Seminário de Teoria Quântica do Campo Topológica   RSS

Sessões anteriores

11/10/2017, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Luis Miguel Pereira, IPFN, Instituto Superior Tecnico

Techniques for the summation of hypergeometric series and the quantum pendulum

In this informal seminar we will give a presentation based on practical examples of some of the several methods that can be used to sum hypergeometric series. These series include several known special functions and almost all combinatorial sums. Questions from the public will be welcomed. The goal of the seminar will ultimately be to set the stage for the preparation of a strategy to attack the problem of finding closed form solutions for the problem of the quantum pendulum (that is, to find closed form solutions for the Fourier coefficients of Mathieu functions), from stationary solutions to this problem in the Wigner formalism that were obtained by the speaker.

14/09/2017, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Luis Miguel Pereira, IPFN, Instituto Superior Tecnico

The quantum pendulum in the Wigner formalism and Mathieu functions

The time-independent Schrödinger equation with the pendulum's potential is the Mathieu equation from 19th century mathematical physics. Though there are many ways to approximate its solutions there are no known closed formulas for these solutions. In this talk we will show that with João Pedro Bizarro's modification of the Wigner-Berry transform it is possible to obtain closed formulas for several families of transforms of stationary observables.

06/09/2017, 16:00 — 17:30 — Sala P4.35, Pavilhão de Matemática
Beatriz Elizaga de Navascués, Instituto de Estructura de la Materia, Madrid

Unitary dynamics as a uniqueness criterion for the quantization of Dirac fields

It is well known that linear canonical transformations are not generally implemented as unitary operators in QFT. Such transformations include the dynamics that arises from linear field equations on the background spacetime. This evolution is specially relevant in nonstationary backgrounds, where there is no time-translational symmetry that can be exploited to select a quantum theory. We investigate whether it is possible to find a Fock representation for the canonical anticommutation relations of a Dirac field, propagating on homogeneous and isotropic cosmological backgrounds, on the one hand, and on tridimensional conformally ultrastatic spacetimes, on the other hand, such that the field evolution is unitarily implementable. First, we restrict our attention to Fock representations that are invariant under the group of symmetries of the system. Then, we prove that there indeed exist Fock representations such that the dynamics is implementable as a unitary operator. Finally, once a convention for the notion of particles and antiparticles is set, we show that these representations are all unitarily equivalent.

21/06/2017, 14:15 — 15:15 — Sala P3.10, Pavilhão de Matemática
Louis H. Kauffman, University of Illinois, Chicago, USA

Majorana Fermions, Braiding and Quantum Computing

We will discuss the mathematics of Majorana Fermions and the structure of representations of the Artin Braid group that are associated with them. We will discuss how one class of representations is related to the Temperley Lieb algebra, and another class of representations is related to a Hamiltonian constructed from the Bell-Basis solution of the Yang-Baxter equation, and the relationship of this Hamiltonian with the Kitaev spin chain. We will discuss how these braiding representations are related to topological quantum computing.

Ver também

presentation (pdf)

07/06/2017, 14:15 — 15:15 — Sala P3.10, Pavilhão de Matemática
Emmanuel Wagner, Université de Bourgogne, France

Trivalent TQFT and applications

MOY calculus has been introduced in the 90s to compute combinatorially the quantum link invariant associated with the Hopf algebra $U_q(\mathfrak{sl}(N))$. It associates to any decorated graph a Laurent polynomial in $q$. I will describe a TQFT-like functor which categorifies the MOY calculus and provides a new description of the $\mathfrak{sl}(N)$-homology.

(joint work with L.-H. Robert)

24/05/2017, 14:15 — 15:15 — Sala P3.10, Pavilhão de Matemática
Paul Wedrich, Imperial College, London

On colored link homologies

Link homology theories are powerful generalizations of classical (and quantum) link polynomials, which are being studied from a variety of mathematical and physical viewpoints. Besides providing stronger invariants, these theories are often functorial under link cobordisms and carry additional topological information. The focus of this talk is on the Khovanov-Rozansky homologies, which categorify the Chern-Simons/Reshetikhin-Turaev $\mathfrak{sl}(N)$ link invariants and their large $N$ limits. I will survey recent results about their behaviour under deformations as well as their stability at large $N$, which together lead to a rigorous proof of a package of conjectures originating in string theory.

10/05/2017, 14:15 — 15:15 — Sala P3.10, Pavilhão de Matemática
Marco Mackaay, Universidade do Algarve

A diagrammatic categorification of the higher level Heisenberg algebras

Khovanov defined a diagrammatic 2-category and conjectured (and partially proved) that it categorifies the level-one Heisenberg algebra. Since then, several interesting generalizations and applications have been found, e.g. Cautis and Licata's generalization involving Hilbert schemes and their construction of categorical vertex operators. However, these are all for level one. In my talk, I will explain Alistair Savage and my results on a generalization of Khovanov's original results for higher level Heisenberg algebras. This is work in progress.

