# Dispersionless and multicomponent BKP hierarchies with quantum torus symmetries

## TL;DR

This paper constructs quantum torus symmetries for the dispersionless BKP hierarchy, revealing how these symmetries are broken from the full hierarchy and defining additional flows for the multicomponent BKP hierarchy.

## Contribution

It introduces the perturbative quantum torus symmetry for the dispersionless BKP hierarchy based on $W_{ty}$ symmetry, and defines new flows for the multicomponent BKP hierarchy.

## Key findings

- Quantum torus symmetry is broken from BKP to dispersionless hierarchy.
- Additional flows form an N-fold product of positive quantum torus symmetries.
- Constructs the symmetry structure based on $W_{ty}$ Lie algebra.

## Abstract

In this article, we will construct the additional perturbative quantum torus symmetry of the dispersionless BKP hierarchy basing on the $W_{\infty}$ infinite dimensional Lie symmetry. These results show that the complete quantum torus symmetry is broken from the BKP hierarchy to its dispersionless hierarchy. Further a series of additional flows of the multicomponent BKP hierarchy will be defined and these flows constitute an $N$-folds direct product of the positive half of the quantum torus symmetries.