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Cohomological Field Theory with vacuum and its Virasoro constraints

Shuai Guo, Qingsheng Zhang

TL;DR

This work develops Virasoro constraints for Cohomological Field Theories with vacuum by formulating ancestor and generalized Virasoro frameworks. It introduces S- and ν-calibrations to define formal descendent potentials, establishes genus-0 results, derives genus-1 simplifications, and proves the full conjectures for semisimple theories. The paper also provides two impactful applications: Virasoro constraints for the ε-deformed negative r-spin theory and a CohFT of extended Grothendieck’s dessins d’enfants, linking Virasoro symmetry to dessins enumeration. The results extend the Virasoro program beyond semisimple flat-unit cases, offering a unified approach via Givental–Teleman reconstruction and calibration data. Overall, the work broadens the landscape of Virasoro constraints in enumerative geometry and highlights new connections to classical combinatorial structures.

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 $ν$-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.

Cohomological Field Theory with vacuum and its Virasoro constraints

TL;DR

This work develops Virasoro constraints for Cohomological Field Theories with vacuum by formulating ancestor and generalized Virasoro frameworks. It introduces S- and ν-calibrations to define formal descendent potentials, establishes genus-0 results, derives genus-1 simplifications, and proves the full conjectures for semisimple theories. The paper also provides two impactful applications: Virasoro constraints for the ε-deformed negative r-spin theory and a CohFT of extended Grothendieck’s dessins d’enfants, linking Virasoro symmetry to dessins enumeration. The results extend the Virasoro program beyond semisimple flat-unit cases, offering a unified approach via Givental–Teleman reconstruction and calibration data. Overall, the work broadens the landscape of Virasoro constraints in enumerative geometry and highlights new connections to classical combinatorial structures.

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.

Paper Structure

This paper contains 37 sections, 27 theorems, 262 equations.

Table of Contents

  1. Introduction
  2. Ancestor Virasoro conjecture
  3. Generalized Virasoro conjecture
  4. Applications
  5. Plan of the paper
  6. CohFT with vacuum
  7. Gromov--Witten theory
  8. Cohomological field theory
  9. Vacuum axiom
  10. Generalized Frobenius manifold
  11. Homogeneity conditions
  12. Semisimple CohFT and reconstruction theorem
  13. Descendents for CohFT
  14. $S$- and $\nu$-calibration of generalized Frobenius manifold
  15. Descendent potentials for a CohFT
  16. ...and 22 more sections

Key Result

Theorem 1

(1). The genus-0 ancestor Virasoro conjecture always holds. (2). For each $m\geq 0$, the genus-1 $L_m^{\tau}$-constraint is equivalent to the following equation: where '$\mathop{\rm str}$' stands for the super-trace defined by $\mathop{\rm str}(A)=\sum_a\eta(\phi_a,A\phi^a)$ for $A\in\mathrm{End}(H)$. (3). The ancestor Virasoro constraints hold for semisimple homogeneous CohFTs.

Theorems & Definitions (77)

  • Definition 1
  • Conjecture 1: Ancestor Virasoro conjecture
  • Remark 2
  • Theorem 1: =Theorem \ref{['thm:anc-vira']}
  • Definition 3
  • Definition 4
  • Conjecture 2: Generalized Virasoro conjecture
  • Theorem 2: $=$Theorem \ref{['thm:vira-des-0']} $+$ Theorem \ref{['thm:vira-des-g']}
  • Remark 5
  • Proposition 1: $=$Proposition \ref{['prop:vira-negative-rspin']}
  • ...and 67 more