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METHODS OF THEORETICAL PHYSICS

Spectral duality in integrable systems from AGT conjecture

A. Mironov^ab, A. Morozov^b, Y. Zenkevich^cbd, A. Zotov<sup>b

a</sup> P. N. Lebedev Physical Institute, Russian Academy of Sciences
^b Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center), Moscow
^c M. V. Lomonosov Moscow State University, Faculty of Physics
^d Institute for Nuclear Research, Russian Academy of Sciences, Moscow

Abstract: 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.

Received: 03.12.2012

Language: English

DOI: 10.7868/S0370274X13010104

 English version: Journal of Experimental and Theoretical Physics Letters, 2013, 97:1, 45–51

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