# Topological Quantum Field Theory Seminar

## Past sessions

Newer session pages: Next 4 3 2 1 Newest

### 20/03/2013, 11:30 — 12:30 — Room P3.10, Mathematics Building

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 — Room P3.10, Mathematics Building

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 — Room P3.10, Mathematics Building

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 — Room P5.18, Mathematics Building

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 — Room P4.35, Mathematics Building

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 — Room P4.35, Mathematics Building

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.

#### See also

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

### 23/11/2012, 11:30 — 12:30 — Room P4.35, Mathematics Building

Jeffrey Morton, *Univ. Hamburg*

###
Classifying Extended TQFT and the Cobordism Hypothesis

An overview of the concept of extended field theories, and a look
at the role of the Cobordism Hypothesis (now more accurately the
Cobordism Theorem) in classification of such theories. Given time
the talk will touch on Jacob Lurie's proof of the Cobordism
Hypothesis.

### 21/11/2012, 15:30 — 16:30 — Room P4.35, Mathematics Building

Louis-Hadrien Robert, *Institut Mathématique de Jussieu, Paris*

###
Indecomposable modules over a Kuperberg-Khovanov algebras

The ${\mathrm{\U0001d530\U0001d529}}_{3}$ polynomial is a quantum invariant for knots.
It has been categorified by Khovanov in 2004 in a TQFT fashion. The
natural way to extend this categorification to an invariant of
tangle is construct a $0+1+1$ TQFT. From this construction emerge
some algebras called Khovanov-Kuperberg algebras (or
${\mathrm{\U0001d530\U0001d529}}_{3}$-web algebras) and some particular projective
modules called web-modules over these algebras. I will give a
combinatorial caracterisation of indecomposable web-modules.

### 21/11/2012, 14:00 — 15:00 — Room P4.35, Mathematics Building

Marco Mackaay, *Univ. Algarve*

###
${\mathrm{\U0001d530\U0001d529}}_{3}$ web algebras, cyclotomic KLR algebras and
categorical quantum skew Howe duality

I will introduce ${\mathrm{\U0001d530\U0001d529}}_{3}$ web algebras $K(S)$, which
involve Kuperberg's ${\mathrm{\U0001d530\U0001d529}}_{3}$ web space $W(S)$ and Khovanov
${\mathrm{\U0001d530\U0001d529}}_{3}$ foams with boundary in $W(S)$. These algebras are
the ${\mathrm{\U0001d530\U0001d529}}_{3}$ analogues of Khovanov's ${\mathrm{\U0001d530\U0001d529}}_{2}$ arc
algebras. I will show how the $K(S)$ are related to cyclotomic
Khovanov-Lauda-Rouquier algebras (cyclotomic KLR algebras, for
short) by a categorification of quantum skew Howe duality. This
talk is closely related to the next one by Robert. In particular, I
will show that the Grothendieck group of $K(S)$ is isomorphic to
$W(S)$ and that, under this isomorphism, the indecomposable
projective $K(S)$-modules, which Robert constructs explicitly,
correspond precisely to the dual canonical basis elements in
$W(S)$.

### 21/11/2012, 11:00 — 12:00 — Room P4.35, Mathematics Building

Volodymyr Mazorchuk, *Univ. Uppsala*

###
The endomorphism category of a cell 2-representation

Fiat 2-categories are 2-analogues of finite dimensional algebras
with involutions. Cell 2-representations of fiat 2-categories are
most appropriate analogues for simple modules over finite
dimensional algebras. In this talk I will try to describe (under
some natural assumptions) a 2-analogue of Schur's Lemma which
asserts that the endomorphism category of a cell 2-representation
is equivalent to the category of vector spaces. This is applicable,
for example to the fiat category of Soergel bimodules in type A.
This is a report on a joint work with Vanessa Miemietz.

