Applications of Moonshine Symmetry in String Theory

Abstract

In this thesis we study the applications of Mathieu moonshine symmetry to compacti cations of supersymmetric string theories. These theories are compacti ed on a 6 dimensional manifold K3 T2. The main ingredient in this study is a topological index called twisted elliptic genus. For a super-conformal eld theory whose target space is a K3 there can be several automorphisms on K3 which are related to Mathieu group M24. Under these automorphisms it was observed that the twining genera of the twisted elliptic genus of K3 could be written in terms of the short and long representations of N = 4 super-conformal algebra and the characters of M24 [1, 2, 3]. We compute the twisted elliptic genus in every sector for 16 of these orbifolds using the results of [2]. Firstly we study the heterotic compacti cations of N = 2 super-symmetric strings compacti ed on orbifolds of K3 T2 and E8 E8 where g0 is an action on K3 corresponding to [M24] along with a 1=N shift on one of the circles of T2. We compute the gauge and gravitational threshold corrections in these theories. Here we need a topological index called the new supersymmetric index. The un-orbifolded result for K3 was known for gauge couplings in [4] and the gravitational ones were computed in [5]. We observe that the di erences in gauge couplings can be written in terms of the twisted elliptic genus of K3 for standard embeddings. For non-standard embeddings we studied two orbifold realizations of K3 as T4=Z2 and T4=Z4 and computed the threshold di erences. The result could be written in terms the twisted elliptic genus of K3 and the elliptic genus of K3. From the gravitational corrections we predict the Gopakumar Vafa invariants and the Euler character for the dual Calabi Yau geometries. We also observe that the conifold singularities of these manifolds are manifested in twisted sectors only and only the genus zero Gopakumar-Vafa invariants at those points are non-zero. Secondly we study the properties of 1/4 BPS dyons in type II string compacti ed on K3 T2 orbifolded with an action of g0 which corresponds to automorphisms of K3 corresponding to the conjugacy classes of Mathieu group M24 and a 1=N shift in one of the circles of T2. For these compacti cations the counting function for these dyons can be computed from Siegel modular forms given by the lift of the twisted elliptic genus. These give the correct sign as predicted from black hole physics as conjectured by Sen [6]. We also study the properties of 1/4th BPS dyons in type II string theory compacti ed on Z2 and Z3 orbifolds on T6 with 1=N shift in one of the S1 and encountered some violations to this conjecture which points to the existence of non-trivial hair modes. We associate mock modular forms corresponding to single centred black holes and extend the work of Dabholkar-Murthy-Zagier [7] to these orbifolds of K3 and also for the toroidal orbifolds. In computing the twisted elliptic genus and new super-symmetric index in various twisted sectors we encounter several identities between some ð€€€0(N) modular forms. With a bit more analysis we determine the exact location of the zeros of some weight 2 Eisenstein series of ð€€€0(N) in the fundamental domain of ð€€€0(N) where N = 2; 3; 5; 7. The location of their zeros were controlled by those of Eisenstein series of weight 4 and 6.