# Seminário de Teoria Quântica do Campo Topológica

## Sessões anteriores

Páginas de sessões mais recentes: Seguinte 2 1 Mais recente

### Actions of 2-groups, moduli spaces in higher gauge theory, and TQFT's

In the context of higher gauge theory (HGT) based on a 2-group, I will discuss how the language of double categories provides a natural description of 2-group actions on a category. One of the motivations is to understand moduli spaces of flat connections modulo gauge transformations in HGT, and this goal is achieved for some simple manifolds. I will also relate these ideas to a class of Topological Quantum Field Theories (TQFT's) for surfaces, obtained from finite groups and 2-groups.

This talk is based on callaborations with João Faria Martins, Jeffrey Morton and Diogo Bragança. It is also intended as preparation for the visit by Urs Schreiber, 15 Jan—14 Feb, 2016.

### Bicategories, classifying spaces and homotopy pullbacks

In this continuation of my last talk I will give an introduction to the homotopy theory of bicategories. First I will present the way to convert bicategories to spaces, and then I will use this to describe homotopy pullbacks of homomorphisms of bicategories.

### Homotopy theory using categories

This is an introductory talk about the homotopy theory of categories. I will present the classifying space of a category and the classical results of Thomason and Quillen for obtaining categorical descriptions of homotopy colimits and homotopy pullbacks.

### Positive energy unitary irreducible representations of $\operatorname{osp}(1|2n)$ superalgebras

Orthosymplectic $\operatorname{osp}(1|2n)$ superalgebras are being considered as alternatives to $d$-dimensional Poincaré/conformal superalgebras and thus have significant potential relevance in various subfields of High Energy Physics and Astrophysics. Yet, due to mathematical difficulties, even the classification of their unitary irreducible representations (UIR's) has not been entirely accomplished. This is also true for the physically most important subclass of positive energy UIR's.

In this talk I will first demonstrate this classification for the $n=4$ case (that corresponds to four dimensional space-time). The classification is obtained by careful analysis of the Verma module structure, which is particularly subtle due to the existence of subsingular vectors. Based on these results I will then conjecture their generalization to the case of arbitrary $n$ (thus also including cases relevant in the string/brane context). In addition, I will show an elegant explicit realization of these UIR's that exists for (half)integer values of the conformal energy and that makes manifest the mathematical connection existing between UIR's of orthogonal and orthosymplectic algebras. The existence of this realization, per se, proves a part of the conjecture.

### Geometrical diffeomorphism invariant observables for General Relativity and their applications

During the talk I will present a recent construction of observables for General Relativity invariant under spatial diffeomorphisms. The construction involves introducing a local structure representing the "observer" (based on arXiv:1403.8062). I will also present how those observables can be used to reduce the phase space of canonical General Relativity (based on arXiv:1506.09164).

If time permits, I will argue that the construction is particularly useful in spherically symmetric situations. This realization lead to a proposal of a scheme of reducing Loop Quantum Gravity to its spherically symmetric sector, which completes the standard, midisuperspace approach (based on arXiv:1410.5609).

swiezewski.pdf

### Physical Mathematics: old and new

In this talk  we will discuss some topics related  to the interaction of physical and mathematical theories that have led to new points of view and new results in mathematics. The area where this is most evident is that of geometric topology of low dimensional manifolds. I coined the term Physical Mathematics to describe this new and fast growing area of research and used it in the title of my paper in Springer's book Mathematics Unlimited: 2001 and beyond.

We will discuss some recent developments in this area. General reference for this talk is my book Topics in Physical Mathematics, Springer (2010).

### Black hole entropy in loop quantum gravity

After briefly introducing the main ingredients of the loop quantum gravity approach, I show how it can applied to the calculation of black hole entropy. I review some well known results and open issues resulting from the interplay with Chern-Simons theory techniques. I then introduce a new analysis of the horizon degrees of freedom in terms of purely LQG methods, which turns out to be dual to a CFT description. I show how this unifying framework allows us to recover the semiclassical Bekenstein-Hawking entropy formula.

