#### Alladi Ramakrishnan Hall

#### Applications of Mathieu Moonshine in N=4 supersymmetric theories

#### Aradhita Chattopadhyaya

##### IISc, Bangalore

*We study the properties of 1/4 BPS dyons in ${\cal N}=4$ type II string compactified on $K3 \times T^2$ orbifolded with an action of automorphisim $g'$ of $K3$*

corresponding to the conjugacy classes of Mathieu group $M_{24}$ and a $1/N$ shift in one of the circles of $T^2$.

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 1/4th BPS dyons agrees with that

predicted from black hole physics as conjectured by Sen. We also study the properties of 1/4th BPS dyons in type II string theory compactified on $\mathbb{Z}_2$

and $\mathbb{Z}_3$ orbifolds on $T^6$ with $1/N$ shift in one of the $S^1$ 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.

Done