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Perspectives on Pfaffians of Heterotic World-sheet Instantons

Gottfried Curio

TL;DR

This paper analyzes the Pfaffian prefactor $Pfaff$ for world-sheet instantons in heterotic compactifications on elliptically fibered Calabi–Yau threefolds. It develops and connects two complementary pictures: a polynomial/determinantal (extrinsic) description and a transcendental(theta-function) (intrinsic) description of the $Pfaff$ factor, linked through the geometry of spectral curves and their Jacobians. The authors map out the vanishing loci of $Pfaff$, including the structural locus $\Sigma$, the explicit locus $\cal R$, and SU(3) special cases with $\lambda=3/2$, and study multiplicities of factors within $(Pfaff)$, with explicit results on how these loci sit inside moduli spaces. They also provide an intrinsic derivation of $Pfaff$ via theta functions on the universal Jacobian, clarifying how the extrinsic and intrinsic pictures encode the same physics and moduli-dependence. The work yields concrete criteria for when instantons contribute to the superpotential and reveals how vector bundle moduli influence holomorphic determinants and their multiplicities, with implications for moduli stabilization in heterotic models.

Abstract

To fix the bundle moduli of a heterotic compactification one has to understand the Pfaffian one-loop prefactor of the classical instanton contribution. For compactifications on elliptically fibered Calabi-Yau spaces X this can be made explicit for spectral bundles and world-sheet instantons supported on rational base curves b: one can express the Pfaffian in a closed algebraic form as a polynomial, or it may be understood as a theta-function expression. We elucidate the connection between these two points of view via the respective perception of the relevant spectral curve, related to its extrinsic geometry in the ambient space (the elliptic surface in X over b) or to its intrinsic geometry as abstract Riemann surface. We identify, within a conceptual description, general vanishing loci of the Pfaffian, and derive bounds on the vanishing order, relevant to solutions of W=dW=0.

Perspectives on Pfaffians of Heterotic World-sheet Instantons

TL;DR

This paper analyzes the Pfaffian prefactor for world-sheet instantons in heterotic compactifications on elliptically fibered Calabi–Yau threefolds. It develops and connects two complementary pictures: a polynomial/determinantal (extrinsic) description and a transcendental(theta-function) (intrinsic) description of the factor, linked through the geometry of spectral curves and their Jacobians. The authors map out the vanishing loci of , including the structural locus , the explicit locus , and SU(3) special cases with , and study multiplicities of factors within , with explicit results on how these loci sit inside moduli spaces. They also provide an intrinsic derivation of via theta functions on the universal Jacobian, clarifying how the extrinsic and intrinsic pictures encode the same physics and moduli-dependence. The work yields concrete criteria for when instantons contribute to the superpotential and reveals how vector bundle moduli influence holomorphic determinants and their multiplicities, with implications for moduli stabilization in heterotic models.

Abstract

To fix the bundle moduli of a heterotic compactification one has to understand the Pfaffian one-loop prefactor of the classical instanton contribution. For compactifications on elliptically fibered Calabi-Yau spaces X this can be made explicit for spectral bundles and world-sheet instantons supported on rational base curves b: one can express the Pfaffian in a closed algebraic form as a polynomial, or it may be understood as a theta-function expression. We elucidate the connection between these two points of view via the respective perception of the relevant spectral curve, related to its extrinsic geometry in the ambient space (the elliptic surface in X over b) or to its intrinsic geometry as abstract Riemann surface. We identify, within a conceptual description, general vanishing loci of the Pfaffian, and derive bounds on the vanishing order, relevant to solutions of W=dW=0.

Paper Structure

This paper contains 33 sections, 155 equations.

Table of Contents

  1. Introduction
  2. Overview and summary
  3. The Pfaffian prefactor
  4. The Pfaffian as a section of a line bundle
  5. The case of base curves in elliptically fibered $X$ and fibrewise semistable bundles
  6. The family of chiral Dirac operators in the elliptic case
  7. Structure of the moduli space of the bundles over $X$
  8. The moduli space of the restricted bundles
  9. Decomposition of ${\cal L}$ relative to a spin choice
  10. Dimension of the moduli space and the divisor $(Pfaff)$
  11. Special loci in the moduli space: $\Sigma \subset (Pfaff)$
  12. A further explicit locus in the moduli space: ${\cal R}\subset \Sigma$
  13. Some remarks about multiplicities
  14. The case of $SU(3)$ bundles
  15. The relation $\; {\cal R}=\Sigma$ for $SU(3)$ bundles
  16. ...and 18 more sections