On exact correlation functions of chiral ring operators in $2d$ $\mathcal{N}=(2, 2)$ SCFTs via localization

TL;DR

This paper develops an extended superlocalization method to compute exact extremal correlation functions of chiral ring operators in 2D $ ext{N}=(2,2)$ SCFTs, with applications to Calabi-Yau geometries and topological theories.

Contribution

It introduces a modified localization algorithm that disentangles operator mixing on $S^2$ and provides geometric interpretation via Griffiths transversality in Calabi-Yau moduli.

Findings

Exact extremal correlators computed as functions of marginal parameters.

Geometric interpretation of the algorithm in Calabi-Yau complex structure moduli.

Alternative formalism for correlators in complete intersections in toric varieties.

Abstract

We study the extremal correlation functions of (twisted) chiral ring operators via superlocalization in $\mathcal{N}=(2, 2)$ superconformal field theories (SCFTs) with central charge $c\geq 3$, especially for SCFTs with Calabi-Yau geometric phases. We extend the method in arXiv:1602.05971 with mild modifications, so that it is applicable to disentangle operators mixing on $S^2$ in nilpotent (twisted) chiral rings of $2d$ SCFTs. With the extended algorithm and technique of localization, we compute exactly the extremal correlators in $2d$ $\mathcal{N}=(2, 2)$ (twisted) chiral rings as non-holomorphic functions of marginal parameters of the theories. Especially in the context of Calabi-Yau geometries, we give an explicit geometric interpretation to our algorithm as the Griffiths transversality with projection on the Hodge bundle over Calabi-Yau complex moduli. We also apply the method to compute extremal correlators in K\"{a}hler moduli, or say twisted chiral rings, of several interesting Calabi-Yau manifolds. In the case of complete intersections in toric varieties, we provide an alternative formalism for extremal correlators via localization onto Higgs branch. In addition, as a spinoff we find that, from the extremal correlators of the top element in twisted chiral rings, one can extract chiral correlators in A-twisted topological theories.