N=2 supersymmetric extensions of relativistic Toda lattice

N = 2 sup ersymmetric extensions of relativistic T o da lattice An ton Gala jinsky T omsk Polyte chnic University, 634050 T omsk, L enin Ave. 30, R ussia e-mail: gala jin@tpu.ru Abstract N = 2 sup ersymmetric extensions of b oth the p erio dic and non–p erio dic relativistic T o da lattice are built within the framework of the Hamiltonian formalism. A geo desic description in terms of a non–metric connection is discussed. Keyw ords: relativistic T o da lattice, N = 2 sup ersymmetry 1. Introduction The range of ph ysical applications of the Calogero and T oda in tegrable systems is excessiv ely broad. It encompasses the fractional statistics, quan tum Hall effect, soliton theory , matrix mo dels, sup ersymmetric gauge theories, and black hole ph ysics. It is kno wn since the work of Ruijsenaars and Sc hneider [1, 2] that b oth the Calogero and T o da mo dels can b e viewed as the non–relativistic limit of a more general in tegrable system, which enjo ys the Poincar ´ e symmetry realized in 1 + 1 dimensions. In contrast to the non–relativistic theories, the Ruijsenaars–Sc hneider systems are describ ed by the equations of motion whic h inv olv e particle velocity . While the non–relativistic Calogero and T o da mo dels received tremendous attention in the past, their relativistic counterparts app ear to b e less p opular. There are sev eral rea- sons to fo cus on them more in tently , though. Firstly , a geometric formulation underlying suc h systems is still missing. Whereas the non–relativistic mo dels can b e consistently em- b edded into the n ull geo desics of a Brinkmann–type metric [3, 4], a similar description of the Ruijsenaars–Sc hneider systems seems problematic. F or one thing, the Hamiltonian do es not hav e a conv en tional quadratic form and the integrals of motion are not p olynomial in momen ta. F or another, even if one is able to rewrite the equations of motion in the geo desic form [5], one reveals a non–metric connection [6]. 1 Secondly , although the thermo dynamic limit of the non–relativistic models is w ell understoo d (see, e.g., [7, 8] and references therein), an exhaustiv e analysis of the relativistic coun terparts is still lac king. Thirdly , an imp ortant asp ect of the studies o v er the past t w o decades has b een the construction of sup ersymmetric extensions (for a review see [9] and references therein). Y et, supersymmetric generalizations of the relativistic man y–b o dy mo dels remain almost completely unexplored. An N = 2 sup ersymmetric extension of the quantum trigonometric Ruijsenaars–Sc hneider mo del was built in [10]. The corresponding eigenfunctions were link ed to the Macdonald su- p erp olynomials. A p eculiar feature of the construction is that the fermionic op erators and their adjoints ob ey the non–standard anticomm utation relations, whic h reduce to the con- v entional ones in the non–relativistic limit only . The Hermitian conjugation of the fermions is realized in the non–standard fashion as w ell. In Ref. [6], the Hamiltonian metho ds were used to construct N = 2 sup ersymmetric extensions of the rational and hyperb olic three– b o dy Ruijsenaars-Sc hneider mo dels. A v arian t of the rational Ruijsenaars–Sc hneider mo del enjo ying an arbitrary even n umber of sup ersymmetries and inv olving extra fermionic de- grees of freedom w as prop osed in [11]. It is worth recalling, though, that the rational mo del describ es a free system in disguise [5]. The goal of this w ork is to extend our recen t analysis in [6] to the case of the relativistic T o da lattice [2]. By making use of the Hamiltonian on–shell formalism, b elow w e construct N = 2 sup ersymmetric extensions of b oth the p erio dic and non–p erio dic relativistic T o da lattice. In contrast to [6], the extension prov es feasible for an arbitrary n um b er of particles. The w ork is organized as follows. 