[Paper Review] Variation of Hodge structures, Frobenius manifolds and Gauge theory

Si Li|arXiv (Cornell University)|Mar 12, 2013

Black Holes and Theoretical Physics18 references3 citations

TL;DR

This paper establishes a precise equivalence between the tree-level generating function of descendant invariants in BCOV theory—derived from Kodaira-Spencer gauge theory on Calabi-Yau manifolds—and the semi-infinite variation of Hodge structures (SIVH) construction by Barannikov. It proves that the genus-zero potential function in BCOV theory matches the Frobenius structure arising from SIVH, generalizing mirror symmetry at genus zero to include gravitational descendants via Feynman diagram integrals with a renormalized propagator.

ABSTRACT

We explain the homological relation between the Frobenius structure on the deformation space of Calabi-Yau manifold and the gauge theory of Kodaira-Spencer gravity. We show that the genus zero generating function of descendant invariants on Calabi-Yau manifolds from Barannikov's semi-infinite variation of Hodge structures is equivalent to the Kodaira-Spencer gauge theory at tree level.

Motivation & Objective

• To establish a mathematical bridge between Kodaira-Spencer gauge theory (BCOV theory) and the Frobenius manifold structure on Calabi-Yau moduli spaces.
• To show that the genus-zero generating function of descendant Gromov-Witten invariants from BCOV theory matches the prepotential constructed via Barannikov's semi-infinite variation of Hodge structures.
• To generalize previous results on mirror symmetry at genus zero by incorporating gravitational descendants and higher-order invariants.
• To provide a rigorous field-theoretic interpretation of the Frobenius potential using Feynman diagram expansions with a singular propagator.
• To demonstrate that the kernel of a certain differential operator in the BCOV framework corresponds to the Lagrangian submanifold defined by the prepotential, thus linking geometry and quantum field theory.

Proposed method

• Uses the symplectic geometry of the space of flat sections in semi-infinite Hodge structures, equipped with a residue pairing and Gauss-Manin connection.
• Applies Givental's loop space formalism to interpret the Frobenius structure via the Lagrangian cone in the symplectic vector space of flat sections.
• Constructs the BCOV theory action $ S^{BCOV} $ as a deformation of the Kodaira-Spencer action, incorporating gravitational descendants via vertex operators.
• Defines the tree-level partition function $ \mathbf{F}^{BCOV}_0 $ as a sum over connected tree Feynman diagrams with propagator $ P = \frac{\bar{\partial}^* \partial}{\Delta} $, and vertex $ S^{BCOV} $.
• Applies the homological perturbation lemma to compute the homotopy inverse of the projection map, enabling the identification of the kernel of the operator $ \eta $ with the Lagrangian submanifold.
• Uses Weyl quantization to define a non-commutative deformation of the classical Lagrangian, with the quantum ideal sheaf in the Fock space encoding the full quantum BCOV theory.

Experimental results

Research questions

• RQ1 Is the tree-level generating function of descendant invariants in BCOV theory equivalent to the Frobenius potential constructed from semi-infinite variation of Hodge structures?
• RQ2 How does the Kodaira-Spencer gauge theory at tree level reproduce the prepotential of the B-model on Calabi-Yau manifolds?
• RQ3 What is the precise field-theoretic realization of the Frobenius manifold structure in terms of Feynman diagrams with gravitational descendants?
• RQ4 How does the kernel of the operator $ \eta $, defined via the BCOV action, relate to the Lagrangian submanifold in the symplectic space of flat sections?
• RQ5 Can the full BCOV theory, including higher-loop diagrams, be consistently renormalized using homological methods, and how does this affect the quantum prepotential?

Key findings

• The tree-level partition function $ \mathbf{F}^{BCOV}_0 $ of BCOV theory is equivalent to the generating function of descendant invariants constructed via Barannikov's semi-infinite variation of Hodge structures.
• The genus-zero potential function of the Frobenius manifold on the Calabi-Yau moduli space is given by the restriction $ \mathbf{F}^{BCOV}_0|_{\operatorname{H}} $, confirming the geometric origin of the prepotential.
• The kernel of the operator $ \eta $, defined as $ \mu \mapsto \partial_\mu S^{BCOV} $, is generated by $ \mu - \Pi \left( \sum_{k\geq 0} (\{S^{BCOV}, -\} G)^k \partial_\mu S^{BCOV} \right) $, which characterizes the Lagrangian submanifold in the symplectic space of flat sections.
• The generating functional $ \mathbf{F}^{BCOV}_0 $ is explicitly computed as a sum over connected tree diagrams with propagator $ P = \frac{\bar{\partial}^* \partial}{\Delta} $, confirming its Feynman diagram interpretation.
• The classical Lagrangian $ \mathcal{L}_X \subset \mathcal{H} $ is quantized via Weyl algebra deformation, with the quantum ideal sheaf in the Fock space encoding the full quantum BCOV theory.
• The renormalization of higher-loop Feynman diagrams in BCOV theory is achieved via Costello's homological perturbation techniques, resolving ultraviolet divergences through a consistent deformation quantization procedure.