26/04/2017, 14:15 — 15:15 — Sala P3.10, Pavilhão de Matemática
Lucile Vandembroucq, Universidade do Minho

Topological Complexity of the Klein Bottle

The notion of topological complexity of a space has been introduced by M. Farber in order to give a topological measure of the complexity of the motion planning problem in robotics. Surprisingly, the determination of this invariant for non-orientable surfaces has turned out to be difficult. A. Dranishnikov has recently established that the topological complexity of the non-orientable surfaces of genus at least 4 is maximal. In this talk, we will determine the topological complexity of the Klein bottle and extend Dranishnikov's result to all the non-orientable surfaces of genus at least 2. This is a work in collaboration with Daniel C. Cohen.

05/04/2017, 14:15 — 15:15 — Sala P3.10, Pavilhão de Matemática
John Huerta, Instituto Superior Técnico

M-theory from the superpoint revisited

The last talk we gave on this topic (in the meeting Iberian Strings 2017) was largely about the physics; here we focus on the mathematics. No prior knowledge will be assumed.

We define the process of invariant central extension: taking central extensions by cocycles invariant under a given subgroup of automorphisms of a Lie superalgebra. We give conditions that allow us to carve out the Lorentz group inside the automorphisms of Minkowski superspacetime, and prove that by successive invariant central extensions of the superpoint, we construct all superspacetimes up to dimension 11.

22/03/2017, 14:15 — 15:15 — Sala P3.10, Pavilhão de Matemática
Pedro Boavida, Dep. Matemática, Instituto Superior Técnico

Operads of genus zero curves and the Grothendieck-Teichmuller group

In Esquisse d’un programme, Grothendieck made the fascinating suggestion that the absolute Galois group of the rationals could be understood via its action on certain geometric objects, the (profinite) mapping class groups of surfaces of all genera. The collection of these objects, and the natural relations between them, he called the "Teichmuller tower”.

In this talk, I plan to describe a genus zero analogue of this story from the point of view of operad theory. The result is that the group of automorphisms of the (profinite) genus zero Teichmuller tower agrees with the Grothendieck-Teichmuller group, an object which is closely related to the absolute Galois group of the rationals. This is joint work with Geoffroy Horel and Marcy Robertson.

08/03/2017, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Aleksandar Mikovic, Universidade Lusófona

Hamiltonian analysis of the BFCG theory for a generic Lie 2-group

We perform a complete Hamiltonian analysis of the BFCG action for a general Lie 2-group by using the Dirac procedure. We show that the resulting dynamical constraints eliminate all local degrees of freedom which implies that the BFCG theory is a topological field theory.

Room 3.10 now confirmed.

15/02/2017, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
José Ricardo Oliveira, Univ. Nottingham

EPRL/FK Asymptotics and the Flatness Problem: a concrete example

Spin foam models are a "state-sum" approach to loop quantum gravity which aims to facilitate the description of its dynamics, an open problem of the parent framework. Since these models' relation to classical Einstein gravity is not explicit, it becomes necessary to study their asymptotics — the classical theory should be obtained in a limit where quantum effects are negligible, taken to be the limit of large triangle areas in a triangulated manifold with boundary.

In this talk we will briefly introduce the EPRL/FK spin foam model and known results about its asymptotics, proceeding then to describe a practical computation of spin foam and asymptotic geometric data for a simple triangulation, with only one interior triangle. The results are used to comment on the "flatness problem" — a hypothesis raised by Bonzom (2009) suggesting that EPRL/FK's classical limit only describes flat geometries in vacuum.

20/01/2017, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática
Urs Schreiber, Czech Academy of Sciences, Prague

Duality in String/M-theory from Cyclic cohomology of Super Lie $n$-algebras

I discuss how, at the level of rational homotopy theory, all the pertinent dualities in string theory (M/IIA/IIB/F) are mathematically witnessed and systematically derivable from the cyclic cohomology of super Lie $n$-algebras. I close by commenting on how this may help with solving the open problem of identifying the correct generalized cohomology theory for M-flux fields, lifting the classification of the RR-fields in twisted K-theory.

This is based on joint work with D. Fiorenza and H. Sati arxiv:1611.06536

Ver também

https://ncatlab.org/schreiber/show/Super+Lie+n-algebra+of+Super+p-branes

07/12/2016, 11:15 — 12:30 — Sala P3.10, Pavilhão de Matemática
Björn Gohla, Grupo de Física Matemática, Universidade de Lisboa

2+1 TQFTs with Defects

We will give an overview of TQFTs with defects along the lines of the preprint 3-dimensional defect TQFTs and their tricategories by Carqueville, Meusburger and Schaumann. In this paper ordinary Atiyah type functorial TQFTs on 2+1 cobordisms are generalized to 2+1 stratified cobordisms with decorations in a certain graph structure. The decoration graph structure together with a given TQFT of this type then give rise to a linear Gray-category with duals. This provides a unifying framework for well known 2+1 TQFTs.

