The solution of the quantum $A_1$ T-system for arbitrary boundary

TL;DR

This paper provides an exact solution to the quantum $A_1$ T-system using quantum networks, revealing Laurent positivity and connecting it to quantum cluster algebras and the quantum lattice Liouville equation.

Contribution

It introduces a novel quantum network approach to solve the quantum $A_1$ T-system and extends the solution to non-commutative cases, linking it to quantum integrable systems.

Findings

Solution demonstrates Laurent positivity for the quantum $A_1$ T-system.

Re-derivation of the quantum $A_1$ $Q$-system solution using quantum networks.

Establishment of a relation between the quantum $T$-system and the quantum lattice Liouville equation.

Abstract

We solve the quantum version of the $A_1$ $T$-system by use of quantum networks. The system is interpreted as a particular set of mutations of a suitable (infinite-rank) quantum cluster algebra, and Laurent positivity follows from our solution. As an application we re-derive the corresponding quantum network solution to the quantum $A_1$ $Q$-system and generalize it to the fully non-commutative case. We give the relation between the quantum $T$-system and the quantum lattice Liouville equation, which is the quantized $Y$-system.