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We introduce the category of holomorphic string algebroids, whose objects are Courant extensions of Atiyah Lie algebroids of holomorphic principal bundles, as considered by Bressler, and whose morphisms correspond to inner morphisms of the underlying holomorphic Courant algebroids in the sense of Severa. This category provides natural candidates for Atiyah Lie algebroids of holomorphic principal bundles for the (complexified) string group and their morphisms. Our main results are a classification of string algebroids in terms of Cech cohomology, and the construction of a locally complete family of deformations of string algebroids via a differential graded Lie algebra.
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The solution of the Calabi Conjecture by Yau implies that every Kahler Calabi-Yau manifold $X$ admits a metric with holonomy contained in $operatorname{SU}(n)$, and that these metrics are parametrized by the positive cone in $H^2(X,mathbb{R})$. In this work we give evidence of an extension of Yaus theorem to non-Kahler manifolds, where $X$ is replaced by a compact complex manifold with vanishing first Chern class endowed with a holomorphic Courant algebroid $Q$ of Bott-Chern type. The equations that define our notion of `best metric correspond to a mild generalization of the Hull-Strominger system, whereas the role of the second cohomology is played by an affine space of `Aeppli classes naturally associated to $Q$ via secondary holomorphic characteristic classes introduced by Donaldson
These lecture notes review the topological string theory and its applications to mathematics and physics. They expand on material presented at the Takagi Lectures of the Mathematical Society of Japan on 21 June 2008 at Department of Mathematics, Kyoto University.
We introduce a moment map picture for holomorphic string algebroids where the Hamiltonian gauge action is described by means of Morita equivalences, as suggested by higher gauge theory. The zero locus of our moment map is given by the solutions of the Calabi system, a coupled system of equations which provides a unifying framework for the classical Calabi problem and the Hull-Strominger system. Our main results are concerned with the geometry of the moduli space of solutions, and assume a technical condition which is fulfilled in examples. We prove that the moduli space carries a pseudo-Kahler metric with Kahler potential given by the dilaton functional, a topological formula for the metric, and an infinitesimal Donaldson-Uhlenbeck-Yau type theorem. Finally, we relate our topological formula to a physical prediction for the gravitino mass in order to obtain a new conjectural obstruction for the Hull-Strominger system.
We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves, and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendiecks Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety. Finally, we prove that every rational foliation, and certain logarithmic foliations have stable tangent sheaves.
The metric-affine and generalized geometries, respectively, are arguably the appropriate mathematical frameworks for Einsteins theory of gravity and the low-energy effective massless oriented closed bosonic string field theory. In fact, mathematical structures in a metric-affine geometry are written on the tangent bundle, which is itself a Lie algebroid; whereas those in generalized geometries introduced as the basis of double field theories, are written on Courant algebroids. The Lie, Courant and the higher Courant algebroids used in exceptional field theories, are all special cases of pre-Leibniz algebroids. Provided with some additional ingredients, the construction of such geometries can all be carried over to regular pre-Leibniz algebroids. We define below the notions of locality structures and locality projectors, which are some such necessary ingredients. In terms of these structures, $E$-metric-connection geometries are constructed with (possibly) a minimum number of assumptions. Certain small gaps in the literature are also filled as we go along. $E$-Koszul connections, as a generalization of Levi-Civita connections, are going to be defined and shown to be helpful for some results including a simple generalization of the fundamental theorem of Riemannian geometry. We also show that metric-affine geometries can be constructed in a unique way as special cases of $E$-metric-connection geometries. Moreover, generalized geometries are shown to follow as special cases, and various properties of linear generalized-connections are proven in the present framework. Similarly, uniqueness of the locality projector in the case of exact Courant algebroids is proven; a result that explains why the curvature operator, defined with a projector in the double field theory literature is a necessity.
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