$$ \mathcal{SW} $$-algebras and strings with torsion

A bstract We explore the connection between super $$ \mathcal{W} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>W</mml:mi> </mml:math> -algebras ( $$ \mathcal{SW} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>SW</mml:mi> </mml:math> -algebras) and G-structures with torsion. The former are realised as symmetry algebras of strings with $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = (1, 0) supersymmetry on the worldsheet, while the latter are associated with generic string backgrounds with non-trivial Neveu-Schwarz flux H . In particular, we focus on manifolds featuring Spin(7), G 2 , SU(2), and SU(3)-structures. We compare the full quantum algebras with their classical limits, obtained by studying the commutators of superconformal and $$ \mathcal{W} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>W</mml:mi> </mml:math> -symmetry transformations — which preserve the action of the (1, 0) non-linear σ -model. We show that, at first order in the string length scale ℓ s , the torsion deforms some of the OPE coefficients corresponding to special holonomy through a scalar torsion class.