Oxford

Abstract: There has been considerable interest recently in the relation between certain 3d supersymmetric Chern-Simons theories, M2-branes, and the AdS_4/CFT_3 correspondence. In this talk I will show that the moduli space of a 3d N=2 Chern-Simons quiver gauge theory always contains a certain branch of the moduli space of a parent 4d N=1 quiver gauge theory. In particular, starting with a 4d quiver theory dual to a Calabi-Yau 3-fold singularity, for certain general choices of Chern-Simons levels this branch of the corresponding 3d theory is a Calabi-Yau 4-fold singularity. This leads to a simple general method for constructing candidate 3d N=2 superconformal Chern-Simons quivers with AdS_4 gravity duals. As simple, but non-trivial, examples, I will identify a family of Chern-Simons quiver gauge theories which are candidate AdS_4/CFT_3 duals to an infinite class of toric Sasaki-Einstein seven-manifolds with explicit metrics.

Abstract: Recent developments in quantum field theory and twistor-string theory have thrown up surprising structures in the perturbative approach to gravity that cry out for a non-perturbative explanation. Firstly the MHV scattering amplitudes, those involving just two left handed and n-2 right handed outgoing gravitons are particularly simple, and a formalism has been proposed that constructs general graviton scattering amplitudes from these MHV amplitudes as building blocks. This formalism is chiral and suggestive of deep links with Ashtekar variables and twistor theory. In this talk, the MHV amplitudes are calculated ab initio by considering scattering of linear gravitons on a fully nonlinear anti-self-dual background using twistor theory, and a twistor action formulation is provided that produces the MHV formalism as its Feynman rules.

Abstract: It is known that many Calabi-Yau manifolds form a connected web. The question of whether all CY manifolds form a single web depends on the degree of singularity that is permitted for the varieties that connect  the distinct families of smooth manifolds. If only conifolds are allowed then, since shrinking two-spheres and three-spheres to points cannot affect the fundamental group, manifolds with different fundamental groups will form disconnected webs. We examine these webs for the tip of the distribution of CY manifolds where the Hodge numbers $(h^{11},h^{21})$ are both small. In the tip of the distribution the quotient manifolds play an important role. We generate via conifold transitions from these quotients a number of new manifolds. These include a manifold with $\chi =-6$, that is an analogue of the $\chi=-6$ manifold found by Yau,  and manifolds with an attractive structure that may prove of interest for string phenomenology.

There is a non-degenerate 2-form on S^6, which is compatible with the almost complex structure that S^6 inherits from its inclusion in the imaginary octonions. Even though this 2-form is not closed, we may still define Lagrangian submanifolds. Surprisingly, they are automatically minimal and are related to calibrated geometry. The focus of this talk will be on the Lagrangian submanifolds of S^6 which are fibered by geodesic circles over a surface. I will describe an explicit classification of these submanifolds using a family of Weierstrass formulae.

We show how the circle method with a suitably chosen Gaussian weight can be used to count unweighted zeros of polynomials. Tschinkel's problem asks for the density of solutions to Diophantine equations with S-unit and integral variables.