LIE ALGEBROIDS AS GAUGE SYMMETRIES IN TOPOLOGICAL FIELD THEORIES ^{a}

M.A. OLSHANETSKY

Max Planck Institute of Mathematics, Vivatsgasse 7, D-53111 Bonn, Germany

and

Institute of Theoretical and Experimental Physics, 117259, Moscow, Russia

To the memory of Mikhail Marinov

The Lie algebroids are generalization of the Lie algebras. They arise, in particular, as a mathematical tool in investigations of dynamical systems with the first class constraints. Here we consider canonical symmetries of Hamiltonian systems generated by a special class of Lie algebroids. The “coordinate part” of the Hamiltonian phase space is the Poisson manifold M and the Lie algebroid brackets are defined by means of the Poisson bivector. The Lie algebroid action defined on M can be lifted to the phase space. The main observation is that the classical BRST operator has the same form as in the case of the Lie groups action. Two examples are analyzed. In the first, M is the space of SL(3, C)-opers on Riemann curves with the Adler-Gelfand-Dikii brackets. The corresponding Hamiltonian system is the W_{3}-gravity. Its phase space is the base of the algebroid bundle. The sections of the bundle are the second order differential operators on Riemann curves. They are the gauge symmetries of the theory. The moduli space of W_{3} geometry of Riemann curves is the symplectic quotient with respect to their action. It is demonstrated that the nonlinear brackets and the second order differential operators arise from the canonical brackets and the standard gauge transformations in the Chern-Simons field theory, as a result of the partial gauge fixing. The second example is M = C^{4} endowed with the Sklyanin brackets. The symplectic reduction with respect to the algebroid action leads to a generalization of the rational Calogero-Moser model. As in the previous example the Sklyanin brackets can be derived from a “free theory.” In this case it is a “relativistic deformation” of the SL(2, C) Higgs bundle over an elliptic curve.

Contents

1 Introduction 3

2 Lie algebroids 5 2.1 Poisson manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 5

aContribution in the Marinov Memorial Volume

2 M. Olshanetsky

2.2 Lie algebroids and Lie groupoids . . . . . . . . . . . . . . . . . 6 2.3 Central extension of Lie algebroids . . . . . . . . . . . . . . . . 7

3 Poisson sigma-model 8 3.1 Poisson sigma-model and Hamiltonian algebroids . . . . . . . . 8 3.2 BRST construction for Hamiltonian algebroids . . . . . . . . . 10

4 W_{3}-gravity 11 4.1 Lie algebroid over SL(3, ℂ)-opers . . . . . . . . . . . . . . . . . 11 4.2 Phase space of W_{3}-gravity and BRST operator . . . . . . . . . 16 4.3 Chern-Simons description . . . . . . . . . . . . . . . . . . . . . 18

5 Sklyanin algebra 20 5.1 Hamiltonian algebroid for Sklyanin algebra . . . . . . . . . . . 20 5.2 Sklyanin algebra from canonical brackets . . . . . . . . . . . . . 21 5.3 Symplectic reduction in the Sklyanin algebra . . . . . . . . . . 25

Acknowledgments 27

Lie algebroids 3

References 27 1 Introduction

In this paper I analyze the special first class constraints in the classical Hamiltonian systems. This subject was one of the main interests of Misha Marinov. I remember very interesting discussions with him when he prepared with Misha Terentev their paper devoted to dynamics on group manifolds.^{1} They used the path integral for the quantization and the Lie group symmetries played essential role in their construction. The aim of this paper is going beyond the group symmetry, though we consider only the classical theory. I hope that Misha would have read to the present paper with interest.^{b}

In fact, the group symmetries by no means exhaust all interesting symmetries in classical and quantum Hamiltonian systems. Generic first class constraints generate transformations that generalize the Lie group action. For example, the structure constants can depend on the points of the phase space. A generalization of this situation leads to the notion of the Lie algebroid and Lie groupoid. Batalin is likely to be the first who explicitly used this construction in gauge theories.^{3} He called these transformations the quasigroups.

