I'll review the recent results on modular properties of generating functions of BPS indices counting states of D4-D2-D0 black holes in Calabi-Yau compactifications of type IIA string theory. These generating functions can be shown to be (higher depth) mock modular forms with a specific non-holomorphic modular completion. This result turns out to have a wider range of applicability than the original framework and has numerous applications ranging from compactifications with higher SUSY to Vafa-Witten theory. In the end, I'll show how one can determine the generating functions explicitly for compact Calabi-Yau threefolds with one Kahler modulus, and how this allows to overcome the usual limitations for computation of GV, DT and PT topological invariants.

Sine-Liouville (SL) theory is a 2d string theory in a background with a tachyon condensate and has a dual description in terms of Matrix Quantum Mechanics. Using this description, almost 20 years ago instanton corrections to the partition function of SL theory have been found. However, this result left several puzzles and it was unclear how it can be reproduced using CFT technique. I reconsider this problem using recent insights about D-instantons in string theory, which allows both to resolve the puzzles and to match the instanton effects in the two formulations.

By exploiting new mathematical relations between Pandharipande-Thomas (PT) invariants, closely related to Gopakumar-Vafa (GV) invariants, and rank 0 Donaldson-Thomas (DT) invariants counting D4-D2-D0 BPS bound states, we rigorously compute the first few terms in the generating series of Abelian D4-D2-D0 indices for compact one-parameter Calabi-Yau threefolds of hypergeometric type. In all cases where GV invariants can be computed to sufficiently high genus, we find striking confirmation that the generating series is modular, and predict infinite series of Abelian D4-D2-D0 indices. Conversely, we use these results to provide new constraints for the direct integration method, which allows to compute GV invariants (and therefore the topological string partition function) to higher genus than hitherto possible. The triangle of relations between GV/PT/DT invariants is powered by a new explicit formula relating PT and rank 0 DT invariants, which is proven in an Appendix by the second named author. As a corollary, we obtain rigorous Castelnuovo-type bounds for PT and GV invariants for CY threefolds with Picard rank one.

I'll explain the role of modular symmetry in the counting of black hole microstates.

I'll give a review of the role that modular symmetry plays in description of BPS black holes and counting of their microstates.

Euclidean D-branes wrapped on non-trivial cycles of a Calabi-Yau threefold are known to affect the metric on the hypermultiplet moduli space determining the effective action of type II strings on such manifolds. After reviewing the existing results on the instanton corrected metric following from a combination of constraints imposed by supersymmetry and dualities, I'll show how the D-instanton effects can be computed by a direct worldsheet approach. This required solving several conceptual and technical problems, which also paved the way for extension of this approach to compactifications with lower supersymmetry. The talk is based on joint work with A. Sen and B. Stefanski.

As shown by T. Bridgeland, a Riemann-Hilbert problem determined by Donaldson-Thomas invariants naturally gives rise to the so-called Joyce structure. It can be characterized by a function known as Plebanski potential, or its close cousin Joyce potential. I'll show that a twistorial solution to the RH problem provides a simple integral expressions for both potentials. Then I'll explain the relation of this solution to the conformal limit of the twistor spacesappearing in gauge and string theories, and physical interpretation acquired bythe two potentials in these setups. For the case of the resolved conifold, I'll present a recipe to make the twistorial solution well-defined despite an infinite BPS spectrum, and trace out the emergence of a tau-function, its relation to topological strings and its behavior under S-duality to the twistorial implementation of instantons in string theory.