# Topological Quantum Field Theory Seminar

## Past sessions

### Multiple zeta values in deformation quantization

In 1997, Kontsevich gave a universal solution to the deformation quantization problem in mathematical physics: starting from any Poisson manifold (the classical phase space), it produces a noncommutative algebra of quantum observables by deforming the ordinary multiplication of functions. His formula is a Feynman expansion whose Feynman integrals give periods of the moduli space of marked holomorphic disks. I will describe joint work with Peter Banks and Erik Panzer, in which we prove that Kontsevich's integrals evaluate to integer-linear combinations of multiple zeta values, building on Francis Brown's theory of polylogarithms on the moduli space of genus zero curves.

Pym slides.pdf

### Categorification of Verma Modules in low-dimensional topology

In this talk I will review the program of categorification of Verma modules and explain their applications to low-dimensional topology, namely to the construction of Khovanov invariants for links in the solid torus via a categorification of the blob algebra.

The material presented spreads along several collaborations with Abel Lacabanne, and Grégoire Naisse.

pedro-vaz-slides.pdf

### Klein TQFT and real Gromov-Witten invariants

In this talk I will explain how the Real Gromov-Witten theory of local 3-folds with base a Real curve gives rise to an extension of a 2d Klein TQFT. The latter theory is furthermore semisimple which allows for complete computation from the knowledge of a few basic elements which can be computed explicitly. As a consequence of the explicit expressions we find in the Calabi-Yau case, we obtain the expected Gopukumar-Vafa formula and relation to SO/Sp Chern-Simons theory.

Penka-Georgieva-slides

### Cyclotomic expansions of the $gl_N$ knot invariants

Newton’s interpolation is a method to reconstruct a function from its values at different points. In the talk I will explain how one can use this method to construct an explicit basis for the center of quantum $gl_N$ and to show that the universal $gl_N$ knot invariant expands in this basis. This will lead us to an explicit construction of the so-called unified invariants for integral homology 3-spheres, that dominate all Witten-Reshetikhin-Turaev invariants. This is a joint work with Eugene Gorsky, that generalizes famous results of Habiro for $sl_2$.

Anna-Beliakova-slides.pdf

### Two dimensional topological field theories and partial fractions

This talk is based on joint work with M. Khovanov and Y. Kononov. By evaluating a topological field theory in dimension $2$ on surfaces of genus $0,1,2$, etc., we get a sequence. We investigate which sequences occur in this way depending on the assumptions on the target category.

victor-ostrik-slides.pdf

Projecto FCT UIDB/04459/2020.

### $3d$ A and B models and link homology

I will discuss some current work (with Garner, Hilburn, Oblomkov, and Rozansky) on new and old constructions of HOMFLY-PT link homology in physics and mathematics, and new connections among them. In particular, we relate the classic proposal of Gukov-Schwarz-Vafa, involving M-theory on a resolved conifold, to constructions in $3d$ TQFT's. In the talk, I will focus mainly on the $3d$ part of the story. I'll review general aspects of $3d$ TQFT's, in particular the "$3d$ A and B models" that play a role here, and how link homology appears in them.

Tudor-Dimofte-slides.pdf

Projecto FCT UIDB/04459/2020.

### Ellipsoidal billiards, extremal polynomials, and partitions

A comprehensive study of periodic trajectories of the billiards within ellipsoids in the $d$-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between the periodic billiard trajectories and the extremal polynomials of the Chebyshev type on the systems of $d$ intervals on the real line. Classification of periodic trajectories is based on a new combinatorial object: billiard partitions.

The case study of trajectories of small periods $T$, $d \leq T \leq 2d$, is given. In particular, it is proven that all $d$-periodic trajectories are contained in a coordinate-hyperplane and that for a given ellipsoid, there is a unique set of caustics which generates $d + 1$-periodic trajectories. A complete catalog of billiard trajectories with small periods is provided for $d = 3$.

The talk is based on the following papers:

• V. Dragović, M. Radnović, Periodic ellipsoidal billiard trajectories and extremal polynomials, Communications Mathematical Physics, 2019, Vol. 372, p. 183-211.
• G. Andrews, V. Dragović, M. Radnović, Combinatorics of the periodic billiards within quadrics, The Ramanujan Journal, DOI: 10.1007/s11139-020-00346-y.

Dragovic-slides.pdf

### The double-centralizer theorem in 2-representation theory and its applications

Finitary birepresentation theory of finitary bicategories is a categorical analog of finite-dimensional representation theory of finite-dimensional algebras. The role of the simples is played by the so-called simple transitive birepresentations and the classification of the latter, for any given finitary bicategory, is a fundamental problem in finitary birepresentation theory (the classification problem).

