# Topological Quantum Field Theory Seminar

## Past sessions

Newer session pages: Next 3 2 1 Newest

### 27/05/2014, 13:30 — 14:30 — Room P3.10, Mathematics Building

Andreas Döring, *Friedrich-Alexander-Universität Erlangen-Nürnberg*

### 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.

### 26/05/2014, 15:30 — 16:30 — Room P3.10, Mathematics Building

Paolo Bertozzini, *Thammasat University, Bangkok, Thailand*

### 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).

### 14/05/2014, 16:00 — 17:00 — Room P4.35, Mathematics Building

Sara Tavares, *University of Nottingham, United Kingdom*

### 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.

### 02/04/2014, 16:00 — 17:00 — Room P4.35, Mathematics Building

Aleksandar Mikovic, *Universidade Lusófona and Grupo de Física Matemática*

### 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

- 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).
- Aleksandar Mikovic, Marko Vojinovic, Poincaré 2-group and quantum
gravity,. Class. Quant. Grav. 29, 165003 (2012).

### 01/04/2014, 11:30 — 12:30 — Room P3.10, Mathematics Building

Zoran Škoda, *University of Zagreb*

### 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.

#### See also

lispr1.pdf

### 12/03/2014, 16:00 — 17:00 — Room P4.35, Mathematics Building

Marko Vojinovic, *Grupo de Fisica Matemática, Universidade de Lisboa*

### 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.

#### See also

https://math.tecnico.ulisboa.pt/seminars/download.php?fid=9

### 26/02/2014, 16:00 — 17:00 — Room P4.35, Mathematics Building

John Huerta, *IST, Lisbon*

### 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.

### 22/01/2014, 16:30 — 17:30 — Room P4.35, Mathematics Building

Marko Vojinovic, *Grupo de Fisica Matemática, Universidade de Lisboa*

### 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.

#### See also

2014-Lisbon-TQFTclub-Renormalization-Lecture.pdf

### 18/12/2013, 16:30 — 17:30 — Room P4.35, Mathematics Building

Marko Vojinovic, *Grupo de Fisica Matemática, Universidade de Lisboa*

### 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.

#### See also

https://math.ist.utl.pt/seminars/download.php?fid=9

### 11/12/2013, 17:00 — 18:00 — Room P4.35, Mathematics Building

Carlos Guedes, *AEI, Golm-Potsdam*

### 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.

### 04/12/2013, 16:30 — 17:30 — Room P4.35, Mathematics Building

Nuno Costa Dias, *Universidade Lusófona and GFM, Universidade de Lisboa*

### 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.

### 27/11/2013, 16:30 — 17:30 — Room P4.35, Mathematics Building

John Huerta, *Instituto Superior Técnico*

### 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.

### 30/09/2013, 15:00 — 16:00 — Room P3.10, Mathematics Building

Nuno Freitas, *Univ. Bayreuth*

### 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.

### 29/08/2013, 15:00 — 16:00 — Room P4.35, Mathematics Building

Travis Willse, *The Australian National University*

### 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.

### 20/06/2013, 11:00 — 12:00 — Room P4.35, Mathematics Building

John Huerta, *IST, Lisbon*

### 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.

### 14/06/2013, 11:00 — 12:30 — Room P3.10, Mathematics Building

John Huerta, *Instituto Superior Técnico*

### QFT IV

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

### 24/04/2013, 11:30 — 12:30 — Room P3.10, Mathematics Building

John Huerta, *Instituto Superior Técnico*

###
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.

### 17/04/2013, 11:30 — 12:30 — Room P3.10, Mathematics Building

John Huerta, *Instituto Superior Técnico*

###
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.

### 10/04/2013, 11:30 — 12:30 — Room P4.35, Mathematics Building

John Huerta, *Instituto Superior Técnico*

###
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.

### 03/04/2013, 11:30 — 12:30 — Room P3.10, Mathematics Building

John Huerta, *Instituto Superior Técnico*

###
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.

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.*