Topological Monodromy of an Integrable Heisenberg Spin Chain

We investigate topological properties of a completely integrable system on $S^2\times S^2 \times S^2$ which was recently shown to have a Lagrangian fiber diffeomorphic to $\mathbb{R} P^3$ not displaceable by a Hamiltonian isotopy [Oakley J., Ph.D. Thesis, University of Georgia, 2014]. This system can be viewed as integrating the determinant, or alternatively, as integrating a classical Heisenberg spin chain. We show that the system has non-trivial topological monodromy and relate this to the geometric interpretation of its integrals.


Introduction
Given a triple of vectors (X, Y, Z) ∈ S 2 × S 2 × S 2 , the Hamiltonian H(X, Y, Z) = X, Y + Y, Z + Z, X models the pairwise interaction of three identical spin vectors fixed to the vertices of an equilateral triangle. Systems of this type -called Heisenberg spin chains -are of interest to physicists as they provide a classical model for quantum spin in a fixed lattice.
A second natural Hamiltonian on S 2 × S 2 × S 2 is the determinant, J(X, Y, Z) = det(X, Y, Z).
Together with a third commuting integral I, these Hamiltonians define a completely integrable system (H, J, I) on S 2 × S 2 × S 2 . Non-displaceability of Lagrangian fibers of the resulting integrable system was recently studied by [15]. In the past few years, similar systems on S 2 × S 2 have provided interesting examples of systems with non-displaceable torus fibers [7, 18,  14]. These systems, which degenerate to toric orbifolds, were shown to contain a continuum of non-displaceable Lagrangian torus fibers which limit to a Lagrangian sphere [7].
In this note we study the topological properties of our integrable system. In particular, we show that there is a line segment of non-degenerate critical values through the interior of the moment map image which we will call the 'critical line' (see Figure  1). Using the results of [19,10], we conclude that the monodromy of this system around the critical line is   1 0 0 0 1 0 0 3 1   , and thus the system does not admit global action-angle coordinates and cannot arise as a toric degeneration. In some sense, the number 3 in this matrix comes from the system's natural Z 3 -symmetry.
In Section 2, we begin by recalling some basic facts and fixing notation. In Section 3, we introduce our integrable system. In Section 4 we describe the moment map image and the set of critical values (or the system's 'bifurcation diagram'). Section 5 shows that the regular level sets are connected 3-tori and Section 6 computes the non-degeneracy of the critical line and deduces the system's topological monodromy.
The author would like to thank his advisor Yael Karshon for her guidance and support and Leonid Polterovich for suggesting the study of integrable spin chains. Special thanks also goes to Joel Oakley for discussing his recent results with the author, Anton Izosimov for explaining his results on non-degenerate singularities, and Peter Crooks for his editorial assistance. The author was supported by NSERC PGS-D and OGS scholarships during the preparation of this work.

Notation and Conventions
Let G be a compact, connected Lie group with semisimple Lie algebra g endowed with an Ad-invariant inner product , . The Kostant-Kirillov-Souriau symplectic structure on an adjoint orbit O Z of Z ∈ g is given by where X, Y ∈ g. Hamilton's equation for a function H : g → R can be written as where ∇H is the gradient vector field defined by the equation Hence, the Poisson bracket of two functions H, F can be conveniently written as A direct sum of semisimple Lie algebras g 1 ⊕ · · · ⊕ g N is endowed with direct sum Lie brackets and Killing forms. An adjoint orbit in g 1 ⊕ · · · ⊕ g N is a product of adjoint orbits O Z 1 × · · · × O Z N and the symplectic structure coincides with the direct sum of their respective symplectic structures, ω = ω 1 ⊕ · · · ⊕ ω N . For example, in so(3) × · · · × so(3) the orbits are products of spheres. Under the Ad-invariant identification of g with its dual, the moment map for the adjoint action of G on an orbit in g is inclusion of the orbit into g. The moment map for the diagonal adjoint action of G on O Z 1 × · · · × O Z N is hence the map (X 1 , . . . , X N ) → X i .

