TAILIEUCHUNG - From moonshine to mock moonshine

The “Monstrous Moonshine” conjecture (now a theorem) of Conway and Nortan has given rise to a large body of new mathematics. This theorem has been extended to other groups revealing unexpected relations to conformal field theory, quantum gravity, black holes, and string theory in physics and to Ramanujan’s mock theta functions and its extensions in mathematics. | Vietnam Journal of Mathematics (2019) 47:183–193 From Moonshine to Mock Moonshine Kishore Marathe1 Received: 21 November 2017 / Accepted: 11 June 2018 / Published online: 28 September 2018 © Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018 Abstract The “Monstrous Moonshine” conjecture (now a theorem) of Conway and Nortan has given rise to a large body of new mathematics. This theorem has been extended to other groups revealing unexpected relations to conformal field theory, quantum gravity, black holes, and string theory in physics and to Ramanujan’s mock theta functions and its extensions in mathematics. We call these results which have been discovered in the last few years “Mock Moonshine.” In this survey article, we will discuss some recent developments connecting diverse areas of mathematics and theoretical physics and indicate directions for future research. Keywords Moonshine · Mock moonshine · Mock modular forms · Black holes · Quantum gravity · String theory Mathematics Subject Classification (2010) 81T30 · 83C45 · 83C57 · 11F46 · 51P05 1 Introduction The theory of permutation groups was developed by Augustin-Louis Cauchy, a French mathematician and physicist who made fundamental contributions to both fields. He felt that groups were a mathematical toy which was not going to have any applications in the mathematical sciences. In fact, the theory of groups has wide-ranging applications in the mathematical sciences for giving a precise description of the symmetries of various objects ranging from geometric figures to fundamental particles and classical and quantum fields. The following historical material is taken from the book [26]. Recall that a group is called simple if it has no proper non-trivial normal subgroups. Thus, an abelian group is simple if and only if it is isomorphic to one of the groups Zp , where p is a prime number. Another infinite family of finite .

• Toán học
• Mock moonshine
• Mock modular forms
• Black holes
• Quantum gravity
• String theory
• Ramanujan’s mock theta functions