Finiteness of volume of moduli spaces

TL;DR

The paper investigates whether the volume of moduli spaces of two-dimensional CFTs, measured by the Zamolodchikov metric, is finite. It develops a physics-based RG argument using a GLSM realization of Calabi-Yau targets (notably the quintic in $\mathbb{C}P^4$) to access the UV metric perturbatively and then argues that RG flow to the IR preserves finiteness under a gap assumption. The UV moduli-space metric on the complex structure sector is shown to be finite after a symplectic reduction from $V=\mathbb{C}^{126}$ to the 101-complex-dimensional moduli space, and the finiteness is argued to persist to the IR when the flow time is finite. The authors discuss subtleties at the discriminant locus and near conifold-like regions, where a vanishing gap could introduce divergences, yet they establish a weak, physically motivated finiteness result for regions with a gap and hint at stronger statements (e.g., preservation of the Kahler class) to enable exact volume computations. This work ties moduli-space geometry to RG dynamics and outlines a path toward estimating attractor counts via moduli-space volume.

Abstract

We give a ``physics proof'' of a conjecture made by the first author at Strings 2005, that the moduli spaces of certain conformal field theories are finite volume in the Zamolodchikov metric, using an RG flow argument.