Аннотация:
We describe relationships between integrable systems with $N$ degrees of freedom arising from the AGT conjecture. Namely, we
prove the equivalence (spectral duality) between the $N$-cite Heisenberg spin chain and a reduced gl$_N$ Gaudin model
both at classical and quantum level. The former one appears on the gauge theory side of the AGT relation in the
Nekrasov–Shatashvili (and further the Seiberg–Witten) limit while the latter one is natural on the CFT side. At the classical
level, the duality transformation relates the Seiberg–Witten differentials and spectral curves via a bispectral involution.
The quantum duality extends this to the equivalence of the corresponding Baxter–Schrödinger equations (quantum spectral
curves). This equivalence generalizes both the spectral self-duality between the $2\times 2$ and $N\times N$ representations
of the Toda chain and the famous AHH duality.