arXiv subjects

Gregor Schaumann

Publications and source records attributed to Gregor Schaumann.

At least 19 recordsLinked to original sources

arXiv2024

Grothendieck-Verdier module categories, Frobenius algebras and relative Serre functors

We develop the theory of module categories over a Grothendieck-Verdier category, i.e. a monoidal category with a dualizing object and hence a duality structure more general than rigidity. Such a category C comes with two monoidal structures which are related by non-invertible morphisms and which we treat on an equal footing. Quite generally, non-invertible structure morphisms play a dominant role in this theory. In any Grothendieck-Verdier module category M we find two important subcategories M' and M''. The internal End of an object in M' that is a C-generator is an algebra such that its category of modules is equivalent to M as a module category. We also introduce a partially defined relative Serre functor S which furnishes an equivalence between M' and M''. Any isomorphism between an object m of M' and S(m) in M'' endows the internal End of m with the structure of a Grothendieck-Verdier Frobenius algebra.

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arXiv2023

Fusion Quivers

We develop a categorical approach to quivers and their modules. Naturally this leads to a notion of an action of a monoidal category on quivers. Using this, we construct for a large class of quivers rigid monoidal structures on their categories of modules. This fusion product on the quiver modules induces a graded ring structure with duality and trace on the moduli spaces of semisimple quiver modules. Our approach allows to consider a class of relations on such fusion quivers that are compatible with the rigid monoidal structure. In particular we obtain a class of preprojective algebras with fusion product on their modules.

math.QA↗

arXiv2023

Grothendieck-Verdier duality in categories of bimodules and weak module functors

Various monoidal categories, including suitable representation categories of vertex operator algebras, admit natural Grothendieck-Verdier duality structures. We recall that such a Grothendieck-Verdier category comes with two tensor products which should be related by distributors obeying pentagon identities. We discuss in which circumstances these distributors are isomorphisms. This is achieved by taking the perspective of module categories over monoidal categories, using in particular the natural weak module functor structure of internal Homs and internal coHoms. As an illustration, we exhibit these concepts concretely in the case of categories of bimodules over associative algebras.

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arXiv2021

Reshetikhin-Turaev TQFTs close under generalised orbifolds

We specialise the construction of orbifold graph TQFTs introduced in Carqueville et al., arXiv:2101.02482 to Reshetikhin-Turaev defect TQFTs. We explain that the modular fusion category ${\mathcal{C}}_{\mathcal{A}}$ constructed in Mulevi\v{c}ius-Runkel, arXiv:2002.00663 from an orbifold datum $\mathcal{A}$ in a given modular fusion category $\mathcal{C}$ is a special case of the Wilson line ribbon categories introduced as part of the general theory of orbifold graph TQFTs. Using this, we prove that the Reshetikhin-Turaev TQFT obtained from ${\mathcal{C}}_{\mathcal{A}}$ is equivalent to the orbifold of the TQFT for $\mathcal{C}$ with respect to the orbifold datum $\mathcal{A}$.

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arXiv2021

Orbifold graph TQFTs

A generalised orbifold of a defect TQFT $\mathcal{Z}$ is another TQFT $\mathcal{Z}_{\mathcal{A}}$ obtained by performing a state sum construction internal to $\mathcal{Z}$. As an input it needs a so-called orbifold datum $\mathcal{A}$ which is used to label stratifications coming from duals of triangulations and is subject to conditions encoding the invariance under Pachner moves. In this paper we extend the construction of generalised orbifolds of $3$-dimensional TQFTs to include line defects. The result is a TQFT acting on 3-bordisms with embedded ribbon graphs labelled by a ribbon category $\mathcal{W}_{\mathcal{A}}$ that we canonically associate to $\mathcal{Z}$ and $\mathcal{A}$. We also show that for special orbifold data, the internal state sum construction can be performed on more general skeletons than those dual to triangulations. This makes computations with $\mathcal{Z}_{\mathcal{A}}$ easier to handle in specific examples.

math.QA↗

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.

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arXiv2019

A modular functor from state sums for finite tensor categories and their bimodules

We construct a modular functor which takes its values in the monoidal bicategory of finite categories, left exact functors and natural transformations. The modular functor is defined on bordisms that are 2-framed. Accordingly we do not need to require that the finite categories appearing in our construction are semisimple, nor that the finite tensor categories that are assigned to two-dimensional strata are endowed with a pivotal structure. Our prescription can be understood as a state-sum construction. The state-sum variables are assigned to one-dimensional strata and take values in bimodule categories over finite tensor categories, whereby we also account for the presence of boundaries and defects. Our construction allows us to explicitly compute functors associated to surfaces and representations of mapping class groups acting on them.

math.QA↗

arXiv2018

Orbifolds of Reshetikhin-Turaev TQFTs

We construct three classes of generalised orbifolds of Reshetikhin-Turaev theory for a modular tensor category $\mathcal{C}$, using the language of defect TQFT from [arXiv:1705.06085]: (i) spherical fusion categories give orbifolds for the "trivial" defect TQFT associated to vect, (ii) $G$-crossed extensions of $\mathcal{C}$ give group orbifolds for any finite group $G$, and (iii) we construct orbifolds from commutative $\Delta$-separable symmetric Frobenius algebras in $\mathcal{C}$. We also explain how the Turaev-Viro state sum construction fits into our framework by proving that it is isomorphic to the orbifold of case (i). Moreover, we treat the cases (ii) and (iii) in the more general setting of ribbon tensor categories. For case (ii) we show how Morita equivalence leads to isomorphic orbifolds, and we discuss Tambara-Yamagami categories as particular examples.

