Quantum Topology and Field Theory Seminar

Time: 4:30-5:30pm

Location: KT 801

Abstract: In recent and upcoming work joint with Anna Wienhard, Zach Greenberg and Merik Niemeyer, we describe a large class of Lie groups as simpler Lie groups “defined over noncommutative rings”, the simplest example expresses the symplectic group SP_2n as as SL_2 over a matrix ring.  We use this description to construct cluster coordinates on moduli spaces of G local systems on surfaces decorated by partial flags of G at the punctures of S. The cluster algebras and varieties which arise this way are noncommutative versions of those coming from Fock and Goncharov cluster coordinates associated to the split Lie group and full flags of this simpler type. I will give an overview of this theory and give an outlook towards some new perspectives on cluster quantization.  

Time: 4:30-5:30pm

Location: KT 801

Abstract:  According to the Gaiotto–Moore–Neitzke algorithm, spectral networks associated to differentials on Riemann surfaces can be used to compute the BPS states of certain supersymmetric quantum field theories. The construction of spectral networks associated with cubic differentials admits a particularly simple description in terms of flat geometry: they appear as graphs of straight trajectories that generate new ones upon intersection under certain conditions. We present the notion of spectral core as a refinement of the classical core concept by Haiden, Katzarkov, and Kontsevich in flat surface theory, and show that it precisely controls the birthing process of spectral networks trajectories. As an application, we describe the spectral networks corresponding to polynomial cubic differentials of degree d=3. Time permitting, we will also discuss the problem of characterizing cubic differentials whose associated spectral networks generated by the algorithm have finite complexity. This work is a collaboration with Omar Kidwai. 

Time: 4:30-5:30pm

Location: KT 801

Abstract:  We define a family of Turaev-Viro type invariants of hyperbolic 3-manifolds with totally geodesic boundary from the 6j-symbols of the modular double of U_q(sl(2; R)), and prove that these invariants decay exponentially with the rate the hyperbolic volume of the manifolds and with the “1-loop term” the adjoint twisted Reidemeister torsion of the double of the manifolds. This is a joint work with Tianyue Liu, Shuang Ming, Xin Sun and Baojun Wu.

Time: 4:30-5:30pm

Location: KT 801

Abstract: We formulate the holomorphic twists of the 6d N=(0,1) and (0,2) abelian superconformal theories as moduli spaces in derived algebraic geometry, using Deligne cohomology as a key tool. This description allows one to mimic the Beilinson-Drinfeld construction of lattice chiral algebras to quantize these 6d theories; their factorization homology on a projective complex 3-fold X relates to Witten's construction of line bundles on intermediate Jacobian of X. This is work in progress with Chris Elliott, Ingmar Saberi, and Brian Williams.

Time: 4:30-5:30pm

Location: KT 801

Abstract:  I will explain a new scheme for construction of conformal blocks for the Virasoro algebra at central charge c=1. One application is a new recipe for producing isomonodromic tau functions. This scheme is joint work with Qianyu Hao. The talk is intended to be self-contained (you don't have to know in advance what a conformal block or a tau function are). 

Time: 4:30-5:30pm

Location: KT 801

Abstract:  Following Harman, Snowden and Snyder we’ll explain the construction of the Delannoy category, its properties and how it leads to a categorification of the ring of integer-valued polynomials. We’ll also discuss a diagrammatic description of that category, in a work in progress with Noah Snyder.

Time: 4:30-5:30pm

Location: KT 801

Abstract:   I will talk about my recent work with Junrong Yan. We proved the convergence of Graph integrals on analytic Kahler manifolds in the sense of Cauchy principal values, which are originally from holomorphic quantum field theories. In particular, this allows us construct geometric invariants of Calabi-Yau metrics. I will also talk about some potential applications of our results.  

References: 

arXiv:2507.09170

arXiv:2401.08113

Time: 4:30-5:30pm

Location: KT 801

Abstract:  In early 2010s, Andersen and Kashaev defined a TQFT based on quantum Teichmuller theory. In particular, they define a partition function for every ordered ideal triangulation of hyperbolic knot complement in $\mathbb{S}^3$ equipped with an angle structure. The Andersen-Kashaev volume conjecture suggests that the partition function can be expressed in terms of a Jones function of the knot which, in its semi-classical limit, decays exponentially with decay rate the hyperbolic volume of the knot complement. In this talk, we will introduce a purely combinatorial condition on triangulations which, together with the geometricity of the triangulations, imply the Andersen-Kashaev volume conjecture and its generalization. This talk is based on the joint work with Fathi Ben Aribi.

Time: 4:30-5:30pm

Location: KT 801

Abstract:  The Kauffman Bracket Skein algebra is a central object in low-dimensional topology. It is an algebra generated by loops on an oriented surface. There have been many attempts to generalize this algebra from various motivations. In this talk, we review one of the generalizations, its motivation, and recent progress on its representation theory. This talk is based on joint work with Hiroaki Karuo and Helen Wong. 

Time: 4:30-5:30pm

Location: KT 801

Abstract:  The AGT correspondence and its extensions posit geometric constructions of vertex algebras and their modules from cohomology of variants of moduli of sheaves on surfaces. Physically, the correspondence has found an explanation through the holomorphic-topological twist of the six dimensional N=(2,0) superconformal field theories. In this talk, I’ll propose a variant of the AGT correspondence coming from the so-called minimal twist of these theories. Instead of vertex algebras, the natural algebras appearing will be holomorphic factorization algebras in three complex dimensions. From this data, I will explain how one extracts an associative algebra and a module which conjecturally agrees with a quantization of moduli of Higgs sheaves on surfaces. In examples, the pair conjecturally admits a Hodge-deRham deformation to the Heisenberg algebra and its action on cohomology of Hilbert schemes of surfaces, constructed in work of Grojnowski-Nakajima.

Time: 4:30-5:30pm

Location: KT 801

Abstract:  TBA