Landau-Ginzburg-Saito theory for descendant Gromov-Witten theory on projective line

TL;DR

The work presents a complete genus-zero mirror framework for descendant Gromov-Witten invariants on $\mathbb{P}^1$ via the Landau-Ginzburg-Saito (LGS) theory. Descendant correlators are defined recursively in the LGS model, shown to obey puncture, divisor, and TRR relations, and linked to GW invariants through a Kontsevich–Manin mirror map. The central theorem asserts equality between GW invariants with descendants and LGS correlators of the mirrored observables; this is supported by Dubrovin reconstruction and explicit TRR matching. The paper then demonstrates concrete computations, including four-, five-, and six-point invariants, Hurwitz numbers, and the polynomiality/integrality structure, illustrating the power and practicality of the LGS mirror approach for $\mathbb{P}^1$ and its descendant theory.

Abstract

We define the correlation functions for the descendants in the Landau-Ginzburg-Saito theory. We show that the correlation functions obey puncture, divisor, dilaton, and topological recursion relations. We formulate the map between the descendant observables in the GW theory on the projective line and the descendant observables in the mirror LGS theory. We prove that the LGS correlation functions of the mirror observables are equal to the GW invariants with descendants.

Theorems & Definitions (45)

• Proposition 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• proof
• Definition 3.1
• Example 3.2
• Definition 3.3