Three-point functions from integrability in $\mathcal{N}=2$ orbifold theories

TL;DR

This work extends the hexagon formalism, originally developed for N=4 SYM, to N=2 Z_M orbifolds by incorporating twist-aware modifications to Bethe equations and hexagon gluing. By performing tree-level checks across multiple SU(2) sectors and two orbifold settings (notably Z_2 bifundamental and adjoint gauges), the authors demonstrate agreement between Wick contractions, spin-chain overlaps, and hexagon predictions, provided twist factors are correctly tracked. The results indicate that structure constants of non-BPS operators in orbifolds can be accessed with integrability techniques at finite coupling, given careful treatment of twisted sectors and multiplet projections; this opens a route toward full finite-coupling determinations. The study also clarifies how translation-like operations interact with orbifold twists and highlights challenges and prospects for generalizing to higher M, other sectors, and including wrapping/gluing corrections guided by localization results. Overall, the paper provides crucial first evidence that the hexagon approach can be adapted to less supersymmetric, orbifolded theories, with significant implications for the broader program of structure constants in integrable AdS/CFT setups.

Abstract

Besides solving the spectral problem of $\mathcal{N}=4$ Super-Yang-Mills (SYM) theory, integrability also provides us with tools to compute the structure constants of the theory, most prominently through the hexagon formalism. We show that, with minor modifications, this formalism can also be applied to orbifolds of $\mathcal{N}=4$ SYM theory, which are integrable theories in their own right. To substantiate this claim, we test our results against a direct gauge-theory calculation at tree-level. We focus here on a family of $\mathcal{N}=2$ supersymmetric $\mathbb{Z}_M$-orbifold theories. BPS correlators in these theories have recently been investigated with independent localisation techniques and a structural matching with wrapping corrections in the hexagon formalism was observed. Together with our weak-coupling evidence, this suggests that a full determination of the structure constants of orbifold theories at finite coupling may be within reach.

Figures (7)

• Figure 1: Structure constants can be evaluated through the hexagon formalism. After cutting the worldsheet, all possible distributions of excitations on the two hexagons have to be summed. Finite-size correction can be captured by inserting full sets of states on the cut edges.
• Figure 2: The "necklace" quiver diagrams associated to $\mathcal{N}=2$$\amsmathbb{Z}_M$-orbifold theories feature $M$ gauge nodes and bifundamental hypermultiplets. All gauge nodes have the same gauge coupling constant $g$.
• Figure 3: The twist operators inserted in the traces correspond to modified boundary conditions of the dual closed string states, which may be represented by twist-lines on the worldsheet Cavaglia:2020hdb. The overall twist within a consistent three-point function has to cancel, resulting in an effective vertex for the twist lines that ensures this cancellation. Alternatively, one may split e.g. $\gamma^m=\gamma^{-k}\gamma^{-l}$ on the boundary and directly connect the twist lines from boundary to boundary.
• Figure 4: Three-point functions can be calculated from spin-chain overlaps summing over all possible contractions. Left: For the transversal excitations only conjugate fields can be contracted. Right: The longitudinal excitations can either be contracted on the translated vacuum or on their conjugate counterpart (contact terms). We also marked the position of the twist operator in a twisted-twisted-untwisted three-point function in the orbifold theory.
• Figure 5: For three-point functions in orbifold theories involving twisted operators, we have to introduce twist operators into the external traces. Here we consider a twisted-twisted-untwisted correlator and extend the twist along the orange line. Moving the twist over the spin chain, it can be moved to either hexagon. Magnons may pick up a twist factor $\omega^k$ when they travel over an edge carrying twist. In scenarios with three twisted operators, the twist lines meet as in Fig. \ref{['Fig:Twistvertex']} and generate the superselection rule \ref{['eq:superselection']}. One may then resolve the vertex into two separate twist lines connecting e.g. $\alpha_1$ with $\alpha_2$ and $\alpha_1$ with $\alpha_3$, respectively.