Generalized structures of ten-dimensional supersymmetric solutions

TL;DR

The work provides a universal ten-dimensional reformulation of type II supersymmetry using a generalized ${\rm ISpin}(7)$ structure on $T\oplus T^*$, encoded by a form $\Phi$ together with two generalized vectors $e_{+_1},e_{+_2}$. The resulting differential system, valid for IIA and IIB, is equivalent to unbroken supersymmetry and includes a key equation $d_H(e^{-\,\phi}\Phi) = -( ilde{K}\wedge + \iota_K)F$; additional equations ensure the full gravitino/dilatino variations are captured. While $\Phi$ alone does not fix the metric and $B$-field, augmenting it with $e_{+_1},e_{+_2}$ yields a complete description of $(g,B,\epsilon_1,\epsilon_2)$ and a corresponding generalized geometry structure. The framework reproduces the pure spinor equations for ${\rm Minkowski}_4$ vacua and yields algebraic constraints in ${\rm Minkowski}_3$ cases, connecting higher‑dimensional SUSY data to known 4D and 3D compactifications and offering a unified route to classify and construct ten‑dimensional SUSY solutions.

Abstract

Four-dimensional supersymmetric type II string theory vacua can be described elegantly in terms of pure spinors on the generalized tangent bundle T+T*. In this paper, we apply the same techniques to any ten-dimensional supersymmetric solution (not necessarily involving a factor with an AdS4 or Minkowski4 metric) in type II theories. We find a system of differential equations in terms of a form describing a "generalized ISpin(7) structure". This system is equivalent to unbroken supersymmetry, in both IIA and IIB. One of the equations reproduces in one fell swoop all the pure spinors equations for four-dimensional vacua.