1 The rational v ariant of the Ruijsenaars-Sc hneider model can b e link ed to a metric connection. Y et, the geo desic motion actually tak es place in a flat space parametrized b y sp ecial curvilinear coordinates [5]. 1 In Sect. 2, N = 2 sup ersymmetric generalizations of the p erio dic T o da lattice are built. W e start with a p ositiv e–definite Hamiltonian and represen t it as the sum of squares of structure functions, which ob ey a non–linear algebra. N = 2 sup ersymmetry charges are in tro duced in the conv en tional (cubic) p olynomial form, the leading order of which is related to the structure functions sp ecifying the b osonic Hamiltonian. Imp osing the commutation relations of the N = 2 sup ersymmetry algebra, w e obtain a set of partial differential equa- tions to fix the b osonic functions en tering the fermionic cubic terms in the sup erchar ges. Two explicit solutions are found which generate consistent N = 2 sup ersymmetric extensions. It is kno wn that the relativistic T o da lattice admits more than one Hamiltonian formulation (see, e.g., [12]). W e then consider an alternative Hamiltonian and build tw o more N = 2 sup ersymmetric generalizations. In Sect. 3, the analysis is rep eated for the non–p erio dic T o da lattice revealing four N = 2 extensions. In Sect. 4, the equations of motion of the relativistic T o da lattice are rewritten in the geodesic form. It is argued that the correspond- ing connection fails to b e deriv able from a metric. Some final remarks are gathered in the concluding Sect. 5. Throughout the pap er no summation ov er repeated indices is understo o d. 2. N = 2 sup ersymmetric extensions of p erio dic relativistic T o da lattice The relativistic T o da lattice is describ ed b y the equations of motion [2] ¨ x i = ˙ x i +1 ˙ x i W ( x i +1 − x i ) − ˙ x i ˙ x i − 1 W ( x i − x i − 1 ) , W ( x − y ) = g 2 e x − y 1 + g 2 e x − y , (1) where i = 1 , . . . , N and g is a coupling constant. The p erio dic case is characterized by the b oundary conditions x 0 = x N , x N +1 = x 1 . (2) As the first step in constructing an N = 2 supersymmetric extension, one in troduces the momen ta p i canonically conjugate to the configuration space v ariables x i and imp oses the P oisson brack ets { x i , p j } = δ i,j , (3) where δ i,j designates the Kronec ker delta. The b oundary conditions (2) imply the relations { x i +1 , p j } = δ i +1 ,j + δ i,N δ j, 1 , { x i − 1 , p j } = δ i − 1 ,j + δ i, 1 δ j,N , (4) whic h are then used to verify that the p ositiv e definite Hamiltonian H B = N X i =1 e p i  1 + g 2 e x i +1 − x i  := N X i =1 λ i λ i , λ i = e p i 2 p 1 + g 2 e x i +1 − x i (5) do es repro duce (1). The structure functions λ i pro ve to obey the non–linear algebra { λ i , λ j } = 1 4 λ i λ j ( W ( x i +1 − x i )[ δ i +1 ,j + δ i,N δ j, 1 ] − W ( x j +1 − x j )[ δ i,j +1 + δ i, 1 δ j,N ]) . (6) 2 As the second step, complex fermionic v ariables ψ i , ( ψ i ) ∗ = ¯ ψ i , i = 1 , . . . , N , are intro- duced whic h ob ey the canonical brack ets { ψ i , ψ j } = 0 , { ψ i , ¯ ψ j } = − i δ i,j , { ¯ ψ i , ¯ ψ j } = 0 . (7) They allo w one to build the Hamiltonian and the supersymmetry c harges in the p olynomial form [6] Q = N X i =1 λ i ψ i + i N X i,j,k =1 f ij k ψ i ψ j ¯ ψ k , ¯ Q = N X i =1 λ i ¯ ψ i + i N X i,j,k =1 f ij k ¯ ψ i ¯ ψ j ψ k , H = H B − 2i N X i,j,k =1 ( f ij k + f kj i + f ikj ) λ k ψ i ¯ ψ j + i N X i,j,k,l ,m,n =1 { f ij l , f mnk } ψ i ψ j ψ k ¯ ψ l ¯ ψ m ¯ ψ n − N X i,j,k,l =1 ( { λ i , f kl j } − { λ l , f ij k } + f ij p f kl p − 4 f pil f