Proof of the refined Dubrovin conjecture for the Lagrangian Grassmanian $LG(2,4)$

TL;DR

This paper provides a new proof of the refined Dubrovin conjecture for the Lagrangian Grassmannian LG(2,4) by explicit computation, confirming the conjecture's predictions for this specific variety.

Contribution

It offers a novel, explicit computational proof of the refined Dubrovin conjecture for LG(2,4), linking quantum cohomology and derived categories.

Findings

Confirmed the refined Dubrovin conjecture for LG(2,4)

Established explicit computational methods for the conjecture

Connected the conjecture to the geometry of LG(2,4)

Abstract

The Dubrovin conjecture predicts a relationship between the monodromy data of the Frobenius manifold associated to the quantum cohomology of a smooth projective variety and the bounded derived category of the same variety. A refinement of this conjecture was given by Cotti, Dubrovin and Guzzetti, which is equivalent to the Gamma conjecture II proposed by Galkin, Golyshev and Iritani. The Gamma conjecture II for quadrics was proved by Hu and Ke. The Lagrangian Grassmanian $LG(2,4)$ is isomorphic to the quadric in $\mathbb P^4$. In this paper, we give a new proof of the refined Dubrovin conjecture for the Lagrangian Grassmanian $LG(2,4)$ by explicit computation.