The BRST approach allows to work with arbitrary forms of constraints. In general situation the BRST operator has an infinite rank (the degree of the ghost momenta), while in the Lie group case it has rank one or less.^{4} Nevertheless, it is possible to modify the Lie group symmetries in a such way that the BRST operator has the same form as for the Lie groups. We consider one of these cases.

To construct these symmetries we carry out the following steps:

1. Consider a Poisson manifold M. The transformations of M depend on the Poisson structure. The local version of them is a particular type of Lie algebroids A_{M}. The space M plays the role of the “coordinate subspace” and the algebroid action is the transformations of the coordinates.

2. To come to the field theory we introduce the space of maps M = D → M, D = {|z| ≤ 1}. The algebroid A_{M} is the analogue of the loop algebra. It can be central extended Â_{M} by a one-cocycle as in the loop algebra case.

3. The transformations coming from the Lie algebroid Â_{M} can be lifted to the phase space R. In addition to the coordinate space M the phase space R contains the conjugate variables. It is endowed with the canonical symplectic structure. The space R along with its symmetries defines the so-called Poisson sigma-model.^{5,6}

4. The infinitesimal form of the symmetry transformations we call the Hamiltonian algebroid A_{R}^{H} related to the Lie algebroid A_{M}. The Hamiltonian algebroid is a generalization of the Lie algebra of symplectic vector fields with respect to the symplectic structure on R. The special feature of these systems is that the BRST operator has the same structure as for the transformations performed by Lie algebras.

b Sections 2-4 of this paper is an updated version of part of Ref. 2.

We present two examples of topological field theories with these symmetries. The first example is the W_{3}-gravity^{7,8,9} and related to this theory generalized deformations of complex structures of Riemann curves by the second order differential operators. This theory is a generalization of 2 + 1-gravity (W_{2}-gravity),^{10} where the space component has a topology of a Riemann curve Σ_{g} of genus g. The Lie algebra symmetries in W_{2}-gravity is the algebra of smooth vector fields on Σ_{g}. After killing the gauge degrees of freedom one comes to the moduli space of projective structures on Σ_{g}. These structures can be described by the BRST method which is straightforward in this case. The case of W_{N}-gravity (N > 2) is more subtle. The main reason is that the gauge symmetries do not generate the Lie group action. This property of W_{N}-gravity was well known.^{7}

We consider here in detail the W_{3} case. The infinitesimal symmetries are carried out by the second order differential operators on Σ_{g} without constant terms. The role of the Poisson manifold M is played by the M_{3} = SL(3, ℂ)- opers^{11,12} endowed with the Adler-Gelfand-Dikii bivector.^{13,14} The space M_{3} is the configuration space of W_{3}-gravity. We define in a canonical way the action of the second order differential operators on SL(3, ℂ)-opers. In this way we come to a Lie algebroid A_{M_{3}} over SL(3, ℂ)-opers. The space of sections Γ(A_{M_{3}}) is the space of second order differential operators on Σ_{g} without constant terms, endowed by a Lie bracket. The algebroid A_{M_{3}} is lifted to the Hamiltonian algebroid A_{R_{3}}^{H} over the phase space R_{3} of W_{3}-gravity. The symplectic quotient of the phase space is the so-called W_{3}-geometry of Σ_{g}. Roughly speaking, this space is a combination of the moduli of generalized complex structures and the spin 2 and 3 fields as the dual variables. Note that we deform the operator of complex structure ¯∂ by symmetric combinations of vector fields (ε∂)^{2}, in contrast with Ref. 15, where the deformations of complex structures are carried out by the polyvector fields. To define the W_{3}-geometry we construct the BRST operator for the Hamiltonian algebroid. As it follows from the general construction, it has the same structure as in the Lie algebra case. It should be noted that the BRST operator for the W_{3}-algebras was constructed in^{16}. But here we construct the BRST operator for the different object - the algebroid symmetries of W_{3}-gravity. We explain our formulae and the origin of the algebroid by the special gauge procedure of the SL(3, ℂ) Chern-Simons theory using an approach developed in Ref. 8.

