Hypermultiplets and Topological Strings

TL;DR

This work provides an off-shell N=2 projective-superspace formulation of the c-map, showing that the hypermultiplet geometry arising from type II CY compactifications is controlled by a single function related to the vector multiplet prepotential. By expressing the tensor-multiplet Lagrangian as a contour integral of H(η) and performing a Legendre transform, the authors derive the hyperkähler potential and the quaternionic-Kähler metric, confirming the c-map at the level of explicit superspace data and twistor structure. They illuminate connections to black-hole physics and the OSV/topological-string framework, linking the Hartle-Hawking wavefunction to topological-string amplitudes via a Legendre transform and circle reduction. Finally, they outline how higher-genus topological string amplitudes should generate corresponding higher-derivative terms in the hypermultiplet sector, providing a coherent route to quantum corrections of hypermultiplet moduli and their role in black-hole wavefunctions.

Abstract

The c-map relates classical hypermultiplet moduli spaces in compactifications of type II strings on a Calabi-Yau threefold to vector multiplet moduli spaces via a further compactification on a circle. We give an off-shell description of the c-map in N=2 superspace. The superspace Lagrangian for the hypermultiplets is a single function directly related to the prepotential of special geometry, and can therefore be computed using topological string theory. Similarly, a class of higher derivative terms for hypermultiplets can be computed from the higher genus topological string amplitudes. Our results provide a framework for studying quantum corrections to the hypermultiplet moduli space, as well as for understanding the black hole wave-function as a function of the hypermultiplet moduli.