After briefly reviewing the basics of birepresentation theory, I will explain an analog of the double centralizer theorem for finitary bicategories, which was inspired by Etingof and Ostrik's double centralizer theorem for tensor categories. As an application, I will show how it can be used to (almost completely) solve the classification problem for Soergel bimodules in any finite Coxeter type.

marco-mackaay-slides.pdf

Projecto FCT UIDB/04459/2020.

### Crossed modules, homotopy 2-types, knotted surfaces and welded knots

I will review the construction of invariants of knots, loop braids and knotted surfaces derived from finite crossed modules. I will also show a method to calculate the algebraic homotopy 2-type of the complement of a knotted surface $\Sigma$ embedded in the 4-sphere from a movie presentation of $\Sigma$. This will entail a categorified form of the Wirtinger relations for a knot group. Along the way I will also show applications to welded knots in terms of a biquandle related to the homotopy 2-type of the complement of the tube of a welded knots.

The last stages of this talk are part of the framework of the Leverhulme Trust research project grant: RPG-2018-029: Emergent Physics From Lattice Models of Higher Gauge Theory.

João Faria Martins slides

Projecto FCT UIDB/04459/2020.

### Quantum modular forms and $3$-manifolds

Quantum modular forms are functions on rational numbers that have rather mysterious weak modular properties. Mock modular forms and false theta functions are examples of holomorphic functions on the upper-half plane which lead to quantum modular forms. Inspired by the $3d-3d$ correspondence in string theory, new topological invariants named homological blocks for (in particular plumbed) three-manifolds have been proposed a few years ago. My talk aims to explain the recent observations on the quantum modular properties of the homological blocks, as well as the relation to logarithmic vertex algebras.

The talk will be based on a series of work in collaboration with Sungbong Chun, Boris Feigin, Francesca Ferrari, Sergei Gukov, Sarah Harrison, and Gabriele Sgroi.

Cheng slides.pdf

Projecto FCT UIDB/04459/2020.

### Cluster realization of quantum groups and higher Teichmüller theory

Quantum higher Teichmüller theory, as described by Fock and Goncharov, endows a quantum character variety on a surface $S$ with a cluster structure. The latter allows one to construct a canonical representation of the character variety, which happens to be equivariant with respect to an action of the mapping class group of $S$. It was conjectured by the authors that these representations behave well with respect to cutting and gluing of surfaces, which in turn yields an analogue of a modular functor. In this talk, I will show how the quantum group and its positive representations arise in this context. I will also explain how the modular functor conjecture is related to the closedness of positive representations under tensor products as well as to the non-compact analogue of the Peter-Weyl theorem. If time permits, I will say a few words about the proof of the conjecture.

This talk is based on joint works with Gus Schrader.

Shapiro talk notes.pdf

Projecto FCT UIDB/04459/2020.

### Counting Monster Potentials

The monster potentials were introduced by Bazhanov-Lukyanov-Zamolodchikov in the framework of the ODE/IM correspondence. They should in fact be in 1:1 correspondence with excited states of the Quantum KdV model (an Integrable Conformal Field Theory) since they are the most general potentials whose spectral determinant solves the Bethe Ansatz equations of such a theory. By studying the large momentum limit of the monster potentials, I retrieve that

1. The poles of the monster potentials asymptotically condensate about the complex equilibria of the ground state potential.
2. The leading correction to such asymptotics is described by the roots of Wronskians of Hermite polynomials.

This allows me to associate to each partition of $N$ a unique monster potential with $N$ roots, of which I compute the spectrum. As a consequence, I prove up to a few mathematical technicalities that, fixed an integer $N$, the number of monster potentials with $N$ roots coincide with the number of integer partitions of $N$, which is the dimension of the level $N$ subspace of the quantum KdV model. In striking accordance with the ODE/IM correspondence.

The talk is based on the preprint https://arxiv.org/abs/2009.14638, written in collaboration with Riccardo Conti (Group of Mathematical Physics of Lisbon University).

Slides of the talk

Projecto FCT UIDB/04459/2020.

### Reps of relative mapping class groups via conformal nets

Given a surface with boundary $\Sigma$, its relative mapping class group is the quotient of $\operatorname{Diff}(\Sigma)$ by the subgroup of maps which are isotopic to the identity via an isotopy that fixes the boundary pointwise. (If $\Sigma$ has no boundary, then that's the usual mapping class group; if $\Sigma$ is a disc, then that's the group $\operatorname{Diff}(S^1)$ of diffeomorphisms of $S^1$.)

Conformal nets are one of the existing axiomatizations of chiral conformal field theory (vertex operator algebras being another one). We will show that, given an arbitrary conformal net and a surface with boundary $\Sigma$, we get a continuous projective unitary representation of the relative mapping class group (orientation reversing elements act by anti-unitaries). When the conformal net is rational and $\Sigma$ is a closed surface (i.e. $\partial \Sigma = \emptyset$), then these representations are finite dimensional and well known. When the conformal net is not rational, then we must require $\partial \Sigma \neq \emptyset$ for these representations to be defined. We will try to explain what goes wrong when $\Sigma$ is a closed surface and the conformal net is not rational.