The next example is the Poisson manifold ℂ^{4} with Sklyanin algebra structure.^{17} As in the previous example, we demonstrate that these algebroid structure can be derived from the canonical brackets. The role of the Chern-Simons theory is played by a free theory in 1+1 dimension. It is a ”relativistic” deformation of the Hitchin model for the elliptic Calogero-Moser system. The Sklyanin algebra arises on an intermediate step of the symplectic reduction. Our construction is closed to the scheme proposed in Ref. 18. In conclusion, we perform the further symplectic reduction with respect to the rest algebroid action and obtain an integrable model with two degrees of freedom following Ref. 19.

2 Lie algebroids

2.1 Poisson manifolds

Let M be a Poisson manifold endowed with the Poisson bivector π. In local coordinates x = (x_{1}, …, x_{n}) π = π^{jk} and

π^{jk}(x) = −π^{kj}(x),
∂_{i}π^{jk}(x)π^{im}(x) + c.p.(j, k, m) = 0.  (2.1)

The Poisson brackets are defined on the space of smooth functions H(M)

c

{f(x), g(x)} = π^{jk}∂_{j}f(x)∂_{k}g(x) := ⟨∂f|π|∂g⟩, (∂_{j} = ∂_{x^{j}}) .

The bivector allows to construct a map from f ∈ H(M) to the vector fields V_{f} ∈ Γ(T M)

f → V_{f} = π^{jk}(x)∂_{j}f(x)∂_{k} = ⟨∂f|π|∂⟩.  (2.2)

Let ε = (ε_{1}, …, ε_{n}) be a section of T^{∗}M. Consider the vector field

V_{ε} = ⟨ε|π|∂⟩.  (2.3)

It acts on H(M) in the standard way

δ_{ε}f = i_{V_{ε}}df = ⟨ε|π|df⟩.

In particular,

δ_{ε}x^{k} = π^{kj}(x)ε_{j}.  (2.4)

One can define the brackets on the sections of T^{∗}M

⌊ε, ε′⌋ = d⟨ε|π(x)|ε′⟩ + ⟨dε|π|ε′⟩ + ⟨ε|π|dε′⟩,  (2.5)

c The sums over repeated indices are understood in Sections 2-4. For simplicity we assume here that M is a finite dimensional manifold, so later we shall consider the infinite dimensional case.

or

⌊ε, ε′⌋_{m} = ⟨ε|∂_{m}π|ε′⟩ + δ_{ε}ε′ − δ_{ε′}ε.

It follows from the Jacobi identity (2.1), that the commutator of the vector fields V_{ε} satisfies the identity

[V_{ε}, V_{ε′}] = V_{⌊ε,ε′⌋}.  (2.6)

2.2 Lie algebroids and Lie groupoids

Now we can define the Lie algebroid over M. As we already mentioned Lie algebroids is a generalization of bundles of Lie algebras over a base M. We present the definition of the Lie algebroid in general case. Details of this theory can be find in Ref. 20, 21, 22. Definition 1 A Lie algebroid over a differential manifold M is a vector bundle A → M with a Lie algebra structure on the space of section Γ(A) defined by the brackets ⌊ε_{1}, ε_{2}⌋, ε_{1}, ε_{2} ∈ Γ(A) and a bundle map (the anchor) δ : A → T M, satisfying the following conditions:

(i) For any ε_{1}, ε_{2} ∈ Γ(A)

[δ_{ε_{1}}, δ_{ε_{2}}] = δ_{⌊ε_{1}ε_{2}⌋},  (2.7)

(ii) For any ε_{1}, ε_{2} ∈ Γ(A) and f ∈ C^{∞}(M)

⌊ε_{1}, fε_{2}⌋ = f⌊ε_{1}, ε_{2}⌋ + (δ_{ε_{1}}f)ε_{2}.  (2.8)

In other words, the anchor defines a representation of the space of the algebroid sections to the Lie algebra of vector fields on the base M. The second condition is the Leibniz rule with respect to the multiplication of the sections by smooth functions.

