Modified Shiffman–Zelditch Conjecture (Part b): Rigorous Polyakov Path Integral via Calabi/Mabuchi Volume Forms

Ascertain whether, for polarized manifolds (M, L) with compact automorphism group Aut(M, L), one can obtain a rigorous definition of the Polyakov path integral over metrics by using either the Calabi metric or the Mabuchi metric to define the associated volume form on Bergman spaces.

Background

Shiffman–Zelditch emphasized that a positive resolution of the finiteness question would enable a rigorous definition of the Polyakov path integral using the Calabi metric-derived volume form.

Given the failure of the original conjecture in non-compact automorphism settings, the authors ask whether such a rigorous formulation is still attainable in the compact automorphism case, possibly using either the Calabi or Mabuchi metric.