2025/02/26 by Shuai Guo, Guo, Shuai, Zhang, Qingsheng · 1 citation
Mathematics · #14N35 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.2502.18895
openalex publication_date 2025/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This is the first part of a series of papers on \it Virasoro constraints for Cohomological Field Theory (CohFT). For a CohFT with vacuum, we introduce the concepts of S-calibration and ν-calibration. Then, we define the (formal) total descendent potential corresponding to a given calibration. Finally, we introduce an additional structure, namely homogeneity, for both the CohFT and the calibrations. After these preliminary introductions, we propose two crucial conjectures: (1) the ancestor version of the Virasoro conjecture for the homogeneous CohFT with vacuum; and (2) the generalized Virasoro conjecture for the (formal) total descendent potential of a calibrated homogeneous CohFT. We verify the genus-0 part of these conjectures and deduce a simplified form of the genus-1 part of these conjectures for arbitrary CohFTs. Additionally, we prove the full conjectures for semisimple CohFTs. As applications, our results yield the Virasoro constraints for the deformed negative r-spin theory. Moreover, by applying the Virasoro constraints, we discover an extension of Grothendieck's dessins d'enfants theory which is widely studied in the literature.