Twisted-factor criterion for inequivalence of β-classes in 𝔅d(G)

Establish that for smooth irreducible projective varieties Y and Y′ with regular generically free actions by a finite Abelian group G, if the sets of Chen–Ruan twisted factors {t(g)} for X=Y/G and {t′(g)} for X′=Y′/G do not coincide, then the Kontsevich–Pestun–Tschinkel β-classes satisfy β(X)≠β(X′) in 𝔅d(G).

Background

The paper proposes a translation between data used to define β-classes in the Burnside-type group 𝔅d(G) and Chen–Ruan cohomological twisted sectors on the orbifold quotient X=Y/G.

Motivated by this correspondence, the authors conjecture a direct implication from non-coincidence of twisted factors to non-equality of β-classes.

References

We propose the following conjecture. Conjecture Let $Y,~Y'$ be smooth irreducible projective varieties with regular generically free group actions by a finite Abelian group $G$. Consider the resulting global quotient orbifolds $X:=Y/G,~X':=Y'/G$. For each global quotient orbifolds $X,~X'$, collect the twisted factors ${\mathrm{t}(g)},{\mathrm{t'}(g)}$. If they do not coincide as sets, $\beta(X)\neq \beta(X')$.

— A Gromov-Witten approach to $G$-equivariant birational invariants (2405.07322 - Cavenaghi et al., 12 May 2024) in Section 5, “A Gromov–Witten reading of β-classes in 𝔅d(G)”