arXiv subjects

Christoph Schweigert

Publications and source records attributed to Christoph Schweigert.

At least 37 records · Page 2Linked to original sources

arXiv2023

Spherical Morita contexts and relative Serre functors

The Morita context provided by an exact module category over a finite tensor category gives a two-object bicategory with duals. Right and left duals of objects in the module category are given by internal Homs and coHoms, respectively. We express the double duals in terms of relative Serre functors, which leads to a Radford isomorphism for module categories. There is a bicategorical version of the Radford $S^4$ theorem: on the bicategory of a Morita context, the relative Serre functors assemble into a pseudo-functor, and the Radford isomorphisms furnish a trivialization of the square of this pseudo-functor, i.e. of the fourth power of the duals. We also show that the Morita bicategories coming from pivotal exact module categories are pivotal as bicategories, leading to the notion of pivotal Morita equivalence. This equivalence of tensor categories amounts to the equivalence of their bicategories of pivotal module categories. Furthermore, we introduce the notion of a spherical module category; it ensures that all categories in the Morita context of a spherical module category are spherical. Our results are motivated by and have applications to topological field theory.

math.QA↗

arXiv2023

A Parallel Section Functor for 2-Vector Bundles

We associate to a 2-vector bundle over an essentially finite groupoid a 2-vector space of parallel sections, or, in representation theoretic terms, of higher invariants, which can be described as homotopy fixed points. Our main result is the extension of this assignment to a symmetric monoidal 2-functor $\operatorname{Par} : \mathbf{2VecBunGrpd} \to \mathbf{2Vect}$. It is defined on the symmetric monoidal bicategory $\mathbf{2VecBunGrpd}$ whose morphisms arise from spans of groupoids in such a way that the functor $\operatorname{Par}$ provides pull-push maps between 2-vector spaces of parallel sections of 2-vector bundles. The direct motivation for our construction comes from the orbifoldization of extended equivariant topological field theories.

math.CT↗

arXiv2023

The differential graded Verlinde Formula and the Deligne Conjecture

A modular category $\mathcal{C}$ gives rise to a differential graded modular functor, i.e. a system of projective mapping class group representations on chain complexes. This differential graded modular functor assigns to the torus the Hochschild complex and, in the dual description, the Hochschild cochain complex of $\mathcal{C}$. On both complexes, the monoidal product of $\mathcal{C}$ induces the structure of an $E_2$-algebra, to which we refer as the differential graded Verlinde algebra. At the same time, the modified trace induces on the tensor ideal of projective objects in $\mathcal{C}$ a Calabi-Yau structure so that the cyclic Deligne Conjecture endows the Hochschild cochain and chain complex of $\mathcal{C}$ with a second $E_2$-structure. Our main result is that the action of a specific element $S$ in the mapping class group of the torus transforms the differential graded Verlinde algebra into this second $E_2$-structure afforded by the Deligne Conjecture. This result is established for both the Hochschild chain and the Hochschild cochain complex of $\mathcal{C}$. In general, these two versions of the result are inequivalent. In the case of Hochschild chains, we obtain a block diagonalization of the Verlinde algebra through the action of the mapping class group element $S$. In the semisimple case, both results reduce to the Verlinde formula. In the non-semisimple case, we recover after restriction to zeroth (co)homology earlier proposals for non-semisimple generalizations of the Verlinde formula.

math.QA↗

arXiv2023

Homotopy Invariants of Braided Commutative Algebras and the Deligne Conjecture for Finite Tensor Categories

It is easy to find algebras $\mathbb{T}\in\mathcal{C}$ in a finite tensor category $\mathcal{C}$ that naturally come with a lift to a braided commutative algebra $\mathsf{T}\in Z(\mathcal{C})$ in the Drinfeld center of $\mathcal{C}$. In fact, any finite tensor category has at least two such algebras, namely the monoidal unit $I$ and the canonical end $\int_{X\in\mathcal{C}} X\otimes X^\vee$. Using the theory of braided operads, we prove that for any such algebra $\mathbb{T}$ the homotopy invariants, i.e. the derived morphism space from $I$ to $\mathbb{T}$, naturally come with the structure of a differential graded $E_2$-algebra. This way, we obtain a rich source of differential graded $E_2$-algebras in the homological algebra of finite tensor categories. We use this result to prove that Deligne's $E_2$-structure on the Hochschild cochain complex of a finite tensor category is induced by the canonical end, its multiplication and its non-crossing half braiding. With this new and more explicit description of Deligne's $E_2$-structure, we can lift the Farinati-Solotar bracket on the Ext algebra of a finite tensor category to an $E_2$-structure at cochain level. Moreover, we prove that, for a unimodular pivotal finite tensor category, the inclusion of the Ext algebra into the Hochschild cochains is a monomorphism of framed $E_2$-algebras, thereby refining a result of Menichi.

