# Seminário de Teoria Quântica do Campo Topológica

## Sessões anteriores

Páginas de sessões mais recentes: Seguinte 1 Mais recente

### 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.

### 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.

### 08/06/2016, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

Hisham Sati, *University of Pittsburgh and New York University*

### M-theory via rational homotopy theory

M-theory has proven to be very rich both from physics and mathematics points of view. We explain how the fields and their dynamics in M-theory can be succinctly captured by the 4-sphere, viewed via spectra and via cohomotopy. Working rationally allows us to have a mathematical handle on the sphere and to bring in interesting techniques and results from rational homotopy theory. Using spheres, we also describe the dynamics arising from various special points in the moduli space of M-theory, including reduction to type IIA and to heterotic string theory, inclusion into the bounding theory, inclusion of M-branes, and lifts. The detailed agreement with the dynamics expected fromthese theories is tantalizing and suggests an emerging deep picture on the mathematical structure of M-theory. This is joint work with Domenico Fiorenza and Urs Schreiber.

### 29/04/2016, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

Kadir Emir, *Eskisehir Osmangazi University, Turkey*

### Simplicial Cocommutative Hopf Algebras

In this talk, we define the Moore complex of any simplicial cocommutative Hopf algebra by using Hopf kernels which are defined quite different from the kernels of groups or various well-known algebraic structures. Furthermore, we will see that these Hopf kernels only make sense in the case of cocommutativity. We also introduce the notion of 2-crossed modules of cocommutative Hopf algebras and continue to talk about its categorical properties such as its relations with simplicial objects, Lie algebras, groups and also the Milnor-Moore theorem, as long as time allows.

Joint work with: João Faria Martins.

### 18/03/2016, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

Urs Schreiber, *Czech Academy of Sciences, Prague*

### Structure Theory for Higher WZW Terms VI

The famous Wess-Zumino-Witten term in 2d conformal field theory turns out to be a fundamental concept in the intersection of Lie theory and differential cohomology: it is a Deligne 3-cocycle on a Lie group whose Dixmier-Douady class is a Lie group 3-cocycle with values in $U(1)$ and whose curvature is the corresponding Lie algebra 3-cocycle, regarded as a left-invariant form. I explain how this generalizes to higher group stacks, and how there is a natural construction from any $(p+2)$-cocycle on any super $L_\infty$ algebra $\mathfrak{g}$ to a WZW-type Deligne cocycle on a higher super group stack $\tilde G$ integrating $\mathfrak{g}$. Every such higher WZW term serves as a local Lagrangian density, hence as an action functional for a $p$-brane sigma model on $\tilde G$ and I explain how the homotopy stabilizer group stacks of such higher WZW terms are the Lie integration of the Noether current algebras of these sigma-models. As an application, we consider the bouquet of iterated super $L_\infty$-extensions emanating from the superpoint, which turns out to be the “old brane scan” of string/M-theory completed by the branes “with tensor multiplet fields”, such as the D-branes and the M5-brane. I explain how applying the general theory of higher WZW terms to the $L_\infty$-extensions corresponding to the M2/M5-brane yields the “BPS charge M-theory super Lie algebra” together with a Lie integration to a super 6-group stack. From running a Serre spectral sequence we read off from this result how the naive nature of M5-brane charge as being in ordinary cohomology gets refined to twisted generalized cohomology in accordance with the conjectures stated in [1] section 6.3, [2] section 2.5.

- Hisham Sati, Geometric and topological structures related to M-branes, part I, Proc. Symp. Pure Math. 81 (2010), 181-236 (arXiv:1001.5020)
- Hisham Sati, Framed M-branes, corners, and topological invariants, (arXiv:1310.1060)

