Sphere-trace class in Lagrangian-core homology

Identify the homology class in the fixed part of the Lagrangian core whose capped-vertex integration realizes the sphere trace functional.

Background

The quantum Hikita isomorphism expresses graded traces geometrically as integration of capped vertex functions against classes in the homology of the fixed part of the Lagrangian core. The sphere trace is a distinguished twisted trace arising in gauge theory and is known analytically for good or ugly theories.

The paper proves existence of a corresponding class abstractly under its conjectural framework but does not identify that class explicitly.

References

It would be interesting to describe this class $a{\mathrm{sph}$.

— The quantum Hikita conjecture via quasimaps  (2608.16746 - Dinkins et al., 17 Aug 2026) in Section 4.4, subsection “An application to graded traces”

Question 8.12. In the finite rigid case, (8.8) can be chosen as a modified trace as discussed in the proof of Theorem 8.11. Beyond the rigid case, this requirement does not make sense because the notion of a modified trace needs rigidity. However, the notion of a trace invariant under the action of the mapping class groupZ2 of the annulus without marked intervals still makes sense and is always fulfilled for modified traces. This suggests that Z2-invariant traces, a notion that always can be defined for the compact projective objects ofRep(V ), could be a reasonable replacement of modified traces beyond the finite rigid case. Which vertex operator algebras have such a trace?

— Modular Functors with Singularities from Vertex Operator Algebras Beyond Rigidity and Finiteness  (2608.28579 - Müller et al., 28 Aug 2026) in Question 8.12, Section 8.4