math.QA↗

arXiv2023

Algebraic structures in two-dimensional conformal field theory

This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics. We review the following algebraic structures which appear in two-dimensional conformal field theory (CFT): The symmetries of two-dimensional conformal field theories (CFTs) can be formalised as chiral algebras, vertex operator algebras or nets of observable algebras. Their representation categories are abelian categories having additional structures, which are induced by properties of conformal blocks, i.e. of vector bundles over the moduli space of curves with marked points, which can be constructed from the symmetry structure. These mathematical notions pertain to the description of chiral CFTs. In a full local CFT one deals in addition with correlators, which are specific elements in the spaces of conformal blocks. In fact, a full CFT is the same as a consistent system of correlators for arbitrary conformal surfaces with any number and type of field insertions in the bulk as well as on boundaries and on topological defect lines. We present algebraic structures that allow one to construct such systems of correlators.

math.QA↗

arXiv2023

A $G$-equivariant String-Net Construction

We develop a string-net construction for the (2,1)-dimensional part of a $G$-equivariant three-dimensional topological field theory based on a $G$-graded spherical fusion category. In this construction, a $G$-equivariant generalization of the Ptolemy groupoid enters. We compute the associated cylinder categories and show that, as expected, the model is closely related to the $G$-equivariant Turaev-Viro theory.

math.QA↗

arXiv2022

Davydov-Yetter cohomology, comonads and Ocneanu rigidity

Davydov-Yetter cohomology classifies infinitesimal deformations of tensor categories and of tensor functors. Our first result is that Davydov-Yetter cohomology for finite tensor categories is equivalent to the cohomology of a comonad arising from the central Hopf monad. This has several applications: First, we obtain a short and conceptual proof of Ocneanu rigidity. Second, it allows to use standard methods from comonad cohomology theory to compute Davydov-Yetter cohomology for a family of non-semisimple finite-dimensional Hopf algebras generalizing Sweedler's four dimensional Hopf algebra.

math.QA↗

arXiv2022

String-net construction of RCFT correlators

We use string-net models to accomplish a direct, purely two-dimensional, approach to correlators of two-dimensional rational conformal field theories. We obtain concise geometric expressions for the objects describing bulk and boundary fields in terms of idempotents in the cylinder category of the underlying modular fusion category, comprising more general classes of fields than is standard in the literature. Combining these idempotents with Frobenius graphs on the world sheet yields string nets that form a consistent system of correlators, i.e. a system of invariants under appropriate mapping class groups that are compatible with factorization. Using markings, we extract operator products of field objects from specific correlators; the resulting operator products are natural algebraic expressions that make sense beyond semisimplicity. We also derive an Eckmann-Hilton relation internal to a braided category, thereby demonstrating the utility of string nets for understanding algebra in braided tensor categories. Finally we introduce the notion of a universal correlator. This systematizes the treatment of situations in which different world sheets have the same correlator and allows for the definition of a more comprehensive mapping class group.

math.QA↗

arXiv2022

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↗

arXiv2022

The Hochschild Complex of a Finite Tensor Category

Modular functors, i.e. consistent systems of projective representations of mapping class groups of surfaces, have been constructed for non-semisimple modular categories already decades ago. Concepts from homological algebra have not been used in this construction although it is an obvious question how they should enter in the non-semisimple case. In the present paper, we elucidate the interplay between the structures from topological field theory and from homological algebra by constructing a homotopy coherent projective action of the mapping class group $\text{SL}(2,\mathbb{Z})$ of the torus on the Hochschild complex of a modular category. This is a further step towards understanding the Hochschild complex of a modular category as a differential graded conformal block for the torus. Moreover, we describe a differential graded version of the Verlinde algebra.

math.QA↗

arXiv2021

Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups

Given a finite modular tensor category, we associate with each compact surface with boundary a cochain complex in such a way that the mapping class group of the surface acts projectively on its cohomology groups. In degree zero, this action coincides with the known projective action of the mapping class group on the space of chiral conformal blocks. In the case that the surface is a torus and the category is the representation category of a factorizable ribbon Hopf algebra, we recover our previous result on the projective action of the modular group on the Hochschild cohomology groups of the Hopf algebra.

math.QA↗

arXiv2021

Duality structures for module categories of vertex operator algebras and the Feigin Fuchs boson

Huang, Lepowsky and Zhang have developed a module theory for vertex operator algebras that endows suitably chosen module categories with the structure of braided monoidal categories. Included in the theory is a functor which assigns to discretely strongly graded modules a contragredient module, obtained as a gradewise dual. In this paper, we show that this gradewise dual endows the module category with the structure of a ribbon Grothendieck-Verdier category. This duality structure is more general than that of a rigid monoidal category; in contrast to rigidity, it naturally accommodates the fact that a vertex operator algebra and its gradewise dual need not be isomorphic as modules and that the tensor product of modules over vertex operator algebras need not be exact. We develop criteria which allow the detection of ribbon Grothendieck-Verdier equivalences and use them to explore ribbon Grothendieck-Verdier structures in the example of the rank $n$ Heisenberg vertex operator algebra or chiral free boson on a not necessarily full rank even lattice with arbitrary choice of conformal vector. We show that these categories are equivalent, as ribbon Grothendieck-Verdier categories, to certain categories of graded vector spaces and categories of modules over a certain Hopf algebra.