pkj ) ψ i ψ j ¯ ψ k ¯ ψ l , (8) where f ij k = − f j ik are real functions to b e fixed b elow. Finally , one v erifies that the generators (8) ob ey the commutation relations of the N = 2 sup ersymmetry algebra { Q, Q } = 0 , { Q, ¯ Q } = − i H , { ¯ Q, ¯ Q } = 0 (9) pro vided the restrictions { λ i , λ j } + 2 N X k =1 f ij k λ k = 0 , { λ k , f nml } + 2 N X p =1 f knp f pml = 0 , { f abc , f mnk } = 0 (10) hold. In the previous formulae the underline/ov erline mark signifies antisymmetrization of the resp ectiv e indices. Comparing (6) with the leftmost equation in (10), one gets f ij k = 1 16 W ( x j +1 − x j )[ δ i,j +1 + δ i, 1 δ j,N ][ aδ i,k λ j + (2 − a ) δ j,k λ i ] − 1 16 W ( x i +1 − x i )[ δ i +1 ,j + δ i,N δ j, 1 ][(2 − a ) δ i,k λ j + aδ j,k λ i ] , (11) where a is an arbitrary real constant. The second equation in (10) yields a (2 − a ) = 0 ⇒ a = 0 or a = 2 , (12) while the third equation in (10) turns out to b e satisfied identically . Note that, in con trast to the N = 2 sup ersymmetric Ruijsenaars-Sc hneider systems studied in [6], the t w o options in (12) do not seem to b e linked to one another b y relab eling the (sup er)particles. 3 Giv en the explicit form of f ij k in (11), (12), one can finally v erify that the six–fermion term en tering the Hamiltonian (8) is equal to zero. The model thus exhibits the p rop erties of the con v entional N = 2 sup ersymmetric man y–b o dy mec hanics in whic h the supersymmetry c harges are at most cubic in the o dd v ariables, while the Hamiltonian is at most quartic in the fermions. It is known that the relativistic T o da lattice admits more than one Hamiltonian for- m ulation (see, e.g., the discussion in [12]). Let us consider an alternative which is giv en b y ˜ H B = N X i =1 e − p i  1 + g 2 e x i − x i − 1  := N X i =1 ˜ λ i ˜ λ i , ˜ λ i = e − p i 2 p 1 + g 2 e x i − x i − 1 . (13) The structure functions ˜ λ i satisfy the algebra { ˜ λ i , ˜ λ j } = 1 4 ˜ λ i ˜ λ j ( W ( x i − x i − 1 )[ δ i − 1 ,j + δ i, 1 δ j,N ] − W ( x j − x j − 1 )[ δ i,j − 1 + δ i,N δ j, 1 ]) , (14) and, similarly to the analysis ab ov e, giv e rise to the phase space functions ˜ f ij k = 1 16 W ( x j − x j − 1 )[ δ i,j − 1 + δ i,N δ j, 1 ][ aδ i,k ˜ λ j + (2 − a ) δ j,k ˜ λ i ] − 1 16 W ( x i − x i − 1 )[ δ i − 1 ,j + δ i, 1 δ j,N ][(2 − a ) δ i,k ˜ λ j + aδ j,k ˜ λ i ] , (15) whic h solve the restrictions (10), pro vided a = 0 or a = 2 . (16) These add t w o more N = 2 models to the list ab o ve. At the moment, it is not clear whether the form ulations based up on ( λ i , f ij k ) and ( ˜ λ i , ˜ f ij k ) can b e connected with one another b y a co ordinate transformation. 3. N = 2 sup ersymmetric extensions of non–p erio dic relativistic T o da lattice The non–p erio dic relativistic T o da lattice is obtained b y imp osing the b oundary conditions x 0 = ∞ , x N +1 = −∞ , (17) whic h bring the equations (1) to the form ¨ x 1 = ˙ x 2 ˙ x 1 W ( x 2 − x 1 ) , ¨ x N = − ˙ x N ˙ x N − 1 W ( x N − x N − 1 ) , ¨ x k = ˙ x k +1 ˙ x k W ( x k +1 − x k ) − ˙ x k ˙ x k − 1 W ( x k − x k − 1 ) , (18) where k = 2 , . . . , N − 1 and, as b efore, W ( x − y ) = g 2 e x − y 1+ g 2 e x − y . 4 Lik e in the preceding case, the Hamiltonian repro ducing (18) is the sum of squares of the structure functions λ i H B = N X i =1 λ i λ i , λ N = e p N 2 p 1 + g 2 e − x N , λ k = e p k 2 p 1 + g 2 e x k +1 − x k , k = 1 , . . . , N − 1 , (19) whic h ob ey the non–linear algebra { λ i , λ j } = 1 4 λ i λ j [ W ( x i +1 − x i ) δ i +1 ,j − W ( x j +1 − x j ) δ i,j +1 ] , (20) with i, j = 1 , . . . , N . Note that the b oundary conditions imply the standard P oisson brack et { x i , p j } = δ i,j and the relations similar to (4) do not o ccur for the case at hand. The