The fundamental solution matrix and relative stable maps

TL;DR

This paper explores the relationship between Givental's Lagrangian cone, the fundamental solution matrix, and relative stable maps, providing new insights and proofs in Gromov-Witten theory.

Contribution

It recasts the construction of Givental's Lagrangian cone within the context of relative stable maps, revealing a connection to the fundamental solution matrix and suggesting a generalization of quantization formalism.

Findings

Identifies the push-forward as the transform of the Lagrangian cone under the fundamental solution matrix.

Establishes a polynomiality property leading to universal relations for Gromov-Witten invariants.

Provides new proofs of foundational results in Gromov-Witten theory.

Abstract

Givental's Lagrangian cone $\mathcal{L}_X$ is a Lagrangian submanifold of a symplectic vector space which encodes the genus-zero Gromov-Witten invariants of $X$. Building on work of Braverman, Coates has obtained the Lagrangian cone as the push-forward of a certain class on the moduli space of stable maps to $X \times \mathbb{P}^1$. This provides a conceptual description for an otherwise mysterious change of variables called the dilaton shift. In this note we recast this construction in its natural context, namely the moduli space of stable maps to $X \times \mathbb{P}^1$ relative the divisor $X \times \infty$. We find that the resulting push-forward is another familiar object, namely the transform of the Lagrangian cone under the action of the fundamental solution matrix. This hints at a generalisation of Givental's quantisation formalism to the setting of relative invariants. Finally, we use a hidden polynomiality property implied by our construction to obtain a sequence of universal relations for the Gromov-Witten invariants, as well as new proofs of several foundational results concerning both the Lagrangian cone and the fundamental solution matrix.