Direct derivation of $\mathcal N=1$ supergravity in ten dimensions to all orders in fermions

TL;DR

The paper provides a direct, all-orders-in-fermions derivation of $\mathscr N=1$ supergravity in ten dimensions within a generalised-geometry framework, yielding a compact action and SUSY transformations that remain remarkably simple even at high fermionic order. By encoding the bosonic and fermionic fields in a generalised metric and half-density, the authors show that the massless string sector admits a natural geometric formulation, with a clean decomposition to the usual fields $g$, $B$, $A$, and $\varphi$. The local supersymmetry of the action is proven to all orders using generalised-curvature identities and the Lichnerowicz-type formula, with sextic terms vanishing through Fierz identities. Moreover, the formulation makes manifest the compatibility of the supergravity equations with Poisson--Lie T-duality, realized via Courant algebroid pullbacks, providing a robust geometric underpinning for duality symmetries in the massless spectrum of string theory. Overall, the work unifies higher-fermion structure, dualities, and ten-dimensional supergravity in a compact, geometrically natural formalism that extends to Yang–Mills couplings and beyond.

Abstract

It has been known for some time that generalised geometry provides a particularly elegant rewriting of the action and symmetries of 10-dimensional supergravity theories, up to the lowest nontrivial order in fermions. By exhibiting the full symmetry calculations in the second-order formalism, we show in the $\mathcal N=1$ case that this analysis can be upgraded to all orders in fermions and we obtain a strikingly simple form of the action as well as of the supersymmetry transformations, featuring overall only five higher-fermionic terms. Surprisingly, even after expressing the action in terms of classical (non-generalised geometric) variables one obtains a simplification of the usual formulae. This in particular confirms that generalised geometry provides the natural set of variables for studying (the massless level of) string theory. We also show how this new reformulation implies the compatibility of the Poisson-Lie T-duality with the equations of motion of the full supergravity theory.