A Lie algebra is a particular case of Lie algebroids if M is a one-point space. If the anchor map δ_{ε} ≡ 0, then we have a bundle of Lie algebras over M.

Let {e^{j}(x)} be a basis of local sections Γ(A). Then the brackets are defined by the structure functions f_{i}^{jk}(x) of the algebroid

⌊e^{j}, e^{k}⌋ = f_{i}^{jk}(x)e^{i}, x ∈ V.  (2.9)

Using the Jacobi identity for the anchor action, we find

C_{j,k,m}^{n}δ_{e_{n}} = 0,  (2.10)

where

C_{n}^{j,k,m} = (f_{i}^{jk}(x)f_{n}^{im}(x) + δ_{e^{m}}f_{n}^{jk}(x) + c.p.(j, k, m))  (2.11)

Thus, (2.10) implies the anomalous Jacoby identity

f_{i}^{jk}(x)f_{n}^{im}(x) + δ_{e^{m}}f_{n}^{jk}(x) + c.p.(j, k, m) = 0  (2.12)

For the Lie algebras bundles the anomalous terms disappear.

In our case A = T^{∗}M with the brackets (2.5) defined on its sections and the structure functions

f_{i}^{jk}(x) = ∂_{i}π^{jk}(x).

The anchor is also has a special form δ_{ε} = V_{ε} (2.3) based on the Poisson bivector. The properties (2.7) and (2.8) follow from (2.6) and from (2.5). We denote the algebroid T^{∗}M as A_{M}. This type of Lie algebroids was introduced in Ref. 24, 23.

In generic case Lie algebroids A can be integrated to Lie groupoid.^{3,20,22} Again we present the general definition. Definition 2 A Lie groupoid G_{M} over a manifold M is a pair of differential manifolds (G, M), two differential mappings l, r : G_{M} → M and a partially defined binary operation (a product) (g, h) ↦ g · h satisfying the following conditions:

(i) It is defined only when l(g) = r(h). (ii) It is associative: (g · h) · k = g · (h · k) whenever the products are defined. (iii) For any g ∈ G_{M} there exist the left and right identity elements l_{g} and r_{g} such that l_{g} · g = g · r_{g} = g. (iv) Each g has an inverse g^{−1} such that g · g^{−1} = l_{g} and g^{−1} · g = r_{g}.

We denote an element of g ∈ G_{M} by the triple ≪ x|g|y ≫, where x = l(g), y = r(g). Then the product g · h is

g · h →≪ x|g · h|z ≫=≪ x|g|y ≫≪ y|h|z ≫ .

An orbit of the groupoid in the base M is defined as an equivalence x ∼ y if x = l(g), y = r(g). There is the isotropy subgroup G_{x} for x ∈ M.

G_{x} = {g ∈ G_{M} | l(g) = x = r(g)} ∼ {≪ x|g|x ≫}.

2.3 Central extension of Lie algebroids

There is a straightforward generalization of this construction to the infinite dimensional case. Consider a disk D = |z| ≤ 1 and the space of anti-meromorphic maps M = {X : D → M}. Define the Lie algebroid A_{M} over the space M. Now the base of the algebroid is infinite-dimensional. For simplicity we do not change the notion of the anchor action

δ_{ε}X = π(X)|ε⟩.  (2.13)

Here ε are the sections of X^{∗}(T^{∗}M).

In the infinite dimensional case it is possible to extend the anchor action (2.13)

δ̂_{ε}f(X) = δ_{ε}f(X) + c(X, ε),  (2.14)

by the one-cocycle

c(X, ε) = \frac{1}{2π} ∮ ⟨ε|¯∂X⟩.  (2.15)

Since

δ_{ε_{1}}c(X, ε_{2}) − δ_{ε_{2}}c(X, ε_{1}) − c(X, ⌊ε_{1}, ε_{2}⌋) = 0,  (2.16)

(the cocycle property) one has

[δ̂_{ε_{1}}, δ̂_{ε_{2}}] = δ̂_{⌊ε_{1},ε_{2}⌋}.