See arxiv.org/abs/1603.01171

16/11/2016, 11:15 — 12:30 — Sala P3.10, Pavilhão de Matemática
Roger Picken, Instituto Superior Técnico

Invariants and TQFTs for cut cellular surfaces from finite groups and $2$-groups

We introduce the notion of a cut cellular surface (CCS), being a surface with boundary, which is cut in a specified way to be represented in the plane, and is composed of $0$-, $1$- and $2$-cells. We obtain invariants of CCS's under Pachner-like moves on the cellular structure, by counting colourings of the $1$-cells with elements of a finite group $G$, subject to a “flatness” condition for each $2$-cell. These invariants are also described in a TQFT setting, which is not the same as the usual $2$-dimensional TQFT framework. We study the properties of functions which arise in this context,associated to the disk, the cylinder and the pants surface, and derive general properties of these functions from topology. One such property states that the number of conjugacy classes of $G$ equals the commuting fraction of $G$ times the order of $G$.

We will comment on the extension of these invariants to 2-groups and their (higher) gauge theory interpretation.

This is work done in collaboration with Diogo Bragança (Dept. Physics, IST).

Ver também

https://arxiv.org/abs/1512.08263

02/11/2016, 11:15 — 12:30 — Sala P3.10, Pavilhão de Matemática
Benjamin Alarcón Heredia, Universidade Nova de Lisboa

On the representations of 2-groups in Baez-Crans 2-vector spaces

In this talk, we review the notions of 2-group, Baez-Crans 2-vector space, and 2-representations of 2-groups. We will study the irreducible and indecomposable 2-representations, and finally we will show that for a finite 2-group $G$ and base field $k$ of characteristic zero, this theory essentially reduces to the representation theory of the first homotopy group of $G$.

19/10/2016, 11:15 — 12:30 — Sala P3.10, Pavilhão de Matemática
Björn Gohla, Grupo de Física Matemática, Universidade de Lisboa

Poincaré Duality as Duality of Categories

We give a construction that associates a small category $\mathcal{C}(X)$ to a CW-decomposition $X$ of a manifold. We obtain interesting families of finite categories from spheres and projective spaces as examples. Under some conditions this category $\mathcal{C}(X)$ seems to represent the homotopy type of $X$. Interestingly, for finite dimensional $X$ the Poincaré dual $\hat{X}$ has associated to it the opposite category $(\mathcal{C}(X))^{\rm{op}}=\mathcal{C}(\hat{X})$.

This is part of a joint project with Benjamin Heredia.

Please note the new date of 19th October - this seminar was originally scheduled for 12th October.

07/07/2016, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática
John Huerta, CAMGSD, Instituto Superior Técnico

M-theory from the superpoint

One mysterious facet of M-theory is how a 10-dimensional string theory can "grow an extra dimension" to become 11-dimensional M-theory. Physically, the process is understood via brane condensation. Mathematically, Fiorenza, Sati, and Schreiber have proposed that brane condensation coincides with extending superspacetime, viewed as a Lie superalgebra, by the cocycle in Lie algebra cohomology which encodes the brane's WZW term. The resulting extension can be regarded as an "extended superspacetime" where still other super p-branes may live, whose condensates yield further extensions, and so on. In this way, all the super p-branes of string theory and M-theory fit into a hierarchy called "the brane bouquet". In this talk, we show how the brane bouquet grows out of the simplest kind of supermanifold, the superpoint.

22/06/2016, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Kowshik Bettadapura, Australian National University

Deformations of Super-Riemann Surfaces

In this talk I will detail aspects of the deformation theory of super-Riemann surfaces, in the spirit of Kodaira and Spencer. The goal is to argue a relation between: (1) deformations of super-Riemann surfaces; and (2) the obstruction theory of these deformations (to be thought of as complex supermanifolds). With this relation the consequences of a vanishing Kodaira-Spencer map of a deformation, in a particular instance, will become apparent and will lead on to further questions.

21/06/2016, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Hisham Sati, University of Pittsburgh and New York University

$L_\infty$-algebra of sphere-valued supercocycles in M-theory

In this talk we describe how super $L_\infty$-algebras capture the dynamics of the various fields and branes encoded in supercocycles associated with super Minkowski spacetimes at the rational level. We illustrate how rational 4-sphere-valued supercocycles in M-theory descend to supercocycles in type IIA string theory and capture the dynamics of the Ramond-Ramond fields predicted by the rational image of twisted K-theory, with the twist given by the usual B-field. We explicitly derive the M2/M5 $\leftrightarrow$ F1/Dp/NS5 correspondence via dimensional reduction of sphere valued $L_\infty$ supercocycles in rational homotopy theory. These results highlight, in the context of M-theory, the observations that supercocycles are still rich even in flat superspace and spectra are still rich even rationally. This is joint work with Domenico Fiorenza and Urs Schreiber.

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Organizadores correntes: Roger Picken, Marko Stošić.

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

CAMGSD FCT