Degenerate integrability of quantum spin Calogero–Moser systems

Reshetikhin, Nicolai

doi:10.1007/s11005-016-0897-8

Cited by 21 publications

(25 citation statements)

References 25 publications

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“…These arise in the context of coset spaces G/G where the denominator groups acts by conjugation. In [49,50] Reshetikhin introduced a matrix version of these A N Calogero-Sutehrland models that is very similar to the matrix BC N models we constructed above. For the A N series these matrix systems were shown to be super (or degenerate) integrable.…”

Section: Discussionmentioning

confidence: 99%

From spinning conformal blocks to matrix Calogero-Sutherland models

Schomerus

Sobko

2018

J. High Energ. Phys.

107

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In this paper we develop further the relation between conformal four-point blocks involving external spinning fields and Calogero-Sutherland quantum mechanics with matrix-valued potentials. To this end, the analysis of [1] is extended to arbitrary dimensions and to the case of boundary two-point functions. In particular, we construct the potential for any set of external tensor fields. Some of the resulting Schrödinger equations are mapped explicitly to the known Casimir equations for 4-dimensional seed conformal blocks. Our approach furnishes solutions of Casimir equations for external fields of arbitrary spin and dimension in terms of functions on the conformal group. This allows us to reinterpret standard operations on conformal blocks in terms of group-theoretic objects. In particular, we shall discuss the relation between the construction of spinning blocks in any dimension through differential operators acting on seed blocks and the action of left/right invariant vector fields on the conformal group.

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Section: Discussionmentioning

confidence: 99%

From spinning conformal blocks to matrix Calogero-Sutherland models

Schomerus

Sobko

2018

J. High Energ. Phys.

107

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“…Certain types of models with spin have recently been studied in [58,59] from the Hamiltonian reduction perspective and their degenerate (or super-) integrability was established. The quantum analogue of [59] finds roots in the work of Harish-Chandra and Kostant and Tirao [60].…”

Section: Applications To 4-dimensional N = 1 Theoriesmentioning

confidence: 99%

Superconformal blocks: general theory

Burić

Schomerus

Sobko

2020

J. High Energ. Phys.

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In this work we launch a systematic theory of superconformal blocks for four-point functions of arbitrary supermultiplets. Our results apply to a large class of superconformal field theories including 4-dimensional models with any number N of supersymmetries. The central new ingredient is a universal construction of the relevant Casimir differential equations. In order to find these equations, we model superconformal blocks as functions on the supergroup and pick a distinguished set of coordinates. The latter are chosen so that the superconformal Casimir operator can be written as a perturbation of the Casimir operator for spinning bosonic blocks by a fermionic (nilpotent) term. Solutions to the associated eigenvalue problem can be obtained through a quantum mechanical perturbation theory that truncates at some finite order so that all results are exact. We illustrate the general theory at the example of d = 1 dimensional theories with N = 2 supersymmetry for which we recover known superblocks. The paper concludes with an outlook to 4-dimensional blocks with N = 1 supersymmetry.

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“…Their reduced flows can be found via the projection method, similarly to the case of H, and all those flows are complete on the full reduced phase space, M red = µ −1 (0)/G. It was shown by Reshetikhin [41,42,43] that the degenerate integrability of the free Hamiltonian H (1.1) is inherited at the reduced level with analytic integrals of motion, at least for generic coadjoint orbits and on a dense open subset of M red . Liouville integrability in the same generic case follows from the results of [31].…”

Section: )mentioning

confidence: 99%

“…These spin Sutherland models can be interpreted as Hamiltonian reductions of free motion on G, relying on the cotangent lift of the conjugation action of G on itself. The reduction can be utilized to show their integrability, and to analyze their quantum mechanics with the aid of representation theory [11,17,18,41,42,43]. Spinless models can be obtained in this way only for G = SU(n), using a minimal coadjoint orbit, for which the T-action on O 0 is transitive.…”

Section: Introductionmentioning

confidence: 99%

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Poisson-Lie analogues of spin Sutherland models

Fehér

2019

Nuclear Physics B

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We present generalizations of the well-known trigonometric spin Sutherland models, which were derived by Hamiltonian reduction of 'free motion' on cotangent bundles of compact simple Lie groups based on the conjugation action. Our models result by reducing the corresponding Heisenberg doubles with the aid of a Poisson-Lie analogue of the conjugation action. We describe the reduced symplectic structure and show that the 'reduced main Hamiltonians' reproduce the spin Sutherland model by keeping only Integrable systems of particles moving in one dimension have been studied intensively for nearly 50 years, beginning with the pioneering papers of Calogero [5], Sutherland [51] and Moser [35]. Thanks to their fascinating mathematics and diverse applications [10,37,38,44,52], the interest in these models shows no sign of diminishing. New connections to mathematics and new applications are still coming to light in the current literature, see e.g. [6,7,22,24,46,53].The richness of these models is also due to their many generalizations and deformations. These are associated with different interaction potentials (from rational to elliptic), root systems and extensions with internal degrees of freedom. We call 'Sutherland models' the systems defined by trigonometric or hyperbolic potentials. For all these systems, classical and quantum mechanical versions are studied separately, and one needs to pay attention to the distinct features of the systems with real particle positions and their complexifications. The investigations of Ruijsenaars-Schneider (RS) type deformations [44,45] is motivated, for example, by relations to solitons, spin chains, special functions and double affine Hecke algebras.The internal degrees of freedom are colloquially called 'spin', and can be of two rather different kinds. First, the point particles can carry spins varying in a vector space, as is the case for the Gibbons-Hermsen models [20] and their RS type generalizations introduced by Krichever and Zabrodin [28]. Second, the models can involve a collective spin variable that typically belongs to a coadjoint orbit, and is not assigned separately to the particles. An example of this second type is the trigonometric spin Sutherland model defined classically by a Hamiltonian of the following form:

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Degenerate integrability of quantum spin Calogero–Moser systems

Cited by 21 publications

References 25 publications

From spinning conformal blocks to matrix Calogero-Sutherland models

From spinning conformal blocks to matrix Calogero-Sutherland models

Superconformal blocks: general theory

Poisson-Lie analogues of spin Sutherland models

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