Handbook for Mirror Symmetry of CalabiYau and Fano Manifolds

In algebraic geometry and theoretical physics, mirror symmetry refers to the relationship between two CalabiYau manifolds which appear very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory. Mathematicians became interested in mirror symmetry around 1990, when it was shown that mirror symmetry could be used to count rational curves on a CalabiYau manifold, thus solving a long-standing problem. Today, mirror symmetry is a fundamental tool for doing calculations in string theory, and it has been used to understand aspects of quantum field theory, the formalism that physicists use to describe elementary particles. Major approaches to mirror symmetry include the homological mirror symmetry program of Maxim Kontsevich and the SYZ conjecture of Andrew Strominger, Shing-Tung Yau, and Eric Zaslow. This handbook surveys recent developments in mirror symmetry. It presents papers based on selected lectures given at a 2014 Taipei conference on CalabiYau Geometry and Mirror Symmetry, along with other contributions from invited authors.