Dualizable tensor categories

Douglas, Christopher L.; Schommer-Pries, Christopher; Snyder, Noah P.

doi:10.48550/arxiv.1312.7188

Cited by 43 publications

(140 citation statements)

References 39 publications

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“…The relation between the Nakayama functor and the categorical Radford S 4 -formula [ENO04,FSS20] yields: Theorem 1.3. For a pivotal finite tensor category C, a non-zero two-sided modified trace on Proj(C) exists if and only if the pivotal structure of C is spherical in the sense of [DSS13].…”

Section: Introductionmentioning

confidence: 99%

Modified traces and the Nakayama functor

Shibata,

Shimizu

2021

Preprint

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We organize the modified trace theory with the use of the Nakayama functor of finite abelian categories. For a linear right exact functor Σ on a finite abelian category M, we introduce the notion of a Σ-twisted trace on the class Proj(M) of projective objects of M. In our framework, there is a one-to-one correspondence between the set of Σ-twisted traces on Proj(M) and the set of natural transformations from Σ to the Nakayama functor of M. Non-degeneracy and compatibility with the module structure (when M is a module category over a finite tensor category) of a Σ-twisted trace can be written down in terms of the corresponding natural transformation. As an application of this principal, we give existence and uniqueness criteria for modified traces. In particular, a unimodular pivotal finite tensor category admits a non-zero two-sided modified trace if and only if it is spherical. Also, a ribbon finite tensor category admits such a trace if and only if it is unimodular.

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Section: Introductionmentioning

confidence: 99%

Modified traces and the Nakayama functor

Shibata,

Shimizu

2021

Preprint

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“…This corresponds to a fusion of the two defects. Fusion of defects have previously been studied in two dimensions [13,14], in supersymmetric theories [15][16][17], and in topological field theories as well as in conformal nets using fusing categories [18][19][20]. In this paper, we will provide three different examples of fusing two defects in four dimensional free theories using the path integral formalism.…”

Section: Introductionmentioning

confidence: 99%

Fusion of conformal defects in four dimensions

Söderberg

2021

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We consider two conformal defects close to each other in a free theory, and study what happens as the distance between them goes to zero. This limit is the same as zooming out, and the two defects have fused to another defect. As we zoom in we find a non-conformal effective action for the fused defect. Among other things this means that we cannot in general decompose the two-point correlator of two defects in terms of other conformal defects. We prove the fusion using the path integral formalism by treating the defects as sources for a scalar in the bulk.

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“…This paper is motivated by the connection, observed in [10], between the Hopf-cyclic theory coefficients and the objects appearing in a twice categorified 1dTFT (one dimensional topological field theory) [7]. We begin to explore this link and its implications with the goal of better understanding and generalizing coefficients, as well as reinterpreting the Hopf cyclic cohomology itself.…”

Section: Introductionmentioning

confidence: 99%

Categorified Chern character and cyclic cohomology

Shapiro

2019

Preprint

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We examine Hopf cyclic cohomology in the same context as the analysis [1,2,3] of the geometry of loop spaces LX in derived algebraic geometry and the resulting close relationship between S 1 -equivariant quasi-coherent sheaves on LX and D X -modules. Furthermore, the Hopf setting serves as a toy case for the categorification of Chern character theory as discussed in [20]. More precisely, this examination naturally leads to a definition of mixed anti-Yetter-Drinfeld contramodules which reduces to that of the usual mixed complexes for the trivial Hopf algebra and generalizes the notion of stable anti-Yetter-Drinfeld contramodules that have thus far served as the coefficients for Hopf-cyclic theories [4]. The cohomology is then obtained as a Hom in this dgcategory between a Chern character object associated to an algebra and an arbitrary coefficient mixed anti-Yetter-Drinfeld contramodule.

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Dualizable tensor categories

Cited by 43 publications

References 39 publications

Modified traces and the Nakayama functor

Modified traces and the Nakayama functor

Fusion of conformal defects in four dimensions

Categorified Chern character and cyclic cohomology

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