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arXiv2018

Eigenvalues of the squared antipode in finite dimensional weak Hopf algebras

We extend Schaumann's theory of pivotal structures on fusion categories matched to a module category and of module traces developed in arXiv:1206.5716 to the case of non-semisimple tensor categories, and use it to study eigenvalues of the squared antipode $S^2$ in weak Hopf algebras. In particular, we diagonalize $S^2$ for semisimple weak Hopf algebras in characteristic zero, generalizing the result of Nikshych in the pseudounitary case. We show that the answer depends only on the Grothendieck group data of the pivotalizations of the categories involved and the global dimension of the fusion category (thus, all eigenvalues belong to the corresponding number field). On the other hand, we study the eigenvalues of $S^2$ on the non-semisimple weak Hopf algebras attached to dynamical quantum groups at roots of $1$ defined by D. Nikshych and the author in arXiv:math/0003221, and show that they depend nontrivially on the continuous parameters of the corresponding module category. We then compute these eigenvalues as rational functions of these parameters. The paper also contains an appendix by G. Schaumann discussing the connection between our generalization of module traces and the notion of an inner product module category introduced in arXiv:1405.5667.

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arXiv2017

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.

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arXiv2017

Line and surface defects in Reshetikhin-Turaev TQFT

A modular tensor category $\mathcal{C}$ gives rise to a Reshetikhin-Turaev type topological quantum field theory which is defined on 3-dimensional bordisms with embedded $\mathcal{C}$-coloured ribbon graphs. We extend this construction to include bordisms with surface defects which in turn can meet along line defects. The surface defects are labelled by $\Delta$-separable symmetric Frobenius algebras and the line defects by "multi-modules" which are equivariant with respect to a cyclic group action. Our invariant cannot distinguish non-isotopic embeddings of 2-spheres, but we give an example where it distinguishes non-isotopic embeddings of 2-tori.

math.QA↗

arXiv2017

Orbifolds of n-dimensional defect TQFTs

We introduce the notion of $n$-dimensional topological quantum field theory (TQFT) with defects as a symmetric monoidal functor on decorated stratified bordisms of dimension $n$. The familiar closed or open-closed TQFTs are special cases of defect TQFTs, and for $n=2$ and $n=3$ our general definition recovers what had previously been studied in the literature. Our main construction is that of "generalised orbifolds" for any $n$-dimensional defect TQFT: Given a defect TQFT $\mathcal{Z}$, one obtains a new TQFT $\mathcal{Z}_{\mathcal{A}}$ by decorating the Poincar\'e duals of triangulated bordisms with certain algebraic data $\mathcal{A}$ and then evaluating with $\mathcal{Z}$. The orbifold datum $\mathcal{A}$ is constrained by demanding invariance under $n$-dimensional Pachner moves. This procedure generalises both state sum models and gauging of finite symmetry groups, for any $n$. After developing the general theory, we focus on the case $n=3$.

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arXiv2016

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↗

arXiv2016

3-dimensional defect TQFTs and their tricategories

We initiate a systematic study of 3-dimensional `defect' topological quantum field theories, that we introduce as symmetric monoidal functors on stratified and decorated bordisms. For every such functor we construct a tricategory with duals, which is the natural categorification of a pivotal bicategory. This captures the algebraic essence of defect TQFTs, and it gives precise meaning to the fusion of line and surface defects as well as their duality operations. As examples, we discuss how Reshetikhin-Turaev and Turaev-Viro theories embed into our framework, and how they can be extended to defect TQFTs.

math.QA↗

arXiv2016

A trace for bimodule categories

We study a 2-functor that assigns to a bimodule category over a finite k-linear tensor category a k-linear abelian category. This 2-functor can be regarded as a category-valued trace for 1-morphisms in the tricategory of finite tensor categories. It is defined by a universal property that is a categorification of Hochschild homology of bimodules over an algebra. We present several equivalent realizations of this 2-functor and show that it has a coherent cyclic invariance. Our results have applications to categories associated to circles in three-dimensional topological field theories with defects. This is made explicit for the subclass of Dijkgraaf-Witten topological field theories.

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arXiv2014

Pivotal tricategories and a categorification of inner-product modules

This article investigates duals for bimodule categories over finite tensor categories. We show that finite bimodule categories form a tricategory and discuss the dualities in this tricategory using inner homs. We consider inner-product bimodule categories over pivotal tensor categories with additional structure on the inner homs. Inner-product module categories are related to Frobenius algebras and lead to the notion of $*$-Morita equivalence for pivotal tensor categories. We show that inner-product bimodule categories form a tricategory with two duality operations and an additional pivotal structure. This is work is motivated by defects in topological field theories.

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arXiv2012

Gray categories with duals and their diagrams

The geometric and algebraic properties of Gray categories with duals are investigated by means of a diagrammatic calculus. The diagrams are three-dimensional stratifications of a cube, with regions, surfaces, lines and vertices labelled by Gray category data. These can be viewed as a generalisation of ribbon diagrams. The Gray categories present two types of duals, which are extended to functors of strict tricategories with natural isomorphisms, and correspond directly to symmetries of the diagrams. It is shown that these functors can be strictified so that the symmetries of a cube are realised exactly. A new condition on Gray categories with duals called the spatial condition is defined. A class of diagrams for which the evaluation for spatial Gray categories is invariant under homeomorphisms is exhibited. This relation between the geometry of the diagrams and structures in the Gray categories proves useful in computations and has potential applications in topological quantum field theory.

math.QA↗

arXiv2012

Traces on Module Categories over Fusion Categories

We consider traces on module categories over pivotal fusion categories which are compatible with the module structure. It is shown that such module traces characterise the Morita classes of special haploid symmetric Frobenius algebras. Moreover, they are unique up to a scale factor and they equip the dual category with a pivotal structure. This implies that for each pivotal structure on a fusion category over the complex numbers there exists a conjugate pivotal structure defined by the canonical module trace.

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