Quantum Topology and Field Theory Seminar

Time: 4:30-5:30pm

Location: KT801

Abstract: Skein modules are algebraic objects that somehow "encode" links in a 3-dimensional manifold, in a way reminiscent of homology; they have many interesting connections to physics, representation theory, knot theory (via the Jones polynomial) and non-commutative algebra. In this talk I will give a broad introduction and overview of the topic and then discuss a new, elementary proof (due to myself and Renaud Detcherry) of finite dimensionality of the Kauffman bracket skein modules, originally done by Gunningham-Jordan-Safronov.

Time: 4:30-5:30pm

Location: KT801

Abstract: We give an algorithm to reduce the number of generators of the Khovanov chain complex of torus braids $(\sigma_1\sigma_2 \dots\sigma_{n−1})^k$ on $n$ strands. I will begin the talk with context on the stable Khovanov homology of torus links leading to the open question of the structure of their homology theory, as well as potential applications to open questions concerning the colored Jones polynomial. Next I will discuss our work, joint with Carmen Caprau, Nicolle Gonzalez, and Radmila Sazdanovic, using Bar-Natan Gaussian elimination, that gives our whittled complex $\mathcal{FT}_n^k$. The whittled complex is homotopy-equivalent to the original Khovanov chain complex but with a reduced number of generators. After sketching the proof, I will end the talk discussing related future projects.

Time: 4:30-5:30pm

Location: KT801

Abstract:  For a Lagrangian submanifold in a CY3, Ekholm and Shende defined a wavefunction living in the HOMFLY-PT skein module of the Lagrangian, which encodes open Gromov-Witten invariants in all genus. In this talk, we study a skein-valued cluster theory that generalizes quantum cluster theory and allows us to compute these wavefunctions in a range of examples. Our results agree with the physical prediction known as the topological vertex. Along the way we introduce a skein dilogarithm and prove a pentagon relation, generalizing previously known forms of the pentagon identity. This talk is based on joint works with Schrader, Zaslow, and Shende.

Time: 4:30-5:30pm

Location: KT801

Abstract:  We will examine the multiplicative structure of two skein algebras---the usual Kauffman bracket skein algebra of a surface (generated by loops) and a generalization of it due to Roger-Yang (generated by loops and arcs).  In joint work with Chloe Marple, we found an unexpected homomorphism between the usual skein algebra for a closed torus and the Roger-Yang skein algebra for a twice-punctured annulus.  In this talk, I’ll discuss how we  used the homomorphism to help compute representations and structural constants of the Roger-Yang skein algebra for a twice-punctured annulus, and  whether there might be similar relationships between skein algebras for other surfaces. 

Time: 4:30-5:30pm

Location: KT801

Abstract:  I will describe a construction of a q-series invariant (BPS q-series, also known as the Z-hat invariant) associated to a 3-manifold decorated by an embedded link. These q-series depend only on the class of the link in the skein module, and hence define a homomorphism from the skein module to the space of q-series. The image of this homomorphism is conjectured to exhibit holomorphic quantum modularity, which suggests a new approach to Langlands duality for skein modules via q-series. 

Time: 4:30-5:30pm

Location: KT801

Abstract:  Finite-N effects in large-N gauge theories, such as trace relations, are expected to be holographically dual to non-perturbative phenomena in string theory, such as Giant Graviton branes. A convenient setting to study these effects are supersymmetric indices of U(N) gauge theories. The finite-N indices can be reproduced by a series of corrections to the infinite-N result, known as the Giant Graviton expansion.

In this talk I will present a generalization of the Molien-Weyl formula computing generating functions of invariants of supergroups U(N|M), which arise as gauge groups of brane/negative brane systems in string theory. The formula leads to a new expansion relating finite-N and infinite-N indices of U(N) gauge theories. I will comment on its relation to Murthy's Giant Graviton expansion and suggest a physical interpretation in terms of branes and negative branes. This talk is based on arXiv:2509.20451 and work in progress with Davide Gaiotto.

