Orientation Twisted Homotopy Field Theories and Twisted Unoriented Dijkgraaf–Witten Theory

Young, Matthew B.

doi:10.1007/s00220-019-03478-5

Cited by 7 publications

(7 citation statements)

References 53 publications

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“…[KT17a, KS14, GKSW15, TvK15, Tac17, DT18, BCH19], and of invertible topological sigma-models, as in [TE18,Tho17]. It should also be straightforward to include time-reversal symmetries using the techniques of [You18].…”

Section: Related Workmentioning

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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Section: Related Workmentioning

confidence: 99%

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

Müller

Szabo

2019

Commun. Math. Phys.

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“…The character theory of (projective) Real 2-representations of finite groups is also studied in [37]. Real 2representations, and their characters, appear naturally in unoriented topological field theory and orientifold string and M-theory [38] and, conjecturally, in Real variants of equivariant elliptic cohomology [37].…”

Section: Introductionmentioning

confidence: 99%

Burnside rings for Real 2-representation theory: The linear theory

Rumynin

Young

2020

Commun. Contemp. Math.

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This paper is a fundamental study of the Real 2-representation theory of 2-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a 2-equivariant Morita bicategory, where a novel construction of induction is introduced. We identify the Grothendieck ring of Real 2-representations as a Real variant of the Burnside ring of the fundamental group of the 2-group and study the Real categorical character theory. This paper unifies two previous lines of inquiry, the approach to 2-representation theory via Morita theory and Burnside rings, initiated by the first author and Wendland, and the Real 2-representation theory of 2-groups, as studied by the second author.

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“…The map τ ref π has already appeared, in the form of its explicit expressions in low degrees, in work on Real 2-representation theory [32] and unoriented topological field theory [33], where its geometry was also foreshadowed. Theorem A gives an a priori construction of τ ref π and τ π , in all degrees, and establishes that they are cochain maps, which is important for applications in [32], [33] and difficult to verify directly in all but the simplest cases. Upon restriction along G → Ĝ, both maps τ ref π , τ π recover Willerton's transgression map τ.…”

Section: Introductionmentioning

confidence: 99%

“…For n = −1, 0, 1, this is a Real U(1)-valued function, a Real U(1)-bundle and a Jandl gerbe on Ĝ, respectively. Jandl gerbes play an important role in unoriented topological field theory [20], [33], orientifold string theory [29], [10] and topological phases of matter with time reversal symmetry [4]. The twisted transgression maps associate to the Jandl n-gerbe λ an ordinary (n − 1)-gerbe τ ref π ( λ) on Λ ref π Ĝ and a Jandl (n − 1)-gerbe τ π ( λ) on Λ π Ĝ.…”

Section: Introductionmentioning

confidence: 99%

Twisted loop transgression and higher Jandl gerbes over finite groupoids

Noohi¹,

Young²

2019

Preprint

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Given a double cover π : G → Ĝ of finite groupoids, we explicitly construct twisted loop transgression maps, τ π and τ ref π , thereby associating to a Jandl n-gerbe λ on Ĝ a Jandl (n− 1)-gerbe τ π ( λ) on the quotient loop groupoid of G and an ordinary (n − 1)-gerbe τ ref π ( λ) on the unoriented quotient loop groupoid of G. For n = 1, 2, we interpret the character theory (resp. centre) of the category of Real λ-twisted n-vector bundles over Ĝ in terms of flat sections of the (n − 1)vector bundle associated to τ ref π ( λ) (resp. the Real (n−1)-vector bundle associated to τ π ( λ)). We relate our results to Real versions of twisted Drinfeld doubles and pointed fusion categories and to discrete torsion in orientifold string and M-theory.

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Orientation Twisted Homotopy Field Theories and Twisted Unoriented Dijkgraaf–Witten Theory

Cited by 7 publications

References 53 publications

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

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

Burnside rings for Real 2-representation theory: The linear theory

Twisted loop transgression and higher Jandl gerbes over finite groupoids

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