BPS spectrum on AdS$_3\times $S$^3 \times $S$^3 \times $S$^1$

TL;DR

This work addresses the BPS spectrum of string theory on $AdS_3\times S^3\times S^3\times S^1$ with pure NS-NS flux by combining a world-sheet WZW analysis and a first-principles supergravity calculation. The central finding is that both the world-sheet spectrum and the 9D supergravity spectrum contain BPS states only for $j^+=j^-$, with explicit bounds and mass-dimension relations confirming the saturation condition. The world-sheet result yields a closed-form bound for the spacetime spin $j$, while the supergravity analysis diagonalizes the scalar fluctuation equations to reproduce the same BPS structure, thereby resolving previous puzzles about quantum corrections in this background. The results have notable implications for the dual CFT, potentially revitalizing the symmetric orbifold of the ${\cal S}_0$ theory as a viable dual, and they enhance the understanding of BPS spectra in AdS$_3$/CFT$_2$ with large ${\cal N}=4$ symmetry.

Abstract

The BPS spectrum of string theory on AdS$_3\times {\rm S}^3 \times {\rm S}^3 \times {\rm S}^1$ is determined using a world-sheet description in terms of WZW models. It is found that the theory only has BPS states with $j^+ = j^-$ where $j^{\pm}$ refer to the spins of the $\mathfrak{su}(2)$ algebras of the large ${\cal N}=4$ superconformal algebra. We then re-examine the BPS spectrum of the corresponding supergravity and find that, in contradistinction to previous claims in the literature, also in supergravity only the states with $j^+=j^-$ are BPS. This resolves a number of long-standing puzzles regarding the BPS spectrum of string theory and supergravity in this background.