For G2 and SU(3) string backgrounds with NS flux, the scalar torsion class controls the first-order deformation of the worldsheet super W-algebra couplings away from special holonomy.
Noncritical strings and W-algebras
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abstract
Singularities of Spin(7) manifolds are considered in the worldsheet approach, and it is argued that the internal CFT describing a singular spin(7) has an enhanced ${\cal SW}({3\over 2},{3\over 2},2)$ algebra tensored with ${\cal N}=1$ linear dilaton CFT, much like singular Calabi-Yau CFTs have a ${\cal N}=2$ linear dilaton tensored with a suitable ${\cal N}=2$ SCFT. Upon adding fundamental strings, these vacua are related to $AdS_3$ vacua with ${\cal N}=1$ supersymmetry, completing the worldsheet classification of $AdS_3$ vacua with NS flux.
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$\mathcal{SW}$-algebras and strings with torsion
For G2 and SU(3) string backgrounds with NS flux, the scalar torsion class controls the first-order deformation of the worldsheet super W-algebra couplings away from special holonomy.