BCOV on the Large Hilbert Space

TL;DR

This work recasts BCOV theory of complex-structure deformations on a Calabi–Yau threefold as a worldline of a spinning particle, addressing the intrinsic non-local kinetic term by introducing auxiliary fields in shifted pictures and making the BV structure explicit. It develops both a small-Hilbert-space formulation with local kinetic terms via a local BCOV action and a large-Hilbert-space formulation that incorporates a compensator for the holomorphic 3-form, organizing fields into long and short multiplets and yielding a BV action compatible with the KS equations. The construction connects picture-changing in the worldline/Ramond-sector context to a geometric BV framework, reproduces the Barannikov–Kontsevich action after a symplectic reduction, and offers a background-field, state–operator perspective that extends BCOV to broader complex manifolds. The approach provides new insights into the role of the compensator, the origin of the divergence operator, and a potential path to quantization via worldline graphs, with clear links to Costello–Li and related BV formalisms. Overall, it offers a novel, geometrically grounded route to BCOV and Kodaira–Spencer gravity using a worldline, large- and small-Hilbert-space perspective that blends string-field–like techniques with target-space deformation theory.

Abstract

We formulate the BCOV theory of deformations of complex structures as a pull-back to the super moduli space of the worldline of a spinning particle. In this approach the appearance of a non-local kinetic term in the target space action has the same origin as the mismatch of pictures in the Ramond sector of super string field theory and is resolved by the same type of auxiliary fields in shifted pictures. The BV-extension is manifest in this description. A compensator for the holomorphic 3-form can be included by resorting to a description in the large Hilbert space.