math.QA↗

arXiv2021

Homotopy Coherent Mapping Class Group Actions and Excision for Hochschild Complexes of Modular Categories

Given any modular category $\mathcal{C}$ over an algebraically closed field $k$, we extract a sequence $(M_g)_{g\geq 0}$ of $\mathcal{C}$-bimodules. We show that the Hochschild chain complex $CH(\mathcal{C};M_g)$ of $\mathcal{C}$ with coefficients in $M_g$ carries a canonical homotopy coherent projective action of the mapping class group of the surface of genus $g+1$. The ordinary Hochschild complex of $\mathcal{C}$ corresponds to $CH(\mathcal{C};M_0)$. This result is obtained as part of the following more comprehensive topological structure: We construct a symmetric monoidal functor $\mathfrak{F}_{\mathcal{C}}:\mathcal{C}\text{-}\mathsf{Surf}^{\mathsf{c}}\to\mathsf{Ch}_k$ with values in chain complexes over $k$ defined on a symmetric monoidal category of surfaces whose boundary components are labeled with projective objects in $\mathcal{C}$. The functor $\mathfrak{F}_{\mathcal{C}}$ satisfies an excision property which is formulated in terms of homotopy coends. In this sense, any modular category gives naturally rise to a modular functor with values in chain complexes. In zeroth homology, it recovers Lyubashenko's mapping class group representations. The chain complexes in our construction are explicitly computable by choosing a marking on the surface, i.e. a cut system and a certain embedded graph. For our proof, we replace the connected and simply connected groupoid of cut systems that appears in the Lego-Teichmüller game by a contractible Kan complex.

math.QA↗

arXiv2021

Equivariant Morita theory for graded tensor categories

We extend categorical Morita equivalence to finite tensor categories graded by a finite group $G$. We show that two such categories are graded Morita equivalent if and only if their equivariant Drinfeld centers are equivalent as braided $G$-crossed tensor categories.

math.QA↗

arXiv2021

Domain walls between 3d phases of Reshetikhin-Turaev TQFTs

We study surface defects in three-dimensional topological quantum field theories which separate different theories of Reshetikhin-Turaev type. Based on the new notion of a Frobenius algebra over two commutative Frobenius algebras, we present an explicit and computable construction of such defects. It specialises to the construction in Carqueville et al., arXiv:1710.10214 if all 3-strata are labelled by the same topological field theory. We compare the results to the model-independent analysis in Fuchs et al., arXiv:1203.4568 and find agreement.

hep-th↗

arXiv2021

Frobenius-Schur Indicators and the Mapping Class Group of the Torus

The Turaev-Viro state sum invariant can be extended to 3-manifolds with free boundaries. We use this fact to describe generalized Frobenius-Schur indicators as Turaev-Viro invariants of solid tori. This provides a geometric understanding of the $\mathrm{SL}(2,\mathbb{Z})$-equivariance of these indicators.

math.QA↗

arXiv2021

CFT Correlators for Cardy Bulk Fields via String-Net Models

We show that string-net models provide a novel geometric method to construct invariants of mapping class group actions. Concretely, we consider string-net models for a modular tensor category ${\mathcal C}$. We show that the datum of a specific commutative symmetric Frobenius algebra in the Drinfeld center $Z(\mathcal{C})$ gives rise to invariant string-nets. The Frobenius algebra has the interpretation of the algebra of bulk fields of the conformal field theory in the Cardy case.

hep-th↗

arXiv2021

Bulk from boundary in finite CFT by means of pivotal module categories

We present explicit mathematical structures that allow for the reconstruction of the field content of a full local conformal field theory from its boundary fields. Our framework is the one of modular tensor categories, without requiring semisimplicity, and thus covers in particular finite rigid logarithmic conformal field theories. We assume that the boundary data are described by a pivotal module category over the modular tensor category, which ensures that the algebras of boundary fields are Frobenius algebras. Bulk fields and, more generally, defect fields inserted on defect lines, are given by internal natural transformations between the functors that label the types of defect lines. We use the theory of internal natural transformations to identify candidates for operator products of defect fields (of which there are two types, either along a single defect line, or accompanied by the fusion of two defect lines), and for bulk-boundary OPEs. We show that the so obtained OPEs pass various consistency conditions, including in particular all genus-zero constraints in Lewellen's list.

hep-th↗