Excitations of bubbling geometries for line defects

Abstract

The half-BPS Wilson line operators in the irreducible representations labeled by the Young diagrams for $\mathcal{N}=4$ $U(N)$ super Yang-Mills theory have gravity dual descriptions. When the number $k$ of boxes of the diagram grows as $k\sim N^2$, the bubbling geometries emerge. We evaluate the spectra of quantum fluctuations on the bubbling geometries from the large $N$ and large $k$ limit of the supersymmetric indices decorated by the Wilson lines. The spectra of excitations over multi-particle $1/8$- and $1/2$-BPS states agree with the numbers of conjugacy classes of general linear group over finite fields while degeneracies of single particle BPS states are given by the general necklace polynomial. The bubbling geometry exhibits a new class of asymptotic degeneracy of states.