# Partition functions of $\mathcal{N}=(2,2)$ supersymmetric sigma models and Special geometry for the two-moduli non-Fermat Calabi-Yau manifold

## TL;DR

This paper verifies the JKLMR conjecture linking partition functions of supersymmetric sigma models on S^2 with special geometry of Calabi-Yau moduli spaces, using mirror symmetry for a two-moduli non-Fermat Calabi-Yau.

## Contribution

It provides an explicit verification of the JKLMR conjecture for a non-Fermat Calabi-Yau manifold via mirror symmetry and special geometry analysis.

## Key findings

- Confirmed the JKLMR conjecture for the non-Fermat Calabi-Yau case
- Constructed the dual GLSM and its mirror manifold
- Explicitly computed the partition functions and moduli space geometry

## Abstract

We study the new case of the application of the JKLMR conjecture on the connection between the exact partition functions of $\mathcal{N}=(2,2)$ supersymmetric gauged linear sigma models (GLSM) on $S^2$ and special K\"ahler geometry on the moduli spaces of Calabi-Yau manifold $Y$. The last ones arise as manifolds of the supersymmetric vacua of the GLSM. We establish this correspondence using the Mirror symmetry in Batyrev's approach. Namely, starting from the two-moduli non-Fermat Calabi-Yau manifold $X$ we construct the dual GLSM with the supersymmetric vacua $Y$, which is the mirror for $X$. Knowing the special geometry on the complex moduli space of $X$ we verify the mirror version of the JKLMR conjecture by explicit computation.