An Integrable Heisenberg Spin Chain
Consider a product of spheres with radius 1, M = S 2 × S 2 × S 2 , whose elements are triples (X, Y, Z) of unit length vectors in R 3 . Define the Hamiltonians 1 where [, ] is the standard cross-product on R 3 , , is the standard inner product, and {e 1 , e 2 , e 3 } is the standard oriented orthonormal basis with [e i , e j ] = ε ijk e k . By ad-invariance of the inner product, the Hamiltonian vector fields of these functions are: Recall that the flow ϕ t [v,X] of a vector field [v, X] acts by rotation of the vector X around the axis v with period 2π/|v|, which we denote as R v t , ϕ t [v,X] X = R v t X. Thus, the Hamiltonian flow of I acts by rotating each sphere around the e 3 -axis with period 2π, ϕ t X I (X, Y, Z) = (R e 3 t X, R e 3 t Y, R e 3 t Z). Where defined, the Hamiltonian flow of H rotates each sphere around the axis X + Y + Z with period 2π, This is perhaps best visualized as rotating the polygon with edges X, Y, Proof. It is a nice exercise to see that this is true based on the geometric description of the Hamiltonians and their flows given above. More algebraically, one can see this using the Lie algebra structure that is present. 1 Note that this H is related to the H in the introduction by a square root. We do this so that the flow of the Hamiltonian vector field is a S 1 -action where it is defined.

3.8.
Remark. The Hamiltonian flow of J is less straightforward to describe, but we can say something about how it acts on the submanifold There the vectors X, Y, and Z are coplanar and the vector field The flow of this vector field acts on L by rotation of each vector around the axis n with constant period.
In [15] it was shown that the fiber L is a non-displaceable Lagrangian submanifold of S 2 × S 2 × S 2 . To see that it is an embedded Lagrangian RP 3 , observe that it is the zero level set for the moment map of the diagonal SO(3)-action, (X, Y, Z) → X + Y + Z, and the diagonal action of SO(3) is free and transitive. It's interesting to observe that the moment map image for our system near this Lagrangian fiber resembles that of the geodesic flow on RP 3 .

3.9.
Remark. Note that one can think of H as a collective function obtained from a Casimir on so(3) via the moment map for the diagonal SO(3)-action, If you view this system as integrating the Hamiltonian H, then complete integrability is not surprising since H has a SO(3)-symmetry, coming from it's definition as a collective function for the SO(3) moment map. This is not typical of classical Heisenberg spin chains considered in the physics literature, where one often studies a 'chain with boundary conditions' of N spin vectors

and the Hamiltonian
where the indices in the sum are considered modulo N. This models N spin vectors placed at the vertices of a regular polygon, if we imagine that each vector X i only interacts with its nearest neighbours X i−1 and X i+1 . The coincidence for N = 3 is that a triangle is also a complete graph, and hence we can re-write H as a Casimir on so(3). Alternately, one might view this system as integrating J, which also has a natural SO(3)-symmetry since it is the determinant, and it might be more natural to generalize this system from the perspective of forms on the Lie algebra so(3) × · · · × so(3), using the algebraic identities at hand as we demonstrated in Proposition 3.7.
3.10. Remark. It is important to note that all three Hamiltonians have a Z 3 -symmetry coming from cyclic permutations (X, Y, Z) → (Z, X, Y ). As we will see, this symmetry provides an explanation for why the system's monodromy matrix contains the number 3.

Image of the Moment Map
The image of the moment map H = (H, J, I) is a solid in R 3 that is symmetric about the (H, I)-plane and about the (H, J)-plane (see Figure 1 and 2). It is obvious that |I| ≤ H, with equality when X + Y + Z ∈ span(e 3 ), and that H ≤ 3 with equality when X = Y = Z.
Observe that where a = X, Y , b = Y, Z and c = Z, X (the second formula is the volume of a parallelpiped). If we maximize |J| with the constraint H = const, then we must have a = b = c (the interior angles between the three vectors are the same). Using this we can deduce that the image of the moment map is bounded by the inequalities and equality is achieved in the first inequality when a = b = c.
To see that the image is the entire region described by these inequalities, we can find a tuple (X, Y, Z) ∈ S 2 ×S 2 ×S 2 for a given value of H and J, then simultaneously rotate the vectors X, Y, and Z in the (e 1 , e 2 )-plane to get the desired value of I.

