Spectral duality in gauge theories, conformal field theories and integrable systems

Spectral duality has been initially put forward in the theory of integrable systems – in this setting it connects pairs of systems with the same spectral curve. One of particular cases of this duality is the Adams–Harnad–Hurtubise duality between two rational Gaudin models with different algebras and different numbers of marked points. We will consider a generalization of this duality to the case of trigonometric Gaudin models, $XXX$ and $XXZ$ spin chains. We will also show how this duality manifests itself in 4d $N=2$ supersymmetric gauge theories. In particular, we will describe its role in the AGT duality relating gauge theories with 2d CFTs. Finally, we will demonstrate how the spectral duality arises naturally in topological string theory on toric Calabi–Yau manifolds.