Time: 4:30-5:30pm

Location: KT801

Abstract:  Poisson sigma models sit at the intersection of deformation theory, geometry, and quantum field theory; specifically, the perturbative expansion of the two-dimensional Poisson sigma model with boundary is known to recover Kontsevich’s deformation quantization formula. In this talk, we introduce a higher-dimensional holomorphic–topological generalization of Poisson sigma models and explain their connections to the deformation quantization (or obstruction) of holomorphic–topological factorization algebras. This construction can be viewed as a field-theoretic incarnation of the higher Deligne conjecture. We also explain how these models relate to the construction of Hopf-type algebras via Koszul duality, and explore examples related to the quantization of Lie bialgebras, W-algebras, and Yangians.

Time: 4:30-5:30pm

Location: KT801

Abstract:  TBA

Time: 4:30-5:30pm

Location: KT801

Abstract:  In this talk I will give an introductory lecture on constructing Topological Quantum Field Theories (TQFTs) from non-semisimple categories. The main goal of the talk is to give a hint of what is needed to extend the Turaev-Viro and Crane-Yetter TQFTs from the useful setting of semisimple categories to the non-semisimple world.  I will do this from an algebraic and categorical point of view.  In particular, I will discuss what kind of structures are needed in non-semisimple categories to give rise to (2+1)-TQFTs.  Then I will remark that any spherical tensor category (in the sense of Etingof, Douglas et al.) has such structures.  This work is joint with Francesco Costantino, Benjamin Haïoun, Bertrand Patureau-Mirand and Alexis Virelizier and based on arXiv:2302.04509 and arXiv:2306.03225.

Time: 4:30-5:30pm

Location: KT801

Abstract:  I will describe recent progress towards understanding the gravitational path integral in AdS_3 quantum gravity and its boundary interpretation. A central question is: which spacetime topologies should be included in the path integral, and why? To address this question, we formulate a "statistical bootstrap" that constrains the universal statistics of CFT data in the boundary theory, imposing crossing symmetry and "typicality" (a generalization of the eigenstate thermalization hypothesis). These constraints are geometrized by iterative surgery moves on bulk manifolds that we refer to as the "gravitational machine," leading to an infinite set of non-handlebody topologies that we argue must be included in the path integral. The machine generates only on-shell (hyperbolic) 3-manifolds, whose partition functions can be computed exactly using Virasoro TQFT. But not all hyperbolic manifolds are produced by this procedure. This reveals a large landscape of consistent sums over topologies. Based on joint work with Alexandre Belin, Lorenz Eberhardt, Diego Liska, and Boris Post. 

Time: 4:30 pm

Location: KT801

Abstract:  I will present a new method to engineer integrable models in 4d with higher genus spectral parameters. The method has a twistorial origin - by working on a branched covering of twistor space, I show how one can derive deformations of holomorphic BF theory on twistor space which descend to elliptic and hyperelliptic models on R^4 via the Penrose transform. I show how one can bootstrap the Penrose transformed actions using symmetry and integrability to find deformations of self-dual Yang-Mills theory. I will also discuss some novel deformations of a BF type description of Hitchin’s equations. This is based on my recent paper: 2509.12486.

Time: 4:30-5:30pm

Location: KT801

Abstract:  A volume conjecture relates a certain asymptotical growth of a given quantum topological invariant of a hyperbolic 3-manifold to the hyperbolic volume of this manifold.  

In this talk I will mention several of these volume conjectures, their common points and differences, notably those associated to the Baseilhac-Benedetti invariants and to the Andersen-Kashaev TQFT.  

A general strategy to prove a volume conjecture is to use the combinatorial properties of a given triangulation of the manifold to simplify the expression of the quantum invariant, and hopefully to successfully apply the saddle point method in the desired asymptotics.  

I will use the figure-eight knot complement as a recurring example, as it is the simplest member of two infinite families, the hyperbolic twist knots and the once-punctured torus bundles over the circle.  No prerequisite in quantum topology or hyperbolic geometry will be needed.  

(This talk will cover joint works with François Guéritaud, Stéphane Baseilhac and Ka Ho Wong)