Fully extended $\boldsymbol{r}$-spin TQFTs

Carqueville, Nils; Szegedy, Lóránt

doi:10.48550/arxiv.2107.02046

Cited by 3 publications

(6 citation statements)

References 8 publications

(19 reference statements)

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“…In case the graded algebra A ⊕ A c is induced by an automorphism φ : A −→ A so that A c = A φ compatible with the multiplication, this bimodule isomorphism can be replaced by an even Frobenius structure on A with Nakayama automorphism (−1) F A •φ. This discussion also applies to Spin r -TFTs [CS21]; in our framework they are given by a Z r -graded superalgebras…”

Section: Classifications Of Some Other Types Of 2d Tftsmentioning

confidence: 97%

“…Hence, all our one dimensional results are theorems. In the bicategorical 2-dimensional setting the cobordism hypothesis has been proven for many tangential structures including framings [Pst22], orientations [SP09], G × SO 2structures [Soz19] and r-spin structures [CS21]. Our approach to the problem is to define the bicategory Bord H2 2,0 through the universal property it satisfies by the cobordism hypothesis.…”

Section: The Cobordism Hypothesis and General Structure Of The Proof ...mentioning

confidence: 99%

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Reflection Structures and Spin Statistics in Low Dimensions

Müller¹,

Stehouwer²

2023

Preprint

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We give a complete classification of topological field theories with reflection structure and spinstatistics in one and two spacetime dimensions. Our answers can be naturally expressed in terms of an internal fermionic symmetry group G which is different from the spacetime structure group. Fermionic groups encode symmetries of systems with fermions and time reversing symmetries. We show that 1-dimensional topological field theories with reflection structure and spin-statistics are classified by finite dimensional hermitian representations of G. In spacetime dimension two we give a classification in terms strongly G-graded stellar Frobenius algebras. Our proofs are based on the cobordism hypothesis. Along the way, we develop some useful tools for the computation of homotopy fixed points of 2-group actions on bicategories.

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Section: Classifications Of Some Other Types Of 2d Tftsmentioning

confidence: 97%

Section: The Cobordism Hypothesis and General Structure Of The Proof ...mentioning

confidence: 99%

Reflection Structures and Spin Statistics in Low Dimensions

Müller¹,

Stehouwer²

2023

Preprint

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“…) × such that the following diagram commutes, where S X is the 2-morphism component of the pseudonatural transformation (2.56) (see [CSz,Eq. (3.27)]):…”

Section: Cobordism Hypothesis the Cobordism Hypothesismentioning

confidence: 99%

“…3.9], which expresses adjoints of evaluation 1-morphisms in terms of coevaluation 1-morphisms and a single factor of S u (or S −1 u ). In particular, the left and right adjoints of adjunction 1-morphisms differ by S 2 u or S −2 u , and one finds that the associated adjunction 2-morphisms involve an even number of λ-factors (see e. g. [CSz,] for more details). In our case of u = x and λ = [I 1 x ] or λ = [I 1 x [1]] these contributions are identities in both cases, because a double shift is trivial both under horizontal and vertical composition.…”

Section: Partition Functions and State Spacesmentioning

confidence: 99%

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Truncated Affine Rozansky–Witten Models as Extended TQFTs

Brunner¹,

Carqueville

Roggenkamp

2023

Commun. Math. Phys.

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We construct extended TQFTs associated to Rozansky–Witten models with target manifolds $$T^*\mathbb {C}^n$$ T ∗ C n . The starting point of the construction is the 3-category whose objects are such Rozansky–Witten models, and whose morphisms are defects of all codimensions. By truncation, we obtain a (non-semisimple) 2-category $$\mathcal C$$ C of bulk theories, surface defects, and isomorphism classes of line defects. Through a systematic application of the cobordism hypothesis we construct a unique extended oriented 2-dimensional TQFT valued in $$\mathcal C$$ C for every affine Rozansky–Witten model. By evaluating this TQFT on closed surfaces we obtain the infinite-dimensional state spaces (graded by flavour and R-charges) of the initial 3-dimensional theory. Furthermore, we explicitly compute the commutative Frobenius algebras that classify the restrictions of the extended theories to circles and bordisms between them.

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Generalized symmetries of topological field theories

Gripaios

Randal-Williams

Tooby-Smith

2023

J. High Energ. Phys.

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We study generalized symmetries in a simplified arena in which the usual quantum field theories of physics are replaced with topological field theories and the smooth structure with which the symmetry groups of physics are usually endowed is forgotten. Doing so allows many questions of physical interest to be answered using the tools of homotopy theory. We study both global and gauge symmetries, as well as ‘t Hooft anomalies, which we show fall into one of two classes. Our approach also allows some insight into earlier work on symmetries (generalized or not) of topological field theories.

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Fully extended $\boldsymbol{r}$-spin TQFTs

Cited by 3 publications

References 8 publications

Reflection Structures and Spin Statistics in Low Dimensions

Reflection Structures and Spin Statistics in Low Dimensions

Truncated Affine Rozansky–Witten Models as Extended TQFTs

Generalized symmetries of topological field theories

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