Ruijsenaars-Schneider three-body models with N = 2 supersymmetry

Journal of High Energy Physics, Apr 2018

Anton Galajinsky

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Ruijsenaars-Schneider three-body models with N = 2 supersymmetry

HJE Ruijsenaars-Schneider three-body models with Anton Galajinsky 0 1 2 0 Lenin Ave. 30 , 634050 Tomsk , Russia 1 School of Physics, Tomsk Polytechnic University 2 [10] J. Arnlind , M. Bordemann, J. Hoppe and C. Lee, Goldfish geodesics and Hamiltonian The Ruijsenaars-Schneider models are conventionally regarded as relativistic generalizations of the Calogero integrable systems. Surprisingly enough, their supersymmetric generalizations escaped attention. In this work, N = 2 supersymmetric extensions of the rational and hyperbolic Ruijsenaars-Schneider three-body models are constructed within the framework of the Hamiltonian formalism. It is also known that the rational model can be described by the geodesic equations associated with a metric connection. We demonstrate that the hyperbolic systems are linked to non-metric connections. Extended Supersymmetry; Integrable Field Theories 1 Introduction 2 such systems stems from the fact that some of them are expected to be relevant for a microscopic description of the extreme black holes [7]. Worth mentioning also is that N -extended supersymmetry in d = 1 exhibits peculiar features which are absent in higher dimensions. Surprisingly enough, supersymmetric extensions of the relativistic counterparts of the Calogero models remain almost completely unexplored. An integrable N = 2 supersymmetric generalization of the quantum trigonometric Ruijsenaars-Schneider model has been reported in [8] whose eigenfunctions were linked to the Macdonald superpolynomials. Note, however, that the fermionic variables in [8] and their adjoints obey the non-standard anticommutation relations which reduce to the conventional ones in the non-relativistic limit only. The goal of this work is to construct N = 2 supersymmetric extensions of the rational and hyperbolic Ruijsenaars-Schneider three-body models within the framework of the Hamiltonian (on-shell) formalism. As is known, the systems admit more than one Hamiltonian description [2, 9]. For a supersymmetric extension to be feasible, we suggest to choose a Hamiltonian each term of which is positive definite. – 1 – The paper is organized as follows. In subsections 2.1, 2.2, and 2.3 we briefly review the basic properties of the rational and hyperbolic Ruijsenaars-Schneider three-body models with a particular emphasis on the issue of (super)integrability. An interesting feature of these systems is that they admit an alternative description in terms of geodesic equations associated with an affine connection [10]. For the rational model the latter is known to be a metric connection and the manifold is actually flat [10]. In subsection 2.4. we demonstrate that the hyperbolic models are linked to non-metric connections. In section 3 for each bosonic variable we introduce a pair of complex conjugate fermionic partners and build novel N = 2 supersymmetric rational and hyperbolic Ruijsenaars-Schneider three-body models. In contrast to the non-relativistic N = 2 Calogero models [11], the supersymmetry which are described by the equations of motion [ 1 ] (2.1) (2.2) x¨i = X x˙ ix˙ j W (xi − xj ), j6=i where W (x) = x2 , sin2h x , or 2 coth x.1 For simplicity of presentation, in what follows we focus on the three-body problem only and assume x1 < x2 < x3. Note that the models hold invariant under the temporal and spatial translations. The rational system is also invariant under independent rescalings of t and xi [9]. 2.1 Rational model as the goldfish model [9]. The equations of motion follow from the Hamiltonian2 The rational Ruijsenaars-Schneider system corresponds to W (x) = x2 which is also known H = ep1 x12x13 + ep2 x12x23 + ep3 x13x23 where xij = xi − xj and (p1, p2, p3) signify momenta canonically conjugate to (x1, x2, x3). The Poisson bracket is chosen in the conventional form {xi, pj } = δij . One of the ways to construct three mutually commuting constants of the motion is to use the Lax matrix [ 1, 2 ] which yields I1 = H, I2 = x˜23ep1 x12x13 + x˜13ep2 x12x23 + x˜12ep3 x13x23 , I3 = x2x3ep1 x12x13 + x1x3ep2 1The so called trigonometric models follow from the hyperbolic systems after the substitution x → ix. In what follows we disregard them. 2The Hamiltonian formulation (2.2) is not unique [9]. One can verify that multiplying each term in (2.2) by an arbitrary constant one does not alter the equations of motion. Keeping in mind the forthcoming construction of an N = 2 supersymmetric extension, we stick to the Hamiltonian each term of which is positive definite. We also do so for the Hamiltonians in subsection 2.2 and 2.3. – 2 – (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) HJEP04(218)79 I0 = p1 + p2 + p3. The third constant of the motion, which ensures the Liouville integrability, reads I2 = ep1+p2 coth +ep2+p3 coth x13 2 x12 2 coth coth x23 2 x13 . 