Summary

The study of black holes (BHs) in string theory has led to the discovery of deep and surprising connections between BHs and modular forms—which are two classical, a priori unrelated, subjects.

It is known from the pioneering work of J. Bekenstein and S. Hawking in the 1970s that BHs have thermodynamic entropy and should, therefore, be made up of a collection of microscopic quantum states. Superstring theory provides a framework wherein one can associate a number of microscopic states that make up the quantum-statistical system underlying a BH, thus explaining their thermodynamic behavior from a more fundamental point of view. The basic connection to modular forms arises from the observation that, in the simplest superstring-theoretic construction, the generating function of the number of microscopic states is a modular form. In one direction, modular symmetry acts as a powerful guide to the calculation of quantum-gravitational effects on the BH entropy. In the other direction, the connection has led to the discovery of surprising relations between Ramanujan’s mock modular forms and a class of string-theoretic BHs, thus providing an infinite number of new examples of mock modular forms.