Talks

Two-dimensional non-linear sigma models on some supergroup manifolds are conformal field theories whether the action includes a Wess-Zumino term or not. These models are relevant for the worldsheet description of string theory in Anti-de Sitter backgrounds with Ramond-Ramond fluxes. The current algebra is an useful tool to study these theories. In these lectures I will review the construction of the current algebra. Then I will discuss some applications to the computation of the spectrum and integrability.

We first discuss quantum measure and integration theory. We then consider various anhomomorphic logics. Finally, we present some connections between the two theories. One connection is transferring a quantum measure to a measure on an anhomomorphic logic. Another is the creation of a reality filter that is stronger than Sorkin's preclusivity. This is accomplished by generating a preclusive coevent from a quantum measure. No prior knowledge of quantum measure theory or anhomomorphic logics will be assumed.

Although inflation is, by far, the best known mechanism to explain the observed properties of our Universe, there is still some room for alternative models, most of which implying a contracting phase preceding the current expanding one. Both phases are connected by a bounce at which the expansion rate must vanish. General relativity can only produce such a phase provided the spatial curvature is positive, in contradiction with the current observations. I will discuss the lines along which one can modify either the matter or the gravity sector (or both) in order to implement a bounce, and show the generic observable cosmological consequences it can induce, in particular in the microwave background.

In this talk, I will show that the five-dimensional Maxwell theory with a Chern-Simons coupling larger than a critical value in the Reissner-Nordstrom black hole geometry has tachyonic modes. This instability has an interesting property that it happens only at non-vanishing momenta, suggesting a spatially modulated phase transition in the holographically dual field theory. The final state after the phase transition has taken place will be discussed in detail in a special limit

We show that the generating function of the equivariant (generalized) Donaldson invariants of ${\bf R}^2 X {\Sigma}$ is captured by the solution of a thermodynamic Bethe ansatz equation. Based on a joint work with S. Shatashvili.

I will discuss a hybrid between Chern-Simons and Rozansky-Witten models. In particular, Wilson loops in this topological field theory are objects of a quantum deformation of the equivariant derived category of coherent sheaves.

I'll give an introduction to twistor-string theory, which is an attempt to reformulate supersymmetric gauge theory in four-dimensional space-time in terms of a certain generalisation of Gromov-Witten theory in twistor space. The resulting theory is closely related to the multi-dimensional residue calculus in G(k,n) (introduced in Cachazo's talk).

In the past year, motivated by physics, a rich structure has emerged from studying certain contour integrals in Grassmannians. Physical considerations single out a natural meromorphic form in G(k,n) with a cyclic structure. The residues obtained from these contour integrals have been shown to be invariants of a Yangian algebra. These residues also control what happens deep inside collisions of protons taking place at colliders like the Large Hadron Collider or LHC at CERN. Applications of the Global Residue Theorem give rise to relations among residues which ensure important physical properties.

I will give an overview of recent work with Davide Gaiotto and Greg Moore. This work relates the phenomenon of ''wall-crossing'' for BPS states in four-dimensional N=2 theories to a new construction of hyperkahler metrics. These metrics include in particular the metrics on moduli spaces of solutions to Hitchin equations. I will also briefly describe some extensions of this work to incorporate line and surface operators in the N=2 theory (in progress).

The study of D-branes at singular points of Calabi-Yau threefolds has revealed interesting connections between certain noncommutative algebras and singular algebraic varieties. In many respects, the choice of an appropriate noncommutative algebra is analogous to finding a resolution of singularities of the variety. We will explain this connection in detail, and outline a program for studying such ''noncommutative resolutions'' globally, for compact algebraic (Calabi--Yau) threefolds.

The Hilbert scheme X[n] of n points on variety X parameterizes length n, zero dimensional subschemes of X. When X is a smooth surface, X[n] is also smooth and a beautiful formula for its motive was determined by Gottsche. When X is a threefold, X[n] is in general singular, of the wrong dimension, and reducible. However if X is a smooth Calabi-Yau threefold, X[n] has a canonical virtual motive --- a motification of the degree zero Donaldson-Thomas invariants. We give a formula analogous to Gottsche's for the virtual motive of X[n]. The key computation gives a q-refinement of the classical formula of MacMahon which counts 3D partitions.