Deformed Shatashvili-Vafa algebra for superstrings on AdS$_3\times {\cal M}_7$

Abstract

String backgrounds of the form M 3 ×M 7 where M 3 denotes 3-dimensional Minkowski space while M 7 is a 7-dimensional G 2 -manifold, are characterised by the property that the world-sheet theory has a Shatashvili-Vafa (SV) chiral algebra. We study the generalisation of this statement to backgrounds where the Minkowski factor M 3 is replaced by AdS 3 . We argue that in this case the world-sheet theory is characterised by a certain N = 1 superconformal W-algebra that has the same spin spectrum as the SV algebra and also contains a tricritical Ising model N = 1 subalgebra. We determine the allowed representations of this W-algebra, and analyse to which extent the special features of the SV algebra survive this generalisation.1 We use 'G 2 -manifold' quite loosely to mean 'manifold with a G 2 -structure'. The 7d geometry should support a nowhere-vanishing Majorana spinor for supersymmetry and, in fact, a connected 7d manifold admits a G 2 -structure if and only if it is both spin and orientable [2]. In the simplest case this structure will be torsion-free leading to a Ricci-flat manifold with holonomy group G 2 , but one should in general anticipate flux backgrounds where this is not the case [3][4][5][6][7][8]. c λ 2