The Spectrum of Excitations of Holographic Wilson Loops

In the holographic framework, a half BPS Wilson loop in $\mathcal{N}=4$ supersymmetric Yang-Mills in the fundamental, symmetric or antisymmetric representation of SU(N), is best described by a fundamental string, a D3 brane or a D5 brane with fluxes in their worldvolumes, respectively. We derive the spectrum of excitations of such D3 brane in $Ad{S}_5\times S^5$ explicitly, considering its action in both the bosonic and the fermionic sectors, and demonstrate that it is organized according to short multiplets of the supergroup $OSp(4^{\ast }\mathrm{\mid }4)$. We also show that the modes of the fundamental string form an ultra-short multiplet of this supergroup. In the case of the D5 brane the bosonic spectrum is only partially known but we argue that it also fills representations of $OSp(4^{\ast }\mathrm{\mid }4)$. This way we provide a step towards a unifying picture for the description of holographic excitations of the circular and straight supersymmetric Wilson loops in arbitrary representations.