An algebraic formula for the index of a 1-form on a real quotient singularity

TL;DR

This paper derives an algebraic formula for the index of a 1-form on real quotient singularities, linking it to the signature of a residue pairing and quantum cohomology in the context of finite abelian group actions.

Contribution

It introduces a novel algebraic formula for the radial index of 1-forms on real quotient singularities, connecting it to residue pairings and quantum cohomology.

Findings

Index equals the signature of the residue pairing on the G-invariant part.

Signature of the residue pairing matches the orbifold index of df.

Provides a link between algebraic indices and quantum cohomology in singularity theory.

Abstract

Let a finite abelian group $G$ act (linearly) on the space $\mathbb{R}^n$ and thus on its complexification $\mathbb{C}^n$. Let $W$ be the real part of the quotient $\mathbb{C}^n/G$ (in general $W \neq \mathbb{R}^n/G$). We give an algebraic formula for the radial index of a 1-form on the real quotient $W$. It is shown that this index is equal to the signature of the restriction of the residue pairing to the $G$-invariant part $\Omega^G_\omega$ of $\Omega_\omega= \Omega^n_{\mathbb{R}^n,0}/\omega \wedge \Omega^{n-1}_{\mathbb{R}^n,0}$. For a $G$-invariant function $f$, one has the so-called quantum cohomology group defined in the quantum singularity theory (FJRW-theory). We show that, for a real function $f$, the signature of the residue pairing on the real part of the quantum cohomology group is equal to the orbifold index of the 1-form $df$ on the preimage $\pi^{-1}(W)$ of $W$ under the natural quotient map.