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In this paper we are begining to explore the complex counterpart of the Chern-Simon-Witten theory. We define the complex analogue of the Gauss linking number for complex curves embedded in a Calabi-Yau 3-fold using the formal path integral that leads to a rigorous mathematical expression. We give an analytic and geometric interpretation of our holomorphic linking following the parallel with the real case. We show in particular that the Green kernel that appears in the explicit integral for the Gauss linking number is replaced by the Bochner-Martinelli kernel. We also find canonical expressions of the holomorphic linking using the Grothendieck-Serre duality in local cohomology, the latter admits a generalization for an arbitrary field.
We offer a new construction of Lagrangian submanifolds for the Gopakumar-Vafa conjecture relating the Chern-Simons theory on the 3-sphere and the Gromov-Witten theory on the resolved conifold. Given a knot in the 3-sphere, its conormal bundle is perturbed to disconnect it from the zero-section and then pulled through the conifold transition. The construction produces totally real submanifolds of the resolved conifold that are Lagrangian in a perturbed symplectic structure and correspond to knots in a natural and explicit way. We prove that both the resolved conifold and the knot Lagrangians in it have bounded geometry, and that the moduli spaces of holomorphic curves ending on the Lagrangians are compact in the Gromov topology.
We consider some elementary aspects of the geometry of the space of probability measures endowed with Wasserstein distance. In such a setting, we discuss the various terms entering Perelman’s shrinker entropy and characterize two new monotonic functionals for the volumenormalized Ricci flow. One is obtained by a rescaling of the curvature term in the shrinker entropy. The second is associated with a gradient flow obtained by adding a curvature-drift to Perelman’s backward heat equation. We show that the resulting Fokker-Planck PDE is the natural diffusion flow for probability measures absolutely continuous with respect to the Ricci-evolved Riemannian measure. We also discuss its exponential trend to equilibrium and its relation with the viscous Hamilton-Jacobi equation.