Topological Strings and Their Applications in Mathematics

In this note, we briefly review the mathematical structures of topological strings and their applications in mathematics. We focus on the discussions of integrality structures in topological strings and large $N$ Chern–Simons/topological string duality. More precisely, prediction of the number of rational curves in quintic, LMOV conjectures of open topological strings and framed knots, derivation of Mariño–Vafa formula and interpretation of Rogers–Ramanujan identities by topological string will be described.

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The author would like to thank professor Kefeng Liu, who suggested him to study the mathematical structures of large $N$ duality, helped him in numerous ways during the past several years. The author is grateful to Qingtao Chen, Kefeng Liu, Pan Peng and Wei Luo for the collaborations, working with them proved to be very stimulating.

Published 8 December 2017