The Symplectic Structure of N=2 Supergravity and its Central Extension

TL;DR

The paper formulates N=2, D=4 supergravity coupled to n_V vector multiplets, highlighting the central role of the symplectic structure of special geometry in determining moduli-dependent electric and magnetic charges. It derives explicit expressions for the central charge Z and matter charges Z_i in terms of the symplectic data and demonstrates their model-independent relations, including BPS mass formulas, via matrices like ${ m M}({ m N})$ and ${ m M}({ m F})$. The authors then specialize the framework to Type IIB string theory on Calabi–Yau threefolds, linking the 4D charges to CY period vectors and to RR charges of the internal theory, with BPS masses set by $M=e^{K/2}|Z^{(3,0)}|$. Overall, the work provides a coherent, duality-friendly bridge between 4D supergravity couplings, their symplectic structure, and string compactifications on CY manifolds, with implications for nonperturbative spectra and black hole physics.

Abstract

We report on the formulation of $N=2$ $D=4$ supergravity coupled to $n_V$ abelian vector multiplets in presence of electric and magnetic charges. General formulae for the (moduli dependent) electric and magnetic charges for the $n_V+1$ gauge fields are given which reflect the symplectic structure of the underlying special geometry. The specification to Type IIB strings compactified on Calabi-Yau manifolds, with gauge group $U(1)^{h_{21}+1}$ is given.