# Integrable Systems of Double Ramification Type

```@article{Buryak2016IntegrableSO,
title={Integrable Systems of Double Ramification Type},
author={Alexandr Buryak and Boris Dubrovin and J'er'emy Gu'er'e and Paolo Rossi},
journal={International Mathematics Research Notices},
year={2016}
}```
• Published 13 September 2016
• Mathematics
• International Mathematics Research Notices
In this paper we study various aspects of the double ramification (DR) hierarchy, introduced by the 1st author, and its quantization. We extend the notion of tau-symmetry to quantum integrable hierarchies and prove that the quantum DR hierarchy enjoys this property. We determine explicitly the genus \$1\$ quantum correction and, as an application, compute completely the quantization of the \$3\$- and \$4\$-KdV hierarchies (the DR hierarchies for Witten’s \$3\$- and \$4\$-spin theories). We then focus…
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