###
19/10/2016, 11:15 — 12:30 — Sala P3.10, Pavilhão de Matemática

Björn Gohla, *Grupo de Física Matemática, Universidade de Lisboa*

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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*

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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*

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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*

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$L_\infty$-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*

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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*

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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*

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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*

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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*

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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*

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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*

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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*

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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*

```
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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*

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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*

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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.

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

Roger Picken, *Instituto Superior Técnico*

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Actions of 2-groups, moduli spaces in higher gauge theory, and TQFT's

In the context of higher gauge theory (HGT) based on a 2-group, I will discuss how the language of double categories provides a natural description of 2-group actions on a category. One of the motivations is to understand moduli spaces of flat connections modulo gauge transformations in HGT, and this goal is achieved for some simple manifolds. I will also relate these ideas to a class of Topological Quantum Field Theories (TQFT's) for surfaces, obtained from finite groups and 2-groups.

This talk is based on callaborations with João Faria Martins, Jeffrey Morton and Diogo Bragança. It is also intended as preparation for the visit by Urs Schreiber, 15 Jan—14 Feb, 2016.

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23/10/2015, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

Benjamin Alarcón Heredia, *Universidade Nova de Lisboa*

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Bicategories, classifying spaces and homotopy pullbacks

In this continuation of my last talk I will give an introduction to the homotopy theory of bicategories. First I will present the way to convert bicategories to spaces, and then I will use this to describe homotopy pullbacks of homomorphisms of bicategories.

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09/10/2015, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

Benjamin Alarcón Heredia, *Universidade Nova de Lisboa*

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Homotopy theory using categories

This is an introductory talk about the homotopy theory of categories. I will present the classifying space of a category and the classical results of Thomason and Quillen for obtaining categorical descriptions of homotopy colimits and homotopy pullbacks.

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01/10/2015, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática

Igor Salom, *Institute of Physics, Belgrade, Serbia*

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Positive energy unitary irreducible representations of $\operatorname{osp}(1|2n)$ superalgebras

Orthosymplectic $\operatorname{osp}(1|2n)$ superalgebras are being considered as alternatives to $d$-dimensional Poincaré/conformal superalgebras and thus have significant potential relevance in various subfields of High Energy Physics and Astrophysics. Yet, due to mathematical difficulties, even the classification of their unitary irreducible representations (UIR's) has not been entirely accomplished. This is also true for the physically most important subclass of *positive energy* UIR's.

In this talk I will first demonstrate this classification for the $n=4$ case (that corresponds to four dimensional space-time). The classification is obtained by careful analysis of the Verma module structure, which is particularly subtle due to the existence of subsingular vectors. Based on these results I will then conjecture their generalization to the case of arbitrary $n$ (thus also including cases relevant in the string/brane context). In addition, I will show an elegant explicit realization of these UIR's that exists for (half)integer values of the conformal energy and that makes manifest the mathematical connection existing between UIR's of orthogonal and orthosymplectic algebras. The existence of this realization, per se, proves a part of the conjecture.

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17/09/2015, 16:30 — 17:30 — Sala P3.10, Pavilhão de Matemática

Jedrzej Swiezewski, *University of Warsaw*

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Geometrical diffeomorphism invariant observables for General Relativity and their applications

During the talk I will present a recent construction of observables for General Relativity invariant under spatial diffeomorphisms. The construction involves introducing a local structure representing the "observer" (based on arXiv:1403.8062). I will also present how those observables can be used to reduce the phase space of canonical General Relativity (based on arXiv:1506.09164).

If time permits, I will argue that the construction is particularly useful in spherically symmetric situations. This realization lead to a proposal of a scheme of reducing Loop Quantum Gravity to its spherically symmetric sector, which completes the standard, midisuperspace approach (based on arXiv:1410.5609).

#### Ver também

swiezewski.pdf