Higher rank Wilson loops from a matrix model

We compute the circular Wilson loop ofN = 4 SYM theory at large N in the rank k symmetric and antisymmetric tensor representations. Using a quadratic Hermitian matrix model we obtain expressions for all values of the ’t Hooft coupling. At large and small couplings we give explicit formulae and reproduce supergravity results from both D3 and D5 branes within a systematic framework. 1. Background and motivation Wilson loops played a key role in the early development of the AdS/CFT correspondence [1]. The identification of the Wilson loop in N = 4 super Yang-Mills (SYM) theory with the dual fundamental string [2, 3] connected naturally with ideas of understanding the ‘confining’ strings of largeN gauge theory as fundamental strings [4]. At finite temperature, thermal Wilson loops enabled the identification of a deconfinement transition in the theory [5] and allowed the computation of the thermal interquark potential and screening length [6–8]. There has recently been a renewed interest in Wilson loop operators in the AdS/CFT correspondence, with an underlying theme of understanding loops in higher representations of SU(N). This has seen a convergence of various different research directions. Firstly, circular Wilson loops are half BPS operators and have been conjectured to be computed exactly, up to instanton corrections [9], to all orders in N and λ = g YMN , from a quadratic Hermitian matrix model [10,11]. This matrix model makes nontrivial predictions for Wilson loops in higher representations which constitute a test of the AdS/CFT correspondence incorporating higher genus string effects [12]. Recently, the matrix model computation of a k winding circular Wilson loop was reproduced using a D3 brane embedded in AdS5 carrying k units of worldvolume flux [13]. This beautiful match lead to further questions as it was not clear which rank k representation of SU(N) the D3 brane was computing. A complete dictionary was later proposed in [14] relating supersymmetric Wilson loops in an arbitrary representation to D3 and D5 branes in the dual geometry. According to this dictionary, a single D3 brane with k units of flux is dual to a Wilson loop in the rank k symmetric representation. It was therefore an open question to explain why the k winding computation of [13], a priori certainly not the same as the rank k symmetric respresentation, reproduced the supergravity result. This confusion lead us to the study of general symmetric representations in this paper. A second line of interest has originated from the description of certain half BPS operators of N = 4 SYM theory carrying angular momentum as ‘bubbling geometries’ [15]. Wilson lines provide another family of half BPS operators and so it is natural to search for a bubbling geometry formalism describing the backreaction of the D3 and D5 branes, as well as a field theory understanding in terms of fermion droplets. Exciting progress in this direction is currently being made [16–18]. This line of interest inspired the succesful matching of a bulk D5 brane computation with a matrix model computation of the rank k antisymmetric Wilson loop [19]. How-