Quantum Elliptic Cohomology From Four-Dimensional Minimal Supersymmetric Gauge Theories
Abstract
In this paper, we study non-perturbative vortex partition functions of four-dimensional N=1 supersymmetric gauge theories on the space-time geometry T2 × D2, and we propose that these partition functions offer a notion of quantum elliptic cohomology. For particular U(1)-gauge theories we calculate these partition functions explicitly by applying equivariant localization methods to Handsaw quiver varieties that realize for this particular class of U(1)-gauge theories the moduli spaces of the non-perturbative vortex sectors. The determined vortex partition functions are annihilated by difference operators, which are interpreted as Ward identities among N=(0,2) BPS surface defects. Compared to lower dimensional gauge theories with four supercharges, anomalies play an essential role for a consistent formulation of the four-dimensional partition functions. In our proposal towards a mathematical formulation of the vortex partition functions in terms of equivariant elliptic cohomology, the gauge theory anomalies relate to geometric properties of the Thom sheaves corresponding to the relevant quasimap moduli spaces. Motivated by the explicit computations we reflect on the existence of a `virtual structure sheaf' on the moduli space of quasimaps for a general mathematical theory of quantum elliptic cohomology.
Related
- Vacuum Structure of Maximally Supersymmetric Two-Dimensional Gauge Theories
- Towards Wedge Construction of Four-Dimensional Non-Supersymmetric Theories and Torsion Classes
- Scattering, bound states, and resonances in the one-dimensional Dirac equation via supersymmetric quantum mechanics
- Supersymmetric semi--infinite cohomology and N = 2 superconformal structures
- Perturbative Holonomy Potentials and Vacua in Chiral Supersymmetric Gauge Theories
BibTeX
@misc{brunner2026quantum,
title = {{Quantum Elliptic Cohomology From Four-Dimensional Minimal Supersymmetric Gauge Theories}},
author = {Ilka Brunner and Peng Cheng and Hans Jockers},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09298},
eprint = {2609.09298},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09298},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Quantum Elliptic Cohomology From Four-Dimensional Minimal Supersymmetric Gauge Theories
AU - Ilka Brunner
AU - Peng Cheng
AU - Hans Jockers
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09298
UR - https://arxiv.org/abs/2609.09298
AB - In this paper, we study non-perturbative vortex partition functions of four-dimensional N=1 supersymmetric gauge theories on the space-time geometry T^2 × D^2, and we propose that these partition functions offer a notion of quantum elliptic cohomology. For particular U(1)-gauge theories we calculate these partition functions explicitly by applying equivariant localization methods to Handsaw quiver varieties that realize for this particular class of U(1)-gauge theories the moduli spaces of the non-perturbative vortex sectors. The determined vortex partition functions are annihilated by difference operators, which are interpreted as Ward identities among N=(0,2) BPS surface defects. Compared to lower dimensional gauge theories with four supercharges, anomalies play an essential role for a consistent formulation of the four-dimensional partition functions. In our proposal towards a mathematical formulation of the vortex partition functions in terms of equivariant elliptic cohomology, the gauge theory anomalies relate to geometric properties of the Thom sheaves corresponding to the relevant quasimap moduli spaces. Motivated by the explicit computations we reflect on the existence of a `virtual structure sheaf' on the moduli space of quasimaps for a general mathematical theory of quantum elliptic cohomology.
ER -