L. Fehér, KFKI RMKI Budapest and University of Szeged Spin Calogero models and dynamical r-matrices

L. Fehér, KFKI RMKI Budapest and University of Szeged Spin Calogero models and dynamical r-matrices Integrable systems of Calogero (Moser, Sutherland, Olshanetsky- Perelomov, Gibbons-Hermsen, Ruijsenaars-Schneider) type are related to many important areas of physics and mathematics. The most general approaches to (classical) integrability use Lax pairs and the projection method based on free systems. Liouville integrability is linked in general (Babelon-Viallet, 90) with an r-matrix in the Poisson brackets of the Lax matrix. The r-matrices of A n -type Calogero models (Avan-Talon 93) are coordinate dependent. They were re-derived in an inspiring way by Avan-Babelon-Billey (94-96), who also related them to the classical dynamical Yang-Baxter equation coming from CFT (Gervais-Neveu 84, Balog-Dabrowski-Fehér 90, Felder 94). 1

Theory of CDYBE was advanced by Etingof-Varchenko (98). Then Li and Xu (2002) developed a method, in a framework relying on Lie algebroids and Lie groupoids, to associate a spin Calogero type model to any dynamical r-matrix in the Etingof- Varchenko sense. This was further elaborated by Li (2003-2005). We wish to contribute to this dynamical chapter of the Yang- Baxter story on integrability by simplifying the Li-Xu method. We also also wish to understand the models corresponding to interesting dynamical r-matrices in terms of the projection ideas. Talk is based mainly on math-ph/0507062 with B.G. Pusztai. 2

Dynamical r-matrix in Etingof-Varchenko sense K G: subalgebra of Lie algebra, Ǩ K open subset, T i : basis of K, [T i, T j ] = f ij k T k, q i q(t i ): components of q K A dynamical r-matrix on K G is a map r : Ǩ G G satisfying [r 12, r 13 ] + T1 i r 23 q i + cycl. perm. = 0, CDYBE [T1 i + T 2 i ji, r(q)] = fk qk r(q), equivariance condition qj where r s = (r + r 21 )/2 is a constant G-invariant. (r 21 = Y a X a for r = X a Y a G G, T i 1 = T i 1 1, r 23 = 1 r) quasi-triangular case: r s = 1 2 T a T a, where G is self-dual with invariant scalar product,, T a, T b = δa b for dual bases of G. triangular case: r s = 0. May also introduce spectral parameter. 3

From dynamical r-matrices to integrable systems Consider phase space M := T Ǩ G Ǩ K G = {(q, p, ξ)} and quasi-lax operator L L : M G, L(q, p, ξ) = p R(q)ξ (1) with R(q) End(G, G) corresponding to r(q) G G. Define χ : M K, χ(q, p, ξ) := (ad K p )T (q) + ξ K and then set ( χ r 12 )(q, p, ξ) := dt d r 12(q + tχ(q, p, ξ)) t=0. Proposition 1. For any dynamical r-matrix in the Etingof- Varchenko sense, one has {L 1, L 2 } = [r 12, L 1 + L 2 ] χ r 12 (2) /Conversely (1)-(2) for an equivariant r also imply the CDYBE./ Basic idea: The G-invariant functions of L yield a Poisson commuting family after introducing the first class constraint χ = 0. 4

Remarks on the fundamental proposition and its history The Poisson brackets on M = T Ǩ G are {q i, p j } = δj i and {ξ a, ξ b } = fc abξc with structure constants fc ab of G, in a basis T a of G extending the basis T i of K. The constraints χ i = 0 are first class, since {χ i, χ j } = f ij k χk. In fact, χ is the momentum map generating the natural action of the group K with Lie algebra K on M. We perform Hamiltonian reduction by setting χ = 0. Thus we are interested only in the gauge invariant functions, i.e, in the reduced phase space. The (spectral parameter dependent variant of the) basic formula (2) first appeared in works by Avan-Babelon-Billey (1994) for examples of quasi-lax operators defined without referring to (1). Proposition 1 was derived by Li-Xu (2002) and Li (2004) in a rather abstract framework based on Lie algebroids. 5

