The Moduli of Reducible Vector Bundles

TL;DR

This work computes the moduli space dimension of reducible, holomorphic vector bundles with structure group $SU(n)\times SU(m)$ on elliptically fibered Calabi–Yau threefolds with base $d\mathbb{P}_9$, arising from small instanton gauge-changing transitions. It combines the spectral cover construction for $SU(n)$ bundles on the threefold, the pull-back of stable $SU(m)$ bundles from the base, and a careful deformation analysis via Leray sequences and index theory to derive a complete moduli count. Key results include explicit formulas for the four deformation sectors I–IV, with cross-terms II and III shown to reduce to a single nonzero contribution for $\lambda \neq 0$, and a final closed-form expression for dim$\mathcal{M}(V\oplus\pi^*M)$ in terms of base data and spectral parameters. The findings supply a concrete, computable framework for tracking how small instanton transitions impact the moduli of reducible bundles, with potential implications for phenomenology in heterotic M-theory and related brane-world constructions.

Abstract

A procedure for computing the dimensions of the moduli spaces of reducible, holomorphic vector bundles on elliptically fibered Calabi-Yau threefolds X is presented. This procedure is applied to poly-stable rank n+m bundles of the form V + pi* M, where V is a stable vector bundle with structure group SU(n) on X and M is a stable vector bundle with structure group SU(m) on the base surface B of X. Such bundles arise from small instanton transitions involving five-branes wrapped on fibers of the elliptic fibration. The structure and physical meaning of these transitions are discussed.