Circling in on correlation functions in $\mathcal{N}=2$ orbifold theories

Abstract

We review the harmonic superspace formalism for 4d $\mathcal{N}=2$ theories and apply it to orbifolds of $\mathcal N=4$ Super-Yang-Mills (SYM) theory. These theories have a remarkably similar description to $\mathcal N=4$ SYM, but their planar supergraphs feature topological twist lines which generate phases when intersecting bifundamental hypermultiplet propagators. These cause twisted sector BPS correlators to depend on the coupling $g$. Twisted Coulomb-branch operators of length $L$ receive a leading perturbative contribution to their two-point function at order $\mathcal{O}(g^{2L})$ from a supergraph involving a wrapped hypermultiplet circle. After performing the complete $D$-algebra, we show that the supergraph reduces to the universal conformal period $W_L=\binom{2L}{L}ζ(2L-1)$. We extend this analysis to multiple nested hypermultiplet circles and show that they reproduce a series of higher-order terms in the exact localisation result. A subsequent evaluation of leading contributions to BPS three-point correlators again shows complete agreement with localisation results in extremal case and generates new perturbative data in the non-extremal case. The latter involves one untwisted BPS descendant, which contains physical hypermultiplet fields and can therefore connect directly to the twist-dependent hypermultiplet circle. This reduces the suppression in $g$, resulting in a contribution $4 g^{2(\ell+1)} W_{\ell+1}$ controlled by a bridge length $\ell$ rather than by an external operator length $L\geq\ell$. These examples demonstrate the feasibility of harmonic superspace calculations in orbifold theories and point to various future applications in the context of non-BPS correlators, integrability and marginal deformations.