Asymptotic Hodge Theory and Instantons

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AbstractAward: DMS-1005761Principal Investigator: Mark A. SternThe PI's proposed research focuses on several problems related to the existence and geometry of connections minimizing the Yang-Mills and related energy functionals of connections on vector bundles over manifolds of special holonomy and on smooth projective varieties. In the first project the PI, in joint work with Bianca Santoro, will study critical points and parabolic flows of the energy, E', given by the L2 norm of the (0,2)component of the curvature tensor. In many special geometries where the de Rham cohomology is all of type (p,p), they will attempt to prove holomorphicity of connections that are stable critical points of E'. They will also study obstructions to the long time existence of solutions to the gradient flow for E'. The primary goal of this project is to improve our understanding of when vector bundles admit holomorphic structures. In a related project, joint with Benoit Charbonneau, the PI will use techniques from geometric quantization to study refined Hodge structures on isomorphism classes of vector bundles on smooth projective varieties. They will study the large k asymptotics of the projection operator onto the kernel of a generalized Diracoperator associated to a bundle F tensored by the kth power of a polarizing line bundle.They have used these asymptotics to define a refined Hodge filtration on bundles and will attempt to exploit the asymptotics further to prove that the Chern character mapping from bundles to de Rham cohomology (equipped with its usual Hodge structure) respects these new Hodge filtrations. Connection analogs of Donaldson's balanced metric conditions arise in this study. The PI will seek corresponding analogs of Gieseker stability. In joint work with Savdeep Sethi, the PI will study the moduli space of instantons on RxT^3, interpolating between flat connections on T^3, with unequal Chern Simons invariants.The physics of the strong, weak, and electromagnetic forces is based on gauge theory, the study of differential calculus on vector bundles. The primary objects in these gauge theories are connections, which may be viewed as rules for defining differentiation. The primary energy guiding the dynamics in these physical theories is the Yang-Mills energy, and the fields dominating the physics are those that minimize the Yang-Mills energy. In dimensions greater than 4, the existence and structure of these optimal connections are poorly understood. In 4-dimensions, optimal connections have been a powerful tool for answering questions in geometry and topology. In higher dimensions they can potentially provide extremely important techniques for attacking questions in differential and algebraic geometry. In particular, they may give techniques for transforming questions involving the solutions of complex partial differential equations to dramatically simpler algebraic problems. As the current theories of quantum gravity all require an understanding of physics in dimensions greater than 4, there is strong interplay between the physical and geometric study of optimal connections. In fact, the antecedents to this project arose from questions posed to the PI by physicists; the answer to these questions then led to new results in geometry.