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 0(N) modular forms. With a bit more analysis we determine
the exact location of the zeros of some weight 2 Eisenstein series of 0(N) in the fundamental domain
of 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.