Kahler Quantization of H3(CY3,R) and the Holomorphic Anomaly

TL;DR

This paper explores the quantization of a quadratic field theory on a Calabi-Yau three-fold, linking the algebra of moduli translation generators to the holomorphic anomaly equations through Dirac quantization.

Contribution

It introduces a Kahler quantization approach for H3(CY3,R) and demonstrates the equivalence of Schrödinger equations to holomorphic anomaly equations using Dirac's method.

Findings

Derived the algebra of moduli translation generators in canonical coordinates.

Established the equivalence between Schrödinger equations and holomorphic anomaly equations.

Showed that proper scaling of the action functional yields the anomaly equations.

Abstract

Studying the quadratic field theory on seven dimensional spacetime constructed by a direct product of Calabi-Yau three-fold by a real time axis, with phase space being the third cohomology of the Calabi-Yau three-fold, the generators of translation along moduli directions of Calabi-Yau three-fold are constructed. The algebra of these generators is derived which take a simple form in canonical coordinates. Applying the Dirac method of quantization of second class constraint systems, we show that the Schr\"{o}dinger equations corresponding to these generators are equivalent to the holomorphic anomaly equations if one defines the action functional of the quadratic field theory with a proper factor one-half.