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Abstract:Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.

The Yang-Mills equations were introduced by theoretical physicists and are now accepted as a basic ingredient in particle theory. In the past decade these equations have become important in mathematics in two separate areas. It was observed early on that the twistor formalism of Penrose and the Atiyah-Singer index theory can be employed to good purpose in order to describe special The theory of holomorphic vector bundles is central to algebraic geometry. The concept of a stable vector bundle was introduced by Mumford in h s study of moduli for bundles [18]. This theme has been pursued by several mathematicians, notably Takemoto, Horrocks, Gieseker, Maruyama and Barth. An important contribution was made by Bogomolov [4], who showed that, for a projective variety M with Pic ( M ) = Z and E a stable bundle on M , the inequality k -1is valid. Here k = rank E , c, and c2 are the first and second Chern classes, and w is a polarization for which E is stable.Bogomolov's work inspired the work of Miyaoka [17] on the inequality 3c, 2 c: for algebraic surfaces of general type. Independently, the second author also gave a proof of this inequality at the same time. While the methods of Miyaoka and Bogomolov are essentially based on algebraic geometry, the method of the second author used the techniques of partial differential equations to

We show that the Reeb vector, and hence in particular the volume, of a Sasaki-Einstein metric on the base of a toric Calabi-Yau cone of complex dimension n may be computed by minimising a function Z on R n which depends only on the toric data that defines the singularity. In this way one can extract certain geometric information for a toric Sasaki-Einstein manifold without finding the metric explicitly. For complex dimension n = 3 the Reeb vector and the volume correspond to the R-symmetry and the a central charge of the AdS/CFT dual superconformal field theory, respectively. We therefore interpret this extremal problem as the geometric dual of a-maximisation. We illustrate our results with some examples, including the Y p,q singularities and the complex cone over the second del Pezzo surface.