On the higher-spin charges of conical defects

TL;DR

The paper addresses matching conical defect higher-spin charges in AdS$_3$/CFT$_2$ by comparing bulk charges computed in the $u$-basis with boundary charges obtained through the quantum Miura transformation. It maps plane boundary currents to the cylinder and derives zero-mode eigenvalues $\tilde{U}_{j,0}$ in terms of elementary symmetric polynomials of $\epsilon_i\cdot(\Lambda+\alpha_0\rho)$, establishing a direct link to bulk charges. In the semi-classical limit, the authors show that the conical defect spectrum exactly matches the boundary CFT spectrum for all high-spin charges, via the identification $\alpha_-\approx i\sqrt{k}$ and $\epsilon_i\cdot(\Lambda_-+\rho)=n_i'$, thereby validating the higher-spin AdS$_3$/CFT$_2$ proposal beyond first few charges. The results extend to the hs$(\lambda)$ theory through an analytic continuation in $N$ to $\lambda$, reinforcing the robustness of the duality. Overall, the work provides a comprehensive charge-matching check that supports the proposed duality and offers a concrete bridge between bulk higher-spin charges and boundary $W_N$-type currents.

Abstract

The conical defect solutions in higher-spin gauge theories on 2+1 dimensional space-times with AdS-asymptotics are conjectured to correspond to certain primary fields in the dual conformal field theory on the boundary. In this note we prove that indeed all higher-spin charges match.