Lemma. For any
Proof. Equivalently, the image of the map f : Next, we turn our attention to the critical set for the system. The critical set consists of several subsets: (1) The sets where H is critical: This set maps to the other two opposite faces of the moment map image. Points in C 1 ∩ C 2 map to edges of the moment map image.

4.5.
Proposition. The critical set for H is The set H(C 1 ∪C 2 ) is the boundary of the image H(M) and the set H( Figure 1).
In the terminology of integrable systems, the set of critical values is the system's 'bifurcation diagram.' This complete description of the bifurcation diagram will be of use to us in Section 6. Note that the three critical spheres S 1 , S 2 , S 3 are permuted by the system's Z 3 -symmetry.
Proof. Throughout we use the fact that df = 0 if and only if X f = 0 (when f is smooth). ( which is true if and only if X + Y + Z ∈ span(e 3 ). Since we now know that our Hamiltonians are independent on an open dense subset of M, we can conclude: 4.10. Corollary. H is a completely integrable system. In particular, the regular level sets of H are homeomorphic to a disjoint union of finitely many 3-tori. 4.11. Corollary. The set of regular values is homotopy-equivalent to S 1 .
In the next section, we will see that the regular level sets are connected and in Section 6 we will describe the structure of the Lagrangian foliation near the critical line. Proof. The proof will have two parts. For H = 1 we can make a general argument. For H = 1 we will provide a direct proof that the fibers of the original system are connected.

Connectedness of Regular Level Sets
Pick a regular value (r, s, t) of H = (H, J, I) with r = 1. Since H generates a Hamiltonian S 1 -action on M \ H −1 (0), it is a Morse-Bott function such that all critical sets have even index [1]. Hence, the regular level sets of H are connected. Since the level sets of H are compact and connected, the symplectic reductions M r ≡ H −1 (r)/S 1 are all compact, connected symplectic manifolds. Since t is a regular value of the reduced HamiltonianĨ, andĨ generates a free S 1 -action, we can reduce once more to obtain the compact and connected manifold M r,t =Ĩ −1 (t)/S 1 . The image of the twice-reduced HamiltonianJ on M r,t is a line segment with the only critical values being the maximum and minimum. A quick computation shows thatJ has exactly two critical points, hence the regular fibersJ −1 (s) are all connected. This implies that the critical fiber H −1 (r, s, t) ⊂ M is connected. Now consider regular values of the form (1, s, t) with s = 0. Since 1 is a critical value of H, the previous argument will not work. Since J does not generate an S 1action on M, we cannot argue similarly using reduction by J to avoid the critical values of H. Instead, we consider the fibres in the reduced system on M t = I −1 (t)/S 1 directly.
First, note that the set is a transverse slice for the S 1 -action of I away from the set H −1 (0), so the reduced manifold M t \H −1 (0) can be identified homeomorphically with S t . Therefore the level sets (H,J) −1 (1, s) of the reduced system are By (3.4), the Hamiltonian flow of the reduced HamiltonianH acts on these level sets by diagonal rotation around the axis √ 1 − t 2 e 1 + te 3 and this action is free. To see that the level set (H,J) −1 (1, s) is connected, we consider the quotient by the S 1 -action generated byH, and show it is homeomorphic to a circle.
For s = 0, a transverse slice for the action generated byH on (H,J) −1 (1, s) can be given by fixing X in the upper half of the (e 1 , e 3 )-plane: , 0, sin(θ)), 0 ≤ θ ≤ π} We now show that Y t is homeomorphic to S 1 for all −1 < t < 1. Let X = (cos(θ), 0, sin(θ)), Y = (y 1 , y 2 , y 3 ) and Z = (z 1 , z 2 , z 3 ). Our set Y t consists of all solutions to the system of equations For a given θ, the system is equivalent to the intersection of two circles in the (y 1 , y 3 )plane for which there are 0, 1, or 2 solutions. As θ varies the circles sweep out two 'bulging' cylinders that intersect in a curve diffeomorphic to S 1 . Since this fiber in the twicereduced system is connected, the original fiber in M is also connected. 5.6. Remark. The image of the invariant Lagrangian L = H −1 (0) in the reduction at 0 by I is a Lagrangian S 2 . 5.7. Remark. In [16] it is shown that if (M 4 , ω, F = (f 1 , f 2 )) is a completely integrable system with two degrees of freedom that has only non-degenerate critical points, and whose bifurcation diagram 2 has no vertical tangent lines, then the system has connected fibers. After rotating the moment map image, (and checking that the boundary of H(M) consists of non-degenerate elliptic critical values except (0, 0, 0)) we can apply this result to deduce connectedness of almost all the fibers of H. Since the reduced system (H,J) has a Lagrangian S 2 fiber, we cannot use this theorem to deduce connectedness of the regular fibers H −1 (1, s, 0), s = 0.