construction of N = 2 sup ersymmetric extensions pro ceeds as ab o ve. Given λ i in (19), it suffices to construct f ij k whic h solve the master equations (10). F rom the leftmost condition in (10) one finds f ij k = 1 16 W ( x j +1 − x j ) δ j +1 ,i [ aδ i,k λ j + (2 − a ) δ j,k λ i ] − 1 16 W ( x i +1 − x i ) δ i +1 ,j [(2 − a ) δ i,k λ j + aδ j,k λ i ] , (21) where a is an arbitrary real constant. The second equation in (10) reveals t w o options a = 0 or a = 2 . (22) The last constraint in (10) turns out to b e satisfied identically . Similarly to the p erio dic case, one can verify that the six–fermion term en tering the Hamiltonian (8) is zero for f ij k displa yed in (21). Concluding this section, let us discuss N = 2 sup ersymmetric extensions associated with the alternativ e Hamiltonian formulation based upon ˜ H B = N X i =1 ˜ λ i ˜ λ i , ˜ λ 1 = e − p 1 2 p 1 + g 2 e x 1 , ˜ λ k = e − p k 2 p 1 + g 2 e x k − x k − 1 , k = 2 , . . . , N . (23) In this case the structure functions satisfy { ˜ λ i , ˜ λ j } = 1 4 ˜ λ i ˜ λ j ( W ( x i − x i − 1 ) δ i − 1 ,j − W ( x j − x j − 1 ) δ i,j − 1 ) . (24) They giv e rise to ˜ f ij k = 1 16 W ( x j − x j − 1 ) δ i,j − 1 [ aδ i,k ˜ λ j + (2 − a ) δ j,k ˜ λ i ] − 1 16 W ( x i − x i − 1 ) δ i − 1 ,j [(2 − a ) δ i,k ˜ λ j + aδ j,k ˜ λ i ] , (25) 5 whic h prov e to b e consisten t with Eqs. (10), provided a = 0 or a = 2. Th us, like in the preceding case, one reveals four N = 2 mo dels generalizing the non–p erio dic relativistic T o da lattice. 4. Geo desic in terpretation As w as demonstrated in [5], the rational v arian t of the Ruijsenaars-Schneider mo del can b e identified with the geo desic equations in a flat space parametrized by sp ecial curvilin- ear co ordinates. The non–existence of a metric c onne ction asso ciated with the hyperb olic Ruijsenaars-Sc hneider systems w as prov en in [6]. In this section, w e carry out a similar analysis for the relativistic T o da lattice. W e start with the non–p erio dic case. Rewriting Eqs. (18) in the form ¨ x i + N X j,k =1 Γ i j k ˙ x j ˙ x k = 0 , (26) one obtains the connection co efficients Γ i j k = − 1 2 ( δ i,j δ k,i +1 + δ i,k δ j,i +1 ) W ( x i +1 − x i ) + 1 2 ( δ i,j δ k,i − 1 + δ i,k δ j,i − 1 ) W ( x i − x i − 1 ) . (27) Let us assume that they are deriv able from a non–degenerate metric g ij Γ i j k = 1 2 g ip ( ∂ j g pk + ∂ k g pj − ∂ p g j k ) , (28) where g ij designate the inv erse metric comp onents. Contracting the last formula with g si , p erm uting the indices ( j, s, k ) → ( s, k , j ), and taking the sum, one gets a coupled set of the partial differen tial equations to fix the metric ∂ j g sk = N X i =1 ( g si Γ i j k + g ki Γ i j s ) = − 1 2 g sj [ δ k,j +1 W ( x j +1 − x j ) − δ k,j − 1 W ( x j − x j − 1 )] − 1 2 g kj [ δ s,j +1 W ( x j +1 − x j ) − δ s,j − 1 W ( x j − x j − 1 )] − 1 2 g sk [ δ j,k +1 W ( x k +1 − x k ) − δ j,k − 1 W ( x k − x k − 1 )] − 1 2 g sk [ δ j,s +1 W ( x s +1 − x s ) − δ j,s − 1 W ( x s − x s − 1 )] . (29) Consider three equations b elonging to the set (29) ∂ 1 g 11 = 0 , ∂ 2 g 11 = − W ( x 2 − x 1 )( g 11 − g 12 ) , ∂ 1 g 12 = − 1 2 W ( x 2 − x 1 )( g 11 − g 12 ) . (30) 6 Computing the deriv ative of the second equation with respect to x 1 and taking in to account the other t wo, one gets  ∂ 1 W ( x 2 − x 1 ) + 1 2 W ( x 2 − x 1 ) 2  ( g 11 − g 12 ) = 0 ⇒ g 11 = g 12 . (31) This is b ecause the first factor en tering the leftmost equation is nonzero for the non–perio dic relativistic T o da lattice. By rep eatedly using the same argument for other comp onen ts of the metric tensor, one can demonstrate that they all are equal to one and the same constant, g ij = const, thus yielding a de gener ate metric. This contradicts to the earlier assumption