The Infinitesimal Moduli Space of Heterotic G 2 Systems

Abstract

Heterotic string compactifications on integrable G 2 structure manifolds Y with instanton bundles (V, A), (T Y,θ) yield supersymmetric three-dimensional vacua that are of interest in physics. In this paper, we define a covariant exterior derivative D and show that it is equivalent to a heterotic G 2 system encoding the geometry of the heterotic string compactifications. This operator D acts on a bundle Q = T * Y ⊕ End(V ) ⊕ End(T Y ) and satisfies a nilpotency conditionĎ 2 = 0, for an appropriate projection of D. Furthermore, we determine the infinitesimal moduli space of these systems and show that it corresponds to the finite-dimensional cohomology groupȞ 1Ď (Q). We comment on the similarities and differences of our result with Atiyah's well-known analysis of deformations of holomorphic vector bundles over complex manifolds. Our analysis leads to results that are of relevance to all orders in the α expansion.

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.…”

“…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.…”

“…It should be stressed that R + fails to satisfy the instanton condition when such α ′ corrections are included. This observation is however crucial for understanding the heterotic moduli problem at O(α ′ ) [12].…”

“…1 In this paper, we use the opposite sign convention for ∇ ± as compared to [12], as it is more conventional in the worldsheet literature. In particular, the connection we call ∇ + is denoted d θ in [12].…”

“…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.…”