# Topological Quantum Field Theory Seminar

## Past sessions

### 10/01/2019, 15:00 — 16:00 — Room P4.35, Mathematics Building

Marco Mackaay, *Universidade do Algarve*

### The 2-representation theory of Soergel bimodules of finite Coxeter type: a road map to the complete classification of all simple transitive 2-representations

I will first recall Lusztig's asymptotic Hecke algebra and its categorification, a fusion category obtained from the perverse homology of Soergel bimodules. For example, for finite dihedral Coxeter type this fusion category is a 2-colored version of the semisimplified quotient of the module category of quantum $\operatorname{sl}(2)$ at a root of unity, which Reshetikhin-Turaev and Turaev-Viro used for the construction of 3-dimensional Topological Quantum Field Theories.

In the second part of my talk, I will recall the basics of 2-representation theory and indicate how the fusion categories above can conjecturally be used to study the 2-representation theory of Soergel bimodules of finite Coxeter type.

This is joint work with Mazorchuk, Miemietz, Tubbenhauer and Zhang.

Please note the unusual time (Thursday 3 p.m.) and that the room has changed from 3.10 to 4.35.

### 05/12/2018, 14:00 — 15:00 — Room P3.10, Mathematics Building

Miguel Tierz, *Grupo de Fisica Matematica, Universidade de Lisboa*

### Random matrix theory in topological gauge theories

We present an overview of analytical tools in random matrix theory and related areas, involving Toeplitz/ Hankel determinants and symmetric functions, with an emphasis on their relevance in the study of topological gauge theory and focussing on some specific Chern-Simons theories and 2d Yang-Mills theories. We will also explain how these methods and results are intertwined with localization results in supersymmetric gauge theories.

### 21/11/2018, 14:00 — 15:00 — Room P3.10, Mathematics Building

Björn Gohla, *CAMGSD, Universidade de Lisboa*

### Internal Strictification

It is well known that ordinary bicategories can always be replaced by bi-equivalent strict 2-categories. Special cases of this are the strictification of monoidal categories and categorical groups. We give an abstract strictification construction for pseudo-monoids in a monoidal 2-category. It is easy to see that bicategories internal to an appropriate category are such pseudo-monoids, and can hence be strictified. (Joint work with Nelson Martins-Ferreira)

### 31/10/2018, 11:30 — 12:30 — Room P3.10, Mathematics Building

Leonardo Santilli, *Grupo de Fisica Matematica, Univ. Lisboa*

### A Chern-Simons view on noncommutative scalar field theory

We present a recent result establishing a bridge between noncommutative scalar field theory in $2$ dimensions and topological field theory in $3$ dimensions.

The content of the seminar is split in two main parts, according to the twofold aspect of the result. In the first half, we show that a version of Abelian gauge theory on $\mathbb{R}^3 _{\lambda}$, when restricted to a single fuzzy sphere, reduces in the large $N$ limit to the Langmann-Szabo-Zarembo (LSZ) matrix model, which originally emerges in the study of scalar field theory on the Moyal plane. Then, throughout the second part, we prove that the LSZ matrix model is actually equivalent to the matrix model of $U(N)$ Chern-Simons theory on the three-sphere. The correspondence holds in a generalized sense: depending on the spectra of the two external matrices of the LSZ model, the Chern-Simons matrix model either describes the Chern-Simons partition function, the unknot invariant, given by quantum dimensions, or the Hopf link invariant. Equivalently, the partition function of the LSZ model can be written in terms of the $S$ and $T$ modular matrices of the WZW model.

Based on: arXiv:1805.10543 [hep-th].

### 17/10/2018, 11:30 — 12:30 — Room P3.10, Mathematics Building

Björn Gohla, *GFM Univ. Lisboa*

### Monoids, Monads and Simplicial Objects

We will present some classical facts about the relationship between monoids and monads. We will use ordinal sums of categories and the join product of topological spaces to define the abstract and topological simplices. Along the way we show how the simplicial identities can be obtained. Time permitting we will indicate a 2-categorical generalization of this circle of ideas.

### 04/07/2018, 11:30 — 12:30 — Room P3.10, Mathematics Building

Björn Gohla, *GFM Univ. Lisboa*

### A Categorical Model for the Hopf Fibration

We give a description up to homeomorphism of $S^3$ and $S^2$ as classifying spaces of small categories, such that the Hopf map $S^3\longrightarrow{}S^2$ is the realization of a functor.

### 20/06/2018, 11:30 — 12:30 — Room P3.10, Mathematics Building

Ricardo Schiappa, *Instituto Superior Técnico*

### Co-equational (i.e. Parametric) Resurgence and Topological Strings

I will briefly review the uses and applications of resurgence applied to topological string theory, with emphasis on nonperturbative completions and the large-order behaviour of enumerative invariants. Due to the nature of the holomorphic anomaly equations, there is a clear need to develop methods of co-equational (i.e. parametric) resurgence in order to achieve a complete description of the topological string transseries.

