Energy functionals for Calabi-Yau metrics

6th International Symposium on Quantum Theory and Symmetries

Headrick, M.^1 ; Nassar, A.^1

Martin Fisher School of Physics, Brandeis University, Waltham, MA 02454, United States^1

关键词: Einstein equations;    Energy functionals;    Functionals;    One parameter family;    Optimization problems;    Orders of magnitude;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/462/1/012019/pdf
DOI  :  10.1088/1742-6596/462/1/012019

来源: IOP

【 摘 要 】

We identify a set of "energy" functionals on the space of metrics in a given Kahler class on a Calabi-Yau manifold, which are bounded below and minimized uniquely on the Ricci-flat metric in that class. Using these functionals, we recast the problem of numerically solving the Einstein equation as an optimization problem. We apply this strategy, using the "algebraic" metrics (metrics for which the Kahler potential is given in terms of a polynomial in the projective coordinates), to the Fermat quartic and to a one-parameter family of quintics that includes the Fermat and conifold quintics. We show that this method yields approximations to the Ricci-flat metric that are exponentially accurate in the degree of the polynomial (except at the conifold point, where the convergence is polynomial), and therefore orders of magnitude more accurate than the balanced metrics, previously studied as approximations to the Ricci-flat metric. The method is relatively fast and easy to implement. On the theoretical side, we also show that the functionals can be used to give a heuristic proof of Yau's theorem.