Emitted radiation and geometry

Abstract

In conformal N = 2 Super Yang-Mills theory, the energy emitted by an accelerated heavy particle is computed by the one-point function of the stress tensor operator in the presence of a Wilson line. In this paper, we consider the theory on the ellipsoid and we prove a conjectured relation between the stress tensor one-point function and the first order expansion of the Wilson loop expectation value in the squashing parameter. To do this, we analyze the behavior of the Wilson loop for a small deformation of the background geometry and, at first order in the deformation, we fix the kinematics using defect CFT constraints. In the final part of the paper, we analyze the consequences of our results for the weak coupling perturbative expansion. In particular, comparing the weakly coupled matrix model with the ordinary Feynman diagram expansion, we find a natural transcendentality driven organization for the latter. Keywords: N = 2 conformal SYM theories, Wilson loops, Brehmsstrahlung, rigid supersymmetry dimensions has no scale, thus the straight Wilson line can be treated as a conformal defect. The emitted radiation is computed by slightly deforming the shape of the defect.As usual, things get harder when one considers non-Abelian gauge theories, which are strongly coupled at low energies and where conformality is broken by quantum corrections. In light of this complexity, it is useful to restrict our attention to those examples of strongly interacting gauge theories which preserve conformality. These cases typically come with a larger symmetry group which includes supersymmetry, making them much more tractable. For the maximally supersymmetric theory in four dimensions, N = 4 Super Yang Mills (SYM) theory, the combination of defect techniques and supersymmetric localization led to the derivation of a beautiful formula for the Bremsstrahlung function associated with the Maldacena Wilson loop [1,2], preserving half of the supercharges. Shortly after, the same result was confirmed by an integrability computation [3,4], providing one of the few examples of a quantity that is accessible to both techniques 1 . The work of [1] heavily relied on the interpretation of the Wilson line as a superconformal defect. In particular, it was pointed out that the Bremsstrahlung function can be computed as the two-point function of an important defect operator, called the displacement operator. Furthermore, the same quantity can be related to the small angle limit of the cusp anomalous dimension, thus providing an interesting connection with massive scattering amplitudes.Similar developments allowed to find an exact expression for the Bremsstrahlung function in ABJM theory [9], a three-dimensional relative of N = 4 SYM. In that case, two superconformal Wilson lines are known (see [10] for a recent review). The immediate generalization of the Maldacena Wilson loop turns out to be 1 6 BPS [11-13] and its Bremsstrahlung function was already proposed in [14]. The maximally supersymmetric case [15], instead, involves also fermionic couplings ...

$$ \mathcal{N} $$ = 2 Conformal SYM theories at large $$ \mathcal{N} $$

“…In the second part of the paper we concentrate on a set of N = 2 theories whose fundamental matter content does not scale with N , so that they have ν = 0. As noted above, these models have a holographic dual [26] and are very close to the N = 4 SYM theory, as confirmed by the fact that some observables, such as the vacuum expectation value of the Wilson loop [42] and the Bremsstrahlung function [23,50], do not deviate from the N = 4 result in the large-N limit. In the present paper we compute the set of observables listed above for the ν = 0 theories, using matrix model techniques, and clarify which observables are different with respect to N = 4 in the planar limit.…”

Analytic bootstrap and Witten diagrams for the ABJM Wilson line as defect CFT1

“…Moreover, for the case of the Wilson line this coefficient is particularly important as it computes the energy emitted by an accelerating heavy probe in a conformal field theory [10,67], often called Bremsstrahlung function. The relation with the two-point function of the displacement operator in the context of superconformal theories in three and four dimensions has allowed for the exact computation of this quantity in a variety of examples [10,28,30,60,64,65,[67][68][69][70][71][72][73][74]. In the present context we have…”

Superconformal surfaces in four dimensions

“…While the localization computation for 2d-4d coupled systems of [46] provides a tool for the exact computation of defect correlation functions, their final expression is too hard to evaluate explicitly leaving the question of a possible coupling dependence unanswered. An alternative recipe to obtain localization results for the stress tensor one-point function is through the relation with a deformation in the background geometry and one could hope to extend the derivation of [23] to the case of surfaces.…”

“…These characters are either solutions of the aforementioned modular linear differential equation or of its conjugate, and the q → 0 limit is controlled by the character with the lowest dimensional state. 23 Thus one can relate the lowest dimension among the characters appearing under the modular transformation, h χ min to the a and c anomalies [101,106], and bound it by the Hofman-Maldacena bounds [18]:…”

“…Its two-point function is an important piece of defect CFT data and, together with the one-point function of the stress tensor, they determine two of the three defect anomaly coefficients featured by a two-dimensional defect [19,20]. Their relation with deformations in the shape of the defect, or in the background geometry [21][22][23], as well as their role in the computation of the emitted radiation [22,24,25], make these two parameters a good starting point for the full characterization of an extended excitation. One of the main results of this paper is to show that for any superconformal surface defect in four dimensions these two quantities are related by a simple, theory independent, numerical factor.…”