M-theory potential from the G 2 Hitchin functional in superspace

Abstract

Abstract:We embed the component fields of eleven-dimensional supergravity into a superspace of the form X × Y where X is the standard 4D, N = 1 superspace and Y is a smooth 7-manifold. The eleven-dimensional 3-form gives rise to a tensor hierarchy of superfields gauged by the diffeomorphisms of Y . It contains a natural candidate for a G 2 structure on Y , and being a complex of superforms, defines a superspace Chern-Simons invariant. Adding to this a natural generalization of the Riemannian volume on X × Y and freezing the (superspin-3 2 and 1) supergravity fields on X, we obtain an approximation to the eleven-dimensional supergravity action that suffices to compute the scalar potential. In this approximation the action is the sum of the superspace Chern-Simons term and a superspace generalization of the Hitchin functional for Y as a G 2 -structure manifold. Integrating out auxiliary fields, we obtain the conditions for unbroken supersymmetry and the scalar potential. The latter reproduces the Einstein-Hilbert term on Y in a form due to Bryant.

Five-dimensional supergravity in N = 1/2 superspace

“…In the eleven-dimensional theory, this 3-form is the M-theory 3-form with all legs in the four-dimensional directions [27]. In the five-dimensional case, the geometrical origin of this form is less apparent, but we will see that there is a complete super-3-form of compensating fields needed for consistency.…”

“…The standard choice is to introduce a chiral scalar superfield [35]. More appropriate to our case, however, is a constrained chiral scalar G = − 1 4D 2 X withX = X a real scalar prepotential transforming under superconformal transformations as [14,27]…”

“…determined up to some completely weightless function F of the field strengths (2.7). The integrand of (3.5) is again a 1-form on Y that can be understood as the superspace volume density dressed up for conformal invariance: The explicit factor of F y may be interpreted as an ein-bein superfield on Y so that the integrand is of the form (Ed 4 x)(F y dy) ∼ E √ g yy d 4 xdy, and the (4, 0)-form field strength G enters in just such a way that we may interpret it as the conformal compensator for (modified) old-minimal supergravity [27].…”

“…We can fix the coefficient of this term from 5D Lorentz invariance [27]. In the superspace Lorentz gauge D α U a = 0, the old-minimal supergravity Lagrangian reduces to −U a 2U a − 1 3Ḡ G, and we simply pick the coefficient of X 2 to match this [15,27]. (TheḠG term is irrelevant to this argument.)…”

“…Nevertheless, it was explicitly shown that it does work for 10D super-Yang-Mills in [24]. More recently, the approach was extended to the far more subtle case of elevendimensional supergravity in [25][26][27][28][29][30].…”