Classical integrability and quantum aspects of the AdS(3) x S(3) x S(3) x S(1) superstring

TL;DR

This work establishes classical integrability of the Green-Schwarz string on $AdS_3\times S^3\times S^3\times S^1$ (and related backgrounds) without fixing $\kappa$-symmetry by constructing a one-parameter flat Lax connection from superisometry currents. It then probes quantum aspects by performing a BMN-like expansion and a one-loop computation of propagators, illustrating how the resulting dispersion relation $E_k=\sqrt{m_k^2+4h^2(\lambda)\sin^2(p/2)}$ depends on an interpolating function $h(\lambda)$ that receives regulator-dependent corrections. The analysis reveals heavy, light, and massless modes with regularization ambiguities analogous to those in $AdS_4/CFT_3$, including a possible heavy-mode decoherence and mass renormalization effects for light modes in certain schemes. The equal-radius case $\phi=\pi/4$ yields a one-loop correction to $h(\lambda)$ matching results from $AdS_4/CFT_3$, highlighting deep connections between these integrable string theories and motivating further work on the complete S-matrix and a regulator that preserves worldsheet supersymmetry. Overall, the paper advances the understanding of nontrivial AdS backgrounds with multiple spheres and their integrable structure, bridging classical constructions with quantum corrections in a controlled setting.

Abstract

In this paper we continue the investigation of aspects of integrability of the type IIA AdS(3) x S(3) x S(3) x S(1) and AdS(3) x S(3) x T(4) superstrings. By constructing a one parameter family of flat connections we prove that the Green-Schwarz string is classically integrable, at least to quadratic order in fermions, without fixing the kappa-symmetry. We then compare the quantum dispersion relation, fixed by integrability up to an unknown interpolating function h(lambda), to explicit one-loop calculations on the string worldsheet. For AdS(3) x S(3) x S(3) x S(1) the spectrum contains heavy, as well as light and massless modes, and we find that the one-loop contribution differs depending on how we treat these modes showing that similar regularization ambiguities as appeared in AdS(4)/CFT(3) occur also here.