#### Ver também

https://ncatlab.org/schreiber/show/Structure+Theory+for+Higher+WZW+Terms

### 15/03/2016, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

Urs Schreiber, *Czech Academy of Sciences, Prague*

### Structure Theory for Higher WZW Terms V

The famous Wess-Zumino-Witten term in 2d conformal field theory turns out to be a fundamental concept in the intersection of Lie theory and differential cohomology: it is a Deligne 3-cocycle on a Lie group whose Dixmier-Douady class is a Lie group 3-cocycle with values in $U(1)$ and whose curvature is the corresponding Lie algebra 3-cocycle, regarded as a left-invariant form. I explain how this generalizes to higher group stacks, and how there is a natural construction from any $(p+2)$-cocycle on any super $L_\infty$ algebra $\mathfrak{g}$ to a WZW-type Deligne cocycle on a higher super group stack $\tilde G$ integrating $\mathfrak{g}$. Every such higher WZW term serves as a local Lagrangian density, hence as an action functional for a $p$-brane sigma model on $\tilde G$ and I explain how the homotopy stabilizer group stacks of such higher WZW terms are the Lie integration of the Noether current algebras of these sigma-models. As an application, we consider the bouquet of iterated super $L_\infty$-extensions emanating from the superpoint, which turns out to be the “old brane scan” of string/M-theory completed by the branes “with tensor multiplet fields”, such as the D-branes and the M5-brane. I explain how applying the general theory of higher WZW terms to the $L_\infty$-extensions corresponding to the M2/M5-brane yields the “BPS charge M-theory super Lie algebra” together with a Lie integration to a super 6-group stack. From running a Serre spectral sequence we read off from this result how the naive nature of M5-brane charge as being in ordinary cohomology gets refined to twisted generalized cohomology in accordance with the conjectures stated in [1] section 6.3, [2] section 2.5.

- Hisham Sati, Geometric and topological structures related to M-branes, part I, Proc. Symp. Pure Math. 81 (2010), 181-236 (arXiv:1001.5020)
- Hisham Sati, Framed M-branes, corners, and topological invariants, (arXiv:1310.1060)

#### Ver também

https://ncatlab.org/schreiber/show/Structure+Theory+for+Higher+WZW+Terms

### 26/02/2016, 11:30 — 12:30 — Sala P4.35, Pavilhão de Matemática

Urs Schreiber, *Czech Academy of Sciences, Prague*

### Structure Theory for Higher WZW Terms IV

The famous Wess-Zumino-Witten term in 2d conformal field theory turns out to be a fundamental concept in the intersection of Lie theory and differential cohomology: it is a Deligne 3-cocycle on a Lie group whose Dixmier-Douady class is a Lie group 3-cocycle with values in $U(1)$ and whose curvature is the corresponding Lie algebra 3-cocycle, regarded as a left-invariant form. I explain how this generalizes to higher group stacks, and how there is a natural construction from any $(p+2)$-cocycle on any super $L_\infty$ algebra $\mathfrak{g}$ to a WZW-type Deligne cocycle on a higher super group stack $\tilde G$ integrating $\mathfrak{g}$. Every such higher WZW term serves as a local Lagrangian density, hence as an action functional for a $p$-brane sigma model on $\tilde G$ and I explain how the homotopy stabilizer group stacks of such higher WZW terms are the Lie integration of the Noether current algebras of these sigma-models. As an application, we consider the bouquet of iterated super $L_\infty$-extensions emanating from the superpoint, which turns out to be the “old brane scan” of string/M-theory completed by the branes “with tensor multiplet fields”, such as the D-branes and the M5-brane. I explain how applying the general theory of higher WZW terms to the $L_\infty$-extensions corresponding to the M2/M5-brane yields the “BPS charge M-theory super Lie algebra” together with a Lie integration to a super 6-group stack. From running a Serre spectral sequence we read off from this result how the naive nature of M5-brane charge as being in ordinary cohomology gets refined to twisted generalized cohomology in accordance with the conjectures stated in [1] section 6.3, [2] section 2.5.

