Infinitesimal moduli of G2 holonomy manifolds with instanton bundles

Abstract

Abstract:We describe the infinitesimal moduli space of pairs (Y, V ) where Y is a manifold with G 2 holonomy, and V is a vector bundle on Y with an instanton connection. These structures arise in connection to the moduli space of heterotic string compactifications on compact and non-compact seven dimensional spaces, e.g. domain walls. Employing the canonical G 2 cohomology developed by Reyes-Carrión and Fernández and Ugarte, we show that the moduli space decomposes into the sum of the bundle moduli H 1 d A (Y, End(V )) plus the moduli of the G 2 structure preserving the instanton condition. The latter piece is contained in H 1 d θ (Y, T Y ), and is given by the kernel of a mapF which generalises the concept of the Atiyah map for holomorphic bundles on complex manifolds to the case at hand. In fact, the mapF is given in terms of the curvature of the bundle and maps, and moreover can be used to define a cohomology on an extension bundle of T Y by End(V ). We comment further on the resemblance with the holomorphic Atiyah algebroid and connect the story to physics, in particular to heterotic compactifications on (Y, V ) when α = 0.

Marginal deformations of heterotic G2 sigma models

“…We do not rely on the standard embedding nor on any artificial assumptions on the flux. We work at tree-level in the α ′ expansion and, in this limit, our results agree with those recently obtained in [10][11][12] using the supergravity perspective.…”

“…This is consistent with the spacetime description, as discussed in section 4 and [11]. In particular, note that…”

“…In particular, it is not known whether higher order α ′ corrections can spoil spacetime supersymmetry and worldsheet conformal invariance, in contrast to what is known about (2, 2) and (1, 1) models [4][5][6]. We do not believe this to be the case, since it is expected that the recent recasting of heterotic geometry in terms of holomorphic structures, extension sequences and Courant algebroids [7][8][9][10][11][12][13] will survive to higher orders, potentially modulo certain field redefinitions.…”

“…These are precisely the (tree-level) relations recently derived from spacetime considerations in [11,12]. The first condition was derived in [40] for type II models in the special case H = 0.…”

“…Moduli indeed occur as massless fields in the spacetime effective theory and are relevant to topological string theory and mirror symmetry on Calabi-Yau manifolds. Lately, focus has shifted to the study of moduli spaces for more general compactifications, including torsion and in diverse dimensions, mostly from a supergravity point of view [7][8][9][10][11][12][31][32][33][34], but also in terms of nonlinear sigma models on the worldsheet [35]. In this note, we initiate a study of infinitesimal moduli of general heterotic G 2 compactifications from the worldsheet point of view.…”