Cohomological Field Theory with vacuum and its Virasoro constraints

TL;DR

This paper introduces new structures and conjectures for Cohomological Field Theory with vacuum, verifies them in genus-0 and genus-1 cases, and applies results to negative r-spin theory and Grothendieck's dessins d'enfants.

Contribution

It proposes and verifies Virasoro constraints and conjectures for homogeneous CohFTs with vacuum, extending the theoretical framework and applications.

Findings

Verified genus-0 and genus-1 parts of the conjectures.

Proved full conjectures for semisimple CohFTs.

Derived Virasoro constraints for deformed negative r-spin theory.

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 $S$-calibration and $\nu$-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.