Elliptic genus of phases of N=2 theories

TL;DR

This paper provides an algebro-geometric framework for understanding the elliptic genus of N=2 theories' phases, connecting physical models with mathematical invariants and establishing invariance properties across phase transitions.

Contribution

It introduces a rigorous definition of elliptic genus for phases of N=2 theories and demonstrates invariance under phase transitions using equivariant McKay correspondence.

Findings

Recovered elliptic genus of LG models from physics literature

Proved invariance of elliptic genus across phase transitions

Established Landau-Ginzburg/Calabi-Yau correspondence for elliptic genus

Abstract

We discuss an algebro-geometric description of Witten's phases of N=2 theories and propose a definition of their elliptic genus provided some conditions on singularities of the phases are met. For Landau-Ginzburg phase one recovers elliptic genus of LG models proposed in physics literature in early 90s. For certain transitions between phases we derive invariance of elliptic genus from an equivariant form of McKay correspondence for elliptic genus. As special cases one obtains Landau-Giznburg/Calabi-Yau correspondence for elliptic genus of weighted homogeneous potentials as well as certain hybrid/CY correspondences.