Handling Handles: Nonplanar Integrability in N=4 Supersymmetric Yang-Mills Theory.

Till Bargheer,Shota Komatsu,Pedro Vieira,João Caetano,Thiago Fleury

doi:10.1103/physrevlett.121.231602

Abstract

We propose an integrability setup for the computation of correlation functions of gauge-invariant operators in N=4 supersymmetric Yang-Mills theory at higher orders in the large N_{c} genus expansion and at any order in the 't Hooft coupling g_{YM}^{2}N_{c}. In this multistep proposal, one polygonizes the string world sheet in all possible ways, hexagonalizes all resulting polygons, and sprinkles mirror particles over all hexagon junctions to obtain the full correlator. We test our integrability-based conjecture against a nonplanar four-point correlator of large 1/2 Bogomol'nyi-Prasad-Sommerfield operators at one and two loops.

Highlights

We propose an integrability setup for the computation of correlation functions of gauge-invariant operators in N 1⁄4 4 supersymmetric Yang-Mills theory at higher orders in the large Nc genus expansion and at any order in the ’t Hooft coupling g2YMNc
Exploiting integrability machinery, the full finite-size spectrum has been obtained at any value of the coupling [3,4,5], yielding the energy spectra of single strings in this curved background or— equivalently—the spectra of anomalous dimensions of single-trace operators in N 1⁄4 4 supersymmetric YangMills (SYM) theory in the planar limit
We are dealing with world sheets with handles

Summary

Published by the American Physical Society

The loop correlator Gk ≡ hQk1Qk2Qk3Qk4i − hQk1Qk2Qk3Qk4itree can be decomposed according to the propagator structures that connect the operators as. The quantum corrections dressing the propagator structures depend on the conformally invariant cross ratios jzj2 1⁄4 x212x234=x213x224 and j1 − zj2 1⁄4 x223x214=x213x224. Other key players are the so-called color factors, which consist of color contractions of four symmetrized traces from the four operators, dressed with insertions of gauge group structure constants. We used the fact that—by their combinatorial nature—the various color factors should be quartic polynomials in k and m (up to boundary cases at extremal values of k or m), which we can fit using the data points at finite k and m. Integrability proposal.—We propose that the connected part of any correlator in the UðNcÞ theory, including the full expansion in 1=Nc, can be recovered from integrability via the formula

CA propagators skeleton labelings bridge graphs fillings

Combinatorial factor