We define a formal Gromov–Witten theory of the quintic threefold via localization on ${\mathbb {P}}^4$. Our main result is a direct geometric proof of holomorphic anomaly equations for the formal quintic in precisely the same form as predicted by B-model physics for the true Gromov–Witten theory of the quintic threefold. The results suggest that the formal quintic and the true quintic theories should be related by transformations which respect the holomorphic anomaly equations. Such a relationship has been recently found by Q. Chen, S. Guo, F. Janda, and Y. Ruan via the geometry of new moduli spaces.
This is the first part of a series of papers on the spectrum of the SYK model, which is a simple model of the black hole in physics literature. In this paper, we will give a rigorous proof of the almost sure convergence of the global density of the eigenvalues. We also discuss the largest eigenvalue of the SYK model.
In this paper, we prove the existence of solutions to the Fu–Yau equation on compact Kähler manifolds. As an application, we give a class of non-trivial solutions of the modified Strominger system.