Title: Holomorphic Anomaly Equations and Crepant Resolution Correspondence for C^n/Z_n
Abstract: In this talk, we present results on the higher genus Gromov-Witten theory of C^n/Z_n by examining its cohomological field theory structure in detail. Holomorphic anomaly equations are recursive partial differential equations predicted by physicists for a Calabi-Yau threefold. We prove holomorphic anomaly equations for C^n/Z_n for any n >= 3. In other words, we demonstrate a phenomenon of holomorphic anomaly equations in arbitrary dimensions, extending beyond the scope considered by physicists. The proof relies on showing that the Gromov-Witten potential of C^n/Z_n lies in a specific polynomial ring. Furthermore, we prove a crepant resolution correspondence for arbitrary genera for C^n/Z_n by showing that its cohomological field theory aligns with that of KP^{n-1}, where KP^{n-1} is the total space of the canonical bundle of P^{n-1}. More precisely, we demonstrate that the Gromov-Witten potential of KP^{n-1} also resides in a similar polynomial ring, and we establish that it matches with the Gromov-Witten potential of C^n/Z_n under an isomorphism of these polynomial rings. This talk is based on joint work with Hsian-Hua Tseng, as detailed in arXiv:2301.08389 and arXiv:2308.00780.