Topological Monodromy Around the Critical Line
It was shown in [6] that the singular foliation of an integrable system by Lagrangian submanifolds is determined in a neighbourhood of a nondegenerate critical point by its 'type' -the conjugacy class of the corresponding Cartan subalgebra in sp(L ⊥ /L) (see Appendix). This was later strengthened in [20] to show that, under a stability condition, the topological foliation is almost completely determined in a neighbourhood of a non-degenerate singular leaf S by its 'type' -the type of any critical point in S of lowest rank. In the case of singularities whose critical type is purely focus-focus, or mixed focus-focus and elliptic, the foliation is completely determined by this data: 6.1. Theorem. [10] Let S be a non-degenerate singular leaf of an integrable system F , which is non-splitting 3 . If the type of S does not contain any hyperbolic components, then in a neighbourhood of S the Lagrangian foliation of F is topologically equivalent to a direct product of elliptic singularities, focus-focus singularities, and the trivial foliation D r × T r .
As was shown in [10], this result also determines the topological monodromy of a given system in a neighbourhood of a non-degenerate singularity. By the computations in the appendix below, the singular fibers H −1 (1, 0, a) over the critical line are rank 1 focus-focus singularities. By Proposition 4.5, each singular fiber contains three critical circles, which are permuted by the system's Z 3 -symmetry. These circles are the intersection of the critical fiber with the three critical spheres S 1 , S 2 , and S 3 . The system's Z 3 -symmetry shows that these singularities are non-splitting (see also Proposition 4.5). Hence, in a neighbourhood of a singular fiber over the critical line, the system is topologically equivalent to the product of the trivial foliation D 1 × S 1 and a focus-focus singularity with three critical points. By Theorem 6.1, the topological monodromy of the system is  6.2. Remark. In the fiberH −1 (1, 0) of the reduced system at I = 0, the three cylinders joining the critical points are the sets {(X, e 1 , −X) ∈ S| X ∈ S 2 \ ±e 1 } , {(X, −X, e 1 ) ∈ S| X ∈ S 2 \ ±e 1 } and {(e 1 , X, −X) ∈ S | Y ∈ S 2 \ {±e 1 }} and the critical points are (e 1 , e 1 , −e 1 ), (e 1 , −e 1 , e 1 ), (−e 1 , e 1 , e 1 ). Hence the critical fiber is explicitly homeomorphic to a torus with three longitudinal pinches. The fiber H −1 (1, 0, 0) is then explicitly homeomorphic toH −1 (1, 0) × S 1 .
6.3. Remark. The fact that this system has non-trivial monodromy should be unsurprising for the following reason: the topology of the H-level sets changes as you pass through the critical value 1. This can be seen directly with Morse theory, but there is also a natural interpretation in terms of the topology of polygon spaces (as introduced by [13]). There is a natural diffeomorphism of the level set H −1 (r) with the manifold M(1, 1, 1, r) of closed 4-gons in R 3 with side lengths 1, 1, 1, and r. When r = 1, it has been observed by Knutson, Hausman [8], and Kapovitch and Millson [13] that the quotient M(1, 1, 1, r)/SO(3) is homeomorphic to S 2 , and that the quotient map π : M(1, 1, 1, r) → S 2 is a principal SO(3)-bundle. Further, the characteristic classes of this principal SO(3)-bundle were described by Knutson and Hausman in their paper [9]. Their result says that for 0 < r < 1 the bundle is trivial, whereas for 1 < r < 3 the bundle is non-trivial.
It was observed in [3] that such a change in the topology of the level set H −1 (r) as r passes through an interior critical value indicates that there must be non-trivial H J 1 −1 Figure 3. Reduced system on M t for −1 < t < 1, t = 0. monodromy around the associated critical fibres, since this forces the pullback of the torus bundle to any circle around the critical line to be non-trivial.
6.4. Remark. For −1 < t < 1, the reduced system on M t = I −1 (t)/S 1 has a focus-focus singularity with three critical points, or after a perturbation, three simple focus-focus singularities. The manifold M t is the blow-up Bl 3 CP 2 , and the moment map image has 3 vertices (see Figure 2). A quick comparison with the list of almost toric systems in [11] shows that this checks out.

