A vanishing associated with irregular MSP fields

TL;DR

This paper proves a vanishing result in the virtual localization of mixed-spin-P fields, clarifying the specific invariants contributing to the phase transition between Gromov-Witten and FJRW invariants of Fermat quintics.

Contribution

It establishes a vanishing of certain terms in the virtual class, confirming that only r-spin invariants with specific insertions influence the phase transition.

Findings

Vanishing of specific terms in the virtual class of mixed-spin-P fields.

Only r-spin invariants with insertions 2/5 contribute to the phase transition.

Verification of the contribution of particular invariants in the GLSM context.

Abstract

In previous work, Mixed-Spin-P field has been introduced and their moduli space $\cal{W}_{g,\gamma,\bf{d}}$ together with a $\mathbb{C}^*$ action is constructed. Applying virtual localization to their virtual classes $[\cal{W}_{g,\gamma,\bf{d}}]^{vir}$, polynomial relations among GW and FJRW invariants of Fermat quintics are derived. In this paper, we prove a vanishing of a class of terms in $[(\cal{W}_{g,\gamma,\bf{d}})^{\mathbb{C}^*}]^{vir}$. This vanishing verifies that in Witten's GLSM only $r$-spin invariants of insertions $2/5$ contribute to the phase transition between GW and FJRW invariants of Fermat quintics.