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 Lett. Math. Phys., 2013, Volume 103, Issue 3, Pages 299–329 (Mi letmp2)

Spectral duality between Heisenberg chain and Gaudin model

A. Mironovab, A. Morozova, B. Runovac, E. Zenkevichad, A. Zotova

a ITEP, Moscow, Russia
b Theory Department, Lebedev Physics Institute, Moscow, Russia
c MIPT, Dolgoprudniy, Moscow, Russia
d Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia

Abstract: In our recent paper we described relationships between integrable systems inspired by the AGT conjecture. On the gauge theory side an integrable spin chain naturally emerges while on the conformal field theory side one obtains some special reduced Gaudin model. Two types of integrable systems were shown to be related by the spectral duality. In this paper we extend the spectral duality to the case of higher spin chains. It is proved that the $N$-site $\mathrm{GL}_k$ Heisenberg chain is dual to the special reduced $k+2$-points $\mathrm{gl}_N$ Gaudin model. Moreover, we construct an explicit Poisson map between the models at the classical level by performing the Dirac reduction procedure and applying the AHH duality transformation.

 Funding Agency Grant Number Federal Agency for Science and Innovations of Russian Federation 14.740.11.0347 Ministry of Education and Science of the Russian Federation NSh-3349.2012.2MK-1646.2011.1 Russian Foundation for Basic Research 10-02-0050910-02-0049912-01-0048212-01-3307112-01- 3138511-02-9045312-02-9210811-01-92612 The work was partially supported by the Federal Agency for Science and Innovations of Russian Federation under contract 14.740.11.0347 (A.Z., B.R. and Y.Z.), by NSh-3349.2012.2 (A. Mir., A. Mor. and B.R.), by RFBR grants 10-02-00509 (A. Mir.), 10-02-00499 (A. Mor. and Y.Z.), 12-01-00482 (A.Z. and B.R.), 12-01-33071 mol a ved (B.R., A.Z. and Y.Z.), “my first grant” 12-01- 31385 (B.R. and Y.Z.) and by joint grants 11-02-90453-Ukr, 12-02-92108-Yaf-a, 11-01-92612-Royal Society. The work of A.Zotov was also supported in part by the Russian President fund MK-1646.2011.1.

DOI: https://doi.org/10.1007/s11005-012-0595-0

Bibliographic databases:

Document Type: Article
MSC: 14H70, 14H81, 81Q99
Revised: 05.11.2012
Accepted:06.11.2012
Language: English