Quantum Topology and Field Theory Seminar

Time: 4:30-5:30pm

Location: KT 101

Abstract: 

H Verlinde suggested in 1980’s to use quantization of the Teichmüller spaces of surfaces to study the spaces of conformal blocks for the Liouville conformal field theory. This suggestion initiated and stimulated the development of quantum Teichmüller theory, and the first major steps were taken by Kashaev and by Chekhov and Fock in 1990’s, where the Chekhov-Fock quantization is generalized later by Fock and Goncharov to quantization of cluster varieties. The modular functor conjecture asserts that these quantum theories of Teichmüller spaces indeed yield a 2-dimensional modular functor, which can be viewed as one axiomatization of conformal field theory. The core part of the conjecture says that, for each punctured surface S and an essential simple loop in S, the Hilbert space associated to S by quantum Teichmüller theory should decompose into the direct integral of the Hilbert spaces associated to the surface obtained by cutting S along the loop and shrinking the holes to punctures. I will give an introduction to this story and present some recent developments, including 2405.14727. 

Time: 4:30-5:30pm

Location: KT 906

Abstract: 

I will give a short review of the physics-inspired approach to Donaldson theory based on topologically-twisted four-dimensional N=2 supersymmetric quantum field theory (SQFT) as propounded by Witten in 1988, and then describe some of my recent and ongoing work. One theme in particular is a generalization of Donaldson-Witten theory to describe invariants of smooth families of smooth, closed, oriented Riemannian 4-manifolds X using methods of SQFT and supergravity, and leading to physical derivations of relevant models of equivariant cohomology. Family Donaldson invariants are cohomology classes of the classifying space of orientation-preserving diffeomorphisms, i.e., elements of H*(BDiff^+(X)), and we propose path integral formulations of their cocycle representatives. Several interesting open questions arise concerning the observables and interpretation of the invariants, calling for a renewed math-physics dialog. I hope to (re?)ignite some interest in these issues through this talk. Time permitting, I will touch upon the second theme, a new perspective of topological twisting using the notion of ‘transfer of structure group’ associated with a continuous homomorphism between topological groups. This approach accounts for the global topology of the structure group of a quantum field theory, rather than just its simply connected cover, and leads to a proposal for twisting more general four-dimensional N=2 SQFTs. The talk will be based on collaborations with G. W. Moore, M. Roček, and R. K. Singh.   

Time: 4:30-5:30pm

Location: KT 101

Abstract: 

I will describe a joint project with Tobias Ekholm, Pietro Longhi, and Vivek Shende constructing a map from the HOMFLYPT skein module of a 3-manifold M to that of its branched cover arising from the projection of a Lagrangian 3-manifold L in the cotangent bundle of M. The map is defined by counting holomorphic curves and is a vast generalization of the quantum UV-IR map of Neitzke and Yan, which is a close cousin of the quantum trace map of Bonahon and Wong. The existence of this map has some interesting consequences in the theory of skein-valued curve counts, and I will discuss some of them if time permits. 

Time: 4:30-5:30pm

Location: KT 101

Abstract: 

The Jones polynomial of a link can be computed diagrammatically by using skein relations which encode the representation theory of SL(2). By considering the vector space spanned by links drawn on a surface and imposing these skein relations, we obtain an algebra known as the Kauffman bracket skein algebra of the surface. Replacing SL(2) by SL(3) or any other higher rank Lie group gives rise to a new skein algebra involving not only links but also certain graphs called webs. In this talk, we will discuss some of the complications involved with studying skein algebras built from webs on surfaces and then discuss how the use of stated skein algebras helps us get around these.

Silde: Click here

Time: 4:30-5:30pm

Location: KT 101

Abstract: 

I will discuss recent progress towards  understanding how the existence of essential surfaces in a 3-manifold reflects on the structure of its Kauffman bracket skein module. 

Time permitting, I will also discuss how this progress can be used to compute the dimension of these skein modules for ``small" 3-manifolds. 

The talk will be based on joint work with  Renaud Detcherry and Adam  Sikora.

Silde: Click here