Spectral Duality in Integrable Systems from AGT Conjecture

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-Schrodinger equations (quantum spectral curves). This equivalence generalizes both the spectral self-duality between the 2x2 and NxN representations of the Toda chain and the famous AHH duality.