Modular Forms and Black Holes in N=4 String Theories

Explore an in-depth 83-minute lecture from Professor Justin David of the Indian Institute of Science, delivered as part of the "Black holes: bridges between number theory and holographic quantum information" event series. Delve into the intricate relationship between modular forms and 1/4 BPS black holes in CHL models within N=4 string theories. Master three fundamental aspects: understand how 1/4 BPS index states function as Fourier coefficients of inverse Siegel modular forms, examine the connection between these forms and K3 elliptic genera twisted by Z_N automorphism, and explore their relationship to the Mathieu group M_24. Learn about Mock-modular form expansions and their role in determining single-centered black hole indices, while understanding how spherical symmetry influences these indices. Discover the modular transformation of Siegel forms and their significance in large charge index expansion, connecting to the Hawking-Bekenstein-Wald entropy theory.