### 06/06/2018, 11:30 — 12:30 — Room P3.10, Mathematics Building

Marco Mackaay, *Universidade do Algarve*

### 2-representation theory

I will give an overview of 2-representation theory, following Mazorchuk and Miemietz' approach. After explaining the general setup, I will sketch the 2-representation theory of dihedral Soergel bimodules as an example.

After the seminar, for those interested we will continue with a discussion of approaches to 2-representation theory.

### 23/05/2018, 11:30 — 12:30 — Room P3.10, Mathematics Building

Ana Bela Cruzeiro, *Department of Mathematics, Instituto Superior Técnico*

### Stochastic Clebsch variational principles

We derive the equations of motion associated with stochastic Clebsch action principles for mechanical systems whose configuration space is a manifold on which a Lie algebra acts transitively. These are stochastic differential equations (spde's in infinite dimensions).

We give the Hamiltonian version of the equations, as well as the corresponding Kolmogorov equations.

This is a joint work with D. D. Holm and T. S. Ratiu.

### 02/05/2018, 11:30 — 12:30 — Room P3.10, Mathematics Building

Marko Stošić, *Instituto Superior Técnico*

### Knots-quivers correspondence and applications

In this talk I shall present the knots-quivers correspondence, as well as some surprising implications in combinatorics involving counting of lattice paths and number theory. The knots-quivers correspondence relates the colored HOMFLY-PT invariants of a knot with the motivic Donaldson-Thomas invariants of the corresponding quiver. This correspondence is made completely explicit at the level of generating series. The motivation for this relationship comes from topological string theory, BPS (LMOV) invariants, as well as categorification of HOMFLY-PT polynomial and A-polynomials. We compute quivers for various classes of knots, including twist knots, rational knots and torus knots.

One of the surprising outcomes of this correspondence is that from the information of the colored HOMFLY-PT polynomials of certain knots we get new expressions for the classical combinatorial problem of counting lattice paths, as well as new integrality/divisibility properties.

The main goal of this talk is to present basic ideas and to present numerous open questions and ramifications coming from knots-quivers correspondence.

(based on joint works with P. Sulkowski, M. Reineke, P. Kucharski, M. Panfil and P. Wedrich).

### 18/12/2017, 14:00 — 15:00 — Room P3.10, Mathematics Building

Ana Ros Camacho, *Univ. Utrecht, Netherlands*

### Strangely dual orbifold equivalence for unimodal and bimodal singularities and Galois groups

In this talk I will introduce orbifold equivalence, an equivalence relation between polynomials satisfying certain conditions, which describe Landau-Ginzburg models (“potentials”). We will review how it relates the potentials associated to simple, (exceptional) unimodal and bimodal singularities, reproducing classical results like strange duality from the classification of singularities from Arnold. In addition, most of these equivalences are controlled by Galois groups. This is joint work with N. Carqueville, I. Runkel, R. Newton et al.

### 06/12/2017, 14:00 — 15:00 — Room P3.10, Mathematics Building

Nora Ganter, *University of Melbourne*

### Examples of categorical groups

Categorical groups formalize situations, where the symmetries of an object are themselves related by symmetries. Their relevance to string theory has been known for a while, and their theory is developing rapidly. We will attempt to fill the theory with life by discussing examples, ranging from the symmetries of the platonic solids with some extra structure to the Schur extensions of the alternating groups, which turn out to be closely linked to the stable three stem, to categorical extensions of sporadic groups and of Lie groups.

### 22/11/2017, 14:00 — 15:00 — Room P3.10, Mathematics Building

João Miguel Nogueira, *Universidade de Coimbra*

### Meridional essential surfaces of unbounded Euler characteristics in knot exteriors

In this talk we will discuss further the existence of knot exteriors with essential surfaces of unbounded Euler characteristics. More precisely, we show the existence of a knot with an essential tangle decomposition for any number of strings. We also show the existence of knots where each exterior contains meridional essential surfaces of simultaneously unbounded genus and number of boundary components. In particular, we construct examples of knot exteriors each of which having all possible compact surfaces embedded as meridional essential surfaces.

### 10/11/2017, 14:00 — 15:00 — Room P3.10, Mathematics Building

Carlos Florentino, *Universidade de Lisboa*

### Geometry, Topology and Arithmetic of character varieties

Character varieties are spaces of representations of finitely presented groups $F$ into Lie groups $G$. When $F$ is the fundamental group of a surface, these spaces play a key role both in Chern-Simons theory and in 2d conformal field theory. In some cases, they are also interpreted as moduli spaces of $G$-Higgs bundles over Kähler manifolds, and were recently studied in connection with the geometric Langlands program, and with mirror symmetry. When $G$ is a complex algebraic group, character varieties are algebraic and have interesting geometry and topology. We can also consider more refined invariants such as Deligne's mixed Hodge structures, which are typically very difficult to compute, but also provide relevant arithmetic information.

In this seminar, we present some explicit computations of the mixed Hodge-Deligne polynomials, and the so-called E-polynomials, of $G$-character varieties of free, and free abelian groups, when $G$ is a group such as $\operatorname{SL}(n,\mathbb{C})$, $(P)\operatorname{GL}(n,\mathbb{C})$ or $\operatorname{Sp}(n,\mathbb{C})$. We also comment on interesting relations between the free case and some explicit formulas by Reineke-Mozgovoy on counting quiver representations over finite fields.

