| Summary: | Abstract We provide a description of the quantum integrable structure behind the Thermodynamic Bethe Ansatz (TBA)-like equation derived by Nekrasov and Shatashvili (NS) for N=2 $$ \mathcal{N}=2 $$ 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. The latter acquires interestingly the deep meaning of a known object in integrability theory, as its two second determinations give the usual non-linear integral equations (nlies) derived from the ‘dual’ Bethe Ansatz equations.
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