Perspectives on Pfaffians of Heterotic World-sheet Instantons

TL;DR

This paper explores the algebraic and geometric aspects of Pfaffians in heterotic world-sheet instantons on elliptically fibered Calabi-Yau spaces, linking spectral curve geometry to Pfaffian vanishing properties.

Contribution

It provides a unified algebraic and geometric framework for understanding Pfaffians of instantons, connecting spectral curve perspectives and identifying vanishing loci.

Findings

Expressed Pfaffian as a polynomial and theta-function

Connected extrinsic and intrinsic geometries of spectral curves

Identified vanishing loci and bounds on vanishing order

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.