Abhiram Kidambi, Tecnical University of Vienna
$\Gamma_0(N)$, quantum black holes and wall crossing

The degeneracies of supersymmetric dyonic black holes are known to be encoded in the Fourier coefficients of certain modular objects. For the case of $N = 4$, $d=4$ theory which I shall discuss, the spectrum of quarter BPS dyons is prone to wall crossing phenomena. The number theory machinery behind wall crossing in $4d$ $N = 4$ theories was described systematically in a comprehensive paper by Atish Dabholkar, Sameer Murthy and Don Zagier. There have also been supergravity localisation calculations thereafter which confirm some of the results that were shown by DMZ.

In this talk, I shall provide some of the number theoretic background for BPS state counting and review some of the key results known so far from both the microscopic and macroscopic side. I shall comment on black hole metamorphosis studied by Sen (and collaborators) and Nampuri et.al from a number theoretic framework. The remainder of the talk will be devoted to the generalisation of the number theory machinery of DMZ to congruence subgroups of $\operatorname{SL}(2,\mathbb{Z})$ i.e. for orbifolded CHL black holes and the supergravity approach for the CHL case.

This talk summarises some of the ongoing work with Sameer Murthy, Valentin Reys, Abhishek Chowdhury and Timm Wrase.