# Topological Quantum Field Theory Seminar

## Past sessions

### A quantization of the Loday-Ronco Hopf algebra

In previous works we considered a model for topological recursion based on the Hopf Algebra of planar binary trees of Loday and Ronco and showed that extending this Hopf Algebra by identifying pairs of nearest neighbour leaves and thus producing graphs with loops we obtain the full recursion formula of Eynard and Orantin. We also discussed the algebraic structure of the spaces of correlation functions in $g=0$ and in $g\gt 0$. By taking a classical and a quantum product respectively we endowed both spaces with a ring structure. Here we will show that the extended algebra of graphs is in fact a Hopf algebra and can be seen as a sort of quantization of the Loday-Ronco Hopf algebra. This is work in progress.

### Research topics in higher gauge theory, knot theory, and anyons

Under the heading higher gauge theory I will briefy discuss:

• gerbes on supergroups (to be described by John Huerta in his talk).
• the relation between 2-group (discretized) connections and transports (with Jeff Morton).
• surface transport in 3D quantum gravity (with Jeanette Nelson).

I will then focus on two projects:

• development of the Eisermann invariant of knots (based on previous work with João Faria Martins).
• (higher) gauge theoretical models for anyons in topological quantum computation.

### Topological Field Theory in dimension $3$

I will give an overview of some aspects of $3d$ TFT, from the Turaev-Viro and Reshetikin-Turaev invariants of oriented $3$-manifolds, to the more recent classifications of fully extended theories in terms of fusion categories and once extended theories in terms of Modular Tensor Categories.

### Quantum theory via (higher) groupoids and quantum measures

The idea of this informal seminar is to present, in cherry-picking fashion, some fascinating work by Ciaglia, Ibort and Marmo, who formulate the Schwinger approach to quantum theory using the contemporary language of groupoids and $2$-groupoids. By bringing in the notion of quantum measures, due to Sorkin, they achieve an elegant description of examples like the qubit or the two-slit experiment.

The aim is for the seminar to be comprehensible also to students with some awareness of quantum theory, and hopefully will be followed up by future seminars around quantum maths topics such as topological quantum computation and entanglement.

### Main references

Ciaglia, Ibort, Marmo: https://arxiv.org/abs/1905.12274, https://arxiv.org/abs/1907.03883, https://arxiv.org/abs/1909.07265

### Eigenvalues of random matrices in the general linear group

I will begin by discussing the two standard prototype random matrix models, one for Hermitian matrices and one for general matrices. For large matrices, the eigenvalues follow the "semicircular law" in the first case and the "circular law" in the second case. Furthermore, there is a simple relationship between these two laws.

I will then discuss two "multiplicative" analogs of these models, in which the random matrices are chosen from the unitary group and the general linear group, respectively. In the unitary case, the limiting eigenvalue distribution was computed by Biane and exhibits an interesting phase transition when a certain scaling parameter equals 4. I will then describe recent results of mine with Driver and Kemp on the general linear case. The limiting distribution again undergoes a phase transition and turns out to have a remarkably simple structure. The talk will be self-contained with lots of pictures and possibly even a few jokes.

### Higher Steenrod squares for Khovanov homology

We describe stable cup-$i$ products on the cochain complex with $\mathbb{F}_2$ coefficients of any augmented semi-simplicial object in the Burnside category. An example of such an object is the Khovanov functor of Lawson, Lipshitz and Sarkar. Thus we obtain explicit formulas for cohomology operations on the Khovanov homology of any link.

Please note the unusual day for the seminar.

### Topological Links and Quantum Entanglement

We present a classification scheme for quantum entanglement based on topological links. This is done by identifying a nonrigid ring to a particle, attributing the act of cutting and removing a ring to the operation of tracing out the particle, and associating linked rings to entangled particles. This analogy naturally leads us to a classification of multipartite quantum entanglement based on all possible distinct links for any given number of rings. We demonstrate the use of this new classification scheme for three and four qubits and its potential in the context of qubit networks.

### 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.

### 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.

### 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)

### 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].

### 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.

### 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.

### 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.

### 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.

### 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.

### 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).

### 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.

Older session pages: Previous 3 4 5 6 7 8 9 10 11 12 13 14 Oldest

Current organizers: José MourãoRoger Picken, Marko Stošić

Mathseminars

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