Elliptic orbifold lines and integrable hierarchies

Abstract

In 2003, Givental and the author constructed Hirota quadratic (or bilinear) equations for simple singularities -- a result later utilized by Fan, Jarvis, and Ruan to establish the generalized Witten conjecture. Whether this construction extends to broader classes of singularities has since remained an open question. In this paper, we resolve this question affirmatively for simple elliptic singularities. More precisely, using mirror symmetry, we construct Hirota quadratic equations for simple elliptic singularities by showing that the Gromov--Witten invariants of the elliptic orbifold lines satisfy a system of bilinear equations.

In version 2: we revised the abstract and the introduction to give a more accurate account of the reserach developments relevant for our work. Some of the references were updated