Irregular vanishing on $\mathbb{P}^2 \times \mathbb{P}^2$

Abstract

In this paper, we describe Mixed-Spin-P(MSP) fields for a smooth CY 3-fold $X_{3,3} \subset \mathbb{P}^2 \times \mathbb{P}^2$. Then we describe $\mathbb{C}^* -$fixed loci of the moduli space of these MSP fields. We prove that any virtual localization term coming from the fixed locus corresponding to an irregular graph does not contribute to the invariant if the graph is not a pure loop, and also prove this vanishing property for the moduli space of N-MSP fields.