Systems built on r-matrices for an Abelian, self-dual K G Suppose G admits an invariant scalar product, that remains non-degenerate on K; yielding identifications G G, K K. Take an r-matrix, which is compatible with the decomposition G = K + K and has vanishing antisymmetric part on K. (The compatibility follows from equivariance if K is maximal Abelian, the vanishing assumption does not lead to loss of generality.) Consider Hamiltonian H(q, p, ξ) = 2 1 L, L. Now χ(q, p, ξ) = ξ K and ξ K = {ξ K, H} = 0. After imposing the constraint ξ K = 0, the evolution equation generated by H implies q = p and the Lax equation L = [RL, L] (3) These are equivalent to the evolution equation if R(q) is invertible on K, since then ṗ and ξ can be recovered from them. 6

Using only the compatibility condition, after imposing ξ K = 0 one obtains H(q, p, ξ) = 1 2 L, L = 1 2 p, p + 1 2 R(q)ξ, R(q)ξ (4) This is a Hamiltonian of spin Calogero type since R(q) is a rational or trigonometric function of q (in all known examples). For any K-valued function κ, gauge transformations operate as (q, p, ξ, L) (q, p, e κ ξ e κ, e κ Le κ ) The spin Calogero Lax equation (3) is gauge equivalent to L = [R(q)L κ, L] for any function κ on the constrained phase space. For H(q, p, ξ) = h(l) with any G-invariant function h, one gets L = {L, H} = [RV, L] ( ξk R)V with V (q, p, ξ) = ( h)(l(q, p, ξ)). 7

We are interested in r-matrices R : Ǩ End(G) whose antisymmetric part vanishes on K and R(q) K is invertible q Ǩ. First, triangular r-matrices of the form R(q) K = ( ad q K ) 1. The inverse exists generically if K is a Cartan in a simple G. Second, the quasi-triangular r-matrices satisfying our assumptions are precisely the maps of the form R θ (q) K = (1 θ 1 e adq K ) 1 where θ is an automorphism of G that preserves also the scalar product, K lies in the fixpoint set of θ, and the inverse that occurs is well-defined for a non-empty open subset Ǩ K. These r-matrices are due to Alekseev and Meinrenken (2003), their uniqueness under the above conditions is a new result. 8

Remarks on degenerate r-matrices If R(q) is an r-matrix that is non-degenerate in our sense, then R v (q) := R(q + v) is another r-matrix of the same type for any constant v. In certain cases v can be taken to infinity in some directions, which yields all degenerate r-matrices known to us. It seems possible to reduce the study of the degenerate spin Calogero models to the non-degenerate ones, but the former are less interesting also since in such examples some components of q do not enter the Hamiltonian and decouple as a free subsystem. For these reasons we focus on the non-degenerate r-matrices. 9

Interpretation as projection of geodesics (quasi-triangular case) Take Lie group G with Lie algebra G and lift θ to automorphism Θ of G. Restrict to open submanifold Ǧ G of regular elements g that admit generalized polar decomposition : g = Θ 1 (ρ)e q ρ 1 with ρ G, q Ǩ. In fact, this parametrization is applicable in a neighbourhood of e q 0 G if and only if R θ (q) is regular at q 0 K. For any such q 0 K, g in a neighbourhood of e q 0 can be uniquely written as g = Θ 1 (e η )e q e η where q varies in K around q 0 and η varies in K around zero. The problem of decomposing g Ǧ enters our solution algorithm. 10

A geodesic in G is a curve g(t) subject to d ( g(t) 1ġ(t) ) = 0, (5) dt whose solutions are g(t) = g 0 e tx with constants g 0 G, X G. Proposition 2. Consider a curve of the product form g(t) = Θ 1 (ρ(t))e q(t) ρ(t) 1 with smooth functions q(t) Ǩ and ρ(t) G. Define the spin variable ξ (t) by ξ (t) := R θ (q(t)) 2 M (t) K, M = M K + M := ρ 1 ρ (6) Then the geodesic equation implies the same time development for the gauge invariant functions of q, q and ξ as does the spin Calogero Lax equation L = [R θ (q)l, L] together with p = q. This result gives rise to a simple solution algorithm. 11

Proof of Proposition 2. One gets directly from the definitions g 1 ġ = ρ( q R θ (q)ξ )ρ 1 = ρlρ 1 where L(q, q, ξ ) = q R θ (q)ξ is just the spin Calogero Lax operator with q = p, ξ K = 0. Then 0 = d dt (g(t) 1 ġ(t)) = ρ( L [L, M])ρ 1 L = [L, M]. This in turn can be spelled out as L = [L, R θ (q) 2 ξ + M K ], which is gauge equivalent to the spin Calogero Lax equation L = [R θ (q)l, L] = [L, R θ (q) 2 ξ 1 2 q]. Proof shows geometric origin of L and gives solution algorithm. 12