Appendix: Non-Degeneracy Computation
In this appendix, we show that the critical line is rank 1 non-degenerate focusfocus. One may refer to [4,5] for introductory details on the topological monodromy of completely integrable systems, and [20,2] for more details on non-degenerate singularities.
Let p be a critical point of rank k of an integrable system (H 1 , . . . , H n ). Without loss of generality, assume that dH 1 , . . . , dH n−k = 0 and the remaining functions H n−k+1 , . . . , H n are independent at p. The operators ω −1 d 2 H 1 , . . . , ω −1 d 2 H n−k form a commutative subalgebra of sp(L ⊥ /L) where the subspace L ⊂ T p M is the span of the vector fields X H n−k+1 , . . . , X Hn and L ⊥ is its symplectic orthocomplement. The point p is non-degenerate if this is a Cartan subalgebra. Equivalently, p is non-degenerate if ω −1 d 2 H 1 , . . . , ω −1 d 2 H n−k are linearly independent and some linear combination of these operators has 2(n−k) distinct eigenvalues. It was shown in [17] that in sp(R, 4) there are four conjugacy classes of Cartan subalgebras corresponding to four possible combinations of eigenvalues for a generic element: (1) elliptic-elliptic: ±iA, ±iB, (2) elliptic-hyperbolic: ±A, ±iB, (3) hyperbolic-hyperbolic: ±A, ±B, and (4) focus-focus: A ± iB, −A ± iB.
Note that considering the induced operators on the vector space L ⊥ /L is equivalent to considering the same question for the reduced system (H 1 , . . . ,H n−k ) on the reduction of M by H n−k+1 , . . . , H n (provided of course, that one can perform this reduction). In Darboux coordinates the operator A H = ω −1 d 2 H p is equal to the linearization of the Hamiltonian vector field X H at p, since We now turn to our system on S 2 × S 2 × S 2 . Consider the cylindrical coordinates (θ, z) ∈ (−π/2, 3π/2) × (−1, 1) with symplectic form dθ i ∧ dz i . The map φ : (−π/2, 3π/2) × (−1, 1) → S 2 given by is a symplectomorphism. Put a Darboux chart on S 2 × S 2 × S 2 with the map Φ = φ × φ × φ, coordinates (θ 1 , z 1 , θ 2 , z 2 , θ 3 , z 3 ), and symplectic form dθ i ∧ dz i . Pulling back our system by Φ we obtain Differentiating, Hamilton's equations tell us that The linearization of X f at a fixed point of the flow of f is then We compute The critical line consists of points p = (1, 0, a), −1 < a < 1 (see Figure 1). As we have seen, the critical points in each fiber H −1 (1, 0, a) are three embedded circles (these are just the intersections of our fiber with the critical spheres S 1 , S 2 , S 3 ). To compute degeneracy of these critical points it is sufficient to compute the degeneracy of a single critical point in a single circle (by Z 3 -symmetry). Let's consider the points p = (0, a, 0, a, π, −a) in cylindrical coordinates. The linearization of X J at p is The linearization of X H at p is where b 2 = 1 − a 2 . These operators are independent and a quick computation shows that for any −1 < a < 1 the operator on L ⊥ /L induced by A J + A H has four distinct complex eigenvalues of the form A ± iB, −A ± iB. Hence the critical point is rank 1 non-degenerate focus-focus.