Meaningfulness of the general motivic formula for moduli spaces of vector bundles

Determine whether the formula expressing the motivic integral for the moduli space of semistable vector bundles of rank r and degree d in terms of motivic classes of lower-rank moduli spaces is meaningful, despite the difficulty of resolving the associated Harder–Narasimhan recursion.

Background

The paper observes that, in principle, the generalized orbifold formula could compute the motivic integral for the moduli space M_{r,d} of semistable vector bundles on a smooth projective curve in terms of motivic classes of moduli spaces M_{r',d'} with r'≤r and d/r=d'/r'. Carrying this out requires resolving the Harder–Narasimhan recursion, and even the leading motivic term [M_{r,d}] is described as difficult to determine. The authors explicitly leave unresolved whether the resulting formula has any meaningful interpretation.

References

In principle, arguments similar to Section 5.5 allow us to compute the right hand side of Proposition \ref{vbf} for all pairs r,d in terms of motivic classes of M_{r',d'} with r' \leq r and d/r = d'/r'. However already the leading term [M_{r,d}] is quite tricky to determine as one has to resolve the Harder-Narasimhan recursion and it is unclear to the authors whether the formula would be meaningful in any way.

— An orbifold formula for algebraic stacks  (2609.10379 - Loeser et al., 9 Sep 2026) in Section Applications, subsection “Stringy E-function for moduli spaces of vector bundles,” paragraph following Proposition \ref{vbf}