- Hisham Sati, Geometric and topological structures related to M-branes, part I, Proc. Symp. Pure Math. 81 (2010), 181-236 (arXiv:1001.5020)
- Hisham Sati, Framed M-branes, corners, and topological invariants, (arXiv:1310.1060)

#### Ver também

https://ncatlab.org/schreiber/show/Structure+Theory+for+Higher+WZW+Terms

### 23/02/2016, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

Urs Schreiber, *Czech Academy of Sciences, Prague*

### Structure Theory for Higher WZW Terms III

The famous Wess-Zumino-Witten term in 2d conformal field theory turns out to be a fundamental concept in the intersection of Lie theory and differential cohomology: it is a Deligne 3-cocycle on a Lie group whose Dixmier-Douady class is a Lie group 3-cocycle with values in $U(1)$ and whose curvature is the corresponding Lie algebra 3-cocycle, regarded as a left-invariant form. I explain how this generalizes to higher group stacks, and how there is a natural construction from any $(p+2)$-cocycle on any super $L_\infty$ algebra $\mathfrak{g}$ to a WZW-type Deligne cocycle on a higher super group stack $\tilde G$ integrating $\mathfrak{g}$. Every such higher WZW term serves as a local Lagrangian density, hence as an action functional for a $p$-brane sigma model on $\tilde G$ and I explain how the homotopy stabilizer group stacks of such higher WZW terms are the Lie integration of the Noether current algebras of these sigma-models. As an application, we consider the bouquet of iterated super $L_\infty$-extensions emanating from the superpoint, which turns out to be the “old brane scan” of string/M-theory completed by the branes “with tensor multiplet fields”, such as the D-branes and the M5-brane. I explain how applying the general theory of higher WZW terms to the $L_\infty$-extensions corresponding to the M2/M5-brane yields the “BPS charge M-theory super Lie algebra” together with a Lie integration to a super 6-group stack. From running a Serre spectral sequence we read off from this result how the naive nature of M5-brane charge as being in ordinary cohomology gets refined to twisted generalized cohomology in accordance with the conjectures stated in [1] section 6.3, [2] section 2.5.

- Hisham Sati, Geometric and topological structures related to M-branes, part I, Proc. Symp. Pure Math. 81 (2010), 181-236 (arXiv:1001.5020)
- Hisham Sati, Framed M-branes, corners, and topological invariants, (arXiv:1310.1060)

#### Ver também

https://ncatlab.org/schreiber/show/Structure+Theory+for+Higher+WZW+Terms

### 15/02/2016, 16:30 — 17:30 — Sala P3.10, Pavilhão de Matemática

Yasuyuki Kawahigashi, *University of Tokyo*

### Moonshine, conformal field theory and operator algebras

The Moonshine conjecture is about mysterious relations between the Monster group and elliptic modular functions. It has been solved in the context of vertex operator algebras, which give an algebraic axiomatization of chiral conformal field theory. Another axiomatization is given in terms of operator algebras. We present our new result going from the former framework to the latter and back. No knowledge of these topics is assumed.

### 29/01/2016, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

Urs Schreiber, *Czech Academy of Sciences, Prague*

### Structure Theory for Higher WZW Terms II

The famous Wess-Zumino-Witten term in 2d conformal field theory turns out to be a fundamental concept in the intersection of Lie theory and differential cohomology: it is a Deligne 3-cocycle on a Lie group whose Dixmier-Douady class is a Lie group 3-cocycle with values in $U(1)$ and whose curvature is the corresponding Lie algebra 3-cocycle, regarded as a left-invariant form. I explain how this generalizes to higher group stacks, and how there is a natural construction from any $(p+2)$-cocycle on any super $L_\infty$ algebra $\mathfrak{g}$ to a WZW-type Deligne cocycle on a higher super group stack $\tilde{G}$ integrating $\mathfrak{g}$. Every such higher WZW term serves as a local Lagrangian density, hence as an action functional for a $p$-brane sigma model on $\tilde{G}$, and I explain how the homotopy stabilizer group stacks of such higher WZW terms are the Lie integration of the Noether current algebras of these sigma-models. As an application, we consider the bouquet of iterated super $L_\infty$-extensions emanating from the superpoint, which turns out to be the “old brane scan” of string/M-theory completed by the branes “with tensor multiplet fields”, such as the D-branes and the M5-brane. I explain how applying the general theory of higher WZW terms to the $L_\infty$-extensions corresponding to the M2/M5-brane yields the “BPS charge M-theory super Lie algebra” together with a Lie integration to a super 6-group stack. From running a Serre spectral sequence we read off from this result how the naive nature of M5-brane charge as being in ordinary cohomology gets refined to twisted generalized cohomology in accordance with the conjectures stated in [1] section 6.3, [2] section 2.5.

