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

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