arXiv research

ETQFTs created from non-semisimple modular categories.

A modular functor is constructed from non-semisimple 3d TFTs.

Mathematical study supports connection between 3D manifolds and modular tensor categories.

Generalizes string-net modular functors to non-spherical categories.

Recent work extends Turaev's modular categories to non-semisimple settings.

We show that once-extended anomalous 3-dimensional topological quantum field theories valued in the 2-category of k-linear categories are in canonical bijection with modular tensor categories equipped with a square root of the global dimension in each factor.

We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a unitary modular category. Many new unitary modular categories are obtained. We also sh…

In this paper, we extend the notion of modular functor and fusion category to what we called $G$ equivariant modular functor and $G$ equivariant fusion category, where $G$ is a finite group, and establish a correspondence between between these notions.

Abstract: Mapping class groups act on cohomology of surfaces via Hochschild cohomology.

New invariants derived from a modular category for links and 3-manifolds.

We define a symmetric monoidal (4,3)-category with duals whose objects are certain enriched multi-fusion categories. For every modular tensor category $\mathcal{C}$, there is a self enriched multi-fusion category $\mathfrak{C}$ giving rise to an object of this symmetric monoidal (4,3)-category. We conjecture that the e…

Homotopy Quantum Field Theories (HQFTs) generalize more familiar Topological Quantum Field Theories (TQFTs). In generalization of the surgery construction of 3-dimensional TQFTs from modular categories, we use surgery to derive 3-dimensional HQFTs from G-modular categories.

Internalizes Turaev's construction for TQFTs using ribbon categories.

The article is devoted to the $q_R$-conformal modular functors, which being ``deformations'' of the conformal modular functor (the projective representation of the category $Train(Diff_+(S^1))$, the train of the group $Diff_+(S^1)$ of all orientation preserving diffeomorphisms of a circle) in the class of all projectiv…

Classical and quantum Chern-Simons with gauge group $\text{U}(1)^N$ were classified by Belov and Moore in \cite{belov_moore}. They studied both ordinary topological quantum field theories as well as spin theories. On the other hand a correspondence is well known between ordinary $(2+1)$-dimensional TQFTs and modular te…

New mapping class group actions on Hochschild complexes for modular categories.

We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d $\mathcal{N}=2$ theories where such structures a priori are not manifest. These modular structures include: mock modular forms, $SL(2,\mathbb{Z})$ Weil representations, quantum mo…

Modular categories are a well-known source of quantum 3-manifold invariants. In this paper we study structures on modular categories which allow to define refinements of quantum 3-manifold invariants involving cohomology classes or generalized spin and complex spin structures. A crucial role in our construction is play…

New modular data from torus bundles via particle-hole equivariantization.

A p-periodic 3-manifold is a 3-manifold that admits a Z_{p}-action whose fixed point set is a circle. We give a congruence relates the quantum invariant of a p-periodic 3-manifold associated to any modular category over an integrally closed ground ring and the corresponding quantum invariant of its orbit space.

We use modified traces to renormalize Lyubashenko's closed 3-manifold invariants coming from twist non-degenerate finite unimodular ribbon categories. Our construction produces new topological invariants which we upgrade to 2+1-TQFTs under the additional assumption of factorizability. The resulting functors provide mon…

We construct modular categories from Hecke algebras at roots of unity. For a special choice of the framing parameter, we recover the Reshetikhin-Turaev invariants of closed 3-manifolds constructed from the quantum groups U_q sl(N) by Reshetikhin-Turaev and Turaev-Wenzl, and from skein theory by Yokota. We then discuss …

A 3-dimensional homotopy quantum field theory (HQFT) can be described as a TQFT for surfaces and 3-cobordisms endowed with homotopy classes of maps into a given space. For a group $π$, we introduce a notion of a modular crossed $π$-category and show that such a category gives rise to a 3-dimensional HQFT with target sp…

We study the behavior of the modular class of a Lie algebroid under general Lie algebroid morphisms by introducing the relative modular class. We investigate the modular classes of pull-back morphisms and of base-preserving morphisms associated to Lie algebroid extensions. We also define generalized morphisms, includin…

Our aim is to introduce and advocate non-$Σ$ (non-symmetric) modular operads. While ordinary modular operads were inspired by the structure of the moduli space of stable complex curves, non-$Σ$ modular operads model surfaces with open strings outputs. An immediate application of our theory is a short proof that the mod…

Motivated by ideas from string theory and quantum field theory new invariants of knots and 3-dimensional manifolds have been constructed from complex algebraic structures such as Hopf algebras (Reshetikhin and Turaev), monoidal categories with additional structure (Turaev and Yetter), and modular functors (Walker and K…

New invariant from Viro's gl(1|1) polynomial distinguishes lens spaces.

Examples of SL(2, Z) actions on differential graded categories are defined and explored.

We develop the general theory for the construction of Extended Topological Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau quantum invariants of closed 3-manifolds. In order to do so, we introduce relative modular categories, a class of ribbon categories which are modeled on representations…

Calculates Dehn twist actions on conformal blocks for modular categories.

Study projective representations from non-semisimple TQFTs on surfaces.

We present an invariant of connected and oriented closed 3-manifolds based on a coribbon Weak Hopf Algebra H with a suitable left-integral. Our invariant can be understood as the generalization to Weak Hopf Algebras of the Hennings-Kauffman-Radford evaluation of an unoriented framed link using a dual quantum-trace. Thi…

3D topological order linked to Seifert manifolds and gauge groups.

The paper explores the pentagon relation and its algebraic forms.

Constructs non-commutative modular vector fields for Poisson manifolds.

Proposes a topological framework to study modular invariants and related concepts.

We calculate the RT-invariants of all oriented Seifert manifolds directly from surgery presentations. We work in the general framework of an arbitrary modular category as in [V. G. Turaev, Quantum invariants of knots and 3--manifolds, de Gruyter Stud. Math. 18, Walter de Gruyter (1994)], and the invariants are expresse…

Defines a new 3D TQFT from non-semisimple categories.

Classifies extended Abelian Chern-Simons theories using quadratic modules.

Paper uses Turaev-Viro TQFT to estimate 3-manifold genus.

A formula connects two algebraic structures derived from a category.

The paper explores how the unit inclusion affects topological quantum field theories in non-semisimple categories.

New 4-manifold invariant defined from trisection diagrams.

This paper focuses on a new task, i.e., transplanting a category-and-task-specific neural network to a generic, modular network without strong supervision. We design an functionally interpretable structure for the generic network. Like building LEGO blocks, we teach the generic network a new category by directly transp…

This paper focuses on a new task, i.e., transplanting a category-and-task-specific neural network to a generic, modular network without strong supervision. We design a functionally interpretable structure for the generic network. Like building LEGO blocks, we teach the generic network a new category by directly transpl…

Constructs dg categories from surfaces using Khovanov homology.

Researchers create projective representations of Hecke groups using TQFT.

A new neural network for efficient density estimation.