Connections in Geometry and Physics - 2009

This talk is a report on joint work with Mohammed Abouzaid and Ludmil Katzarkov about mirror symmetry for blowups, from the perspective of the Strominger-Yau-Zaslow conjecture. Namely, we first describe how to construct a Lagrangian torus fibration on the blowup of a toric variety X along a codimension 2 subvariety S contained in a toric hypersurface. Then we discuss the SYZ mirror and its instanton corrections, to provide an explicit description of the mirror Landau-Ginzburg model (possibly up to higher order corrections to the superpotential). This construction allows one to recover geometrically the predicted mirrors in various interesting settings: pairs of pants, curves of arbitrary genus, etc.

This is joint work with Francois Lalonde. Using an analogue of Seidel's homomorphism in Lagrangian Floer homology for one Lagrangian, we give a condition for a diffeomorphism on a Lagrangian to extend to a Hamiltonian diffeomorphism on the whole symplectic manifold.

In this talk we will discuss the (local) construction of a calibrated G_2 structure on the 7-dimensional quotient of an 8-dimensional quaternion-Kahler (QK) manifold M under the action of a group S^1 of isometries. The idea is to construct explicitly a 3-form of type G_2, using the data associated to the S^1 action and to the QK structure on M. In the same spirit, we can consider the level sets of the QK moment-map square-norm function on M, and again take the S^1 quotient: we will discuss in this case the construction of half-flat metrics in dimension 6, under suitable circumstances. This talk is based on a joint work with F. Lonegro, Y. Nagatomo and S. Salamon, still in progress.

I will discuss the existence problem of extremal Kahler metrics (in the sense of Calabi) on the total space of a holomorphic projective bundle P(E) over a compact complex curve. The problem is not solved in full generality even in the case of a projective plane bundle over CP^1. However, I will show that sufficiently ``small'' Kahler classes admit extremal Kahler metrics if and only if the underlying vector bundle E can be decomposed as a sum of stable factors. This result can be viewed as a ``Hitchin-Kobayashi correspondence'' for projective bundles over a curve, but in the context of the search for extremal Kahler metrics. The talk will be based on a recent work with D. Calderbak, P. Gauduchon and C. Tonnesen-Friedman.

I will describe how the geometry of supersymmetric AdS solutions of type IIB string theory may be rephrased in terms of the geometry of generalized (in the sense of Hitchin) Calabi-Yau cones. Calabi-Yau cones, and hence Sasaki-Einstein manifolds, are a special case, and thus the geometrical structure described may be considered a form of generalized Sasaki-Einstein geometry. Generalized complex geometry naturally describes many features of the AdS/CFT correspondence. For example, a certain type changing locus is identified naturally with the moduli space of the dual CFT. There is also a generalized Reeb vector field, which defines a foliation with a transverse generalized Hermitian structure. For solutions with non-zero D3-brane charge, the generalized Calabi-Yau cone is also equipped with a canonical symplectic structure, and this captures many quantities of physical interest, such as the central charge and conformal dimensions of certain operators, in the form of Duistermaat-Heckman type integrals.

"Sasakian geometry is often described as an odd dimensional counterparts of K\""ahler geometry. There is a natural Riemannian metric on the space of Sasakian metrics, which in turn gives a geodesic equation on this space. It can be viewed as parallel case of a well-known geodesic equation for the space of K\""ahler metrics. The equation is connected to some interesting geometric properties of Sasakian manifolds. It is a complicated complex Monge-Amp\`ere type involving gradient terms. We discuss the problem of existence and regularity of solutions of this type of equations. This is a joint work with Xi Zhang."

Quite a bit of progress has been achieved over the past seven years in understanding from a rigorous mathematical perspective the long time dynamics of waves in the Kerr geometry of a rotating black hole in equilibrium. A proof of the Penrose process for scalar waves has notably been given in this context. I will review some of these results, obtained in collaboration with Felix Finster, Joel Smoller and Shing-Tung Yau. I will also indicate a number of open problems.

Inspired by homological mirror symmetry, Seidel and Thomas constructed braid group actions on derived categories of coherent sheaves of various varieties and proved faithfulness of such actions for braid groups of type A. I will discuss joint work with Hugh Thomas giving some faithfulness results for derived braid group actions of types D and E

I will describe some combinatorial problems which arise when computing various types of partition functions for the Donaldson-Thomas theory of a space with a torus action. The problems are all variants of the following: give a generating function which enumerates the number of ways to pile n cubical boxes in the corner of a room. Often the resulting generating functions are nice product formulae, as predicted by the recent wall-crossing formulae of Kontsevich-Soibelman. There are now a variety of techniques, both geometric and combinatorial, to compute these formula. My work uses the entirely combinatorial techniques, namely vertex operators and the planar dimer model; these techniques can be applied essentially "bare-handed" and rely very little upon the underlying algebraic geometry.

"In this joint work with Jingyi Chen and Weiyong He, we prove existence of longtime smooth solutions to mean curvature flow of entire Lipschitz Lagrangian graphs. A Bernstein type result for translating solitons is also obtained."