In this talk I will describe how supersymmetric gauged sigma-models and mirror symmetry can be used to compute the quantum equivariant cohomology ring of toric manifolds in a simple way.

I shall present the construction of a certain geometric representation of KO-homology, the homological theory associated with KO-theory, inspired by the work of Paul Baum and Ronald Douglas on K-homology. I'll also discuss its application to the construction and classification of D-brane charges in the absence of a background B-field.

In this talk I present a new formalism for computing unambiguously open and closed topological string amplitudes. The formalism is based on a recursive method for computing invariants of algebraic curves recently proposed by Eynard and Orantin. The resulting amplitudes are non-perturbative in both the closed and the open moduli. The computational efficiency of this formalism can be dramatically increased by rewriting the amplitudes in a form that makes their transformation properties under the modular group manifest. As an application of this method I will illustrate how to compute open Gromov-Witten invariants for the $\mathbb{C}^3/\mathbb{Z}_3$ orbifold.

Berkovits' pure spinor string (type II) will be discussed in a general superspace background. The derivation of the supergravity constraints from classical BRST invariance will carefully be reviewed and new aspects added. Then it will be sketched how to extract the supersymmetry transformations of the physical fields.

Magnetic monopoles develop "abelian" singularities in rather general situations. At first, this was seen as a somewhat embarrassing fact by physicists, yet singular monopoles have become important in recent applications of gauge theory to geometry. In this talk, I will review singular monopoles in euclidean space, and explain how they are described by a complex curve with a signed intersection divisor. I will also discuss how these data can be calculated in the case of "nonabelian" charge two, as well as some features of their behaviour under deformations of the singularity locus.

In this talk, we will briefly revise the algebraic formulation of classical and quantum physical systems in terms of $C^\ast$-algebras. We will check that, in the case of atomic systems, this description is actually better motivated than the usual Dirac-von Neumann axiomatic structure of Quantum Theory, which becomes nearly inevitable in this context — except for the so-called measurement problem and the reduction of the wave packet, concerning the interaction between the quantum system and the measuring apparatus, which will be briefly analyzed.

Video talks @ Strings 2002
http://www.damtp.cam.ac.uk/strings02/avt/rastelli/
http://www.damtp.cam.ac.uk/strings02/avt/taylor/

We review recently introduced geometric tools used to characterize supersymmetric flux backgrounds, with special emphasis on Generalised Complex Geometry as developed by Hitchin. We then proceed to describe Exceptional Generalised Geometry, an extension of the latter formalism based on the exceptional Lie group $E_7(7)$ which achieves a full geometrization of all fluxes, including Ramond-Ramond fluxes. First we present the formal structure of the theory: the Exceptional Generalized Tangent bundle endowed with a non-trivial twisted topology and a corresponding gerbe structure, an  Exceptional Courant bracket and an Exceptional Generalized Metric in which the bosonic degrees of freedom of $11 D$ supergravity (the traditional metric and the fluxes) enter on equal footing. We then show how this formalism may be used to rewrite part of $11 D$ SUGRA in the language of $N=1$ $D=4$ supersymmetry and express the corresponding effective superpotential in a manifestly $E_7(7)$ form.

Video talk: Colloquium by Dijkgraaf @ KITP 2008 Mathematics and the Quantum Universe

Recently, several new techniques have been developed which allow one to much more efficiently calculate scattering amplitudes in gauge theories. The general lesson from these techniques seems to be that for gauge theories, Feynman diagram expansions are often not the most efficient tool to calculate scattering amplitudes. Instead, one should use structures such as the helicity configuration of the amplitude and analyticity in the external momenta. In this talk, I discuss how many of these observations carry over directly from gauge theories to open string theory, and how they can be used to calculate alpha'-corrections to effective field theory amplitudes. It turns out that, viewed as an effective quantum field theory, open string theory may actually be much simpler and more elegant than one would expect.

Large black holes in an asymptotically Anti-de Sitter spacetime have a dual description in terms of approximately thermal states in the boundary CFT. The reflecting boundary conditions of AdS prevent such black holes from evaporating completely. However, this situation is not ideal for the formulation of the black hole information paradox, which requires the ability of the black hole to evaporate. This can be accomplished by making the boundary of AdS partially absorptive, corresponding to a particular deformation of the gauge theory in view of the AdS/CFT correspondence. I will discuss a simple construction that produces the necessary changes on the boundary conditions. The model couples a scalar field in $1+1$ dimensions to a scalar field on AdS. The interaction is localized at the boundary and leads to partial transmission into the auxiliary space. Evaporation of the large black hole corresponds to cooling down the CFT by transferring energy to an external sector. 

A new point of view on the calculation of the exact spectrum of two dimensional sigma models in finite volume is presented. We will start from the bootstrap S-matrix, asymptotic Bethe ansatz equations, pass through the construction of bound states and Y-system (TBA), and finally use the integrable T-system (Hirota equation) to end up with convenient Destri-de-Vega-like integral equations to solve the problem numerically. The approach will be demonstrated on the $O(4)$-sigma model.