Solution algorithm (quasi-triangular case) First, take an initial value (q 0, p 0 = q 0, (ξ ) 0 ) and define L 0 out of these data. Second, set ρ 0 = e G and construct geodesic g(t) = e q 0e tl 0. Third, decompose this geodesic as g(t) = Θ 1 (ρ(t))e q(t) ρ 1 (t) with ρ(t) = e η(t), η(t) K Finally, determine ξ (t) and L(t) using this decomposition. This procedure yields the solution of spin Calogero equation of motion (constrained by ξ K = 0) up to a time dependent gauge transformation, which is sufficient for gauge invariant objects. Procedure must be equivalent to factorization algorithm of Li (2004) for overlap cases, but much simpler. Consistent with Reshetikhin s (2003) derivation of the principal trigonometric spin Calogero models (θ = id) by Hamiltonian reduction of T Ǧ. 13

Non-degenerate triangular r-matrices and geodesics on G Now consider curve X(t) G of the product form This gives rise to X(t) = ρ(t)q(t)ρ(t) 1, ρ(t) G, q(t) Ǩ. Ẋ = ρlρ 1, where R(q) K = (ad q K ) 1 and L = q R(q)ξ with ξ = R(q) 2 M, M = M K +M = ρ 1 ρ Then d2 X(t) dt 2 = 0 translates into the geodesic Lax equation L = [L, M] = [R(q)L M K, L], which is gauge equivalent to the spin Calogero Lax equation. This gives solution algorithm. Consistent with Nekrasov s (1999) treatment of principal rational spin Calogero models, based on Cartan K of compact simple G, by Hamiltonian reduction of T G. 14

Hamiltonian reduction of T G by twisted conjugations Geodesic motion corresponds to Hamiltonian system (T G, Ω, H). Twisted conjugations, Ad Θ k (g) := Θ 1 (k)gk 1, define symmetry. Can reduce (T G, Ω, H) by this Θ-twisted adjoint action of G. If O is a coadjoint orbit through µ G, and ψ, ψ are moment maps, then one has the Marsden-Weinstein reduced phase spaces (T G) ψ=µ /G µ ( T G O ) ψ=0 that give the symplectic leaves of T G/G. /G (7) Main result: Let Ǧ G contain the Θ-twisted regular conjugacy classes parametrized by eǩ (where R θ is well-defined on Ǩ K). Then ( T Ǧ O ) ψ=0 /G is isomorphic to the spin Calogero phase space defined by imposing ξ K = 0 on T Ǩ O M := T Ǩ G. The kinetic energy H on T Ǧ yields the Calogero Hamiltonian. 15

Remarks: 1. For a compact, connected Lie group G, K is the Cartan subalgebra of the Lie algebra of the fixpont set G Θ. The regular conjugacy classes are represented by a Weyl alcove (fundamental domain of an affine Weyl group) in K. 2. For general reductive G, non-conjugate Cartan subalgebras and Cartan subgroups in G Θ exist. The reduction of the regular part of T G leads to disconnected pieces corresponding to spin Calogero models based on non-conjugate Cartan subalgebras. 3. It would be interesting to see if the (singular) reduction of the entire T G is related to questions about spin Calogero models. 4. Only elements of Aut(G)/Int(G) may correspond to different integrable systems. If Θ := Θ I h with inner automorphism I h : G g hgh 1 for a fixed h G, then Ad Θ k = L h 1 Ad Θ k L h with L h : g hg. Hence Θ, Θ give isomorphic reduced systems. 16

Proof of main result: Trivializing T G by right translations and using G G, we obtain T G G G = {(g, J) g G, J G}, Ω = d J, (dg)g 1, H = J, J /2. The G-action on T G O Ãd Θ k : T G O (g, J, ξ) (Θ 1 (k)gk 1, Θ 1 (k)jθ 1 (k 1 ), kξk 1 ) is generated by the momentum map ψ(g, J, ξ) = θ(j) g 1 Jg +ξ. Bringing g Ǧ to its diagonal representative e q with q Ǩ, the constraint ψ = 0 is solved by ξ K = 0 and θ(j) = J K R θ (q)ξ. This gives the gauge slice, S, of a partial gauge fixing S := {(e q, J K θ 1 R θ (q)ξ, ξ ) q Ǩ, J K K, ξ O K } for which ( T Ǧ O ) ψ=0 /G S/K Ǩ K (O K )/K. The statement follows since (Ω + ω O ) S = d J K, dq + ω O O K also results from symplectic form of T Ǩ O by imposing ξ K = 0 and setting p := J K. /Here ω O belongs to O and K = exp(k)./ 17

