Higher spin $${{\mathfrak {s}}}{{\mathfrak {l}}}_2$$ s l 2 R -matrix from equivariant (co)homology
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Higher spin sl2 R-matrix from equivariant (co)homology Dmitri Bykov1,2,3 · Paul Zinn-Justin4 Received: 3 August 2019 / Revised: 1 April 2020 / Accepted: 13 June 2020 © Springer Nature B.V. 2020
Abstract We compute the rational sl2 R-matrix acting in the product of two spin- 2 ( ∈ N) representations, using a method analogous to the one of Maulik and Okounkov, i.e., by studying the equivariant (co)homology of certain algebraic varieties. These varieties, first considered by Nekrasov and Shatashvili, are typically singular. They may be thought of as the higher spin generalizations of A1 Nakajima quiver varieties (i.e., cotangent bundles of Grassmannians), the latter corresponding to = 1. Keywords R-matrix · Spin chain · Nakajima variety · Stable envelope Mathematics Subject Classification 14L24 · 55N91 · 81R12 · 16T25 · 81T60
1 Introduction In [23,28], the cohomology of quiver varieties introduced in [21,22] was endowed with the action of a Yangian. More precisely, one obtains arbitrary tensor products of fundamental representations this way. In [20], Maulik and Okounkov reinterpreted this construction in the formalism of quantum integrable systems: Via the stable envelope,
PZJ was supported by ARC Grant FT150100232. DB wishes to thank D. Lüst and A. A. Slavnov for support. PZJ wishes to thank A. Knutson, Y. Yang, G. Zhao for valuable discussions. Computerized checks of the results of this paper were performed with the help of Macaulay2 [11].
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Paul Zinn-Justin [email protected] Dmitri Bykov [email protected]; [email protected]; [email protected]
1
Max-Planck-Intitut für Physik, Föhringer Ring 6, 80805 Munich, Germany
2
Arnold Sommerfeld Center for Theoretical Physics, Theresienstrasse 37, 80333 Munich, Germany
3
Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina Str. 8, Moscow, Russia 119991
4
School of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
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D. Bykov, P. Zinn-Justin
they defined the R-matrix which in turn allows to define the Yangian action thanks to the RT T approach, as first advocated in [14] (see also [7,8]). It is known that any finite-dimensional irreducible representation of the Yangian of a simple Lie algebra can be obtained as a subquotient of tensor products of fundamental representations [3, Cor. 12.1.13]. One could argue that defining geometrically such tensor products of fundamental representations is all that is therefore needed. We believe that this point of view is unsatisfactory: One should be able to obtain directly such general finite-dimensional representations as the cohomology of some appropriate varieties generalizing Nakajima quiver varieties. The relation to tensor products of fundamental representations (usually called fusion procedure in the language of quantum integrable systems) should come about as a Lagrangian correspondence, which itself should be a degenerate limit of the stable envelope construction. (See also [33] where fusion between different Nakajima quiver varieties is considered.) Thi
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