Non-holomorphic deformations of special geometry and their applications

TL;DR

This paper introduces non-holomorphic deformations of special geometry, highlighting their role in N=2 supergravity and black hole physics, and explores their connections to duality invariance and topological string theory.

Contribution

It presents a theorem linking arbitrary point-particle Lagrangians to a complex function F, extending special geometry to non-holomorphic cases with applications in field theory and string theory.

Findings

Formulation of Lagrangians using a complex function F

Introduction of duality covariant variables

Expansion of the Hesse potential in external parameters

Abstract

The aim of these lecture notes is to give a pedagogical introduction to the subject of non-holomorphic deformations of special geometry. This subject was first introduced in the context of N=2 BPS black holes, but has a wider range of applicability. A theorem is presented according to which an arbitrary point-particle Lagrangian can be formulated in terms of a complex function F, whose features are analogous to those of the holomorphic function of special geometry. A crucial role is played by a symplectic vector that represents a complexification of the canonical variables, i.e. the coordinates and canonical momenta. We illustrate the characteristic features of the theorem in the context of field theory models with duality invariances. The function F may depend on a number of external parameters that are not subject to duality transformations. We introduce duality covariant complex variables whose transformation rules under duality are independent of these parameters. We express the real Hesse potential of N=2 supergravity in terms of the new variables and expand it in powers of the external parameters. Then we relate this expansion to the one encountered in topological string theory. These lecture notes include exercises which are meant as a guidance to the reader.