that g ij is in vertible. The p erio dic relativistic T o da lattice is treated lik ewise. T aking into account the bound- ary conditions (2), the equations of motion (1) can b e put into the geo desic form in whic h the connection co efficien ts read Γ 1 j k = − 1 2 ( δ j, 1 δ k, 2 + δ k, 1 δ j, 2 ) W ( x 2 − x 1 ) + 1 2 ( δ j, 1 δ k,N + δ k, 1 δ j,N ) W ( x 1 − x N ) , (32) Γ i j k = − 1 2 ( δ i,j δ k,i +1 + δ i,k δ j,i +1 ) W ( x i +1 − x i ) + 1 2 ( δ i,j δ k,i − 1 + δ i,k δ j,i − 1 ) W ( x i − x i − 1 ) , Γ N j k = − 1 2 ( δ j, 1 δ k,N + δ k, 1 δ j,N ) W ( x 1 − x N ) + 1 2 ( δ j,N δ k,N − 1 + δ k,N δ j,N − 1 ) W ( x N − x N − 1 ) , with i = 2 , . . . , N − 1, and j, k = 1 , . . . , N . Considering triples of equations similar to (30), one can again verify that all comp onen ts of the metric are equal to one and the same constan t, which con tradicts to the assumption that the metric is inv ertible. W e th us conclude that, similarly to the hyperb olic Ruijsenaars-Sc hneider systems, the relativistic T o da mo dels are link ed to non–metric connections. 5. Conclusion T o summarize, in this work we ha ve constructed v arious N = 2 sup ersymmetric general- izations of the relativistic T o da lattice b oth for the p erio dic and non–p erio dic versions. In con trast to the h yp erb olic Ruijsenaars-Schneider mo dels, for which only the three–b o dy case has b een w orked out in full detail [6], the description abov e is v alid for an arbitrary n umber of particles. Both the sup ercharges and the Hamiltonian w ere sho wn to hav e the con ven tional p olynomial form in the fermionic degrees of freedom. A possible geo desic interpretation has b een discussed. It was demonstrated that, although the equations of motion of the relativis- tic T o da lattice can b e formally rewritten in the geo desic form, the resulting connection fails to b e a metric connection. T urning to p ossible further developmen ts, it w ould b e interesting to extend the present study to the N = 4 case and to reveal what would b e the analog of the Witten–Dijkgraaf– V erlinde–V erlinde equation. The construction of an off–shell sup erfield Lagrangian form ula- tion is an in teresting op en problem. A generic description of sup ersymmetric mechanics on spaces endo wed with a non–metric connection is a c hallenge. 7 One more imp ortant question to study concerns quantization of the N = 2 mo dels. F or the relativistic T o da lattice the conv en tional strategy is to analyse the sp ectral problem asso ciated with the full set of commuting quan tum integrals of motion and to attain the separation of v ariables (see [13] for the original consideration and [14] for a recen t alterna- tiv e treatment). Proceeding to N = 2 case, one first has to conv ert the brack ets (7) into those sp ecifying the fermionic creation/annihilation op erators and then prop erly modify the original b osonic quantum integrals of motion in such a wa y that they commute with the N = 2 Hamiltonian in (8). This p oint seems mostly tec hnical. A more sev ere problem is to pro ve that the separation of v ariables is still feasible. W e leav e these issues for further study . Ac kno wledgements This w ork was supported by the Russian Science F oundation, grant No 19-11-00005. References [1] S. Ruijsenaars, H. Schneider, A new class of inte gr able systems and its r elation to solitons , Annals Ph ys. 170 (1986) 370. [2] S. Ruijsenaars, R elativistic T o da systems , Commun. Math. Ph ys. 133 (1990) 217. [3] A. Gala jinsky , Higher r ank Kil ling tensors and Calo ger o mo del , Phys. Rev. D 85 (2012) 085002, [4] M. Cariglia, G.W. Gibb ons, Gener alise d Eisenhart lift of the T o da chain , J. Math. 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