### 28/09/2012, 14:00 — 15:00 — Room P4.35, Mathematics Building

Nuno Freitas, *Univ. Barcelona*

###
Fermat-type equations of signature \((13,13,p)\) via Hilbert
cuspforms

In this talk I will give an introduction to the modular approach to
Fermat-type equations via Hilbert cuspforms and discuss how it can
be used to show that certain equations of the form ${x}^{13}+{y}^{13}=C{z}^{p}$ have no solutions $(a,b,c)$ such that $\mathrm{gcd}(a,b)=1$ and $13\nmid c$ if $p>4992539$. We will first relate a putative
solution of the previous equation to the solution of another
Diophantine equation with coefficients in $Q(\sqrt{13})$. Then we
attach Frey curves $E$ over $Q(\sqrt{13})$ to solutions of the
latter equation. Finally, we will discuss on the modularity of $E$
and irreducibility of certain Galois representations attached to
it. These ingredients enable us to apply a modular approach via
Hilbert newforms to get the desired arithmetic result on the
equation.

### 09/05/2012, 16:15 — 17:15 — Room P3.10, Mathematics Building

Anne-Laure Thiel, *Instituto Superior Técnico*

###
Diagrammatic categorification of extended Hecke algebra and quantum
Schur algebra of affine type A

Joint work with Marco Mackaay.

### 09/05/2012, 14:00 — 16:00 — Room P3.10, Mathematics Building

Marco Mackaay, *Univ. Algarve*

### \(\mathfrak{sl}_3\) web algebras

This is joint work with Weiwei Pan and Daniel Tubbenhauer from
Gottingen University, Germany.

### 12/01/2012, 11:30 — 12:30 — Room P12, Mathematics Building

Jeffrey C. Morton, *Instituto Superior Técnico*

### Groupoidification and Khovanov's Categorification of the Heisenberg Algebra

The aim of this talk is to describe the connection between two approaches to categorification of the Heisenberg algebra. The groupoidification program of Baez and Dolan has been used to give a representation of the quantum harmonic oscillator in the category Span(Gpd) where the Fock space is represented by the groupoid of finite sets and bijections. This naturally gives a combinatorial interpretation of the (one-variable) Heisenberg algebra in the endomorphisms of this groupoid. On the other hand, Khovanov has given a categorification in which the integral part of the (many variable) Heisenberg algebra is recovered as the Grothendieck ring of a certain monoidal category described in terms of a calculus of diagrams. I will describe how an extension of the groupoidification program to a 2-categorical form of Span(Gpd) recovers the relations used by Khovanov's construction, and how to interpret them combinatorially in terms of the groupoid of finite sets.

### 06/01/2012, 16:00 — 17:00 — Room P3.10, Mathematics Building

Nuno Freitas, *Universitat de Barcelona*

### From Fermat's Last Theorem to some generalized Fermat equations

The proof of Fermat's Last Theorem was initiated by Frey,
Hellegouarch, Serre, further developed by Ribet and ended with
Wiles' proof of the Shimura-Tanyama conjecture for semi-stable
elliptic curves. Their strategy, now called the modular approach,
makes a remarkable use of elliptic curves, Galois representations
and modular forms to show that ${a}^{p}+{b}^{p}={c}^{p}$ has no solutions,
such that $(a,b,c)=1$ if $p\ge 3$. Over the last 17 years, the
modular approach has been continually extended and allowed people
to solve many other Diophantine equations that previously seemed
intractable. In this talk we will use the equation ${x}^{p}+{2}^{\alpha}{y}^{p}={z}^{p}$ as the motivation to introduce informally the
original strategy ($\alpha =0$) and illustrate one of its first
refinements (for $\alpha =1$). Then we will discuss some further
generalizations that recently led to the solution of equations of
the form ${x}^{5}+{y}^{5}=d{z}^{p}$.

