# Topological Quantum Field Theory Seminar

## Past sessions

### Categorifying the Knizhnik-Zamolodchikov connection

(Joint with Lucio Simone Cirio, Max Planck Institute for Mathematics)
In the context of higher gauge theory, we categorify the Knizhnik-Zamolodchikov connection in the configuration space of $n$ particles in the complex plane by means of a differential crossed module of (totally symmetric) horizontal 2-chord diagrams, categorifying the 4-term relation.

We carefully discuss the representation theory of differential crossed modules in chain-complexes of vector spaces, inside which we formulate the notion of infinitesimal 2-R matrix, an infinitesimal counterpart of some of the relations satisfied by braid cobordisms.

We present several open problems.
FCT, CAMGSD, New Geometry and Topology.

### Seifert graphs, Eulerian subgraphs, and partial duality

I will describe how to characterize the Seifert graphs of link diagrams, and how, using the notion of partial duality in ribbon graphs, this leads to a generalization of the classical result that a plane graph is Eulerian if and only if its geometric dual is bipartite. This is joint work with Iain Moffatt and Natalia Virdee.
FCT, CAMGSD, New Geometry and Topology.

### The Schur quotient of the Khovanov-Lauda categorification of quantum ${\mathrm{sl}}_{n}$ and colored HOMFLY homology

In my talk I will first remind everyone of the relation between quantum $\mathrm{sln}$, the $q$-Schur algebra and colored HOMFLY homology. After that I will explain how these objects and the relation between them have been categorified.
Support: FCT, CAMGSD, New Geometry and Topology

### Hecke algebras and a categorification of the Heisenberg algebra

In this talk, we will present a graphical category in terms of certain planar braid-like diagrams. The definition of this category is inspired by the representation theory of Hecke algebras of type A (which are certain deformations of the group algebra of the symmetric group). The Heisenberg algebra (in infinitely many generators), which plays an important role in the description of certain quantum mechanical systems, injects into the Grothendieck group of our category, yielding a "categorification" of this algebra. We will also see that our graphical category acts on the category of modules of Hecke algebras and of general linear groups over finite fields. Additionally, other algebraic structures, such as the affine Hecke algebra, appear naturally.

We will assume no prior knowledge of Hecke algebras or the Heisenberg algebra. The talk should be accessible to graduate students. This is joint work with Anthony Licata and inspired by work of Mikhail Khovanov.
Support: FCT, CAMGSD, New Geometry and Topology.

### A short introduction to string theory II

This will be a very introductory (and informal) mini-course on string theory. No prior exposure to string theory will be expected. Although I will use physicists language, the course is mainly addressed at math-students / postdocs who are welcome to help me translating expressions into math-language during the lectures. I will try to make at least some connection to higher gauge fields and also say a few words on how non-commutative geometry arises from open strings. In the first session I will concentrate on the bosonic string, while in the second I intend to discuss the superstring.
Room QA1.2 Torre Sul

### Quantum gravity and spin foams II

In the first lecture I will explain what is the problem of quantum gravity from a physics and a mathematics perspective. In the second lecture I will concentrate on the spin foam approach and explain its basic features.
Room QA1.2 Torre Sul

### A short introduction to string theory I

This will be a very introductory (and informal) mini-course on string theory. No prior exposure to string theory will be expected. Although I will use physicists language, the course is mainly addressed at math-students / postdocs who are welcome to help me translating expressions into math-language during the lectures. I will try to make at least some connection to higher gauge fields and also say a few words on how non-commutative geometry arises from open strings. In the first session I will concentrate on the bosonic string, while in the second I intend to discuss the superstring.
Room QA1.4 Torre Sul

### Quantum gravity and spin foams I

In the first lecture I will explain what is the problem of quantum gravity from a physics and a mathematics perspective. In the second lecture I will concentrate on the spin foam approach and explain its basic features.
Room QA1.4 Torre Sul

