# Topological Quantum Field Theory Seminar

## Past sessions

### 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.

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.

### What can higher categories do for physics? Part II

In this follow up to last year's talk, we briefly review the cobordism hypothesis that formed the subject of our first part, and then outline its use for the existence and construction of field theories, in particular Chern-Simons theory, as discussed in a 2009 paper of Freed, Hopkins, Lurie and Teleman.

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

In the previous talk we gave an overview of the renormalization procedure in Quantum Field Theory. In this lecture we will demonstrate that abstract procedure on a simple explicit example, the so-called ${\varphi }^{4}$ theory of a single real scalar field. We will illustrate the construction of a renormalized state sum using two different regularization schemes, construct the renormalization group equations, and discuss some of their properties.

2014-Lisbon-TQFTclub-Renormalization-Lecture.pdf

### Introduction to renormalization in QFT

We will give an overview of the renormalization procedure in Quantum Field Theory. The emphasis will be on the general idea of constructing a finite QFT from the one plagued by divergencies, in the standard perturbative approach, and discussing the uniqueness of the resulting QFT. The lecture does not assume much background knowledge in QFT, and should be accessible to a wide audience.

### The non-commutative Fourier transform for Lie groups

The phase space given by the cotangent bundle of a Lie group appears in the context of several models for physical systems. In quantum mechanics on the Euclidean space, the standard Fourier transform gives a unitary map between the position representation -- functions on the configuration space -- and the momentum representation -- functions on the corresponding cotangent space. That is no longer the case for systems whose configuration space is a more general Lie group. In this talk I will introduce a notion of Fourier transform that extends this duality to arbitrary Lie groups.

arXiv:1301.7750

### Quantum mechanics in phase space: The Schrödinger and the Moyal representations

I will present some recent results on the dimensional extension of pseudo-differential operators. Using this formalism it is possible to generalize the standard Weyl quantization and obtain, in a systematic way, several phase space (operator) representations of quantum mechanics. I will present the Schrodinger and the Moyal phase space representations and discuss some of their properties, namely in what concerns the relation with deformation quantization.

### What can higher categories do for physics?

We describe Baez and Dolan's cobordism hypothesis - a deep connection between topological quantum field theory, higher categories, and manifolds. Physically, this encodes the idea that quantum field theories, even "topological" ones, should be local: no matter how we cut up the spacetime on which they are defined in order to perform the path integral, the net result must be the same. Recently, this hypothesis was formulated and proved by Jacob Lurie using the tools of homotopy theory. We describe the version of the hypothesis he proved. Finally, we touch on Freed, Hopkins, Lurie and Teleman's recent work on Chern-Simons theory, and on Urs Schreiber's ideas for using Lurie's toolkit in full-fledged quantum field theory.

### The Fermat equation over totally real number fields

Jarvis and Meekin have shown that the classical Fermat equation $$x^p + y^p = z^p$$ has no non-trivial solutions over $$\mathbb{Q}(\sqrt{2})$$. This is the only result available over number fields. Two major obstacles to attack the equation over other number fields are the modularity of the Frey curves and the existence of newforms in the spaces obtained after level lowering.

In this talk, we will describe how we deal with these obstacles, using recent modularity lifting theorems and level lowering. In particular, we will solve the equation for infinitely many real quadratic fields.

### Groups of type ${G}_{2}$ and exceptional geometric structures in dimensions 5, 6, and 7

Several exceptional geometric structures in dimensions 5, 6, and 7 are related in a striking panorama grounded in the algebra of the octonions and split octonions. Considering strictly nearly Kähler structures in dimension 6 leads to prolonging the Killing-Yano (KY) equation in this dimension, and the solutions of the prolonged system define a holonomy reduction to a group of exceptional type ${G}_{2}$ of a natural rank-7 vector bundle, which can in turn be realized as the tangent bundle of a pseudo-Riemannian manifold, which hence relates this construction to exceptional metric holonomy. In the richer case of indefinite signature, a suitable solution $\omega$ of the KY equation can degenerate along a (hence 5-dimensional) hypersurface $\Sigma$, in which case it partitions the underlying manifold into a union of three submanifolds and induces an exceptional geometric structure on each. On the two open manifolds (which have common boundary $\Sigma$), $\omega$ defines asymptotically hyperbolic nearly Kähler and nearly para-Kähler structures. On $\Sigma$ itself, $\omega$ determines a generic $2$-plane field, the type of structure whose equivalence problem Cartan investigated in his famous Five Variables paper. The conformal structure this plane field induces via Nurowski's construction is a simultaneous conformal infinity for the nearly (para-)Kähler structures.

This project is a collaboration with Rod Gover and Roberto Panai.

Older session pages: Previous 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.