Preface The conference ‘String-Math 2013’ was held June 17–21, 2013 at the Simons Center for Geometry and Physics. This was the third in a series of large meetings exploring the interface of mathematics and string theory. This volume presents the proceedings of that conference. The nature of interactions between mathematicians and physicists has been thoroughly transformed in recent years. String theory and quantum field theory have contributed a series of profound ideas which gave rise to entirely new math- ematical fields and revitalized older ones. The influence flows in both directions, with mathematical techniques and ideas contributing crucially to major advances in string theory. There is now a large and rapidly growing number of both mathe- maticians and physicists working at the string-theoretic interface between the two academic fields. For mathematics, string theory has been a source of many significant inspira- tions, ranging from Seiberg-Witten theory in four-manifolds, to enumerative ge- ometry and Gromov-Witten theory in algebraic geometry, to work on the Jones polynomial in knot theory, to recent progress in the geometric Langlands program and the development of derived algebraic geometry and n-category theory. In the other direction, mathematics has provided physicists with powerful tools, ranging from powerful differential geometric techniques for solving or analyzing key par- tial differential equations, to toric geometry, to K-theory and derived categories in D-branes, to the analysis of Calabi-Yau manifolds and string compactifications, to the use of modular forms and other arithmetic techniques. The depth, power and novelty of the results obtained in both fields thanks to their interaction is truly mind-boggling. In the past few years, the annual String-Math conferences have become the cen- tral venue for these profound and wide-ranging interactions. They bring together the leading mathematicians and mathematically-minded physicists working in this interface. These meetings promote and publicize such interactions, giving atten- dees greater opportunities to cross cultural boundaries, learn aspects of other fields relevant for their research, and advertise important developments to audiences that might not otherwise hear of them or appreciate their importance. The 2013 conference was organized by Alexander Abanov, Michael Douglas, Ljudmila Kamenova, Claude LeBrun, John Morgan, Nikita Nekrasov, Leonardo Rastelli, and Martin Roˇ cek. The Steering Committee consisted of Ron Donagi, Dan Freed, Nigel Hitchin, Sheldon Katz, Maxim Kontsevich, David Morrison, Edward Witten, and Shing-Tung Yau. The meeting covered a wide array of topics at the interface of mathematics and high energy physics, including: vii

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