Define elliptic Gromov–Witten invariants and establish quasimap–stable-map wall crossing

Determine whether elliptic Gromov–Witten invariants can be defined for moduli spaces of stable maps in four-dimensional supersymmetric gauge theories, and, if so, establish a wall-crossing description relating the corresponding stable-map invariants to quasimap invariants in order to compute them explicitly.

Background

The paper studies elliptic vortex partition functions using quasimap moduli spaces, which are interpreted as encoding ultraviolet data of the associated gauge theories. In cohomology and K-theory, quasimap invariants can be related to stable-map or Gromov–Witten invariants through wall-crossing.

The authors leave unresolved whether an analogous theory of elliptic Gromov–Witten invariants exists for stable-map moduli spaces in the four-dimensional setting, and whether a quasimap-to-stable-map wall-crossing formalism can be developed and used for explicit calculations.

References

This observation triggers the question whether elliptic Gromov--Witten invariants can also be defined for the moduli space of stable maps in the context of four-dimensional supersymmetric gauge theories. If so, it would be interesting to know if one can formulate a wall-crossing description between the quasimap moduli spaces and the stable map moduli spaces along the lines of ref., in order to calculate elliptic Gromov--Witten invariants explicitly.

— Quantum Elliptic Cohomology From Four-Dimensional Minimal Supersymmetric Gauge Theories  (2609.09298 - Brunner et al., 8 Sep 2026) in Section 6, Conclusions