Related papers: Some asymptotics of TQFT via skein theory

This paper resolves the unicity conjecture of Bonahon and Wong for the Kauffman bracket skein algebras of all oriented finite type surfaces at all roots of unity. The proof is a consequence of a general unicity theorem that says that the…

We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group $G$, the derived and the stable categories…

In this paper, we study the Kauffman bracket skein module of closed oriented three-manifolds at a non-multiple-of-four roots of unity. Our main result establishes that the localization of this module at a maximal ideal, which corresponds to…

We obtain the topological expansion of the hermitian matrix model using its representation as a CFT on a hyperelliptic Riemann surface. To each branch point of the Riemann surface we associate an operator which represents a twist field…

In this lecture, we survey a number of recent results and developments regarding the representation theory of infinite-dimensional quantum groups (quantum affine algebras and related algebras), as well as their connections with cluster…

Invariants of 3-manifolds from a non semi-simple category of modules over a version of quantum $sl(2)$ were obtained by the last three authors in arXiv:1202.3553 . They are invariants of $3$-manifolds together with a cohomology class which…

Quantum Field Theory (QFT) represents a vast generalization of Quantum Mechanics (QM), as it deals with systems that have an infinite number of degrees of freedom. The Stone-von Neumann theorem, which establishes the equivalence of…

This paper is a short account of the construction of a new class of the infinite-dimensional representations of the quantum groups. The examples include finite-dimensional quantum groups $U_q(\mathfrak{g})$, Yangian $Y(\mathfrak{g})$ and…

The mapping class group of a surface with one boundary component admits numerous interesting representations including as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class…

We relate the Kauffman bracket stated skein modules to two independent constructions of quantum representation spaces of Habiro and Van der Veen with the second author. We deduce from this relation a description of the classical limit of…

The twisted q-Yangians are coideal subalgebras of the quantum affine algebra associated with gl(N). We prove a classification theorem for finite-dimensional irreducible representations of the twisted q-Yangians associated with the…

We prove various finiteness and representability results for cohomology of finite flat abelian group schemes. In particular, we show that if $f\colon X\rightarrow \mathrm{Spec}(k)$ is a projective scheme over a field $k$ and $G$ is a finite…

We address two linked problems at the interface of quantum topology and number theory: deriving asymptotic expansions of the Witten--Reshetikhin--Turaev invariants for 3-manifolds and establishing quantum modularity of false theta…

Let $F$ be a $p$-adic field of characteristic zero and odd residual characteristic. Let $\mathbf{Sp}_{2n}(F)$ denote the symplectic group defined over $F$, where $n\geq 2$. We prove that the Speh representations $\mathcal{U}(\delta,2)$,…

In the framework of quantum field theory (QFT) on noncommutative (NC) space-time with $SO(1,1)\times SO(2)$ symmetry, which is the feature arising when one has only space-space noncommutativity ($\theta_{0i}=0$), we prove that the…

We produce 2-representations of the positive part of affine quantum enveloping algebras on their finite-dimensional counterparts in type $A_n$. These 2-representations naturally extend the right-multiplication 2-representation of…

We combine I. background independent Loop Quantum Gravity (LQG) quantization techniques, II. the mathematically rigorous framework of Algebraic Quantum Field Theory (AQFT) and III. the theory of integrable systems resulting in the invariant…

We classify the cosemisimple Hopf algebras whose corepresentation semi-ring is isomorphic to that of GL(2). This leads us to define a new family of Hopf algebras which generalize the quantum similitude group of a non-degenerate bilinear…

Consider a topological surface $\Sigma$. We introduce the spectrum of a representation from the fundamental group of $\Sigma$ to SL(2,R), which is a subset of projective measured lamination on the surface, which captures the directions…

Let $\Sigma_{g,n}$ be the orientable genus $g$ surface with $n$ punctures, where $2-2g-n<0$. Let $$\rho: \pi_1(\Sigma_{g,n})\to GL_m(\mathbb{C})$$ be a representation. Suppose that for each finite covering map $f: \Sigma_{g', n'}\to…