# Seminário de Teoria Quântica do Campo Topológica

## Sessões anteriores

Páginas de sessões mais recentes: Seguinte 5 4 3 2 1 Mais recente

### 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.

### Categorification, Feynman Integrals, and Quantization - II

We review the general notion of categorification in order to study quantization -- in the sense of deformation quantization -- and Feynman integrals from the viewpoint of category theory. The fairly abstract setting that we propose leads to a rather down to earth approach to these often elusive notions.

Support: FCT, CAMGSD, New Geometry and Topology

### Quantum $\left({\mathrm{sl}}_{n},\wedge {V}_{n}\right)$ link invariant and matrix factorizations.

M. Khovanov and L. Rozansky constructed a homology for a link diagram whose Euler characteristic is the quantum link invariant associated to the quantum group $\mathrm{Uq}\left({\mathrm{sl}}_{n}\right)$ and its vector representation ${V}_{n}$ by using matrix factorizations. In my thesis, I study a generalization of the Khovanov-Rozansky homology for the quantum link invariant associated to $\mathrm{Uq}\left({\mathrm{sl}}_{n}\right)$ and its fundamental representations $\wedge {V}_{n}$. In this talk, I will define a new link invariant derived from the generalization of Khovanov-Rozansky homology.
Support: FCT, CAMGSD, New Geometry and Topology

### 2-Linearization, Discrete and Smooth

This talk will introduce the construction "2-linearization", an extension of the "groupoidification" program of Baez and Dolan which seeks to interpret constructions in linear algebra in terms of groupoids and spans of groupoids. The 2-linearization construction finds this as a special case of a 2-functor $\Lambda :\mathrm{Span}\left(\mathrm{Gpd}\right)\to 2\mathrm{Vect}$, where $2\mathrm{Vect}$ is the 2-category of Kapranov-Voevodsky 2-vector spaces. This 2-functor is constructed in terms of pairs of ambi-adjoint functors associated to each groupoid homomorphism, the "push" and "pull" operations, closely related to restriction and induction maps in representation theory, Grothendieck's 6-operation framework for sheaves, among other examples. We will begin with the discrete case, and consider generalizations to smooth groupoids. Finally we will consider some applications.
Support: FCT, CAMGSD, New Geometry and Topology

### Categorification, Feynman Integrals, and Quantization - I

We review the general notion of categorification in order to study quantization -- in the sense of deformation quantization -- and Feynman integrals from the viewpoint of category theory. The fairly abstract setting that we propose leads to a rather down to earth approach to these often elusive notions.

Support: FCT, CAMGSD, New Geometry and Topology

### Introduction to nonabelian bundle 2-gerbes

We introduce nonabelian bundle 2-gerbes related to Lie 2-crossed modules and discuss their properties. Such bundle 2-gerbes are defined in terms of bundle 1-gerbes, which play a role similar to that of transition functions in the theory of principal bundles.

### Introduction to Abelian and non-Abelian gerbes

We shall use the language of Lie groupoids and Morita equivalence to introduce gerbes, and connections on them. The advantage of this presentation is that it makes it possible to use ordinary differential geometry, without going to "higher" objects - the default being the need to check the invariance of all constructions under Morita equivalence. I shall try to relate this construction with previous ones, especially the constructions of Breen and Messing.

(see Lecture 1)

(see Lecture 1)

(see Lecture 1)

### The “Crossed” technology and its applications

In this short series of lectures I will introduce crossed modules and crossed complexes, both algebraically and topologically, will look at their relevance for combinatorial group theory, the theory of syzygies and group cohomology, and then will head for higher order objects namely 2-crossed modules, and related complexes. In the final parts I will introduce some of the constructions of non-Abelian cohomology, sheaves, torsors and Bitorsors and consider the interaction between the crossed gadgetry of the earlier lectures and this area.
##### References

Schedule:
Lecture 1: Wednesday, 3rd December, 2008, 11h30 - 13
Lecture 2: Wednesday, 3rd December, 2008, 16h30 - 18
Lecture 3: Thursday, 4th December, 2008, 14h30 - 16
Lecture 4: Thursday, 4th December, 2008, 16h30 - 18

### Topological 2-Groups and Their Classifying Spaces

Categorifying the concept of topological group, one obtains the notion of a topological 2-group. This in turn allows a theory of "principal 2-bundles" generalizing the usual theory of principal bundles. It is well-known that under mild conditions on a topological group $G$ and a space $M$, principal $G$-bundles over $M$ are classified by either the Cech cohomology ${H}^{1}\left(M,G\right)$ or the set of homotopy classes $\left[M,BG\right]$, where $BG$ is the classifying space of $G$. Here we review work by Bartels, Jurco, Baas-Bökstedt-Kro, and others generalizing this result to topological 2-groups. We explain various viewpoints on topological 2-groups and the Cech cohomology ${H}^{1}\left(M,G\right)$ with coefficients in a topological 2-group $G$, also known as "nonabelian cohomology". Then we sketch a proof that under mild conditions on $M$ and $G$ there is a bijection between ${H}^{1}\left(M,G\right)$ and $\left[M,B\mid G\mid \right]$, where $B\mid G\mid$ is the classifying space of the geometric realization of the nerve of $G$. Applying this result to the ''string 2-group" $\mathrm{String}\left(G\right)$ of a simply-connected compact simple Lie group $G$, we obtain a theory of characteristic classes for principal $\mathrm{String}\left(G\right)$-2-bundles.

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Organizadores correntes: Roger Picken, Marko Stošić.

Projecto FCT PTDC/MAT-GEO/3319/2014, Quantization and Kähler Geometry.