#### Ver também

Pranzetti_Black_Hole_in_Loop_Quantum_Gravity_Slides_20150422.pdf

### Introduction to Loop Quantum Gravity (part 3)

This series of lectures is intended to give an elementary introduction to the topic of the canonical quantization of the gravitational field, in the context of the Loop Quantum Gravity approach.

In the third lecture we will finish the construction of the spin-knot space and introduce the loop transform. Then we move on to the analysis of geometric observables (distance, area and volume) and the structure of the scalar constraint. Finally, matter coupling will be introduced. If time permits, we will also give a short review of two applications of the formalism: calculation of the black hole entropy, and the Big Bounce model of Loop Quantum Cosmology.

### Introduction to Loop Quantum Gravity (part 2)

This series of lectures is intended to give an elementary introduction to the topic of the canonical quantization of the gravitational field, in the context of the Loop Quantum Gravity approach.

The second lecture is devoted to the canonical quantization procedure within the LQG framework. We will begin by a short introduction to the notion of background independence, and differences between perturbative and nonperturbative quantization. We will then rewrite general relativity in the canonical space+time formulation and introduce Ashtekar variables, as preparation for the canonical quantization. Then the main step is the quantization itself, and the construction of the appropriate Hilbert space of the theory based on the notions of spin networks and spin-knots.

Please note the change of date from Wednesday 11th March to Wednesday 18th March.

### Introduction to Loop Quantum Gravity (part 1)

This series of lectures is intended to give an elementary introduction to the topic of the canonical quantization of the gravitational field, in the context of the Loop Quantum Gravity approach.

The first lecture will be devoted to the formulation of the problem of quantization of the gravitational field. We will give an overview of perturbative quantization, discuss the issue of nonrenormalizability, and provide a general classification of most prominent approaches to constructing a theory of quantum gravity. One such approach is Loop Quantum Gravity, which will be studied in more detail in subsequent lectures.

### Twisted actions of categorical groups

We develop a theory of twisted actions of categorical groups using the notion of semidirect product of categories. I will present many examples of semi-direct product of categories. If time permits I will also work-out an example of twisted action involving the Poincaré 2-group. Specializing to the case of representations, where the the category on which categorical group acts has some kind of a vector space structure, we will establish a categorical analogue of Schur's lemma.

This is a joint work with A. Lahiri and A. Sengupta.

### Holography for parallel conformal data

The Fefferman-Graham ambient metric construction, with some technical asterisks, positively resolves the Dirichlet problem for compactification of asymptotically hyperbolic Einstein metrics, the compactification that occurs in the AdS/CFT correspondence. We show that data on the conformal boundary parallel with respect to Cartan's normal conformal connection — which is nearly the same thing as a holonomy reduction of the conformal structure — can be extended (again with an asterisk) to data parallel with respect to a natural connection on a corresponding bundle over the bulk, which in particular enables holographic study of such data. As an application, we use this extension result to construct metrics of exceptional holonomy.

Note unusual day/time and room

### Grothendieck topologies for C*-algebras

We investigate a (contravariant) functor from C*-algebras to toposes and geometric morphisms that generalizes the Gelfand spectrum in the commutative case. The functor produces a locale, presented by means of a Grothendieck topology on an inf-semilattice of 'Gelfand' opens $[U;a]$.

