Week 22.11.2009 – 28.11.2009

A new framework within which an integrable defect may be described is introduced in the classical context (type II defect). It relays on the introduction of a new degree of freedom located on the defect and coupled to the discontinuity of the fields across the defect. Its features will be analyzed and few concrete examples will be provided. In particular, this new scheme allows to provide a description of the Tzitzeica model with a defect, which was not possible in the previous approach (type I defect).

Choose at random a matrix from the unitary group and cut a square block from it. The obtained matrix is a random contraction. Such matrices appear in a variety of mathematical contexts and are also used for modelling physical processes. After surveying some of these applications, I will talk about universal statistical patterns in the distribution of eigenvalues of random contractions in the limit of infinite matrix dimension.

An introductory review of the foundations of the Attractor Mechanism in extremal black holes in d=4 supergravity theories will be given. The issues of the classification of black hole attractors, of their stability and of the definition of an effective potential in the scalar manifold will be addressed. Finally, some recent developments will be shortly considered, such as the charge orbits and the moduli spaces of attractor solutions in N=2 and extended supergravities, and the generalization of the Attractor Mechanism to intersecting configurations of black branes in higher dimensions.

We study the sub-structure of the heterotic Kahler moduli space due to the presence of non-Abelian internal gauge fields from the perspective of the four-dimensional effective theory. Internal gauge fields can be supersymmetric in some regions of the Kahler moduli space but break supersymmetry in others. In the context of the four-dimensional theory, we investigate what happens when the Kahler moduli are changed from the supersymmetric to the non-supersymmetric region. Our results provide a low-energy description of supersymmetry breaking by internal gauge fields as well as a physical picture for the mathematical notion of bundle stability. Specifically, we find that at the transition between the two regions an additional anomalous U(1) symmetry appears under which some of the states in the low-energy theory acquire charges. We compute the associated D-term contribution to the four-dimensional potential which contains a Kahler-moduli dependent Fayet-Iliopoulos term and contributions from the charged states. We show that this D-term correctly reproduces the expected physics. Several mathematical conclusions concerning vector bundle stability are drawn from our arguments. We also discuss possible physical applications of our results to heterotic model building and moduli stabilization.

I will discuss recent progress in understanding non-geometric string backgrounds. These are inherently stringy constructions that are globally well- defined only up to T-dualities. One approach in analyzing such backgrounds is through the corresponding effective gauged supergravities. The classification of the latter in terms of embedding tensors provides us with a picture of the landscape of such backgrounds, realized in terms of non-geometric fluxes. Also, it is by now evident that a doubling of the coordinates of the underlying geometry is necessary for describing these backgrounds, since their stringy nature implies that winding modes should be treated on equal footing with the momentum modes. This leads us to doubled geometries and in particular twisted doubled tori, that are expected to underlie general supergravity gaugings. Of particular interest is the worldsheet description of such doubled geometries which involves chiral boson models. Although these models are not Lorentz invariant in general, for the twisted doubled tori Lorentz invariance is restored. Furthermore, the conditions for conformal invariance of those models agree with the conditions from minimizing the corresponding supergravity potential. This result firmly establishes that chiral boson models with twisted doubled tori target spaces are the worldsheet theories underlying a very broad class of supergravity gaugings.