I'll discuss (from a mathematician's perspective) the process of extracting an effective quiver gauge theory from a D3-brane probing a singularity in a Calabi-Yau 3-fold.

Given two conformal field theories, one of which is defined on the upper half plane and the other on the lower half plane, one can ask for conformally invariant ways to join them along the real line. The resulting interface is called a defect line. These defects contain interesting information about the CFT, such as its symmetries, order-disorder dualities and T-dualities. They also provide relations between string theories on different target spaces.

Berry's phase is a beautiful and simple idea in quantum mechanics, with application to many areas of condensed matter physics. After reviewing the non-Abelian version of Berry's phase, I will explain how this concept naturally fits together with supersymmetry. I will then show how to compute Berry's phase in D-brane systems, using both traditional quantum mechanics, as well as AdS/CFT techniques.

I will reconsider toroidal compactifications of bosonic string theory with particular regard to the phases (cocycles) necessary for a consistent definition of the vertex operators, the boundary states, and the T-duality rules. I will then use these ingredients to compute the planar multi-loop partition function describing the interaction among magnetised or intersecting D-branes. Finally I will examine the degeneration limit of the 2-loop partition function and show that one obtains known and new tree-level 3-point correlators between twist fields.

We will review in simple terms how mirror symmetry works for general D-brane configurations, and in particular discuss how abstract mathematical concepts can be realized by physical LG models based on matrix factorizations. As for an application of these methods, we will explain how to explicitly compute exact, instanton-corrected effective superpotentials. (Directions to the room can be found on the triangle website http://brahms.mth.kcl.ac.uk/cgi-bin/main.pl?action=triangle)

Pure Yang-Mills theory (with no matter) in 2+1-dimensions can be thought of as a system of 1+1 integrable field theories coupled together. These theories decouple in an anisotropic limit. This fact makes confinement and the mass gap simple to understand. This is the only analytic approach to this problem which does not rely on strong-coupling assumptions. Exact knowledge of the S-matrix and form factors of these integrable theories can be used to reveal details of the static potential between quarks and the mass spectrum. If a further assumption is made, the isotropic case should also be accessible to this technique.

N=4 SYM is known to have a confinement-deconfinement type phase transition in finite volume as the temperature is raised. This phase transition has been conjectured to smoothly become the Hawking-Page transition between hot AdS space and an AdS black hole as the 't Hooft coupling becomes larger. I show that this phase transition at weak coupling is actually a topology changing transition for the VEVs of the scalar fields and Polyakov loop. This means that the high temperature phase cannot be, as previously thought, the black hole in the dual. I then argue for the existence of a new second order phase transition at a higher temperature to a new phase which has the right symmetries to be identified with the black hole. This is work based on the recent paper hep-th/0703100 with Umut Gursoy, Sean Hartnoll and Prem Kumar.

(the second lecture will be on Wednesday morning)

This talk continues the discussion of hep-th/0605038, applying the holographic formulation of self-dual theory to the Ramond-Ramond fields of type II supergravity. We formulate the RR partition function, in the presence of nontrivial H-fields, in terms of the wavefunction of an 11-dimensional Chern-Simons theory. Using the methods of hep-th/0605038 we show how to formulate an action principle for the RR fields of both type IIA and type IIB supergravity, in the presence of RR current. We find a new topological restriction on consistent backgrounds of type IIA supergravity, namely the fourth Wu class must have a lift to the H-twisted cohomology.

The solution of 2D Sigma models on superspaces (-groups,-cosets, etc.) is a problem with various potential applications in condensed matter theory and string theory, in particular in the context of the AdS/CFT correspondence. At the example of the PSU(1,1/ 2) sigma model, I shall illustrate some of the intriguing novel features of such theories, review a few recent results and assess the prospects for a complete or partial solution.