### The Spectral Presheaf as the Spectrum of a Noncommutative Operator Algebra

The spectral presheaf of a nonabelian von Neumann algebra or C*-algebra was introduced as a generalised phase space for a quantum system in the so-called topos approach to quantum theory. Here, it will be shown that the spectral presheaf has many features of a spectrum of a noncommutative operator algebra (and that it can be defined for other classes of algebras as well). The main idea is that the spectrum of a nonabelian algebra may not be a set, but a presheaf or sheaf over the base category of abelian subalgebras. In general, the spectral presheaf has no points, i.e., no global sections. I will show that there is a contravariant functor from the category of unital C*-algebras to a category of presheaves that contains the spectral presheaves, and that a C*-algebra is determined up to Jordan *-isomorphisms by its spectral presheaf in many cases. Moreover, time evolution of a quantum system can be described in terms of flows on the spectral presheaf, and commutators show up in a natural way. I will indicate how combining the Jordan and Lie algebra structures can lead to a full reconstruction of nonabelian C*- or von Neumann algebra from its spectral presheaf.

### Higher Categories of Operator Algebras

A satisfactory marriage between “higher” categories and operator algebras has never been achieved: although (monoidal) C*-categories have been systematically used since the development of the theory of superselection sectors, higher category theory has more recently evolved along lines closer to classical higher homotopy.

We present axioms for strict involutive $n$-categories (a vertical categorification of dagger categories) and a definition for strict higher C*-categories and Fell bundles (possibly equipped with involutions of arbitrary depth), that were developed in collaboration with Roberto Conti, Wicharn Lewkeeratiyutkul and Noppakhun Suthichitranont.

In order to treat some very natural classes of examples arising from the study of hypermatrices and hyper-C*-algebras, that would be otherwise excluded by the standard Eckmann-Hilton argument, we suggest a non-commutative version of exchange law and we also explore alternatives to the usual globular and cubical settings.

Possible applications of these non-commutative higher C*-categories are envisaged in the algebraic formulation of Rovelli's relational quantum theory, in the study of morphisms in Connes' non-commutative geometry, and in our proposed “modular” approach to quantum gravity (arXiv: 1007.4094).

Note: unusual time/day and room

### Two-dimensional state sum models and spin structures

Topological field theories are very special in two dimensions: they have been classified and provide a rich class of examples. In this talk I will discuss a new state sum construction for these models that considers not just the topology of surfaces but also their spin structure. Emphasis is given to partition functions: I will detail general properties of these manifold invariants and discuss some non-trivial examples.

### 2-BF Theories

We will describe 2-BF topological field theories, which are categorical generalization of the BF theories. We will also explain how to construct invariants of manifolds by using 2-BF theory path integrals.

#### References

1. João Faria Martins, Aleksandar Mikovic,. Lie crossed modules and gauge-invariant actions for 2-BF theories. Adv. Theor. Math. Phys. Volume 15, Number 4, 1059-1084 (2011).
2. Aleksandar Mikovic, Marko Vojinovic, Poincaré 2-group and quantum gravity,. Class. Quant. Grav. 29, 165003 (2012).

### Coherent states for quantum groups

Quantum groups at roots of unity appear as hidden symmetries in some conformal field theories. For this reason I. Todorov has (in 1990s) used coherent state operators for quantum groups to covariantly build the field operators in Hamiltonian formalism. I tried to mathematically found his coherent states by an analogy with the Perelomov coherent states for Lie groups. For this, I use noncommutative localization theory to define and construct the noncommutative homogeneous spaces, and principal and associated bundles over them. Then, in geometric terms, I axiomatize the covariant family of coherent states which enjoy a resolution of unity formula, crucial for physical applications. Even the simplest case of quantum $\operatorname{SL2}$ is rather involved and the corresponding resolution of unity formula involves the Ramanujan's $q$-beta integral. The correct covariant family differs from ad hoc proposed formulas in several published papers by earlier authors.

#### Ver também

lispr1.pdf
Note: unusual time/day and room

### Introduction to renormalization in QFT (part III)

In the previous talk we discussed the renormalization procedure on the example ${\varphi }^{4}$ scalar field theory. In this lecture we will conclude the analysis of that example, construct the final renormalized state sum, and discuss the renormalization group equations. At the end we will give some final general remarks about renormalization in QFT.