More on Superstrings in AdS(3) x N

TL;DR

This work addresses the problem of identifying necessary constraints on the internal CFT $\mathcal N$ for string theory on $AdS_3\times\mathcal N$ to realize boundary superconformal algebras with $N=2,3,4$ SUSY. It develops a unified framework linking spacetime supercharges to worldsheet spin fields, BRST consistency, and affine current structures, deriving explicit necessary conditions: for $N=2$, $\mathcal N$ must contain an affine $U(1)$ and the coset $\mathcal N/U(1)$ must carry an $N=2$ worldsheet algebra; for $N=3$, $\mathcal N$ must include an affine $SU(2)_{4k}$ and the quotients by the three $SU(2)$ supercurrents must yield $N=2$ subalgebras with suitable $U(1)$ currents $M^a$; for $N=4$, the small and large cases require an $SU(2)$ R-symmetry and appropriate level relations, e.g., $1/k=1/k'+1/k''$. The paper demonstrates that these conditions are not merely sufficient but necessary and, in many cases, sufficient, and discusses explicit backgrounds, GSO choices, and coset/orbifold constructions that realize the various amounts of boundary SUSY. This work clarifies the precise link between spacetime supersymmetry and worldsheet symmetries in $AdS_3$ backgrounds and provides a catalog of viable models and techniques for spectrum analysis in $AdS_3$/CFT$_2$.

Abstract

We study superstring theories on AdS(3) x N backgrounds yielding N=2,3,4 extended superconformal symmetries in the dual boundary CFT. In each case the necessary constraints on the internal worldsheet theory N are found.