Topological Quantum Field Theory Seminar

Past sessions

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.

QFT V

In the final lecture of our gentle introduction to quantum field theory, we discuss the renormalization of phi cubed theory at one loop.
Please note that the seminar lasts an hour and a half, 11h00 - 12h30.

QFT IV

We will introduce Feynman diagrams by studying finite-dimensional Gaussian integrals and their perturbations, leading up to phi-cubed theory.

QFT III

Last time, we talked about quantization of the free scalar field by replacing the modes of the field by quantum oscillators. Now, we put this field into the form used by physicists, and talk about the Wightman axioms, which allow a rigorous treatment of free fields.

QFT II

We continue our gentle introduction to quantum field theory for mathematicians. We discuss the Klein-Gordon equation, and how it decomposes into oscillators. We quantize this system by quantizing the oscillators, obtaining the free scalar field, the simplest quantum field there is.

QFT I

This series of lectures will be a gentle introduction to quantum field theory for mathematicians. In our first lecture, we give a lightning introduction to quantum mechanics and discuss the simplest quantum system: the harmonic oscillator. We then sketch how this system is used to quantize the free scalar field.
Note: room 4.35 is on the 4th floor at the end of the main corridor

Anomalies IV

We will introduce the notion of stable isomorphism for gerbes, and talk about how stable isomorphism classes are in one-to-one correspondence with Deligne cohomology classes. We define WZW branes and discuss how the basic gerbe on a group trivializes when restricted to the brane.

Older session pages: Previous 5 6 7 8 9 10 11 12 Oldest

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.