Supersymmetry in complex geometry: PDF files

4-9 January 2009, IPMU, Japan

Home | Venue | Schedule | Program | Poster | PDF

PDF files of the talks

Cecilia Albertsson
Doubled geometry and string theory (PDF file, 42 Mb)

Doubled geometry is a geometric way to describe non-geometric spaces that are relevant when compactifying string theory to render it consistent with observations. The formalism of doubled geometry bears many superficial similarities to that of generalised geometry, which has prompted several attempts to uncover any actual relations between the two. To date, however, such relations, if they exist, remain elusive. I will review the motivation and construction of doubled geometry, and illustrate its application in string theory.

Ryushi Goto
Holomorphic Poisson structures and deformations of generalized Kahler structures

Lecture 1 (PDF, 144 Kb) | Lecture 2 (PDF, 160 Kb)

I would like to talk my recent works about unobstructed deformations of generalized Kahler structures. Given small deformations {J_t} of generalized complex structures on a compact generalized Kahler manifolds with one pure spinor, there exist corresponding deformations {J_t, psi_t} of generalized Kahler structures. This is an analogous to the fact that small deformations of a compact Kahler manifold is still Kahlerian. A holomorphic Poisson structure on a compact Kahler manifold provides deformations of generalized complex structures. Thus applying our theorem, we obtain deformations of non-trivial generalized Kahler structures starting from the ordinary Kahler structure by using Poisson structure. Unobstructed deformations of generalized Kahler structures on Fano surfaces, Hirzebruch surfaces and several 3-folds will be discussed. I also obtain a correspondence between generalized Kahler submanifolds and Poisson submanfolds.

Geo Grantcharov
Neutral hypercomplex structures (PDF file, 112 Kb)

A neutral hypercomplex structure consists of a complex structure and a couple of product structures, mutually anti-commuting. Such structure is naturally associated with a metric of neutral signature, which in dimension four is anti-self-dual. A special case of this structure also is the hypersymplectic structure introduced by N.Hitchin. We notice first that most of the compact complex surfaces with vanishing first Chern class admit such a structure. Then we will discuss the reduction of neutral hypercomplex structures and its relation with generalized geometry.

Akito Futaki
Hilbert series and obstructions to asymptotic semistability (PDF file, 148 Kb)

Given a polarized manifold there are obstructions for asymptotic semistability described as integral invariants. One of them is an obstruction to the existence for the first Chern class of the polarization to admit a constant scalar curvature Kahler (cscK) metric. A natural question is whether or not the other obstructions are linearly dependent on the obstruction to the existence of a cscK metric. The purpose of this talk is to show that this is not the case by exhibiting toric Fano threefolds in which these obstructions span at least two dimension. To see this we show that on toric Fano manifolds these obstructions are obtained as derivatives of the Hilbert series.

Luca Martucci
Flux vacua in string theory, generalized calibrations
and supersymmetry breaking (PDF file, 1648 Kb)

I will discuss how supersymmetric type II string vacua with generic fluxes can by completely characterized in terms of appropriately defined generalized calibrations, that naturally fit into the context of Generalized Complex Geometry. These structures turn out to be crucial to prove that supersymmetric flux compactifications to 4D are solutions of the full set of 10D supergravity equations of motion with D-branes and orientifolds as localized sources. Furthermore, I will show how, within this same framework, one can construct non-supersymmetric vacua where supersymmetry is broken in a controlled way, violating only part of the underlying generalized calibration structures, but still allowing for a drastic simplification of the 10D supergravity equations.

Liviu Ornea
Holomorphic maps in generalized complex geometry (PDF file, 384 Kb)

We introduce a natural notion of holomorphic map between generalized complex manifolds and we prove some related results on Dirac structures and generalized Kaehler manifolds.

In addition to the regular research lectures, a number of review talks was prepared. We aim to give a clear and elementary introduction to the physics and mathematics of generalized complex and generalized Kahler geometry for non-specialists.

Liviu Ornea
Introduction to Dirac and generalized complex geometry (PDF file, 412 Kb)

I shall provide the necessary background for (real) Dirac structures and for generalized complex structures, in particular for generalized Kahler structures.

Institute for the Physics and Mathematics of the Universe