### 06/12/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building

John Huerta, *Australian National University, Canberra*

### Superstrings, higher gauge theory, and division algebras

Recent work on higher gauge theory suggests the presence of 'higher symmetry' in superstring theory. Just as gauge theory describes the physics of point particles using Lie groups, Lie algebras and bundles, higher gauge theory is a generalization that describes the physics of strings and membranes using categorified Lie groups, Lie algebras and bundles. In this talk, we will summarize the mathematics of a higher gauge theory. We then show how to construct the categorified Lie algebras relevant to superstring theory by a systematic use of the normed division algebras. At the end, we will touch on how this leads to a categorified supergroup extending the Poincare supergroup in the mysterious dimensions where the classical superstring makes sense — 3, 4, 6 and 10.

### 09/11/2011, 14:00 — 15:00 — Room P4.35, Mathematics Building

Pedro Vaz, *Instituto Superior Técnico*

### Categorified $q$-Schur algebra and the BMW algebra

In 1989 François Jaeger showed that the the Kauffman polynomial of a link $L$ can be obtained as a weighted sum of HOMFLYPT polynomials on certain links associated to $L$. In this talk I will explain how to use a version of Jaeger's theorem to stablish a connection between the $\mathrm{SO}(2N)-\mathrm{BMW}$ and the $q$-Schur algebras. I will then present a subcategory of the Schur category which categorifies the $\mathrm{SO}(2N)-\mathrm{BMW}$ algebra (joint with E. Wagner).

### 28/10/2011, 10:30 — 11:30 — Room P3.10, Mathematics Building

Jeffrey Morton, *Instituto Superior Técnico*

### Extended TQFT in a Bimodule 2-Category

I will describe an extended (2-categorical) topological QFT with target 2-category consisting of C*-algebras and bimodules. The construction is explained as factorizable into a classical field theory valued in groupoids, and a quantization functor, as in the program of Freed-Hopkins-Lurie-Teleman. I will explain the Lagrangian action functional in terms of cohomological twisting of the groupoids in the classical part of the theory, and describe how this is incorporated into the quantization functor. This project is joint work with Derek Wise.

### 28/07/2011, 15:30 — 16:30 — Room P3.10, Mathematics Building

Andreas Döring, *Univ. of Oxford*

### Towards Noncommutative Gel'fand Duality

Gel'fand-Naimark duality (1943) between the categories of unital commutative ${C}^{*}$-algebras and compact Hausdorff spaces is a key insight of 20th century mathematics, providing an enormously useful bridge between algebra on the one hand and topology and geometry on the other. Many generalisations and related dualitites exist, in logical, localic and constructive forms. Yet, all this is for *commutative* algebras (and distributive lattices of projections, or opens), while in quantum theory and in a large variety of mathematical situations, *noncommutative* algebras play a central role. A good, generally useful notion of spectra of noncommutative algebras is still lacking. Clearly, such spectra will be of considerable interest for physics and Noncommutative Geometry. I will report on recent progress towards defining spectra of noncommutative operator algebras, mostly for von Neumann algebras. This work comes from the approach using topos theory to reformulate quantum physics (C. Isham, AD), where a presheaf or sheaf topos is assigned to each noncommutative operator algebra, together with a distinguished spectral object. It will be shown that this assignment is functorial, and that the spectral object determines the algebra up to Jordan isomorphisms (J. Harding, AD). Progress on characterising the action of the unitary group of an algebra - relating to Lie group and Lie algebra aspects - is presented. Moreover, recently established connections with Zariski geometries from geometric model theory will be sketched (B. Zilber, AD). This is joint work with John Harding, Boris Zilber, and Chris Isham.

### 28/07/2011, 14:00 — 15:00 — Room P3.10, Mathematics Building

Yasuyoshi Yonezawa, *Univ. of Bonn*

### A specialized Kauffman polynomial using 4-valent planar diagrams

I want to discuss about specialized Kauffman polynomial using 4-valent planar diagrams. A problem is can we categorify this polynomial.

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.*