# Topological Quantum Field Theory Seminar

## Past sessions

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

presentation (pdf)

### 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)

### 24/05/2017, 14:15 — 15:15 — Room P3.10, Mathematics BuildingPaul Wedrich, Imperial College, London

Link homology theories are powerful generalizations of classical (and quantum) link polynomials, which are being studied from a variety of mathematical and physical viewpoints. Besides providing stronger invariants, these theories are often functorial under link cobordisms and carry additional topological information. The focus of this talk is on the Khovanov-Rozansky homologies, which categorify the Chern-Simons/Reshetikhin-Turaev $\mathfrak{sl}(N)$ link invariants and their large $N$ limits. I will survey recent results about their behaviour under deformations as well as their stability at large $N$, which together lead to a rigorous proof of a package of conjectures originating in string theory.

### A diagrammatic categorification of the higher level Heisenberg algebras

Khovanov defined a diagrammatic 2-category and conjectured (and partially proved) that it categorifies the level-one Heisenberg algebra. Since then, several interesting generalizations and applications have been found, e.g. Cautis and Licata's generalization involving Hilbert schemes and their construction of categorical vertex operators. However, these are all for level one. In my talk, I will explain Alistair Savage and my results on a generalization of Khovanov's original results for higher level Heisenberg algebras. This is work in progress.

### Topological Complexity of the Klein Bottle

The notion of topological complexity of a space has been introduced by M. Farber in order to give a topological measure of the complexity of the motion planning problem in robotics. Surprisingly, the determination of this invariant for non-orientable surfaces has turned out to be difficult. A. Dranishnikov has recently established that the topological complexity of the non-orientable surfaces of genus at least 4 is maximal. In this talk, we will determine the topological complexity of the Klein bottle and extend Dranishnikov's result to all the non-orientable surfaces of genus at least 2. This is a work in collaboration with Daniel C. Cohen.

### M-theory from the superpoint revisited

The last talk we gave on this topic (in the meeting Iberian Strings 2017) was largely about the physics; here we focus on the mathematics. No prior knowledge will be assumed.

We define the process of invariant central extension: taking central extensions by cocycles invariant under a given subgroup of automorphisms of a Lie superalgebra. We give conditions that allow us to carve out the Lorentz group inside the automorphisms of Minkowski superspacetime, and prove that by successive invariant central extensions of the superpoint, we construct all superspacetimes up to dimension 11.

### Operads of genus zero curves and the Grothendieck-Teichmuller group

In Esquisse d’un programme, Grothendieck made the fascinating suggestion that the absolute Galois group of the rationals could be understood via its action on certain geometric objects, the (profinite) mapping class groups of surfaces of all genera. The collection of these objects, and the natural relations between them, he called the "Teichmuller tower”.

In this talk, I plan to describe a genus zero analogue of this story from the point of view of operad theory. The result is that the group of automorphisms of the (profinite) genus zero Teichmuller tower agrees with the Grothendieck-Teichmuller group, an object which is closely related to the absolute Galois group of the rationals. This is joint work with Geoffroy Horel and Marcy Robertson.

### Hamiltonian analysis of the BFCG theory for a generic Lie 2-group

We perform a complete Hamiltonian analysis of the BFCG action for a general Lie 2-group by using the Dirac procedure. We show that the resulting dynamical constraints eliminate all local degrees of freedom which implies that the BFCG theory is a topological field theory.

Room 3.10 now confirmed.

### EPRL/FK Asymptotics and the Flatness Problem: a concrete example

Spin foam models are a "state-sum" approach to loop quantum gravity which aims to facilitate the description of its dynamics, an open problem of the parent framework. Since these models' relation to classical Einstein gravity is not explicit, it becomes necessary to study their asymptotics — the classical theory should be obtained in a limit where quantum effects are negligible, taken to be the limit of large triangle areas in a triangulated manifold with boundary.

In this talk we will briefly introduce the EPRL/FK spin foam model and known results about its asymptotics, proceeding then to describe a practical computation of spin foam and asymptotic geometric data for a simple triangulation, with only one interior triangle. The results are used to comment on the "flatness problem" — a hypothesis raised by Bonzom (2009) suggesting that EPRL/FK's classical limit only describes flat geometries in vacuum.

### Duality in String/M-theory from Cyclic cohomology of Super Lie $n$-algebras

I discuss how, at the level of rational homotopy theory, all the pertinent dualities in string theory (M/IIA/IIB/F) are mathematically witnessed and systematically derivable from the cyclic cohomology of super Lie $n$-algebras. I close by commenting on how this may help with solving the open problem of identifying the correct generalized cohomology theory for M-flux fields, lifting the classification of the RR-fields in twisted K-theory.

This is based on joint work with D. Fiorenza and H. Sati arxiv:1611.06536

https://ncatlab.org/schreiber/show/Super+Lie+n-algebra+of+Super+p-branes

### 2+1 TQFTs with Defects

We will give an overview of TQFTs with defects along the lines of the preprint 3-dimensional defect TQFTs and their tricategories by Carqueville, Meusburger and Schaumann. In this paper ordinary Atiyah type functorial TQFTs on 2+1 cobordisms are generalized to 2+1 stratified cobordisms with decorations in a certain graph structure. The decoration graph structure together with a given TQFT of this type then give rise to a linear Gray-category with duals. This provides a unifying framework for well known 2+1 TQFTs.

### Invariants and TQFTs for cut cellular surfaces from finite groups and $2$-groups

We introduce the notion of a cut cellular surface (CCS), being a surface with boundary, which is cut in a specified way to be represented in the plane, and is composed of $0$-, $1$- and $2$-cells. We obtain invariants of CCS's under Pachner-like moves on the cellular structure, by counting colourings of the $1$-cells with elements of a finite group $G$, subject to a “flatness” condition for each $2$-cell. These invariants are also described in a TQFT setting, which is not the same as the usual $2$-dimensional TQFT framework. We study the properties of functions which arise in this context,associated to the disk, the cylinder and the pants surface, and derive general properties of these functions from topology. One such property states that the number of conjugacy classes of $G$ equals the commuting fraction of $G$ times the order of $G$.

We will comment on the extension of these invariants to 2-groups and their (higher) gauge theory interpretation.

This is work done in collaboration with Diogo Bragança (Dept. Physics, IST).