Construct the virtual object in equivariant elliptic cohomology

Construct a virtual fundamental object, or equivalently a virtual structure sheaf, for quasimap moduli spaces in equivariant elliptic cohomology so that elliptic vortex partition functions can be defined by push-forward along the quasimap evaluation map.

Background

The paper formulates four-dimensional vortex partition functions using equivariant elliptic cohomology of quasimap moduli spaces. In the cohomological and K-theoretic settings, the corresponding virtual fundamental classes or virtual structure sheaves are available and enter the push-forward defining the vortex partition function.

For equivariant elliptic cohomology, push-forwards require twists by Thom sheaves, and the relevant quasimap moduli spaces are generally singular and require virtual data. The authors use the Thom sheaf of the obstruction bundle as a candidate, but do not provide a general construction of the required virtual object.

References

We subsequently extrapolate the arguments to the elliptic case and conjecture the form of the vortex partition function, which in the case of elliptic cohomology becomes a meromorphic section of a Thom sheaf. We use the same toolbox as for K~theory. In particular, symmetry and localization arguments are still available, though different since localization requires a twist by the Thom sheaf. To conclude our argument, we conjecture that a virtual object exists also in elliptic cohomology. The precise definition or form does not affect our arguments.

— Quantum Elliptic Cohomology From Four-Dimensional Minimal Supersymmetric Gauge Theories  (2609.09298 - Brunner et al., 8 Sep 2026) in Section 5, “Vortex Partition Function and Quasimaps”; Section 5.4, “Elliptic Vortex Partition Function and Anomalies”