Algebraic curve for a cusped Wilson line

Abstract

We consider the classical limit of the recently obtained exact near-BPS result for the anomalous dimension of a cusped Wilson line with the insertion of an operator with L units of R-charge at the cusp in planar N = 4 SYM. The classical limit requires taking both the 't Hooft coupling and L to infinity. Since the formula for the cusp anomalous dimension involves determinants of size proportional to L, the classical limit requires a matrix model reformulation of the result. Building on results of Gromov and Sever, we construct such a matrix model-like representation and find the corresponding classical algebraic curve. Using this we find the classical value of the cusp anomalous dimension and the 1-loop correction to it. We check our results against the energy of the classical solution and numerically by extrapolating from the quantum regime of finite L.

Bremsstrahlung function, leading Lüscher correction at weak coupling and localization

“…Moreover one can use localization in a suitable limit to obtain the exact form of the infamous Bremsstrahlung function [25], that controls the near-BPS behavior of the cusp anomalous dimension (see also [26] for a different derivation). The same result has been later directly recovered from the TBA equations [27,28] and QSC method [24,29]. It is clear that the generalized cusp anomalous dimension Γ(θ, ϕ) represents, in N = 4 SYM, a favorable playground in which the relative domains of techniques as integrability and localization overlap.…”

“…Two-dimensional Yang-Mills theory on the sphere has an exact solution, even at finite N [39,40]: on the other hand, the construction of the vacuum expectation values of Wilson loops with local operator insertions has not been studied in the past, at least to our knowledge. The matrix model [29], computing the generalized Bremsstrahlung function, strongly suggests that these two-dimensional observables, in the zero-instanton sector, should be obtained by extending the techniques of [14]. One could expect that also a finite N answer is possible, as in the case of L = 0.…”

“…For θ = 0 it was reproduced at any coupling in [27] by the analytic solution of the TBA, which also produced a new prediction for arbitrary L. The extension to the case with arbitrary (θ − ϕ) and any L is presented in [28]. The near-BPS results for L ≥ 1 can be rewritten as a matrix model partition function whose classical limit was investigated in [29] giving the corresponding classical spectral curve. A related approach based on a QSC has been proposed in [24], reproducing the above results.…”

Quantum Spectral Curve for a cusped Wilson line in N = 4 $$ \mathcal{N}=4 $$ SYM

“…For θ = 0 it was reproduced at any coupling in [32] by a direct analytic solution of the TBA, which also gave a new prediction for the case with arbitrary L. This analytic solution was extended to the case with arbitrary θ ∼ φ and any L in [33]. The near-BPS results for L ≥ 1 organize in a curious matrix model type partition function whose classical limit was investigated in [32,34] giving the corresponding classical spectral curve (see also [35,36,37]). In addition, the TBA was solved to two loops at weak coupling for finite φ, θ 1 the scalars inserted at the cusp should be orthogonal to the combinations n · Φ and n θ · Φ which couple to the Wilson lines [38], reproducing direct perturbative predictions which are also known at up to four loops [39,40,41,42].…”

Exact Bremsstrahlung and effective couplings

“…This setup is described by TBA equations very similar to the ones of the spectral problem [10,11]. However, these TBA equations are simpler and can be recast in terms of a matrix model [62][63][64] with a spectral curve that can be mapped to the classical string algebraic curve. For N = 2 SCFTs, it is currently not clear what happens beyond the SU (2, 1|2) sector, but it is worth thinking whether it is possible to derive TBA equations for supersymmetric Wilson loops with local fields from the SU (2, 1|2) sector inserted at the cusp.…”