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Black holes and dualities in string theory compactifications

Khalil Bendriss

TL;DR

This thesis advances three interconnected problems at the intersection of string theory and mathematical structures. It first computes NS5-instanton corrections to the hypermultiplet moduli space in type II string theory on Calabi–Yau threefolds using twistorial methods, with data guided by mirror symmetry and S-duality; the corrections are organized by D4–D2–D0 BPS indices, connected to rank-0 Donaldson–Thomas invariants and wall-crossing. Second, it formulates a quantum Riemann–Hilbert problem by refining BPS indices via a non-commutative Moyal product, constructs perturbative solutions, and identifies adjoint representations that yield a generating function for coordinates on the quantum twistor space. Third, it analyzes the modular properties of D4–D2–D0 generating functions, develops a tree-based framework to simplify modular anomaly kernels, and provides explicit (up to holomorphic modular ambiguities) solutions using indefinite theta series and Jacobi-like forms, while also extending the lattice and introducing a refinement to obtain all-order structures. Collectively, these results illuminate the non-perturbative geometry of HM moduli spaces, connect black-hole microstate counting to mock modular forms, and offer concrete predictions for string amplitudes in NS5 backgrounds, with rigorous cross-checks in rigid CY limits and small-g_s regimes. The work thus deepens the bridge between non-perturbative string dynamics and advanced modular/theta-formalism, with potential implications for quantum geometry and holographic counts of BPS states.

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.

Black holes and dualities in string theory compactifications

TL;DR

This thesis advances three interconnected problems at the intersection of string theory and mathematical structures. It first computes NS5-instanton corrections to the hypermultiplet moduli space in type II string theory on Calabi–Yau threefolds using twistorial methods, with data guided by mirror symmetry and S-duality; the corrections are organized by D4–D2–D0 BPS indices, connected to rank-0 Donaldson–Thomas invariants and wall-crossing. Second, it formulates a quantum Riemann–Hilbert problem by refining BPS indices via a non-commutative Moyal product, constructs perturbative solutions, and identifies adjoint representations that yield a generating function for coordinates on the quantum twistor space. Third, it analyzes the modular properties of D4–D2–D0 generating functions, develops a tree-based framework to simplify modular anomaly kernels, and provides explicit (up to holomorphic modular ambiguities) solutions using indefinite theta series and Jacobi-like forms, while also extending the lattice and introducing a refinement to obtain all-order structures. Collectively, these results illuminate the non-perturbative geometry of HM moduli spaces, connect black-hole microstate counting to mock modular forms, and offer concrete predictions for string amplitudes in NS5 backgrounds, with rigorous cross-checks in rigid CY limits and small-g_s regimes. The work thus deepens the bridge between non-perturbative string dynamics and advanced modular/theta-formalism, with potential implications for quantum geometry and holographic counts of BPS states.

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.

Paper Structure

This paper contains 104 sections, 10 theorems, 392 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Background on Type II Superstring Theory
  4. Type II Superstring Theory
  5. The massless bosonic spectrum
  6. D-branes and NS5-branes
  7. Symmetries
  8. Calabi-Yau compactifications
  9. Definition and properties
  10. Moduli spaces of Kähler and complex structures
  11. Mirror symmetry
  12. Moduli spaces
  13. Type IIA
  14. Type IIB
  15. Low energy effective action
  16. ...and 89 more sections

Key Result

Proposition 1

Let $\varphi(\tau,z)$ be a Jacobi-like form of modular weight $w$ and index $m$. Then is a Jacobi-like form of the same weight and vanishing index, and the coefficients of its Laurent expansion $\tilde{\varphi}(\tau,z)=\sum_{n=n_0}^\infty h_n(\tau) z^n$ are modular forms of weight $w+n$.

Figures (8)

  • Figure 1: A representation of the identity \ref{['cyc3']} in terms of rooted trees. Here $T_k$'s are any rooted subtrees.
  • Figure 2: The rooted trees corresponding to the expansions of $\mathsf{X}_0$ in $\mathsf{X}^{\rm ref}_i$ and $\mathsf{X}^{\rm sf}_i$, respectively.
  • Figure 3: Three examples of Schröder trees, using the notation $\gamma_{i+j}=\gamma_i+\gamma_j$. The tree a) belongs to $\mathds{T}^{\rm S}_2$ and the trees b) and c) belong to $\mathds{T}^{\rm S}_5$. The trees a) and c) have $n_T=1$ and the tree b) has $n_T=3$.
  • Figure 4: An example of Schröder tree contributing to $R_8$. Near each vertex we showed the corresponding factor using the shorthand notation $\gamma_{i+j}=\gamma_i+\gamma_j$.
  • Figure 5: Unrooted trees constructed from the same three subtrees.
  • ...and 3 more figures

Theorems & Definitions (16)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 1
  • Proposition 3
  • Definition 1
  • Lemma 1
  • Conjecture 1
  • Proposition 4
  • ...and 6 more