Mirror symmetry is the name given to a broad collection of mathematical conjectures arising from a type of duality in physics. Roughly it states that certain information about a smooth variety M should by expressible in terms of different properties of a ``mirror manifold’’ W. Rephrasing questions about M in terms of W has led to striking solutions to many previously intractable problems in geometry.
In this talk I will give a friendly introduction to the mathematics of mirror symmetry, and outline some ways in which given a manifold M one can construct its mirror W. Not all of these constructions agree however, and sometimes one is left with different spaces which should be mirror to the same M. This ``multiple mirror’’ phenomenon has interesting mathematical implications. In particular I will show that in certain settings, the mirror manifolds which arise are in fact birational to one another.
In the first part of the talk I will discuss the Eynard-Orantin theory to the family of spectral curves of Hitchin fibrations over a smooth base curve C of genus at least 2. In particular, we study rank 2 holomorphic Higgs field and we identify the spectral curve of the Eynard-Orantin as the spectral curve of Hitchin fibration.
In the second part we generalize this construction to meromorphic Higgs fields with possibly singular spectral curve in the compactified cotangent bundle. We identify the spectral curve of the Eynard-Orantin recursion with a divisor in a ruled surface over the curve C. We further apply recursion to quantize spectral curve.
We present as simple examples of our theory meromorphic Higgs bundles over the rational curve, and we obtain well-known classical equations as Airy function and Catalan recursion. This talk is based on my joint work with Motohico Mulase, U.C. Davis.
