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

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 $W\ne {\mathbb{R}}^n/G$). 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 ${\Omega }_{\omega }^G$ of ${\Omega }_{\omega }={\Omega }_{{\mathbb{R}}^n,0}^n/\omega \wedge {\Omega }_{{\mathbb{R}}^n,0}^{n-1}$. 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.

arXiv comment19 pages