Instanton moduli in string theory

Expressions for the number of moduli of arbitrary SU(n) vector bundles constructed via Fourier-Mukai transforms of spectral data over Calabi- Yau threefolds are derived and presented. This is done within the context of simply connected, elliptic Calabi-Yau threefolds with base F-r, but the methods have wider applicability. The condition for a vector bundle to possess the minimal number of moduli for fixed r and n is discussed and an explicit formula for the minimal number of moduli is presented. In addition, transition moduli for small instanton phase transitions involving non-positive spectral covers are defined, enumerated and given a geometrical interpretation.