On the Quantization of Special K\"ahler Manifolds

TL;DR

This paper explores the quantization of special Kähler manifolds, develops wavefunction representations for coherent states, and connects these to the BCOV holomorphic anomaly equation, providing explicit solutions.

Contribution

It introduces a quantization framework for special Kähler manifolds and links it to the holomorphic anomaly equation, extending previous theoretical work.

Findings

Constructed wavefunction representations for coherent states.

Derived the generalization of the BCOV holomorphic anomaly equation.

Provided explicit solutions to the holomorphic anomaly equation.

Abstract

We show how affine and projective special K\"ahler manifolds emerge from the structure of quantization. We quantize them and construct natural (wavefunction) representations for the corresponding coherent states. These in turn are shown to satisfy the precise generalizations of the BCOV holomorphic anomaly equation (hep-th/9309140), thus extending the work in hep-th/9306122. As a byproduct of the analysis we construct the explicit general solution to the holomorphic anomaly equation.