# Seminário de Teoria Quântica do Campo Topológica

## Sessões anteriores

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

### 26/02/2014, 16:00 — 17:00 — Sala P4.35, Pavilhão de Matemática

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 — Sala P4.35, Pavilhão de Matemática

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.

#### Ver também

2014-Lisbon-TQFTclub-Renormalization-Lecture.pdf

### 18/12/2013, 16:30 — 17:30 — Sala P4.35, Pavilhão de Matemática

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.

#### Ver também

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

### 11/12/2013, 17:00 — 18:00 — Sala P4.35, Pavilhão de Matemática

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 — Sala P4.35, Pavilhão de Matemática

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 — Sala P4.35, Pavilhão de Matemática

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 — Sala P3.10, Pavilhão de Matemática

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 — Sala P4.35, Pavilhão de Matemática

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 — Sala P4.35, Pavilhão de Matemática

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 — Sala P3.10, Pavilhão de Matemática

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 — Sala P3.10, Pavilhão de Matemática

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 — Sala P3.10, Pavilhão de Matemática

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 — Sala P4.35, Pavilhão de Matemática

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 — Sala P3.10, Pavilhão de Matemática

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.

### 20/03/2013, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

Aleksandar Mikovic, *Univ. Lusófona*

###
Categorification of Spin Foam Models

We briefly review spin foam state sums for triangulated
manifolds and motivate the introduction of state sums based on
2-groups. We describe 2-BF gauge theories and the construction of
the corresponding path integrals (state sums) in the case of
Poincaré 2-group.

#### References

- J. F. Martins and A. Mikovic,
*Lie crossed modules and
gauge-invariant actions for 2-BF theories*, Adv. Theor. Math.
Phys. 15 (2011) 1059,
arxiv:1006.0903
- A. Mikovic and M. Vojinovic,
*Poincaré 2-group and quantum
gravity*, Class. Quant. Grav. 291 (2012) 165003, arxiv:1110.4694

### 13/03/2013, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

John Huerta, *Instituto Superior Técnico*

###
Anomalies III

We continue examining Gawedzki and Reis's paper:
WZW branes and gerbes,
http://arxiv.org/abs/hep-th/0205233

We define a gerbe, and show gerbes can be "transgressed" to give
line bundles over loop space. Trivial gerbes give trivial bundles
on loop space, whose sections are thus mere functions. Any compact,
simply connected Lie group comes with a god-given gerbe whose
curvature is the canonical invariant 3-form. Restricting this gerbe
to certain submanifolds, we get trivial gerbes who thus transgress
to trivial line bundles, "cancelling" the anomaly of a nontrivial
line bundle.

### 27/02/2013, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

John Huerta, *Instituto Superior Técnico*

###
Anomalies II

We continue our informal discussion of anomalies by talking about
global anomalies on branes, and their relationship with gerbes.

### 06/02/2013, 14:00 — 15:00 — Sala P5.18, Pavilhão de Matemática

John Huerta, *Instituto Superior Técnico*

###
Introduction to anomalies

In physics, an "anomaly" is the failure of a classical symmetry at
the quantum level. Anomalies play a key role in assessing the
consistency of a quantum field theory, and link up with cohomology
in mathematics, a general tool by which mathematicians understand
whether a desired construction is possible. In this informal series
of talks, we aim to understand what physicists mean by an "anomaly"
and their mathematical interpretation.

### 19/12/2012, 11:30 — 12:30 — Sala P4.35, Pavilhão de Matemática

Sara Tavares, *Univ. of Nottingham*

###
Observables in 2D BF theory

BF theory in two dimensions has been the subject of intensive study
in the last twenty five years. I will readdress it by highlighting
the TQFT interpretation of the spinfoam approach to its
quantisation. I will also introduce the mathematical model that
allows us to treat surfaces with inbuilt topological defects and
how we expect them to relate to operators in the quantum field
theory.

### 28/11/2012, 11:30 — 12:30 — Sala P4.35, Pavilhão de Matemática

Louis H. Kauffman, *Univ. of Illinois at Chicago*

###
Non-Commutative Worlds and Classical Constraints

This talk shows how discrete measurement leads to commutators and
how discrete derivatives are naturally represented by commutators
in a non-commutative extension of the calculus in which they
originally occurred. We show how the square root of minus one ($i$)
arises naturally as a time-sensitive observable for an elementary
oscillator. In this sense the square root of minus one is a clock
and/or a clock/observer. This sheds new light on Wick rotation,
which replaces $t$ (temporal quantity) by $it$. In this view, the
Wick rotation replaces numerical time with elementary temporal
observation. The relationship of this remark with the Heisenberg
commutator $[P,Q]=i\hslash $ is explained. We discuss iterants - a
generalization of the complex numbers as described above. This
generalization includes all of matrix algebra in a temporal
interpretation. We then give a generalization of the Feynman-Dyson
derivation of electromagnetism in the context of non-commutative
worlds. This generalization depends upon the definitions of
derivatives via commutators and upon the way the non-commutative
calculus mimics standard calculus. We examine constraints that link
standard and non-commutative calculus and show how asking for these
constraints to be satisfied leads to some possibly new physics.

#### Ver também

https://www.math.ist.utl.pt/seminars/qci/index.php.en?action=show&id=3243

Páginas de sessões mais antigas: Anterior 5 6 7 8 9 10 11 12 Mais antiga

Organizadores correntes: Roger Picken, Marko Stošić.

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