Models built on involutive diagram automorphisms Consider complex simple Lie algebra A with Cartan H, simple roots Π = {ϕ k }, positive roots Φ +, basis T ϕk := [X ϕk, X ϕk ] and X ±ϕ for ϕ Φ +. Taking G := A, θ := id and K := H, expanding ξ = ϕ Φ ξ ϕ X ϕ, one gets principal trigonometric spin Calogero Hamiltonian H(q, p, ξ) = 1 2 p, p 1 4 ϕ Φ + ξ ϕ ξ ϕ sinh 2 ϕ(q) 2 real forms induced, e.g., by compact/split real forms of (H, A). For A = A n, this was introduced by Gibbons-Hermsen (1984). Described also by Li-Xu (2002), Reshetikhin (2003), Li (2004). 18

Now choose θ as involutive automorphism induced from Dynkin diagram symmetry of A {A m, D m, E 6 }. Denote eigensubspaces by H ± and A ±. Recall that A + is simple with Cartan H + and A is irreducible module of A + with multiplicity 1 non-zero weights. = {α}: roots of (H +, A + ) with associated root vectors X α + Γ = {λ}: non-zero weights of (H +, A ) with weight vectors Xλ Taking K := H + and expanding ξ as ξ = ξh + ξ α + X α + + ξλ X λ α λ Γ we obtain the Hamiltonian H = 1 2 p, p 1 4 α + + ξα ξ α + sinh 2 α(q) 2 + 1 4 λ Γ + ξλ ξ λ cosh 2 λ(q) 2 + 1 8 ξ H, ξ H These (new) systems as well as their normal and compact real forms are presented in detail in our paper math-ph/0507062. 19

Roots and weights for involutive diagram automorphisms If A = D n+1, then A + = B n and A spans its defining irrep: + = {e k ± e l, e m 1 k < l n, 1 m n }, Γ + = {e m 1 m n }. One may take H + q = diag(q 1,..., q n, 0, 0, q n,..., q 1 ) and e m : q q m If A = A 2n 1, then A + = C n with Γ + = {e k ± e l 1 k < l n } and + = {e k ± e l, 2e m 1 k < l n, 1 m n }. Now H + q = diag(q 1,..., q n, q n,..., q 1 ) and e m : q q m For A = A 2n one has A + = B n and Γ + = {e k ± e l, e m, 2e m 1 k < l n, 1 m n }. 20

Remarks on further examples First, it is reasonable to consider Cartan involutions of simple real Lie algebras. New systems are expected if the Cartan involution is an outer automorphism of a non-split real form. This holds for su (2n), so(2p + 1, 2q + 1) (p q) and some real forms of E 6. Second, a rich set of examples result from direct sums G := A A A, for which θ: G G, θ(u 1, u 2,..., u N ) := (τu N, τu 1,..., τu N 1 ), defines an automorphism of G out of any automorphism τ of A. We studied the simplest case τ = id, reproducing (for A = A n ) certain generalized spin Calogero models introduced by Blom- Langmann (1998) and Polychronakos (1999) by different means. 21

Non-Abelian r-matrices give nothing new (in all reductive cases) Take, e.g., the extended versions of the Alekseev-Meinrenken r- matrices given on full fixpoint set, G 0 G, of θ as R θ ext = 1 2 +Rθ ext, where Rext θ : Ǧ 0 End(G) is defined, using f(z) = 1 2 coth 2 z 1 z, by Rext θ (q) := f(ad q ) on G ( ) ( 0 1 θe ad q G + 1 θe ad q 0 G 1) on G 0 0 1 2 Now we start with a quasi-lax operator on T Ǧ 0 G Ǧ 0 G 0 G and impose the first class constraints χ(q, p, ξ) = [q, p] + ξ G0 = 0. By partial gauge fixing (conjugating q into Cartan) we arrive at T Ǩ G constrained by ξ K = 0. Hence, eventually we find the same reduced systems as from the Abelian starting point. 22