### Toward synthetic noncommutative geometry

In the first part of the talk we outline the basic ideas of synthetic approach to differential geometry. The main idea of this approach, which originates from considerations of Sophus Lie is very simple: All geometric constructions are performed within a suitable base category in which space forms are objects. In the second part we indicate how a synthetic method could be employed in the context of Noncommutative Differential Geometry.
This talk is addressed to mathematicians who have some very basic familiarity with general category theory culture and are familiar with elementary concepts of geometry and algebra. The aim is to explain synthetic approach to commutative and noncommutative geometry on two examples of geometric notions. First we explain all categorical ingredients that enter the synthetic definition of a principal bundle (in classical geometry) and then we show that noncommutative generalisation of this definition yields in particular principal comodule algebras or faithfully flat Hopf-Galois extensions.
This seminar is transmitted by videoconference from Oporto

### The asymptotic expansion of the Witten-Reshetikhin-Turaev invariants

Witten's influential invariants for links in 3-manifolds given in terms of a non-rigorous Feynman path integral have been rigorously defined first by Reshetikhin and Turaev. Their combinatorial definition based on the axioms of topological quantum field theory is expected to have an asymptotic expansion in view of the perturbation theory of Witten's path integral with leading order term (the semiclassical approximation) given by formally applying the method of stationary phase. Furthermore, the terms in this asymptotic expansion are expected to be well-known classical invariants like the Chern-Simons invariant, spectral flow, the Rho invariant and Reidemeister torsion. I will present new results on the expansion for finite order mapping tori, whose leading order terms we identified with classical topological invariants. Joint with Jørgen E. Andersen.
Support: FCT, CAMGSD, New Geometry and Topology.

### Categorifying Measure Theory II

This lecture will be a continuation of my October lecture on a program to categorify measure theory. After a quick reminder of some of the basic concepts like the finite Grothendieck topology and cosheaves, and the basic theorem on the (bi)representability of the the cosheaf category, I will try to demonstrate that all the work gone into setting up the machinery was not in vain by first, giving categorified versions of some basic theorems of measure theory (e.g. Fubini and Fubini-Tonelli) and second, by explaining some of the rich structure underlying categorified measure theory, a blend of analysis, algebraic geometry and topos theory.
Support: FCT, CAMGSD, New Geometry and Topology

### Tri-connections and trifunctors

It is well known that connections on a principal G-bundle over a manifold M can be represented by group homomorphisms LM->G, where LM is the loop group of M. Similarly, 2-connections can be seen as 2-functors from a 2-groupoid of points, paths and bigons in M to a 2-group G. In passing to dimension 3, that is, considering tri-connections, 3-groupoids are too rigid, instead one needs to consider Gray-groupoids, their morphisms and transformations of higher dimension. We will describe a groupoid structure for certain functors between a pair of Gray-categories and certain transformations between them.
Support: FCT, CAMGSD, New Geometry and Topology.

### Khovanov and Lauda's diagrammatic calculus for ${\mathrm{sl}}_{n}$

In this talk I plan to fill in some details of what I sketched in my previous talk. In particular I want to explain how one can obtain Khovanov and Lauda\'s calculus from bimodule maps corresponding to MOY-movies. In the end I will sketch the extended sl2 calculus for divided powers. References: 1) A. Lauda A categorification of quantum sl2, arXiv:0803.3652. 2) M. Khovanov and A. Lauda, A diagrammatic approach to categorification of quantum groups I-III, arXiv:0803.4121, arXiv:0804.2080, arXiv:0807.3250. 3) M. Khovanov, A. Lauda, M. Mackaay, M. Stosic, Extended graphical calculus for categorified quantum sl2, arXiv:1006.2866. 4) M. Mackaay, M. Stosic, P. Vaz, A diagrammatic categorification of the q-Schur algebra, arXiv:1008.1348.
Support: FCT, CAMGSD, New Geometry and Topology.