This is joint work with A. Nozad, J. Silva and A. Zamora.

Please note unusual day (Friday).

### 30/10/2017, 14:00 — 15:00 — Room P3.10, Mathematics Building

Alissa Crans, *Loyola Marymount University, USA*

### Crossed Modules of Racks

A rack is a set equipped with two binary operations satisfying axioms that capture the essential properties of group conjugation and algebraically encode two of the three Reidemeister moves. We will begin by generalizing Whitehead's notion of a crossed module of groups to that of a crossed module of racks. Motivated by the relationship between crossed modules of groups and strict 2-groups, we then will investigate connections between our rack crossed modules and categorified structures including strict 2-racks and trunk-like objects in the category of racks. We will conclude by considering topological applications, such as fundamental racks. This is joint work with Friedrich Wagemann.

Please note unusual day (Monday).

### 11/10/2017, 14:00 — 15:00 — Room P3.10, Mathematics Building

Luis Miguel Pereira, *IPFN, Instituto Superior Tecnico*

### Techniques for the summation of hypergeometric series and the quantum pendulum

In this informal seminar we will give a presentation based on practical examples of some of the several methods that can be used to sum hypergeometric series. These series include several known special functions and almost all combinatorial sums. Questions from the public will be welcomed. The goal of the seminar will ultimately be to set the stage for the preparation of a strategy to attack the problem of finding closed form solutions for the problem of the quantum pendulum (that is, to find closed form solutions for the Fourier coefficients of Mathieu functions), from stationary solutions to this problem in the Wigner formalism that were obtained by the speaker.

### 14/09/2017, 15:00 — 16:00 — Room P3.10, Mathematics Building

Luis Miguel Pereira, *IPFN, Instituto Superior Tecnico*

### The quantum pendulum in the Wigner formalism and Mathieu functions

The time-independent Schrödinger equation with the pendulum's potential is the Mathieu equation from 19th century mathematical physics. Though there are many ways to approximate its solutions there are no known closed formulas for these solutions. In this talk we will show that with João Pedro Bizarro's modification of the Wigner-Berry transform it is possible to obtain closed formulas for several families of transforms of stationary observables.

### 06/09/2017, 16:00 — 17:30 — Room P4.35, Mathematics Building

Beatriz Elizaga de Navascués, *Instituto de Estructura de la Materia, Madrid*

### Unitary dynamics as a uniqueness criterion for the quantization of Dirac fields

It is well known that linear canonical transformations are not generally implemented as unitary operators in QFT. Such transformations include the dynamics that arises from linear field equations on the background spacetime. This evolution is specially relevant in nonstationary backgrounds, where there is no time-translational symmetry that can be exploited to select a quantum theory. We investigate whether it is possible to find a Fock representation for the canonical anticommutation relations of a Dirac field, propagating on homogeneous and isotropic cosmological backgrounds, on the one hand, and on tridimensional conformally ultrastatic spacetimes, on the other hand, such that the field evolution is unitarily implementable. First, we restrict our attention to Fock representations that are invariant under the group of symmetries of the system. Then, we prove that there indeed exist Fock representations such that the dynamics is implementable as a unitary operator. Finally, once a convention for the notion of particles and antiparticles is set, we show that these representations are all unitarily equivalent.

### 21/06/2017, 14:15 — 15:15 — Room P3.10, Mathematics Building

Louis H. Kauffman, *University of Illinois, Chicago, USA*

### Majorana Fermions, Braiding and Quantum Computing

We will discuss the mathematics of Majorana Fermions and the structure of representations of the Artin Braid group that are associated with them. We will discuss how one class of representations is related to the Temperley Lieb algebra, and another class of representations is related to a Hamiltonian constructed from the Bell-Basis solution of the Yang-Baxter equation, and the relationship of this Hamiltonian with the Kitaev spin chain. We will discuss how these braiding representations are related to topological quantum computing.

#### See also

presentation (pdf)

### 07/06/2017, 14:15 — 15:15 — Room P3.10, Mathematics Building

Emmanuel Wagner, *Université de Bourgogne, France*

### Trivalent TQFT and applications

MOY calculus has been introduced in the 90s to compute combinatorially the quantum link invariant associated with the Hopf algebra $U_q(\mathfrak{sl}(N))$. It associates to any decorated graph a Laurent polynomial in $q$. I will describe a TQFT-like functor which categorifies the MOY calculus and provides a new description of the $\mathfrak{sl}(N)$-homology.

(joint work with L.-H. Robert)

Older session pages: Previous 2 3 4 5 6 7 8 9 10 11 12 Oldest

Current organizers: Roger Picken, Marko Stošić.

FCT Projects PTDC/MAT-GEO/3319/2014, *Quantization and Kähler Geometry*, PTDC/MAT-PUR/31089/2017, *Higher Structures and Applications.*