Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cohomological representations of quantum tau functions

Xavier Blot, Danilo Lewański, Sergey Shadrin

Abstract

In 2016, Buryak and Rossi introduced the quantum Double Ramification (DR) hierarchies which associate a quantum integrable hierarchy to any Cohomological Field Theory (CohFT). Shortly after, they introduced, in collaboration with Dubrovin and Guéré, the quantum tau functions of these hierarchies. In this work, we study quantum tau functions associated to a specific solution called the topological solution. We provide two cohomological representations for the correlators of these tau functions. The first representation involves an analog in the quantum setting of the $A$-class of the DR-DZ equivalence. The second representation, valid for CohFT of low degree, involves the so-called $Ω$-classes. Furthermore, we establish the string and dilaton equations for these tau functions, and present certain vanishing of their correlators.

Cohomological representations of quantum tau functions

Abstract

In 2016, Buryak and Rossi introduced the quantum Double Ramification (DR) hierarchies which associate a quantum integrable hierarchy to any Cohomological Field Theory (CohFT). Shortly after, they introduced, in collaboration with Dubrovin and Guéré, the quantum tau functions of these hierarchies. In this work, we study quantum tau functions associated to a specific solution called the topological solution. We provide two cohomological representations for the correlators of these tau functions. The first representation involves an analog in the quantum setting of the -class of the DR-DZ equivalence. The second representation, valid for CohFT of low degree, involves the so-called -classes. Furthermore, we establish the string and dilaton equations for these tau functions, and present certain vanishing of their correlators.

Paper Structure

This paper contains 41 sections, 22 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. The topological tau function of the quantum DR hierarchy
  4. Cohomological Field Theories with unit
  5. Differential polynomials and local functionals
  6. Hamiltonian of the DR hierarchies
  7. The star product
  8. Integrability and tau symmetry of the quantum DR hierarchies
  9. Topological tau function of the quantum DR hierarchies
  10. Background on Omega-classes
  11. Acknowledgements
  12. Main statements
  13. The quantum A-class and quantum correlators
  14. Quantum correlators and the Hej-class
  15. Level structure for the correlators
  16. ...and 26 more sections

Key Result

Proposition 1.5

Fix a differential polynomial $f \in\mathcal{A}_{N}$ and a local functional $\overline{g}\in\overline{\mathcal{A}}_{N}$. Then, the commutator of the star product $\left[f,\overline{g}\right]$ is a differential polynomial, that is an element of $\mathcal{A}_{N}$.

Theorems & Definitions (59)

  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Proposition 1.5: BR16-quantum
  • Proposition 1.6: Integrability, BR16
  • Proposition 1.7: Tau symmetry, BDGR20
  • Definition 1.8: Two point function
  • Lemma 1.9
  • Remark 1.10
  • Remark 1.11
  • ...and 49 more