Cohomological Field Theory with vacuum and its Virasoro constraints
Abstract
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 -calibration and -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 -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.
Keywords
Cite
@article{arxiv.2502.18895,
title = {Cohomological Field Theory with vacuum and its Virasoro constraints},
author = {Shuai Guo and Qingsheng Zhang},
journal= {arXiv preprint arXiv:2502.18895},
year = {2025}
}