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Christoph Schweigert

Publications and source records attributed to Christoph Schweigert.

At least 55 records · Page 3Linked to original sources
arXiv2021

The Trace Field Theory of a Finite Tensor Category

Given a finite tensor category $\mathcal{C}$, we prove that a modified trace on the tensor ideal of projective objects can be obtained from a suitable trivialization of the Nakayama functor as right $\mathcal{C}$-module functor. Using a result of Costello, this allows us to associate to any finite tensor category equipped with such a trivialization of the Nakayama functor a chain complex valued topological conformal field theory, the trace field theory. The trace field theory topologically describes the modified trace, the Hattori-Stallings trace, and also the structures induced by them on the Hochschild complex of $\mathcal{C}$. In this article, we focus on implications in the linear (as opposed to differential graded) setting: We use the trace field theory to define a non-unital homotopy commutative product on the Hochschild chains in degree zero. This product is block diagonal and can be described through the handle elements of the trace field theory. Taking the modified trace of the handle elements recovers the Cartan matrix of $\mathcal{C}$.

math.QA↗
arXiv2021

Matrix product operator symmetries and intertwiners in string-nets with domain walls

We provide a description of virtual non-local matrix product operator (MPO) symmetries in projected entangled pair state (PEPS) representations of string-net models. Given such a PEPS representation, we show that the consistency conditions of its MPO symmetries amount to a set of six coupled equations that can be identified with the pentagon equations of a bimodule category. This allows us to classify all equivalent PEPS representations and build MPO intertwiners between them, synthesising and generalising the wide variety of tensor network representations of topological phases. Furthermore, we use this generalisation to build explicit PEPS realisations of domain walls between different topological phases as constructed by Kitaev and Kong [Commun. Math. Phys. 313 (2012) 351-373]. While the prevailing abstract categorical approach is sufficient to describe the structure of topological phases, explicit tensor network representations are required to simulate these systems on a computer, such as needed for calculating thresholds of quantum error-correcting codes based on string-nets with boundaries. Finally, we show that all these string-net PEPS representations can be understood as specific instances of Turaev-Viro state-sum models of topological field theory on three-manifolds with a physical boundary, thereby putting these tensor network constructions on a mathematically rigorous footing.

quant-ph↗
arXiv2020

Internal natural transformations and Frobenius algebras in the Drinfeld center

For M and N finite module categories over a finite tensor category C, the category Rex_C(M,N) of right exact module functors is a finite module category over the Drinfeld center Z(C). We study the internal Homs of this module category, which we call internal natural transformations. With the help of certain integration functors that map C-C-bimodule functors to objects of Z(C), we express them as ends over internal Homs and define horizontal and vertical compositions. We show that if M and N are exact C-modules and C is pivotal, then the Z(C)-module Rex_C(M,N) is exact. We compute its relative Serre functor and show that if M and N are even pivotal module categories, then Rex_C(M,N) is pivotal as well. Its internal Ends are then a rich source for Frobenius algebras in Z(C).

math.CT↗
arXiv2020

Module Eilenberg-Watts calculus

The categorical formulation of the Eilenberg-Watts calculus relates, for any pair of finite categories M and N, the finite categories Fun^{le}(N,M) and Fun^{re}(N,M) of linear left or right exact functors and the Deligne product \bar N \boxtimes M by adjoint equivalences. We establish a variant of this calculus for the case that the finite categories M and N are module categories over a finite tensor category. This provides in particular canonical and explicitly computable equivalences between categories of left or right exact module functors (or, more generally, balanced functors) and certain twisted centers of bimodule categories.

math.CT↗
arXiv2019

Eilenberg-Watts calculus for finite categories and a bimodule Radford $S^4$ theorem

We obtain Morita invariant versions of Eilenberg-Watts type theorems, relating Deligne products of finite linear categories to categories of left exact as well as of right exact functors. This makes it possible to switch between different functor categories as well as Deligne products, which is often very convenient. For instance, we can show that applying the equivalence from left exact to right exact functors to the identity functor, regarded as a left exact functor, gives a Nakayama functor. The equivalences of categories we exhibit are compatible with the structure of module categories over finite tensor categories. This leads to a generalization of Radford's $S^4$-theorem to bimodule categories. We also explain the relation of our construction to relative Serre functors on module categories that are constructed via inner Hom functors.

math.RT↗
arXiv2019

On isotypic decompositions for non-semisimple Hopf algebras

In this paper we study the isotypic decomposition of the regular module of a finite-dimensional Hopf algebra over an algebraically closed field of characteristic zero. For a semisimple Hopf algebra, the idempotents realizing the isotypic decomposition can be explicitly expressed in terms of characters and the Haar integral. In this paper we investigate Hopf algebras with the Chevalley property, which are not necessarily semisimple. We find explicit expressions for idempotents in terms of Hopf-algebraic data, where the Haar integral is replaced by the regular character of the dual Hopf algebra. For a large class of Hopf algebras, these are shown to form a complete set of orthogonal idempotents. We give an example which illustrates that the Chevalley property is crucial.

math.QA↗
arXiv2019

Extended Homotopy Quantum Field Theories and their Orbifoldization

We present a precise definition of extended homotopy quantum field theories and develop an orbifold construction for these theories when the target space is the classifying space of a finite group $G$, i.e. for $G$-equivariant topological field theories. More precisely, we use a bicategorical version of the parallel section functor to associate to an extended equivariant topological field theory an ordinary extended topological field theory. Thereby, we give a unification, geometric underpinning and vast generalization of algebraic concepts of orbifoldization. In the special case of 3-2-1-dimensional equivariant topological field theories, we investigate the equivariant modular structure on the categories that such theories yield upon evaluation on the circle. By means of our orbifold construction this equivariant modular structure will be related to the modular structure on the orbifold category. We also generalize our orbifold construction to a pushforward operation along an arbitrary morphism of finite groups and hence provide a valuable tool for the construction of extended homotopy quantum field theories.

