Applications of Mathieu Moonshine in N=4 supersymmetric theories (Quantum Spacetime Seminar Series)

We study the properties of 1/4 BPS dyons in N=4 type II string compactified on K3 x T2  orbifolded with an action of automorphism g' of K3 corresponding to the conjugacy classes of Mathieu group M24 and a 1/N shift in one of the circles of T2. The partition function of dyons in these compactifications are given in terms of inverse of Siegel modular forms constructed from the lift of the twisted elliptic genus. The sign of the supersymmetric index counting 1/4 BPS dyons agrees with that predicted from black hole physics as conjectured by Sen. We also study the properties of 1/4 BPS dyons in type II string theory compactified on Z2 and Z3 orbifolds on T6 with 1/N shift in one of the S1 and encounter some violations to this conjecture. These point to the existence of possible non-trivial hair modes for 1/4 BPS dyons in these theories. We associate mock modular forms corresponding to single centred black holes and extend the work of Dabholkar-Murthy-Zagier to these orbifolds of K3 and also for the toroidal orbifolds.