Topological Quantum Field Theory Seminar   RSS

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.


  • 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
Room 3.10 is confirmed

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,

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.

Room 3.10 is now confirmed

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.
Room 3.10 is confirmed

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
, 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 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
Note also another seminar session by the same speaker on Friday 30th November

23/11/2012, 11:30 — 12:30 — Room P4.35, Mathematics Building
, 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
, Institut Mathématique de Jussieu, Paris

Indecomposable modules over a Kuperberg-Khovanov algebras

The 𝔰𝔩 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 𝔰𝔩 3 -web algebras) and some particular projective modules called web-modules over these algebras. I will give a combinatorial caracterisation of indecomposable web-modules.
Categorification mini-workshop

21/11/2012, 14:00 — 15:00 — Room P4.35, Mathematics Building
Marco Mackaay, Univ. Algarve

𝔰𝔩 3 web algebras, cyclotomic KLR algebras and categorical quantum skew Howe duality

I will introduce 𝔰𝔩 3 web algebras K(S), which involve Kuperberg's 𝔰𝔩 3 web space W(S) and Khovanov 𝔰𝔩 3 foams with boundary in W(S). These algebras are the 𝔰𝔩 3 analogues of Khovanov's 𝔰𝔩 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).
Categorification mini-workshop

21/11/2012, 11:00 — 12:00 — Room P4.35, Mathematics Building
, 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.
Categorification mini-workshop

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 =Cz p have no solutions (a,b,c) such that gcd(a,b)=1 and 13 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(13 ). Then we attach Frey curves E over Q(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.
Duration 90 minutes or slightly less

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.
Second of two talks in an Informal Categorication Afternoon about current research projects in the area of categorification

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.

First of two talks in an Informal Categorication Afternoon about current research projects in the area of categorification.

12/01/2012, 11:30 — 12:30 — Room P12, Mathematics Building
, 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 p3. 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 αy p=z p as the motivation to introduce informally the original strategy (α=0) and illustrate one of its first refinements (for α=1). Then we will discuss some further generalizations that recently led to the solution of equations of the form x 5+y 5=dz p.

06/12/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
, 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 SO(2N)BMW and the q-Schur algebras. I will then present a subcategory of the Schur category which categorifies the SO(2N)BMW algebra (joint with E. Wagner).
Support: FCT, CAMGSD, New Geometry and Topology.. (Room P4.35 still to be confirmed)

28/10/2011, 10:30 — 11:30 — Room P3.10, Mathematics Building
, 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.
Support: FCT, CAMGSD, New Geometry and Topology

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.
FCT, CAMGSD, New Geometry and Topology.

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.
FCT, CAMGSD, New Geometry and Topology.

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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.