Integrable Wilson loops

TL;DR

The work develops an integrable framework for the generalized quark-antiquark potential and cusp anomalous dimension in planar N=4 SYM by mapping cusped Wilson loops with operator insertions to open psu(2|2)^2 spin-chains. It introduces boundary reflections, boundary scalar factors, and two-angle twists to encode the cusp geometry, and formulates a twisted boundary BTBA to compute the exact spectrum, including wrapping corrections. The approach reproduces known weak-coupling results and provides a nonperturbative route to the cusp data at all couplings, with potential numerical implementation and extensions to related open-spin-chain systems and other holographic setups. This framework links gauge theory observables to open-string dynamics in AdS5×S5 and offers avenues to cross-check with semiclassical string results and known cusp-formulae such as BES.

Abstract

The generalized quark-antiquark potential of N=4 supersymmetric Yang-Mills theory on S^3 x R calculates the potential between a pair of heavy charged particles separated by an arbitrary angle on S^3 and also an angle in flavor space. It can be calculated by a Wilson loop following a prescribed path and couplings, or after a conformal transformation, by a cusped Wilson loop in flat space, hence also generalizing the usual concept of the cusp anomalous dimension. In AdS_5 x S^5 this is calculated by an infinite open string. I present here an open spin-chain model which calculates the spectrum of excitations of such open strings. In the dual gauge theory these are cusped Wilson loops with extra operator insertions at the cusp. The boundaries of the spin-chain introduce a non-trivial reflection phase and break the bulk symmetry down to a single copy of psu(2|2). The dependence on the two angles is captured by the two embeddings of this algebra into \psu(2|2)^2, i.e., by a global rotation. The exact answer to this problem is conjectured to be given by solutions to a set of twisted boundary thermodynamic Bethe ansatz integral equations. In particular the generalized quark-antiquark potential or cusp anomalous dimension is recovered by calculating the ground state energy of the minimal length spin-chain, with no sites. It gets contributions only from virtual particles reflecting off the boundaries. I reproduce from this calculation some known weak coupling perturtbative results.

Figures (11)

• Figure 1: A cusped Wilson loop ($a$) in ${\mathcal{N}}=4$ SYM on $\mathbb{R}^4$ (and by analytic continuation also on $\mathbb{R}^{3,1}$) is related to a pair of antiparallel lines ($b$) on $\mathbb{S}^3\times\mathbb{R}$. In the dual string theory this is calculated by a string world--sheet in $AdS_5\times S^5$ ending along the two lines on the boundary ($c$).
• Figure 2: A generalization of Figure \ref{['fig:cylinder']} allows an arbitrary adjoint valued local operator to be inserted at the apex of the cusp ($a$) in $\mathbb{R}^4$. After the conformal transformation this is a pair of antiparallel lines ($b$) on $\mathbb{S}^3\times\mathbb{R}$, seemingly like in Figure \ref{['fig:cylinder']}$b$, but in fact the details of the operator ${\mathcal{O}}$ are represented by nontrivial boundary condition ${\left| {\psi} \right>}$ at past and future infinity. The dual string solution in $AdS_5\times S^5$ still ends along the two lines, but is in an excited state (in the figure the sphere is suppressed).
• Figure 3: Yet another picture of a Wilson loop with two cusps (with possible local insertions) connected by arcs. It is related to that in Figure \ref{['fig:excited']}$a$ by a conformal transformation mapping the point at infinity to finite distance. If the distance between the cusps is $d$, the expectation value of this Wilson loop is ${\left< {W} \right>}\propto 1/d^{2V(\lambda,\phi,\theta)}$, where the logarithmic divergences in (\ref{['G']}) are interpreted, as usual as renormalizing the classical dimension.
• Figure 4: Sample planar Feynman graph calculations for a Wilson loop with two insertion of operators made of five scalar fields at the cusps At leading order there are only five free propagators (dashed lines) so the classical conformal dimension is five ($a$). At one loop there are nearest neighbor interactions among the scalar fields (like $b$), which are identical to those of single trace local operators, but also interaction between the last scalar and the Wilson loop ($c$). At this order planar graphs connecting the loop to itself are restricted to each of the arcs, they are finite and do not modify the conformal dimension. Finally The two arcs interact by "wrapping effects", which arise in this example only at six loop order ($d$). Up to this order the dimension of an operator made of the scalars $Z$ and $Y$ does not depend on the cusp angles $\phi$ and $\theta$.
• Figure 5: Using the reflection trick a semi-infinite open spin--chain can be replaced with a spin--chain on the entire line. The original magnons ($a$) carry representations of ${\mathfrak{psu}}(2|2)_R\times{\mathfrak{psu}}(2|2)_L$ and momentum $p$, which gets reflected to momentum $-p$. In the doubled picture ($b$) the ${\mathfrak{psu}}(2|2)_R$ label is carried by a magnon of momentum $-p$ on the right side, which gets scattered off the magnon of momentum $p$ and a ${\mathfrak{psu}}(2|2)_L$ representation. As usual, the momentum does not get modified by the scattering and the magnon with momentum $-p$ continues on the left side, now carrying a ${\mathfrak{psu}}(2|2)_L$ representation