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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.
Related
- Integrable hierarchies with zero dispersion and elliptic curves
- Elliptic spin Ruijsenaars-Schneider integrable models from 5d 𝒩=1 gauge theories
- Spin generalizations of the classical elliptic Calogero--Inozemtsev model and integrable models of interacting tops associated with the BCn root system
- Exact orbifold fillings of contact manifolds
- Virasoro Constraints for Orbifold Curves
BibTeX
@misc{milanov2026elliptic,
title = {{Elliptic orbifold lines and integrable hierarchies}},
author = {Todor Milanov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.08909},
eprint = {2606.08909},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.08909},
}
RIS
TY - UNPB
TI - Elliptic orbifold lines and integrable hierarchies
AU - Todor Milanov
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2606.08909
UR - https://arxiv.org/abs/2606.08909
AB - 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.
ER -