Black holes and dualities in string theory compactifications

TL;DR

This thesis explores the mathematical and physical structures of black holes and dualities in string theory compactifications, focusing on moduli space corrections, BPS invariants, and modular properties.

Contribution

It provides new computations of instanton corrections, formulates a quantum deformation of the Riemann-Hilbert problem, and offers solutions to modular completion equations in string theory.

Findings

Computed NS5-instanton corrections to hypermultiplet moduli space.

Formulated a perturbative solution to the quantum-deformed Riemann-Hilbert problem.

Derived solutions to modular completion equations using theta series and Jacobi forms.

Abstract

This thesis addresses three problems arising in type II string theory compactified on a Calabi-Yau manifold. In the first one we study the hypermultiplet moduli space (HM), by working on its twistor space. Using data derived via mirror symmetry and S-duality, we compute NS5-instanton corrections to the HM metric in the one-instanton approximation. These corrections are weighted by D4-D2-D0 BPS indices, which coincide with rank 0 Donaldson-Thomas invariants and count the (signed) number of BPS black hole microstates. These invariants exhibit wall-crossing behavior and induce a Riemann-Hilbert problem. This problem can describe many setups, including the D-instanton corrected twistor space of the HM in type II string theory and is of independent mathematical interest. We consider a quantum deformation of the RH problem, induced by the refined BPS indices. Using a formulation of the problem in terms of a non-commutative Moyal star product, we provide a perturbative solution to it. From the adjoint form of this solution, we identify a generating function for coordinates on the still mysterious quantum analog of the twistor space. Finally, we study the modular properties of the D4-D2-D0 BPS indices, more precisely of their generating functions. It was previously argued, using S-duality, that the generating functions are higher depth mock modular forms. Moreover, they satisfy a modular completion equation, which fixes their shadow in terms of other (lower rank) generating functions. We start by bringing about a significant simplification to these equations and recovering subtle contributions that were overlooked. Then, we provide (a recipe for) solutions to these modular completion equations, up to all the holomorphic modular ambiguities that need to be fixed independently. For this, we use indefinite generalized theta series and Jacobi-like forms to write the solutions.