Vector Bundle Moduli and Small Instanton Transitions

TL;DR

This paper provides a general prescription to compute vector bundle moduli for irreducible, stable $SU(n)$ holomorphic bundles with positive spectral covers on elliptically fibered Calabi–Yau threefolds, with explicit results for base $B={\mathbb F}_{r}$. It defines and enumerates transition moduli arising from chirality-changing small instanton transitions, showing they originate from deformations of the spectral cover restricted to the lift of the M5-brane horizontal curve, and offering an alternative description as sections of a rank-$n$ bundle on the M5-brane curve. The transition moduli are shown to localize on the lift $\pi^{*}z$, and the authors derive concrete counts (e.g. $n_{tm}=14,10$ in examples) through both spectral-cover and push-forward formalisms, including explicit formulas for $\pi_{*}{\mathcal L}_{a}$ on the M5-brane curve. The results connect to physical applications by identifying vector bundle moduli as gauge singlet fields and by informing non-perturbative effects in 4D effective theories, with implications for cosmology in heterotic M-theory scenarios.

Abstract

We give the general presciption for calculating the moduli of irreducible, stable SU(n) holomorphic vector bundles with positive spectral covers over elliptically fibered Calabi-Yau threefolds. Explicit results are presented for Hirzebruch base surfaces B=F_r. The transition moduli that are produced by chirality changing small instanton phase transitions are defined and specifically enumerated. The origin of these moduli, as the deformations of the spectral cover restricted to the ``lift'' of the horizontal curve of the M5-brane, is discussed. We present an alternative description of the transition moduli as the sections of rank n holomorphic vector bundles over the M5-brane curve and give explicit examples. Vector bundle moduli appear as gauge singlet scalar fields in the effective low-energy actions of heterotic superstrings and heterotic M-theory.