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Vector Bundle Moduli Superpotentials in Heterotic Superstrings and M-Theory

Evgeny I. Buchbinder, Ron Donagi, Burt A. Ovrut

TL;DR

The paper develops a precise algebraic framework to compute non-perturbative vector-bundle moduli superpotentials in heterotic string theory and its M-theory duals by relating the Pfaffian of the Dirac operator to the determinant of a moduli-dependent map $f_C$. Vector bundles are constructed from spectral data via a Fourier–Mukai transform on elliptically fibered Calabi–Yau threefolds, and the relevant moduli are encoded in the restricted spectral cover $C$ and its image on a chosen curve $z$. The central result is that ${ m Pfaff}({ m D}_{-})$ and hence the superpotential $W$ are proportional to ${ m det}f_C$, with zeros of the Pfaffian corresponding exactly to the zeros of ${ m det}f_C$; explicit determinant computations are carried out for four isolated-curve examples, yielding concrete polynomial expressions in the vector-bundle and CY moduli. This enables explicit, holomorphic expressions for the moduli-dependent superpotential, facilitating analyses of vacuum stability, small instanton transitions, and related cosmological scenarios within heterotic frameworks. The work also provides a detailed algebraic framework and appendices giving the spectral data, direct-image formulas, and concrete polynomial structures needed for practical calculations of $W$ across moduli spaces.

Abstract

The non-perturbative superpotential generated by a heterotic superstring wrapped once around a genus-zero holomorphic curve is proportional to the Pfaffian involving the determinant of a Dirac operator on this curve. We show that the space of zero modes of this Dirac operator is the kernel of a linear mapping that is dependent on the associated vector bundle moduli. By explicitly computing the determinant of this map, one can deduce whether or not the dimension of the space of zero modes vanishes. It is shown that this information is sufficient to completely determine the Pfaffian and, hence, the non-perturbative superpotential as explicit holomorphic functions of the vector bundle moduli. This method is illustrated by a number of non-trivial examples.

Vector Bundle Moduli Superpotentials in Heterotic Superstrings and M-Theory

TL;DR

The paper develops a precise algebraic framework to compute non-perturbative vector-bundle moduli superpotentials in heterotic string theory and its M-theory duals by relating the Pfaffian of the Dirac operator to the determinant of a moduli-dependent map . Vector bundles are constructed from spectral data via a Fourier–Mukai transform on elliptically fibered Calabi–Yau threefolds, and the relevant moduli are encoded in the restricted spectral cover and its image on a chosen curve . The central result is that and hence the superpotential are proportional to , with zeros of the Pfaffian corresponding exactly to the zeros of ; explicit determinant computations are carried out for four isolated-curve examples, yielding concrete polynomial expressions in the vector-bundle and CY moduli. This enables explicit, holomorphic expressions for the moduli-dependent superpotential, facilitating analyses of vacuum stability, small instanton transitions, and related cosmological scenarios within heterotic frameworks. The work also provides a detailed algebraic framework and appendices giving the spectral data, direct-image formulas, and concrete polynomial structures needed for practical calculations of across moduli spaces.

Abstract

The non-perturbative superpotential generated by a heterotic superstring wrapped once around a genus-zero holomorphic curve is proportional to the Pfaffian involving the determinant of a Dirac operator on this curve. We show that the space of zero modes of this Dirac operator is the kernel of a linear mapping that is dependent on the associated vector bundle moduli. By explicitly computing the determinant of this map, one can deduce whether or not the dimension of the space of zero modes vanishes. It is shown that this information is sufficient to completely determine the Pfaffian and, hence, the non-perturbative superpotential as explicit holomorphic functions of the vector bundle moduli. This method is illustrated by a number of non-trivial examples.

Paper Structure

This paper contains 18 sections, 450 equations.

Table of Contents

  1. Introduction:
  2. Non-Perturbative Superpotentials:
  3. The Moduli of Stable SU(n) Vector Bundles:
  4. Parameters of ${\mathcal{C}}$ and the Space of Line Bundles ${\cal N}$:
  5. Vector Bundles Restricted to $\pi^{*}z$ and their Moduli:
  6. Vector Bundles on the Curve $z$:
  7. Vector Bundle Moduli on $z$:
  8. The Vector Bundle Restricted to $z$:
  9. General Structure of the Zeros of the Pfaffian:
  10. The Superpotential and the Pfaffian:
  11. Vanishing of the Pfaffian:
  12. Computing $detf_C$:
  13. Expressions for the Superpotential:
  14. Appendix A: Evaluation of $\int {\cal D}g e^{-S_{WZW0}}$
  15. Appendix B: Spectral Data and $SU(n)$ Vector Bundles
  16. ...and 3 more sections