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Register a new user$T$-Dual solutions and infinitesimal moduli of the $G_2$-Strominger system
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We consider $G_2$-structures with torsion coupled with $G_2$-instantons, on a compact $7$-dimensional manifold. The coupling is via an equation for $4$-forms which appears in supergravity and generalized geometry, known as the Bianchi identity. First studied by Friedrich and Ivanov, the resulting system of partial differential equations describes compactifications of the heterotic string to three dimensions, and is often referred to as the $G_2$-Strominger system. We study the moduli space of solutions and prove that the space of infinitesimal deformations, modulo automorphisms, is finite dimensional. We also provide a new family of solutions to this system, on $T^3$-bundles over $K3$ surfaces and for infinitely many different instanton bundles, adapting a construction of Fu-Yau and the second named author. In particular, we exhibit the first examples of $T$-dual solutions for this system of equations.
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We consider $G_2$ structures with torsion coupled with $G_2$-instantons, on a compact $7$-dimensional manifold. The coupling is via an equation for $4$-forms which appears in supergravity and generalized geometry, known as the Bianchi identity. The resulting system of partial differential equations can be regarded as an analogue of the Strominger system in $7$-dimensions. We initiate the study of the moduli space of solutions and show that it is finite dimensional using elliptic operator theory. We also relate the associated geometric structures to generalized geometry.
We construct new examples of solutions of the Hull-Strominger system on non-Kahler torus bundles over K3 surfaces, with the property that the connection $ abla$ on the tangent bundle is Hermite-Yang-Mills. With this ansatz for the connection $ abla$, we show that the existence of solutions reduces to known results about moduli spaces of slope-stable sheaves on a K3 surface, combined with elementary analytical methods. We apply our construction to find the first examples of T-dual solutions of the Hull-Strominger system on compact non-Kahler manifolds with different topology.
We construct new smooth solutions to the Hull-Strominger system, showing that the Fu-Yau solution on torus bundles over K3 surfaces can be generalized to torus bundles over K3 orbifolds. In particular, we prove that, for $13 leq k leq 22$ and $14leq rleq 22$, the smooth manifolds $S^1times sharp_k(S^2times S^3)$ and $sharp_r (S^2 times S^4) sharp_{r+1} (S^3 times S^3)$, have a complex structure with trivial canonical bundle and admit a solution to the Hull-Strominger system.
We consider the heterotic string on Calabi-Yau manifolds admitting a Strominger-Yau-Zaslow fibration. Upon reducing the system in the $T^3$-directions, the Hermitian Yang-Mills conditions can then be reinterpreted as a complex flat connection on $mathbb{R}^3$ satisfying a certain co-closure condition. We give a number of abelian and non-abelian examples, and also compute the back-reaction on the geometry through the non-trivial $alpha$-corrected heterotic Bianchi identity, which includes an important correction to the equations for the complex flat connection. These are all new local solutions to the Hull-Strominger system on $T^3timesmathbb{R}^3$. We also propose a method for computing the spectrum of certain non-abelian models, in close analogy with the Morse-Witten complex of the abelian models.
Suppose $(X, g)$ is a compact, spin Riemannian 7-manifold, with Dirac operator $D$. Let $G$ be SU$(m)$ or U$(m)$, and $Eto X$ be a rank $m$ complex bundle with $G$-structure. Write ${mathcal B}_E$ for the infinite-dimensional moduli space of connections on $E$, modulo gauge. There is a natural principal ${mathbb Z}_2$-bundle $O^D_Eto{mathcal B}_E$ parametrizing orientations of det$,D_{{rm Ad }A}$ for twisted elliptic operators $D_{{rm Ad }A}$ at each $[A]$ in ${mathcal B}_E$. A theorem of Walpuski shows $O^D_E$ is trivializable. We prove that if we choose an orientation for det$,D$, and a flag structure on X in the sense of Joyce arXiv:1610.09836, then we can define canonical trivializations of $O^D_E$ for all such bundles $Eto X$, satisfying natural compatibilities. Now let $(X,varphi,g)$ be a compact $G_2$-manifold, with d$(*varphi)=0$. Then we can consider moduli spaces ${mathcal M}_E^{G_2}$ of $G_2$-instantons on $Eto X$, which are smooth manifolds under suitable transversality conditions, and derived manifolds in general, with ${mathcal M}_E^{G_2}subset{mathcal B}_E$. The restriction of $O^D_E$ to ${mathcal M}_E^{G_2}$ is the ${mathbb Z}_2$-bundle of orientations on ${mathcal M}_E^{G_2}$. Thus, our theorem induces canonical orientations on all such $G_2$-instanton moduli spaces ${mathcal M}_E^{G_2}$. This contributes to the Donaldson-Segal programme arXiv:0902.3239, which proposes defining enumerative invariants of $G_2$-manifolds $(X,varphi,g)$ by counting moduli spaces ${mathcal M}_E^{G_2}$, with signs depending on a choice of orientation. This paper is a sequel to Joyce-Tanaka-Upmeier arXiv:1811.01096, which develops the general theory of orientations on gauge-theoretic moduli spaces, and gives applications in dimensions 3,4,5 and 6. A third paper Cao-Gross-Joyce arXiv:1811.09658 studies orientations on moduli spaces in dimension 8.
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