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-Carri\'on and Fern\'andez and Ugarte, we show that the moduli space decomposes into the sum of the bundle moduli $H^1_{\check{d}_A}(Y,\mathrm{End}(V))$ plus the moduli of the $G_2$ structure preserving the instanton condition. The latter piece is contained in $H^1_{\check{d}_\theta}(Y,TY)$, and is given by the kernel of a map ${\cal\check F}$ which generalises the concept of the Atiyah map for holomorphic bundles on complex manifolds to the case at hand. In fact, the map ${\cal\check F}$ is given in terms of the curvature of the bundle and maps $H^1_{\check{d}_\theta}(Y,TY)$ into $H^2_{\check{d}_A}(Y,\mathrm{End}(V))$, and moreover can be used to define a cohomology on an extension bundle of $TY$ by $\mathrm{End}(V)$. 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 $\alpha'=0$.
Highlights
Manifolds with special holonomy have, since long, been used to construct supersymmetric lower-dimensional vacuum solutions of string and M theory
We describe the infinitesimal moduli space of pairs (Y, V ) where Y is a manifold with G2 holonomy, and V is a vector bundle on Y with an instanton connection
We have studied the infinitesimal deformations of a pair (Y, V ), where Y is a manifold of G2 holonomy, and V is a vector bundle with a connection satisfying the G2 instanton condition
Summary
Manifolds with special holonomy have, since long, been used to construct supersymmetric lower-dimensional vacuum solutions of string and M theory. Due to the heterotic anomaly condition, which relates the gauge field strength, tangent bundle curvature to the H-flux of the Kalb-Ramond B-field, the infinitesimal moduli space is restricted to a more intricate nested kernel in Dolbeault cohomology, which is most conveniently encoded as a holomorphic structure on an extension bundle This N = 1 result is of importance for the development of a generalised geometry for the heterotic string [49,50,51,52,53,54,55,56]. We mention that when finalising the current paper, an article appeared on ArXiv [57], wherein the authors compute the infinitesimal moduli space of sevendimensional heterotic compactifications and show by means of elliptic operator theory that the resulting space is finite dimensional They relate the resulting geometric structures to generalised geometry in a similar fashion to the six-dimensional Strominger system [49]. This result is discussed from the perspective of extension bundles
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