Quantum integrability of $\mathcal{N}=2$ 4d gauge theories

Understand

We provide a description of the quantum integrable structure behind the Thermodynamic Bethe Ansatz (TBA)-like equation derived by Nekrasov and Shatashvili (NS) for $\mathcal{N}=2$ 4d Super Yang-Mills (SYM) theories.

• In this regime of the background, -- we shall show --, the instanton partition function is characterised by the solution of a TQ-equation.
• Exploiting a symmetry of the contour integrals expressing the partition function, we derive a 'dual' TQ-equation, sharing the same T-polynomial with the former.
• This fact allows us to evaluate to $1$ the quantum Wronskian of two dual solutions (for $Q$) and, then, to reproduce the NS TBA-like equation.