Small deformations of supersymmetric Wilson loops and open spin-chains

TL;DR

We address how adjoint insertions deform circular/straight Wilson loops in $\mathcal{N}=4$ SYM and classify these insertions by $SL(2,\mathbb{R})$ representations to assign conformal dimensions. At one loop the mixing matrix maps to an open $SU(2)$ spin-chain with Neumann boundaries, solvable by Bethe ansatz, with spectra matching the string dual in $AdS_5\times S^5$ in BMN and thermodynamic limits. The dual string description uses open-string solutions with one or two angular momenta; the BMN fluctuations yield $\Delta-J=\omega_n=\sqrt{1+\lambda n^2/(4J^2)}$, and the two-spin case reproduces folded-string elliptic-integral relations, reflecting the open-chain thermodynamics. These results affirm a defect-CFT-like framework for Wilson loops and point to broad extensions to higher loops, different insertions, and connections to Hubbard models and other defect setups.

Abstract

We study insertions of composite operators into Wilson loops in N=4 supersymmetric Yang-Mills theory in four dimensions. The loops follow a circular or straight path and the composite insertions transform in the adjoint representation of the gauge group. This provides a gauge invariant way to define the correlator of non-singlet operators. Since the basic loop preserves an SL(2,R) subgroup of the conformal group, we can assign a conformal dimension to those insertions and calculate the corrections to the classical dimension in perturbation theory. The calculation turns out to be very similar to that of single-trace local operators and may also be expressed in terms of a spin-chain. In this case the spin-chain is open and at one-loop order has Neumann boundary conditions on the type of scalar insertions that we consider. This system is integrable and we write the Bethe ansatz describing it. We compare the spectrum in the limit of large angular momentum both in the dilute gas approximation and the thermodynamic limit to the relevant string solution in the BMN limit and in the full AdS_5 x S^5 metric and find agreement.

Figures (4)

• Figure 1: The tree-level diagram for a circular Wilson loop (the dotted line) with the insertion of two words each made of three scalars. The Wilson loop sets the order at which the gauge indices are contracted and only the depicted diagram is planar.
• Figure 2: The planar one-loop graphs that do not involve the Wilson loop are the same as for single trace local operators. The self energy diagrams like (a) includes all possible fields going around the loop. The H-diagrams, like (b), involves the exchange of a gluon between nearest-neighbors. The X-diagrams, like (c), involves the quartic scalar term and leads to the permutation term in the spin-chain Hamiltonian. Note though, that the H and X interaction graphs involving the first and last scalars are not planar and therefore are not included.
• Figure 3: At the planar one-loop level only the external most lines may interact with the Wilson loop. These diagrams come from expanding the holonomy to first order bringing down the gauge field and $\Phi_6$. This gauge field can then be contracted with the outermost scalars. Note that the scalars $Z$ and $X$ do not contract with $\Phi_6$.
• Figure 4: A depiction of the string solution on $AdS_5\times S^5$. The string fills an $AdS_2$ subspace of $AdS_5$ (on the left). Near the boundary of $AdS$ it is located at the north pole of an $S^2\subset S^5$ (on the right). Away from the boundary it is no longer at the north pole and rotates around the sphere and as it gets close to the center of $AdS$ it approaches the equator. The dotted circles represent the region one zooms on to get the BMN limit.