Geometry and Physics of non-Kähler Calabi-Yau

In this talk, I will first explain how SU(3) structures arise naturally in the compactification of superstrings. In particular, under the presence of flux, such theories seek for various SU(3) structures with torsion, which motivates the study of non-Kahler Calabi-Yau 3-folds. In the second half of my talk, I will introduce a novel geometric flow to study Type IIA superstring equations. We show that this Type IIA flow is well-imposed, establish the basic estimates, and give some of its applications in symplectic geometry. This is based on joint work with D.H. Phong, S. Picard and X.W. Zhang.

I review the geometry of heterotic string compactifications leading to supersymmetric gauge theories in 4 dimensions. The data of these compactifications are specified by a quadruple (X, V, TY, H) where X is a 6 dimensional manifold with a G-structure (a certain SU(3) structure in 6 dimensions), V is a vector bundle over X with a Yang Mills connection which satisfies instanton constraints, TX is the tangent bundle over X with an instanton connection, and H is a three form on Y defined in terms of the B-field and the Chern-Simons forms for the connections on V and TX (the so called anomaly condition). We recast all the constraints on the geometry of these compactifications in terms of an extension bundle Q over X which admits a nilpotent operator which squares to zero. We show that the tangent space of the moduli space is then given by the first cohomology group with values in Q.

Time permitting, we discuss the fact that all our results can be reproduced from a superpotential.

I will introduce the Calabi system for a bundle over a complex manifold: a coupled system of equations that generalizes both the Calabi problem (when the bundle is trivial) and the Hull-Strominger system (when the manifold is Calabi-Yau). To do this, I will first recall the Calabi problem and the Hull-Strominger system, then give a crash course on generalized geometry and finally see how it can be used to state and advance in the understanding of the Calabi system. This talk is based in several joint works with M. García-Fernandez, C. Shahbazi and C. Tipler.

I will discuss some properties of the class of compact complex manifolds whose first Chern class vanishes in Bott-Chern cohomology. This can be thought of as a generalization of the Calabi-Yau condition to possibly non-Kähler manifolds, which preserves several of the good properties that hold in the Kähler case. I will discuss the existence of special Hermitian metrics on these manifolds, including theorems and conjectures that generalize Yau’s solution of the Calabi Conjecture to this setting.