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Register a new userThe heterotic $rm{G}_2$ system on contact Calabi--Yau $7$-manifolds
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We obtain non-trivial solutions to the heterotic $rm{G}_2$ system, which are defined on the total spaces of non-trivial circle bundles over Calabi--Yau $3$-orbifolds. By adjusting the $S^1$ fibres in proportion to a power of the string constant $alpha$, we obtain a cocalibrated $rm{G}_2$-structure the torsion of which realises an arbitrary constant (trivial) dilaton field and an $H$-flux with nontrivial Chern--Simons defect. We find examples of connections on the tangent bundle and a non-flat $rm{G}_2$-instanton induced from the horizontal Calabi--Yau metric which satisfy together the anomaly-free condition, also known as the heterotic Bianchi identity. The connections on the tangent bundle are $rm{G}_2$-instantons up to higher order corrections in $alpha$.
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We construct balanced metrics on the family of non-Kahler Calabi-Yau threefolds that are obtained by smoothing after contracting $(-1,-1)$-rational curves on Kahler Calabi-Yau threefold. As an application, we construct balanced metrics on complex manifolds diffeomorphic to connected sum of $kgeq 2$ copies of $S^3times S^3$.
In this note we initiate a program to obtain global descriptions of Calabi-Yau moduli spaces, to calculate their Picard group, and to identify within that group the Hodge line bundle, and the closely-related Bagger-Witten line bundle. We do this here for several Calabi-Yaus obtained in [DW09] as crepant resolutions of the orbifold quotient of the product of three elliptic curves. In particular we verify in these cases a recent claim of [GHKSST16] by noting that a power of the Hodge line bundle is trivial -- even though in most of these cases the Picard group is infinite.
We prove that the categorical entropy of the autoequivalence $T_{mathcal{O}}circ(-otimesmathcal{O}(-1))$ on a Calabi-Yau manifold is the unique positive real number $lambda$ satisfying $$ sum_{kgeq 1}frac{chi(mathcal{O}(k))}{e^{klambda}}=e^{(d-1)t}. $$ We then use this result to construct the first counterexamples of a conjecture on categorical entropy by Kikuta and Takahashi.
We formulate a Calabi-Yau type conjecture in generalized Kahler geometry, focusing on the case of nondegenerate Poisson structure. After defining natural Hamiltonian deformation spaces for generalized Kahler structures generalizing the notion of Kahler class, we conjecture unique solvability of Gualtieris Calabi-Yau equation within this class. We establish the uniqueness, and moreover show that all such solutions are actually hyper-Kahler metrics. We furthermore establish a GIT framework for this problem, interpreting solutions of this equation as zeros of a moment map associated to a Hamiltonian action and finding a Kempf-Ness functional. Lastly we indicate the naturality of generalized Kahler-Ricci flow in this setting, showing that it evolves within the given Hamiltonian deformation class, and that the Kempf-Ness functional is monotone, so that the only possible fixed points for the flow are hyper-Kahler metrics. On a hyper-Kahler background, we establish global existence and weak convergence of the flow.
In this paper we study Higgs and co-Higgs $G$-bundles on compact Kahler manifolds $X$. Our main results are: (1) If $X$ is Calabi-Yau, and $(E,,theta)$ is a semistable Higgs or co-Higgs $G$-bundle on $X$, then the principal $G$-bundle $E$ is semistable. In particular, there is a deformation retract of ${mathcal M}_H(G)$ onto $mathcal M(G)$, where $mathcal M(G)$ is the moduli space of semistable principal $G$-bundles with vanishing rational Chern classes on $X$, and analogously, ${mathcal M}_H(G)$ is the moduli space of semistable principal Higgs $G$-bundles with vanishing rational Chern classes. (2) Calabi-Yau manifolds are characterized as those compact Kahler manifolds whose tangent bundle is semistable for every Kahler class, and have the following property: if $(E,,theta)$ is a semistable Higgs or co-Higgs vector bundle, then $E$ is semistable.
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