2 + ep1+p3 coth coth x23 2 One of the ways to obtain (2.9) is to use the Lax matrix [ 1, 2 ]. It is readily verified that (I0, I1, I2) are mutually commuting and functionally independent. 2.3 Hyperbolic model II The second Ruijsenaars-Schneider hyperbolic model is associated with W (x) = 2 coth x. We choose the Hamiltonian in the form is conserved. Other constants of the motion are built by considering the elementary monomials Mp = X i1<···<ip xi1 . . . xip , {Mp, H} = Ip, where p = 1, . . . , 3, such that MiIj − Mj Ii are conserved quantities. For the case at hand it suffices to consider I4 = x2x3x˜23ep1 x12x13 + x1x3x˜13ep2 x12x23 + x1x2x˜12ep3 x13x23 = M1I3 − M3I1. It is straightforward to verify that Ik, k = 0, . . . , 4, are functionally independent which implies the three-body problem (2.2) is maximally superintegrable. where x˜ij = xi + xj . These are functionally independent. translation invariant, the total momentum The rational model is known to be maximally superintegrable [12]. Since (2.1) is which is chosen such that each term is positive definite (recall x1 < x2 < x3). Like its rational counterpart (2.2), the system (2.7) is invariant under the spatial translation, x′i = xi + a, which results in the conservation of the total momentum ep2 – 3 – ep1 + ally commuting and functionally independent integrals of motion include (2.10) the total momentum and The simplest way to obtain (2.12) is to use the Lax matrix [ 1, 2 ]. tions on a manifold which is parametrized by the local coordinates xi and equipped with the affine connection (no summation over repeated indices) [10] HJEP04(218)79 Γijk = δji wik + δkiwij , W (xi − xk), , i 6= k i = k wik = −   1 2 0 gij = ∂Mp ∂Mp ∂xi ∂xj , For the rational model (2.13) turns out to be a metric connection associated with [10] (2.11) (2.12) (2.13) (2.14) (2.15) (2.16) (2.17) (2.18) where the functions Mp are given in (2.5) with p = 1, . . . , n. Since (2.14) is the Kronecker delta in curvilinear coordinates, the transformation x′i = Mi(x) links the rational Ruijsenaars-Schneider model to a free particle propagating in a flat space. Let us examine whether the hyperbolic choices of W (x) result in metric connections. Assuming a metric is non-degenerate and (2.13) can be represented in the conventional form Γijk = 2 1 gip (∂j gpk + ∂kgpj − ∂pgjk) , contracting with gsi, permuting the indices (j, s, k) → (s, k, j), and taking the sum, one gets a coupled set of partial differential equations ∂j gsk = wjk(gsj − gsk) + wjs(gkj − gks). It turns out that (2.16) leads to a contradiction as it yields a degenerate metric whose all components are equal to one and the same constant, gij = const. In order to see this, it suffices to consider three equations belonging to the set (2.16) ∂1g11 = 0, ∂2g11 = 2w12(g11 − g12), ∂1g12 = w12(g11 − g12). Computing the derivative of the second equation with respect to x1 and taking into account the other two, one gets w1′2 − w122 (g11 − g12) = 0, – 4 – W (x) = x2 where w′ = dwd(xx) . Since for the hyperbolic models w1′2 − w122 6= 0, one obtains of gij are equal to each other. The left hand side of (2.16) then implies gij = const. Thus, in contrast to the rational model, the hyperbolic Ruijsenaars-Schneider systems are linked to non-metric connections. While in the former case all components of the Riemann tensor vanish identically, in the latter case the curvature tensor is non-trivial. 3 N = 2 supersymmetric extension of Ruijsenaars-Schneider models As was emphasized above, the Hamiltonian formulations for the Ruijsenaars-Schneider models were chosen so that each term in the Hamiltonian was positive definite. In order to construct N = 2 supersymetric extensions, we first represent the original bosonic Hamiltonian in the form where the phase space functions λi, i = 1, 2, 3, are given above in table 1. They prove to obey the quadratic algebra (no summation over repeated indices and i 6= j) HB = λiλi, 1 4 {λi, λj } = W (xi − xj )λiλj . Note that this algebra holds invariant under the rescalings λi → αiλi (no sum), where αi are arbitrary real constants. This transformation links to the arbitrariness in the choice of the Hamiltonian mentioned above. Then we introduce the complex fermionic partners ψi, i = 1, 2, 3, for the bosonic coordinates xi, and impose the canonical brackets {ψi, ψj } = 0, {ψi, ψ¯j } = δij , {ψ¯i, ψ¯j } = 0, where ψ¯i stands for the complex conjugate of ψi. (3.1) (3.2) (3.3) – 5 – HJEP04(218)79 Two supersymmetry charges are chosen in the polynomial form Q = λiψi + ifijkψiψj ψ¯k, Q¯ = λiψ¯i + ifijkψ¯iψ¯j ψk, where fijk = −fjik are real functions. The latter are determined from the condition that the supersymmetry charge is nilpotent {Q, Q} = 0: {λi, λj } + 2fijkλk = 0, {λk, fnml} + 2fknpfpml = 0, {fabc, fmnk} = 0, (3.5) where the underline/overline mark signifies antisymmetrization of the respective indices. The Hamiltonian which governs the dynamics of an N = 2 supersymmetric extension HJEP04(218)79 follows from the superalgebra which yields {Q, Q¯} = −iH, H = λiλi − 2i(fijk + fkji + fikj )λkψiψ¯j + i{fijl, fmnk}ψiψj ψkψ¯lψ¯mψ¯n −({λi, fklj } − {λl, fijk} + fijpfklp − 4fpilfpkj )ψiψj ψ¯kψ¯l. Comparing (3.2) with the leftmost equation in (3.5), one gets a 8 b 8 c 8 where (a, b, c) are arbitrary real constants, while other components of fijk prove to vanish. Substituting (3.8) into the second equation in (3.5), one obtains the quadratic algebraic bc = 0, a(1 − c) = 0, (1 − a)(1 − b) = 0, which imply that two options are available equations or 1 8 (3.4) (3.6) (3.7) (3.8) (3.9) (3.10) (3.11) (3.12) (3.13) 1 8 It is straightforward to verify that the second possibility is linked to the first by relabelling x1 ↔ x3, p1 ↔ p3, ψ1 ↔ ψ3, ψ¯1 ↔ ψ¯3, which gives λ1 ↔ λ3, λ2 ↔ λ2. For the three-body problem the rightmost equation in (3.5) holds automatically. Thus, N = 2 supersymmetric extensions of the Ruijsenaars-Schneider models build upon λi, which are exposed above in table 1, and the structure functions f121 = − W (x1 − x2)λ2, f133 = − W (x1 − x3)λ1, f232 = − W (x2 − x3)λ3. a = 1, a = 0, b = 0, b = 1, c = 1, = 2 supersymmetric extensions of the nonrelativistic Calogero model [11], the supersymmetry charges involve contributions cubic in the fermionic variables. Thus, provided one is focused on a Hamiltonian each term of which is positive definite, the N = 2 supersymmetric extension is essentially unique. It proves instructive to expose the (complex) supersymmetry charge and the Hamiltonian in terms of λi and the prepotential W (x) Q = λ1ψ1 + λ2ψ2 + λ3ψ3 − W (x1 − x2)λ2ψ1ψ2ψ¯1 − W (x1 − x3)λ1ψ1ψ3ψ¯3 H = λ21 + λ22 + λ32 + W (x1 − x2)λ1λ2(ψ1ψ¯2 − ψ2ψ¯1) + W (x1 − x3)λ1λ3(ψ1ψ¯3 − ψ3ψ¯1) where W ′(x) = dWdx(x) . Curiously enough, for the three-body models the six-fermion term present in (3.7) proves to be zero. We failed to demonstrate that it also vanishes for n > 3 on account of eqs. (3.5). 4 Conclusion The construction of the N = 2 supersymmetric rational and hyperbolic RuijsenaarsSchneider three-body models reported in this work can be continued in several directions. First of all, it is worth extending the present analysis to the case of arbitrary number of particles. For the rational model an optimal strategy might be to switch to the geodesic formulation associated with the metric (2.14). One can first implement a coordinate transformation which brings the model to the free form, supersymmetrize the free system, and then apply the inverse transformation. A canonical transformation linking such a system to (2.2) for n = 3 is of interest. For the hyperbolic models the construction may break beyond n = 3. For the case of n particles the structure functions fijk involve nCn2 components, where Cmk are the binomial coefficients. The first, second, and third equations in (3.5) yield Cn2, nCn3, and Cn2Cn4 conditions, respectively. For n > 3 the set of restrictions is overcomplete. In particular, some of them may turn out to be incompatible with the form of the prepotential W (x) chosen. Secondly, it is interesting to construct an off-shell superfield Lagrangian formulation for the on-shell component Hamiltonian (3.7) and to study its peculiarities. Thirdly, an N = 4 supersymmetric generalization is an intriguing open problem. The key point is to reveal an analogue of the Witten-Dijkgraaf-Verlinde-Verlinde equation [4]. As was mentioned above, the hyperbolic Ruijsenaars-Schneider models can be described – 7 – in terms of the geodesic equations associated with a non-metric connection. The description of many-body mechanics with extended supersymmetry on such spacetimes in purely geometric terms is a challenge. Finally, it would be interesting to understand whether supersymmetric extensions of the Ruijsenaars-Schneider models may be relevant for the study of the space of vacua of supersymmetric gauge theories (see the discussion in [13] and references therein). Acknowledgments ment program. This work was supported by the Tomsk Polytechnic University competitiveness enhance This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] S.N.M. Ruijsenaars and H. Schneider , A new class of integrable systems and its relation to


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Anton Galajinsky. Ruijsenaars-Schneider three-body models with N = 2 supersymmetry, Journal of High Energy Physics, 2018, 79, DOI: 10.1007/JHEP04(2018)079