orthogonal group of symmetries, i.e. we study 't Hooft anomalies. In other words, the orthogonal group almost acts: There is a lack of coherence at the top level. This lack of coherence is captured by a "higher (central) extension" of the orthogonal group, obtained via a modification of the obstruction theory of Etingof-Nikshych-Ostrik-Meir [ENO10]. This extension tautologically acts on the given TQFT/fusion category, and this precisely classifies a projective (equivalently anomalous) TQFT. We explain the sense in which this is an analogue of the classical spin representation. This is an instance of a phenomenon emphasized by Freed [Fre23]: Quantum theory is projective.In the appendices we establish a general relationship between the language of projectivity/anomalies and the language of topological symmetries. We also identify a universal anomaly associated with any theory which is appropriately "simple". JACKSON VAN DYKE 3.7. Module structures on 3d TQFTs 31 Appendix A. TQFT and category theory 32 A.1. The Cobordism Hypothesis 32 A.2. Relative theories 34 A.3. Invertible theories 35 Appendix B. Topological symmetry 35 B.1. TQFTs associated to π-finite spaces 35 B.2. Module structures 38 B.3. Reduction of topological symmetry 39 Appendix C. Anomalies 41 C.1. Definitions 42 C.2. The anomaly associated to a projectivity class 43 C.3. Projective theories 45 C.4. Anomalies and sandwiches 46 C.5. A universal anomaly 47 References 48

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On the Brauer Groups of Symmetries of Abelian Dijkgraaf–Witten Theories

Cited by 33 publications

References 26 publications

Three natural subgroups of the Brauer-Picard group of a Hopf algebra with applications

Three natural subgroups of the Brauer-Picard group of a Hopf algebra with applications

’t Hooft Anomalies of Discrete Gauge Theories and Non-abelian Group Cohomology

Two and Three Dimensional Calculus

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