arXiv research

We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with certain algebraic properties determine topological invariants. We prove that fusion categories of nonzero global dimension are 3-dualizable, and therefore provide 3-di…

We construct certain tensor categories that are dominated by finitely many simple objects. Objects in these categories are modules over rings of algebra integers. We show how to obtain TQFTs defined over algebra integers from these categories.

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

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…

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

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

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…

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.

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

The paper defines new TQFTs from non-semisimple categories and proves spherical categories are chromatic.

Study of generalized Legendrian racks and their GL-structures.

Category theory enhances understanding of group-equivariant neural networks.

Develops graphical calculus for monoidal categories with twisted pivotal structures.

The paper constructs Yang-Baxter solutions using categorical augmented racks.

Propose a model-independent axiomatic framework for derived skein theory.

We show that the renormalized quantum invariants of links and graphs in the 3-sphere, derived from tensor categories in ["Modified quantum dimensions and re-normalized link invariants", arXiv:0711.4229] lead to modified 6j-symbols and to new state sum 3-manifold invariants. We give examples of categories such that the …

We categorify the notion of an infinitesimal braiding in a linear strict symmetric monoidal category, leading to the notion of a (strict) infinitesimal 2-braiding in a linear symmetric strict monoidal 2-category. We describe the associated categorification of the 4-term relation, leading to six categorified relations. …

TQFT invariants are either easy or hard to compute, depending on the TQFT type.

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

In this paper, we study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. Our aim is to construct knot homologies categorifying Reshetikhin-Turaev invariants of knots for arbitrary representations, which will be done in a fo…

We define the Hopf superalgebra U_T sl(1,1), which is a variant of the quantum supergroup U_q sl(1,1), and its tensor product representations V_1^{\otimes n} for n>0. We construct families of DG algebras A, B and R_n, and consider the DG categories DGP(A), DGP(B) and DGP(R_n), which are full DG subcategories of the cat…

Introduces tensor product for quiver representations and applies to stable bundles and character varieties.

New skein categories for non-semisimple settings, extending existing theory.

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

We use super $q$-Howe duality to provide diagrammatic presentations of an idempotented form of the Hecke algebra and of categories of $\mathfrak{gl}_N$-modules (and, more generally, $\mathfrak{gl}_{N|M}$-modules) whose objects are tensor generated by exterior and symmetric powers of the vector representations. As an ap…

Classifies extended Abelian Chern-Simons theories using quadratic modules.

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…

We show that in the analytic category, given a Riemannian metric $g$ on a hypersurface $M\subset \Z$ and a symmetric tensor $W$ on $M$, the metric $g$ can be locally extended to a Riemannian Einstein metric on $Z$ with second fundamental form $W$, provided that $g$ and $W$ satisfy the constraints on $M$ imposed by the …

Joint analysis of data from multiple sources has the potential to improve our understanding of the underlying structures in complex data sets. For instance, in restaurant recommendation systems, recommendations can be based on rating histories of customers. In addition to rating histories, customers' social networks (e…

In this paper we construct invariants of 3-manifolds "à la Reshetikhin-Turaev" in the setting of non-semi-simple ribbon tensor categories. We give concrete examples of such categories which lead to a family of 3-manifold invariants indexed by the integers. We prove this family of invariants has several notable features…

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

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

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

Affine structures on a Lie groupoid, including affine $k$-vector fields, $k$-forms and $(p,q)$-tensors are studied. We show that the space of affine structures is a 2-vector space over the space of multiplicative structures. Moreover, the space of affine multivector fields has a natural graded strict Lie 2-algebra stru…

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl_2 and sl_3 and by Mazorchuk-Stroppel and Sussan for sl_n. Our technique is to study 2-representations of 2-quantum gr…

Introduces P-tensors for generalized higher-order message passing in graph neural networks.

In previous work we showed that the contact category algebra of a quadrangulated surface is isomorphic to the homology of a strand algebra from bordered sutured Floer theory. Being isomorphic to the homology of a differential graded algebra, this contact category algebra has an A-infinity structure, allowing us to comb…

New (3+1) TQFTs created from non-semisimple categories.

The paper classifies virtual knot polynomials and trivalent graph invariants using skein theory.

Defines odd Khovanov homology via categorification of q-Schur algebra.

Study mapping class groups and their representations linking to algebraic structures.

Using quantum skew-Howe duality, we study the category $\operatorname{Rep}(\mathfrak{gl}(m|n))$ of tensor products of exterior powers of the standard representation of $U_q(\mathfrak{gl}(m|n))$, and prove that it is equivalent to a category of ladder diagrams modulo one extra family of relations. We then construct a ca…

The paper constructs a Poisson algebra bundle for multilocal observables.

Data collected at very frequent intervals is usually extremely sparse and has no structure that is exploitable by modern tensor decomposition algorithms. Thus the utility of such tensors is low, in terms of the amount of interpretable and exploitable structure that one can extract from them. In this paper, we introduce…

Given a semisimple stable autonomous tensor category over a field $K$, to any group presentation with finite number of generators we associate an element $Q(P)\in K$ invariant under the Andrews-Curtis moves. We show that in fact, this is the same invariant as the one produced by the algorithm of Frank Quinn. The new de…

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…

It has been conjectured that every $(2+1)$-TQFT is a Chern-Simons-Witten (CSW) theory labelled by a pair $(G,λ)$, where $G$ is a compact Lie group, and $λ\in H^4(BG;Z)$ a cohomology class. We study two TQFTs constructed from Jones' subfactor theory which are believed to be counterexamples to this conjecture: one is the…

The goal of this paper is to find a close to isomorphic presentation of 3-manifolds in terms of Hopf algebraic expressions. To this end we define and compare three different braided tensor categories that arise naturally in the study of Hopf algebras and 3-dimensional topology. The first is the category \Cob of connect…