The material presented in this talk is partially based on my paper arXiv:1409.8672 with Arthur Bartels and Chris Douglas.

Projecto FCT UIDB/04459/2020.

### Homotopy Quantum Field Theories

Homotopy quantum field theories (HQFTs) generalize topological quantum field theories (TQFTs) by replacing manifolds by maps from manifolds to a fixed target space $X$. For example, any cohomology class in $H^3(X)$ defines a 3-dimensional HQFT with target $X$. If $X$ is aspherical, that is $X = K(G, 1)$ for some group $G$, then this cohomological HQFT is related to the Dijkgraaf-Witten invariant and is computed as a Turaev-Viro state sum via the category of $G$-graded vector spaces. More generally, the state sum Turaev-Viro TQFT and the surgery Reshetikhin-Turaev TQFT extend to HQFTs (using graded fusion categories) which are related via the graded categorical center.

This is joint work with V. Turaev.

Slides of the talk

Projecto FCT UIDB/04459/2020.

### Gluing local gauge conditions in BV quantum field theory

In supersymmetric sigma models, there is frequently no global choice of Lagrangian submanifold for BV quantization. I will take the superparticle, a toy version of the Green Schwarz superstring, as my example, and show how to extend the light-cone gauge to the physically relevant part of phase space. This involves extending a formula of Mikhalkov and A. Schwarz that generalizes the prescription of Batalin and Vilkovisky for the construction of the functional integral.

This is joint work with S. Pohorence.

Slides of the talk

Projecto FCT UIDB/04459/2020.

### Galois symmetries of knot spaces

I’ll describe how the absolute Galois group of the rationals acts on a space which is closely related to the space of all knots. The path components of this space form a finitely generated abelian group which is, conjecturally, a universal receptacle for integral finite-type knot invariants. The added Galois symmetry allows us to extract new information about its homotopy and homology beyond characteristic zero. I will then discuss some work in progress concerning higher-dimensional variants.

This is joint work with Geoffroy Horel.

Slides of the talk

Projecto FCT UIDB/04459/2020.

### The homotopy type of associative and commutative algebras

Given a topological space, how much of its homotopy type is captured by its algebra of singular cochains? The experienced rational homotopy theorist will argue that one should consider instead a commutative algebra of forms. This raises the more algebraic question

Given a dg commutative algebra, how much of its homotopy type (quasi-isomorphism type) is contained in its associative part?

Despite its elementary formulation, this question turns out to be surprisingly subtle and has important consequences.

In this talk, I will show how one can use operadic deformation theory to give an affirmative answer in characteristic zero.

We will also see how the Koszul duality between Lie algebras and commutative algebras allows us to use similar arguments to deduce that under good conditions Lie algebras are determined by the (associative algebra structure of) their universal enveloping algebras.

Joint with Dan Petersen, Daniel Robert-Nicoud and Felix Wierstra and based on arXiv:1904.03585.

Projecto FCT UIDB/04459/2020.

### Mirror symmetry for Painlevé surfaces

This talk will survey aspects of mirror symmetry for ten families of non-compact hyperkähler manifolds on which the dynamics of one of the Painlevé equations is naturally defined. They each have a pair of natural realisations: one as the complement of a singular fibre of a rational elliptic surface and another as the complement of a triangle of lines in a (singular) cubic surface. The two realisations relate closely to a space of stability conditions and a cluster variety of a quiver respectively, providing a perspective on SYZ mirror symmetry for these manifolds. I will discuss joint work in progress with Helge Ruddat studying the canonical basis of theta functions on these cubic surfaces.

Slides of the talk

Projecto FCT UIDB/04459/2020.

### Rational and algebraic links and knots-quivers correspondence

I will start with a brief overview of knots-quivers correspondence, where colored HOMFLY-PT (or BPS) invariants of the knot are expressed as motivic Donaldson-Thomas invariants of a corresponding quiver.

In this talk I will focus on showing that the knots-quivers correspondence holds for rational links, as well as much larger class of arborescent links (algebraic links in the sense of Conway). This is done by extending the correspondence to tangles, and showing that the set of tangles satisfying tangles-quivers correspondence is closed under the tangle addition operation.

This talk is based on joint work with Paul Wedrich.

Slides of the talk

Projecto FCT UIDB/04459/2020.

### Introduction to foam evaluation

Foam evaluation was discovered by Louis-Hardrien Robert and Emmanuel Wagner slightly over three years ago. It's a remarkable formula assigning a symmetric function to a foam, that is, to a decorated 2-dimensional CW-complex embedded in three-space. We'll explain their formula in the 3-color case in the context of unoriented foams and discuss its relation to Kronheimer-Mrowka homology of graphs and the four-color theorem.

Slides of the talk

Projecto FCT UIDB/04459/2020.

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