*In this talk I will discuss some features of interacting conformal higher spin theories in 2+1 dimensions. This is done in the context of Chern-Simons theory (giving e.g. the complete spin 2 covariant spin 3 sector) and a higher spin coupled unfolded equation for the scalar singleton. One motivation for studying these theories is that their non-linear properties are rather poorly understood contrary to the situation for the Vasiliev type theory in this dimension which is under much better control. Another reason for the interest in these theories comes from AdS4/CFT3 and the possibility that Neumann/mixed bc for bulk higher spin fields may lead to conformal higher spin fields governed by Chern-Simons terms on the boundary. These theories generalise the spin 2 gauged BLG-ABJ(M) theories found a few years ago to higher spins than 2.*

# Past String Theory Seminar

*A Fueter map between two hyperKaehler manifolds is a solution of a Cauchy-Riemann-type equation in the quaternionic context. In this talk I will describe relations between Fueter maps, generalized Seiberg-Witten equations, and Yang-Mills instantons on G2-manifolds (so called G2-instantons).*

Calabi-Yau spaces provide well-understood examples of supersymmetric vacua in supergravity. The supersymmetry conditions on such spaces can be rephrased as the existence and integrability of a particular geometric structure. When fluxes are allowed, the conditions are more complicated and the analogue of the geometric structure is not well understood.

In this talk, I will review work that defines the analogue of Calabi-Yau geometry for generic D=4, N=2 supergravity backgrounds. The geometry is characterised by a pair of structures in generalised geometry, where supersymmetry is equivalent to integrability of the structures. I will also discuss the extension AdS backgrounds, where deformations of these geometric structures correspond to exactly marginal deformations of the dual field theories.

M5-branes on 4-manifolds M_4 realized as co-associatives in G_2 give rise to 2d (0,2) superconformal theories. In this talk I will propose a duality between these 2d (0,2) theories and 4d topological theories, which are sigma-models from M_4 into the Nahm moduli space.

*I will review the recently exposed connection between N=2 superconformal field theories in four dimensions and vertex operator algebras (VOAs). I will outline some general features of the VOAs that arise in this manner and describe the manner in which they reflect four-dimensional operations such as gauging and Higgsing. Time permitting, I will also touch on the modular properties of characters of these VOAs.*

Pure spinor space, a cone over orthogonal Grassmannian OGr(5,10), is a central concept in the Berkovits formulation of string theory. The space of states of the beta-gamma system on pure spinors is tensor factor in the Hilbert space of string theory . This is why it would be nice to have a good definition of this space of states. This is not a straightforward task because of the conical singularity of the target. In the talk I will explain a strategy for attacking conical targets. In the case of pure spinors the method gives a formula for partition function of pure spinors.

Generalised Geometry is a very powerful tool to study gravity duals of strongly coupled gauge theories. In this talk I will discuss how Exceptional Geometry can be used to study marginal deformations of N=1 SCFT's in 4 and 3 dimensions.

Heterotic vacua, defined with a holomorphic bundle and connection satisfying hermitian Yang-Mills, realise four-dimensional chiral gauge theories. We exploit the rich interplay between four-dimensional physics, supersymmetry and geometry to construct a natural Kaehler metric for the moduli space, with a shockingly simple Kaehler potential. Along the way, we discover a natural geometric structure for the heterotic moduli.

Recently, there has been some progress in examining mirror symmetry beyond Calabi-Yau threefolds. I will discuss how this is related to flux vacua of type II supergravity on eight-dimensional manifolds equipped with SU(4)-structure. It will be shown that the natural framework to describe such vacua is generalized complex geometry. Two classes of type IIB solutions will be given, one of which is complex, the other symplectic, and I will describe in what sense these are mirror to one another.