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Ray-Singer Torsion, Topological Strings and Black Holes

Cumrun Vafa

Abstract

Genus one amplitude for topological strings on Calabi-Yau 3-folds can be computed using mirror symmetry: The partition function at genus one gets mapped to a holomorphic version of Ray-Singer torsion on the mirror Calabi-Yau. On the other hand it can be shown by a physical argument that this gives a curvature squared correction term to the gravitational action. This in paticular leads to an effective quantum gravity cutoff known as the species scale, which varies over moduli space of Calabi-Yau manifolds. This resolves some of the puzzles associated to the entropy of small black holes when there are a large number of light species of particles. Thus Ray-Singer torsion, via its connection to topological strings at genus one, provides a measure of light degrees of freedom of four dimensional N=2 supergravity theories. Based on a talk given on May 12th, 2023 at the Singer Memorial Conference, MIT.

Ray-Singer Torsion, Topological Strings and Black Holes

Abstract

Genus one amplitude for topological strings on Calabi-Yau 3-folds can be computed using mirror symmetry: The partition function at genus one gets mapped to a holomorphic version of Ray-Singer torsion on the mirror Calabi-Yau. On the other hand it can be shown by a physical argument that this gives a curvature squared correction term to the gravitational action. This in paticular leads to an effective quantum gravity cutoff known as the species scale, which varies over moduli space of Calabi-Yau manifolds. This resolves some of the puzzles associated to the entropy of small black holes when there are a large number of light species of particles. Thus Ray-Singer torsion, via its connection to topological strings at genus one, provides a measure of light degrees of freedom of four dimensional N=2 supergravity theories. Based on a talk given on May 12th, 2023 at the Singer Memorial Conference, MIT.
Paper Structure (10 sections, 31 equations, 7 figures)

This paper contains 10 sections, 31 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Ray-Singer Torsion in Compact Kahler and Calabi-Yau Manifolds
  3. Planck Scale as a UV Quantum Gravity Cutoff
  4. Basics of Black Hole Entropy
  5. Light Species Puzzle for Black Holes
  6. Species Scale
  7. Dualities and Moduli Space
  8. Topological Strings on CY 3-folds and dependence of Species scale on moduli
  9. Application to Bounds on positive potentials
  10. Conclusion

Figures (7)

  • Figure 1: The minimal size of the black hole which can be reliably treated using the effective action is $1/\Lambda_s$ which is larger than the Planck length $1/M_{pl}$.
  • Figure 2: At extreme corners of the moduli space a light tower of particle appears whose mass scales down exponentially with distance. The effective number of light fields $N$ increases as we approach the boundaries. These light fields lead to the dual description of the theory.
  • Figure 3: Gromov-Witten invariants 'count' holomorphic maps from curves to Calabi-Yau manifolds.
  • Figure 4: The genus 1 Gromov-Witten invariants get mapped using mirror symmetry to one loop amplitudes of Kodaira-Spencer theory, which in turn is given by a holomorphic version of Ray-Singer torsion.
  • Figure 5: A typical behaviour of $\Lambda_s^2$ as a function of moduli fields: exponential decay at infinity and a maximum of order 1 in the interior.
  • ...and 2 more figures