### On higher analogues of symplectic fibrations

One can obtain simple models for abstract field theories by looking at closed forms defined on the total space of a fibration. For instance, in the case of hamiltonian fibrations, this is how Weinstein recovered Sternberg's minimal coupling, yielding a geometrical context for classical Yang-Mills theories. More generally, one can introduce 2-plectic fibrations and interpret the equations for coupling in terms of higher analogues of connections.
Support: FCT, CAMGSD, New Geometry and Topology.

### Categorifying Measure Theory

Measure theory is the study of measures and their siamese twins, integrals. Its central position in analysis is partly explained because it provides us with large families of Banach spaces. In this seminar I will try to explain the why, the what and the how of categorifying measure theory. It will consist mostly in laying the groundwork so as to be able to explain the construction of the category of \"categorified integrable functionsänd the integral functor. Time permitting, I will also give a categorified Radon- Nikodym theorem. In a second, future seminar, I will give categorified versions of other basic theorems of measure theory (e.g. Fubini and Fubini-Tonelli on the equality of iterated integrals) and explain some new phenomena peculiar to the categorified setting and with no counterpart in ordinary measure theory.
Support: FCT, CAMGSD, New Geometry and Topology

### Groupoidification in Physics

The Baez-Dolan groupoidification program describes linear-algebraic structures as reflections of those found in a category whose objects are groupoids, and whose morphisms are spans of groupoids. An extension of this program represents those structures in a 2-category of 2-vector spaces. Since quantum physics relies heavily on the category of Hilbert spaces, it has been possible to describe certain toy physical systems in this setting. The talk will describe these in terms of groupoids and spans with intrinsic combinatorial and geometric interest, and discuss how the 2-categorical extension applies in these models, including exotic statistics, and applications to 3D quantum gravity.
Support: FCT, CAMGSD, New Geometry and Topology

### Categorifications of Hecke algebras, $q$-Schur algebras and quantum groups

This is meant to be a (rather incomplete) survey talk of the rapidly developing field of categorification. I will only concentrate on two of the various now existing approaches: the one using (singular) Soergel bimodules, due to Soergel (1992) and Williamson (2008), and the one using diagrams due to Khovanov and Lauda (2008) and Elias and Khovanov (2009). My intention is to explain the relation between the two approaches, as was worked out by Stosic, Vaz and myself.
Support: CAMGSD, FCT

### Categorification of singular braid monoids and of virtual braid groups

Rouquier categorified braid groups, in the sense that he associated to each braid a complex of Soergel bimodules such that complexes associated to isotopic braids are homotopy equivalent. I will start with recalling Rouquier's construction and then extend this result to singular braids and to virtual braids.
Support: CAMGSD, FCT

### 3-colorings of cubic graphs and operators

We consider a monoidal category consisting of cubic graphs with free ends as morphisms between finite sets of points on a line (in the same sense as the category of tangles). We give a linear representation of this category that codifies the number of edge 3-colorings of cubic graphs. Some developments and conjectures will be presented.
Support: CAMGSD, FCT

### Lie crossed modules and gauge-invariant actions for 2-BF theories

We generalize the BF theory action to the case of a general Lie crossed module $\left(H\to G\right)$, where $G$ and $H$ are non-abelian Lie groups. Our construction requires the existence of $G$-invariant non-degenerate bilinear forms on the Lie algebras of $G$ and $H$ and we show that there are many examples of such Lie crossed modules by using the construction of crossed modules provided by short-complexes of vector spaces. We construct two gauge-invariant actions for 2-flat and fake-flat 2-connections with auxiliary fields. The first action is of the same type as the BFCG action introduced by Girelli, Pfeiffer and Popescu for a special class of Lie crossed modules, where $H$ is abelian. The second action is an extended BFCG action which contains an additional auxiliary field. We also construct a two-parameter deformation of the extended BFCG action, which we believe to be relevant for the construction of non-trivial invariants of knotted surfaces embedded in the four-sphere. http://arxiv.org/abs/1006.0903
Support: FCT, CAMGSD, New Geometry and Topology.Support: FCT, CAMGSD, New Geometry and Topology.

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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.