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Critical points and supersymmetric vacua, III: String/M models

Michael R. Douglas, Bernard Shiffman, Steve Zelditch

TL;DR

This work provides the first rigorous results for counting supersymmetric vacua in type IIb Calabi–Yau flux compactifications by combining lattice-point analysis with holomorphic-geometry of the moduli space. It proves counting formulas with remainder estimates (extending Ashok–Douglas and Denef–Douglas) and develops a van der Corput-type framework to relate discrete flux sums to continuum integrals. The paper derives explicit density formulas for nondegenerate critical points via a Hessian push-forward and a distortion operator $\Lambda_Z$, and connects these densities to Hodge-theoretic and curvature data (Weil–Petersson, Chern forms). Overall, it demonstrates equidistribution of vacua under the tadpole constraint and provides a road map for quantifying the string landscape, including concrete examples for small $h^{2,1}$ and heuristic scaling with $b_3$. The results bridge statistical algebraic geometry with string theory phenomenology, offering concrete asymptotics and remainder terms that sharpen prior physics heuristics about vacuum multiplicities and their moduli-space distribution.

Abstract

A fundamental problem in contemporary string/M theory is to count the number of inequivalent vacua satisfying constraints in a string theory model. This article contains the first rigorous results on the number and distribution of supersymmetric vacua of type IIb string theories compactified on a Calabi-Yau 3-fold $X$ with flux. In particular, complete proofs of the counting formulas in Ashok-Douglas and Denef-Douglas are given, together with van der Corput style remainder estimates. We also give evidence that the number of vacua satisfying the tadpole constraint in regions of bounded curvature in moduli space is of exponential growth in $b_3(X)$.

Critical points and supersymmetric vacua, III: String/M models

TL;DR

This work provides the first rigorous results for counting supersymmetric vacua in type IIb Calabi–Yau flux compactifications by combining lattice-point analysis with holomorphic-geometry of the moduli space. It proves counting formulas with remainder estimates (extending Ashok–Douglas and Denef–Douglas) and develops a van der Corput-type framework to relate discrete flux sums to continuum integrals. The paper derives explicit density formulas for nondegenerate critical points via a Hessian push-forward and a distortion operator , and connects these densities to Hodge-theoretic and curvature data (Weil–Petersson, Chern forms). Overall, it demonstrates equidistribution of vacua under the tadpole constraint and provides a road map for quantifying the string landscape, including concrete examples for small and heuristic scaling with . The results bridge statistical algebraic geometry with string theory phenomenology, offering concrete asymptotics and remainder terms that sharpen prior physics heuristics about vacuum multiplicities and their moduli-space distribution.

Abstract

A fundamental problem in contemporary string/M theory is to count the number of inequivalent vacua satisfying constraints in a string theory model. This article contains the first rigorous results on the number and distribution of supersymmetric vacua of type IIb string theories compactified on a Calabi-Yau 3-fold with flux. In particular, complete proofs of the counting formulas in Ashok-Douglas and Denef-Douglas are given, together with van der Corput style remainder estimates. We also give evidence that the number of vacua satisfying the tadpole constraint in regions of bounded curvature in moduli space is of exponential growth in .

Paper Structure

This paper contains 41 sections, 283 equations.

Table of Contents

  1. Introduction
  2. Background to the results
  3. Statement of the problem
  4. Statement of the results
  5. Special geometry and critical point density
  6. Index density
  7. Relations to prior results in the physics and mathematics literature
  8. Background on Calabi-Yau manifolds and complex geometry
  9. Geometry of Calabi-Yau manifolds
  10. Variational derivatives and covariant derivatives
  11. $\mathcal{C}$ as the moduli space of complex structures on $X \times T^2$
  12. Hodge-Riemann form and inner products
  13. Critical points of superpotentials
  14. Flux superpotentials as holomorphic sections
  15. Critical points and Hessians of holomorphic sections
  16. ...and 26 more sections

Theorems & Definitions (9)

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