math.QA↗
arXiv2019

Full Logarithmic Conformal Field theory - an Attempt at a Status Report

Logarithmic conformal field theories are based on vertex algebras with non-semisimple representation categories. While examples of such theories have been known for more than 25 years, some crucial aspects of local logarithmic CFTs have been understood only recently, with the help of a description of conformal blocks by modular functors. We present some of these results, both about bulk fields and about boundary fields and boundary states. We also describe some recent progress towards a derived modular functor. This is a summary of work with Terry Gannon, Simon Lentner, Svea Mierach, Gregor Schaumann and Yorck Sommerhäuser.

hep-th↗
arXiv2018

Non-abelian gerbes and some applications in string theory

We review a systematic construction of the 2-stack of bundle gerbes via descent, and extend it to non-abelian gerbes. We review the role of non-abelian gerbes in orientifold sigma models, for the anomaly cancellation in supersymmetric sigma models, and in a geometric description of so-called non-geometric T-duals.

hep-th↗
arXiv2018

The logarithmic Cardy case: Boundary states and annuli

We present a model-independent study of boundary states in the Cardy case that covers all conformal field theories for which the representation category of the chiral algebra is a - not necessarily semisimple - modular tensor category. This class, which we call finite CFTs, includes all rational theories, but goes much beyond these, and in particular comprises many logarithmic conformal field theories. We show that the following two postulates for a Cardy case are compatible beyond rational CFT and lead to a universal description of boundary states that realizes a standard mathematical setup: First, for bulk fields, the pairing of left and right movers is given by (a coend involving) charge conjugation; and second, the boundary conditions are given by the objects of the category of chiral data. For rational theories our proposal reproduces the familiar result for the boundary states of the Cardy case. Further, with the help of sewing we compute annulus amplitudes. Our results show in particular that these possess an interpretation as partition functions, a constraint that for generic finite CFTs is much more restrictive than for rational ones.

math.QA↗
arXiv2018

Orbifold Construction for Topological Field Theories

An equivariant topological field theory is defined on a cobordism category of manifolds with principal fiber bundles for a fixed (finite) structure group. We provide a geometric construction which for any given morphism $G \to H$ of finite groups assigns in a functorial way to a $G$-equivariant topological field theory an $H$-equivariant topological field theory, the pushforward theory. When $H$ is the trivial group, this yields an orbifold construction for $G$-equivariant topological field theories which unifies and generalizes several known algebraic notions of orbifoldization.

math.QA↗
arXiv2017

A GNS construction of three-dimensional abelian Dijkgraaf-Witten theories

We give a detailed account of the so-called "universal construction" that aims to extend invariants of closed manifolds, possibly with additional structure, to topological field theories and show that it amounts to a generalization of the GNS construction. We apply this construction to an invariant defined in terms of the groupoid cardinality of groupoids of bundles to recover Dijkgraaf-Witten theories, including the vector spaces obtained as a linearization of spaces of principal bundles.

math.QA↗
arXiv2017

Frobenius algebras and homotopy fixed points of group actions on bicategories

We explicitly show that symmetric Frobenius structures on a finite-dimensional, semi-simple algebra stand in bijection to homotopy fixed points of the trivial SO(2)-action on the bicategory of finite-dimensional, semi-simple algebras, bimodules and intertwiners. The results are motivated by the 2-dimensional Cobordism Hypothesis for oriented manifolds, and can hence be interpreted in the realm of Topological Quantum Field Theory.

math.QA↗
arXiv2017

Hochschild Cohomology and the Modular Group

It has been shown in previous work that the modular group acts projectively on the center of a factorizable ribbon Hopf algebra. The center is the zeroth Hochschild cohomology group. In this article, we extend this projective action of the modular group to an arbitrary Hochschild cohomology group of a factorizable ribbon Hopf algebra, in fact up to homotopy even to a projective action on the entire Hochschild cochain complex.

math.RA↗
arXiv2017

Consistent systems of correlators in non-semisimple conformal field theory

Based on the modular functor associated with a -- not necessarily semisimple -- finite non-degenerate ribbon category $\mathcal D$, we present a definition of a consistent system of bulk field correlators for a conformal field theory which comprises invariance under mapping class group actions and compatibility with the sewing of surfaces. We show that when restricting to surfaces of genus zero such systems are in bijection with commutative symmetric Frobenius algebras in $\mathcal D$, while for surfaces of any genus they are in bijection with modular Frobenius algebras in $\mathcal D$. This provides additional insight into structures familiar from rational conformal field theories and extends them to rigid logarithmic conformal field theories.

math.QA↗
arXiv2016

On unrolled Hopf algebras

We show that the definition of unrolled Hopf algebras can be naturally extended to the Nichols algebra $\mathcal{B}$ of a Yetter-Drinfeld module $V$ on which a Lie algebra $\mathfrak g$ acts by biderivations. Specializing to Nichols algebras of diagonal type, we find unrolled versions of the small, the De Concini-Procesi and the Lusztig divided power quantum group, respectively.

math.QA↗
arXiv2016

Coends in conformal field theory

The idea of "summing over all intermediate states" that is central for implementing locality in quantum systems can be realized by coend constructions. In the concrete case of systems of conformal blocks for a certain class of conformal vertex algebras, one deals with coends in functor categories. Working with these coends involves quite a few subtleties which, even though they have in principle already been understood twenty years ago, have not been sufficiently appreciated by the conformal field theory community.

math.QA↗