New trends in generalized geometry and Poisson geometry

ICMAT, Madrid

12-15 November, 2014











icmat
Instituto de Ciencias Matemáticas
C/ Nicolás Cabrera, 13-15
  Campus Cantoblanco - UAM
  28049 Madrid, Spain
  www.icmat.es
  Telephone: +34 91 2999700

 

Schedule

 

Wednesday Thursday Friday Saturday
10:00 Bursztyn Rubio Bailey
11:00 Coffee Break Coffee Break Coffee Break
11:30 Marcut De la Ossa Pym
12:00 - 13:00 - Registration
13:00 - 15:00 - Lunch
Lunch Lunch
15:00 Cavalcanti Zambon Strickland-Constable
16:00 Coffe Break Coffe Break Coffe Break
16:30 Gualtieri Iglesias

 

 

 

Programme

 

Michael Bailey (University of Utrech, The Netherlands)

Local analytic structure of generalized complex structures

Abstract: Globally, generalized complex manifolds may look very different from complex manifolds. However, about a point, generalized complex structures look like a product of a symplectic manifold with a holomorphic Poisson structure vanishing at the point. In this decomposition, the holomorphic Poisson structure is unique up to diffeomorphism. Furthermore, in a generalized complex manifold, the locus of points of complex type have the structure of a complex analytic space. I will explain these facts in detail and describe some applications.

 

Henrique Bursztyn (IMPA, Rio de Janeiro, Brasil)

Poisson geometry and Lie theory of vector bundles

Abstract: In this talk I will discuss the connection between Poisson geometry and certain "double structures" that arise in the theory of Lie groupoids and algebroids. I will focus the discussion on VB-groupoids and VB-algebroids, which are vector bundles in the world of Lie groupoids and algebroids. Just as Lie groupoids are common generalizations of manifolds and Lie groups, VB-groupoids generalize vector bundles and linear representations. I will explain the Lie theory relating VB-groupoids and VB-algebroids, including examples and applications to more general double structures.
Connections between double structures and generalized geometry will be pointed out along the way. This is joint work with A. Cabrera and M. del Hoyo.

 

Gil Cavalcanti (University of Utrech, The Netherlands)

Examples and counter-examples of log-symplectic manifolds

Abstract: We study topological properties of log-symplectic structures and produce examples of compact manifolds with such structures. Notably we show that several symplectic manifolds do not admit log-symplectic structures and several log-symplectic manifolds do not admit symplectic structures, for example #mCP2#nCP2 has log-symplectic structures if and only if m,n > 0 while they only have symplectic structures for m = 1. We introduce surgeries that produce log-symplectic manifolds out of symplectic manifolds and show that for any simply connected 4-manifold M, the manifolds M#(S2 × S2) and M#CP2#CP2 have log-symplectic structures and any compact oriented log-symplectic four-manifold can be transformed into a collection of symplectic manifolds by reversing these surgeries.

 

Xenia De la Ossa (University of Oxford, UK)

Moduli space of heterotic string compactifications

 

David Iglesias Ponte (La Laguna, Spain)

Symplectic groupoids of Poisson homogeneous spaces

Abstract: Symplectic groupoids were introduced as a tool to quantize Poisson manifolds. Although the precise obstructions to integrating Poisson manifolds have been obtained, integrability of large classes of Poisson structures (Poisson Lie groups, quotients of integrable Poisson manifolds, log symplectic manifolds...) has been studied, giving a precise description of the corresponding symplectic groupoids. In this talk, it will be shown that any Poisson homogeneous space \(G/H\) admits a symplectic groupoid \(\mathcal{G}(G/H)\). Showing that the cotangent Lie algebroid \(T^*(G/H)\) can be obtained as the reduction of a Hamiltonian action on a certain Dirac structure \(\mathfrak{l}_G\) on \(G\), the symplectic groupoid \(\mathcal{G}(G/H)\) is obtained performing a reduction procedure on a presymplectic groupoid \(\Sigma\) which integrates \(\mathfrak{l}_G\).

 

Marco Gualtieri (University of Toronto, Canada)

Log symplectic vs generalized complex geometry

 

Ioan Marcut (Radboud University, Nijmegen, The Nederlands)

On source-compact Poisson manifold

Abstract: I will talk about the class of Poisson manifolds whose Weinstein groupoid have compact source-fibers. These ``source-compact'' Poisson manifolds have a very interesting, but manageable deformation theory. I will also explain the main technical tool in their study, which is the embedding theorem into rigid Poisson manifolds.

 

Brent Pym (Oxford University)

Quantum deformations of projective three-space

Abstract: Complex projective space admits no nontrivial deformations as a complex manifold, but it can be deformed as a generalized complex manifold by introducing a holomorphic Poisson bracket. D-branes in these generalized complex manifolds are then closely linked with noncommutative algebraic geometry via deformation quantization. I will describe the classification of these Poisson brackets and their quantizations in low dimensions.

 

Roberto Rubio (IMPA, Rio de Janeiro, Brasil)

Generalized geometry of type Bn

Abstract: The Courant algebroid T+T* is the starting point of generalized geometry. By adding a trivial line bundle 1 to it, we get a new Courant algebroid: T+T*+1, with a canonical pairing of signature (n+1,n). Its group of orthogonal transformations is SO(n+1,n), of Lie type Bn. The bundle T+T*+1 is thus the starting point of generalized geometry of type Bn. In this setting, some aspects of generalized geometry become more complex or interesting. Presenting these aspects, and comparing them with usual generalized geometry, is the main aim of this talk. For instance, the symmetries of T+T*+1 include not only the diffeomorphisms and B-fields, but also the A-fields. More interestingly, Bn-generalized complex structures are defined in both even and odd-dimensional manifolds, and type change can occur already on surfaces. On the other hand, the bundle T+T*+1 is the simplest instance of a transitive Courant algebroid. In the last part of the talk I will mention ongoing joint work with Mario Garcia-Fernandez and Carl Tipler where we approach the Strominger system by means of an admissible metric for a suitable transitive Courant algebroid. This approach shows interesting links with the moduli problem for holomorphic Courant algebroids on Calabi-Yau manifolds.

 

Charles Strickland-Constable (University of Hamburg, Germany)

Generalised geometry, supergravity and supersymmetric backgrounds

Abstract: I will present work on the description of subsectors of supergravity theories in the language of generalised geometry. The construction rewrites the bosonic sector as an analogue of Einstein gravity in generalised geometry, while the operators appearing in the fermionic equations and supersymmetry transformations are formulated in terms of projections of a torsion-free generalised connection. The latter of these features gives rise to an elegant description of supersymmetric Minkowski backgrounds of string/M-theory, with all fluxes switched on, as the analogues of spaces with special holonomy.

 

Marco Zambon (KU Leuven)

Simultaneous deformations and Poisson geometry

Abstract: Given two interrelated (geometric, algebraic,..) structures, we consider their simultaneous deformations. We establish conditions under which the deformation problem is governed by an L-infinity algebra, and construct the latter explicitly. The main tool is Ted Voronov's derived bracket construction. We than apply the above to a few instances in Poisson and generalized complex geometry. In particular, we will discuss the deformations of an ordinary complex structure to generalized complex structures in arbitrary exact Courant algebroids. This is joint work with Yael Fregier.

 


 Organized by:

Nigel Hitchin
(Oxford), chair
Mario Garcia-Fernandez
(ICMAT)
Oscar García-Prada
(ICMAT)

 Contact:






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