2025 Theses Doctoral

Gromov-Witten theory of smooth Calabi-Yau hypersurfaces in weighted ℙ⁴

Lei, Patrick

We study the all-genus Gromov-Witten theory of the Calabi-Yau threefolds 𝑍₆ ⊂ ℙ(1,1,1,1,2), 𝑍₈ ⊂ ℙ(1,1,1,1,4), and 𝑍₁₀ ⊂ ℙ(1,1,1,2,5). In these examples, where the threefold satisfies 𝒉²=1, there are several important structural predictions which arise from translating structural results in B-model Kodaira-Spencer gravity (which is not defined mathematically) to Gromov-Witten theory using mirror symmetry.

The first major result of this thesis is the finite generation conjecture of Yamaguchi-Yau, which states that a normalization P_{𝑔,𝑛} of the genus-𝑔 Gromov-Witten potential of the Calabi-Yau threefold 𝑍 is a polynomial in five power series 𝐴, 𝐵, 𝐵₂, 𝐵₃, 𝑋 defined using genus-0 data. We then prove the Feynman rule of Bershadsky-Cecotti-Ooguri-Vafa, which recursively determines 𝑃_{𝑔,𝑛} from all 𝑃_{𝒉<𝑔, 𝑚} up to a polynomial of degree 3𝑔-3+𝑛 in 𝑋, which is known as the 𝘩𝘰𝘭𝘰𝘮𝘰𝘳𝘱𝘩𝘪𝘤 𝘢𝘮𝘣𝘪𝘨𝘶𝘪𝘵𝘺 in the literature. This Feynman rule determines a canonical expression in the five generators and implies the modular anomaly equation conjectured by Yamaguchi-Yau.

Our main computational tool is the theory of Mixed-Spin-P (MSP) fields developed originally by Chang-Li-Li-Liu and Chang-Guo-Li-Li for the quintic threefold. In order to apply it to our situation, we give an explicit description of the moduli space and write an explicit form of the virtual localization formula in our setting. The key geometric input is a generalization of the irregular vanishing theorem of Chang-Li. Systematically applying the idea of Givental to package MSP virtual localization into the action of an 𝑅-matrix, we prove first the Yamaguchi-Yau finite generation conjecture and then the BCOV Feynman rule conjecture as corollaries of increasingly detailed results about this 𝑅-matrix.