- Hisham Sati, Geometric and topological structures related to M-branes, part I, Proc. Symp. Pure Math. 81 (2010), 181-236 (arXiv:1001.5020)
- Hisham Sati, Framed M-branes, corners, and topological invariants, (arXiv:1310.1060)

#### Ver também

https://ncatlab.org/schreiber/show/Structure+Theory+for+Higher+WZW+Terms

### 26/01/2016, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

Urs Schreiber, *Czech Academy of Sciences, Prague*

### Structure Theory for Higher WZW Terms I

The famous Wess-Zumino-Witten term in 2d conformal field theory turns out to be a fundamental concept in the intersection of Lie theory and differential cohomology: it is a Deligne 3-cocycle on a Lie group whose Dixmier-Douady class is a Lie group 3-cocycle with values in $U(1)$ and whose curvature is the corresponding Lie algebra 3-cocycle, regarded as a left-invariant form. I explain how this generalizes to higher group stacks, and how there is a natural construction from any $(p+2)$-cocycle on any super $L_\infty$ algebra $\mathfrak{g}$ to a WZW-type Deligne cocycle on a higher super group stack $\tilde{G}$ integrating $\mathfrak{g}$. Every such higher WZW term serves as a local Lagrangian density, hence as an action functional for a $p$-brane sigma model on $\tilde{G}$, and I explain how the homotopy stabilizer group stacks of such higher WZW terms are the Lie integration of the Noether current algebras of these sigma-models. As an application, we consider the bouquet of iterated super $L_\infty$-extensions emanating from the superpoint, which turns out to be the “old brane scan” of string/M-theory completed by the branes “with tensor multiplet fields”, such as the D-branes and the M5-brane. I explain how applying the general theory of higher WZW terms to the $L_\infty$-extensions corresponding to the M2/M5-brane yields the “BPS charge M-theory super Lie algebra” together with a Lie integration to a super 6-group stack. From running a Serre spectral sequence we read off from this result how the naive nature of M5-brane charge as being in ordinary cohomology gets refined to twisted generalized cohomology in accordance with the conjectures stated in [1] section 6.3, [2] section 2.5.

- Hisham Sati, Geometric and topological structures related to M-branes, part I, Proc. Symp. Pure Math. 81 (2010), 181-236 (arXiv:1001.5020)
- Hisham Sati, Framed M-branes, corners, and topological invariants, (arXiv:1310.1060)

#### Ver também

https://ncatlab.org/schreiber/show/Structure+Theory+for+Higher+WZW+Terms

### 22/01/2016, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

Marko Stosic, *CAMGSD, Instituto Superior Técnico*

### Homological knot invariants, A-polynomial and integrality properties

The theory of homological knot invariants - the categorification of polynomial knot invariants - appeared 15 years ago, and has been very active ever since. As in the case of the the quantum polynomial knot invariants, they turned out to be related with numerous different fields of mathematics (including topology, quantum groups, representation theory, homological algebra, von Neumann algebras, etc.). In this talk I'll present a basic overview of this categorification in the case of the HOMFLY-PT invariants - both concerning their definition and their properties. Finally, a particular recent application will be shown related to the physics interpretation via BPS invariants, which implies some surprising integrality properties of a pure number theoretical interest.

### 11/12/2015, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

Marko Stosic, *CAMGSD, Instituto Superior Técnico*

### Homological knot invariants, A-polynomial and integrality properties

The theory of homological knot invariants - the categorification of polynomial knot invariants - appeared 15 years ago, and has been very active ever since. As in the case of the the quantum polynomial knot invariants, they turned out to be related with numerous different fields of mathematics (including topology, quantum groups, representation theory, homological algebra, von Neumann algebras, etc.). In this talk I'll present a basic overview of this categorification in the case of the HOMFLY-PT invariants - both concerning their definition and their properties. Finally, a particular recent application will be shown related to the physics interpretation via BPS invariants, which implies some surprising integrality properties of a pure number theoretical interest.

Páginas de sessões mais antigas: Anterior 3 4 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*.