We denote the Lie algebroid over M with the anchor (2.14) as Â_{M}. This construction is the generalization of the bundle of the central extended loop Lie algebras with the loop parameter z̄.

Consider the two-cocycle c(X; ε_{1}, ε_{2}) on Γ(A_{M})

δ_{ε_{1}}c(X; ε_{2}, ε_{3}) − δ_{ε_{2}}c(X; ε_{1}, ε_{3}) = 0.  (2.17)

+δ_{ε_{3}}c(X; ε_{1}, ε_{2}) − c(X; ⌊ε_{1}, ε_{2}⌋, ε_{3}) + c(X; ⌊ε_{1}, ε_{3}⌋, ε_{2}) − c(X; ⌊ε_{2}, ε_{3}⌋, ε_{1}).

It allows to construct the central extensions of brackets on Γ(A)

⌊(ε_{1}, k_{1}), (ε_{2}, k_{2})⌋_{c.e.} = (⌊ε_{1}, ε_{2}⌋, c(x; ε_{1}, ε_{2})).  (2.18)

The cocycle condition (2.17) means that the new brackets ⌊ , ⌋_{c.e.} satisfies AJI (2.12). The exact cocycles leads to the splitted extensions.

3 Poisson sigma-model

3.1 Poisson sigma-model and Hamiltonian algebroids

The manifold M serves as the target space for the Poisson sigma model. The space-time of the sigma model is the disk D.

Consider the one-form on D taking values in the pull-back by X of the cotangent bundle T^{∗}M:

ξ = (ξ_{1}, …, ξ_{n})

Endow the space of fields R = {X, ξ} with the canonical symplectic form

ω = \frac{1}{2π} ∮ ⟨Dξ ∧ DX⟩.  (3.19)

Consider the set of the first-class constraints

F^{j} := ¯∂X^{j} + (π(X)|ξ⟩)^{j} = 0.  (3.20)

They generate the canonical transformations of ω

δ̂_{ε}ξ_{m} = \frac{δ}{δX^{m}} c(X, ε) + ⟨ε|\frac{δ}{δX^{m}} π|ξ⟩ = ¯∂ϵ_{m} + ⟨ε|\frac{δ}{δX^{m}} π|ξ⟩,  (3.21)

and (2.13). It means that we lift the anchor action on R by means of the cocycle (2.15). Equivalently, the canonical transformations of a smooth functionals can be described as

δ_{ε_{j}}f(X, ξ) = {h_{ε_{j}}, f(X, ξ)}.

Here the Poisson brackets are inverse to the symplectic form ω (3.19) and

h_{ε_{j}} = \frac{1}{2π} ∫ ε_{j}F^{j},  (3.22)

(no summation on j). Again, due to (2.16)

{h_{ε}, h_{ε′}} = h_{⌊ε,ε′⌋}.  (3.23)

Summarizing, we have defined the symplectic manifold R{ξ, X} and the bundle A_{R}^{H} over R with the sections ε ∈ Γ(A_{R}^{H}).

For a general symplectic manifold R this construction leads to the Hamiltonian algebroid^{2} Definition 3 A_{R}^{H} is a Hamiltonian algebroid over a symplectic manifold R if there is a bundle map from A_{R}^{H} to the Lie algebra on C^{∞}(R): ε → h_{ε}, (i.e. fε → fh_{ε} for f ∈ C^{∞}(R)) satisfying the following conditions:

(i) For any ε_{1}, ε_{2} ∈ Γ(A_{R}^{H}) and x ∈ R

{h_{ε_{1}}, h_{ε_{2}}} = h_{⌊ε_{1},ε_{2}⌋}.  (3.24)

(ii) For any ε_{1}, ε_{2} ∈ Γ(A_{R}^{H}) and f ∈ C^{∞}(R)

⌊ε_{1}, fε_{2}⌋ = f⌊ε_{1}, ε_{2}⌋ + {h_{ε_{1}}, f}ε_{2}.  (3.25)

The both conditions are similar to the defining properties of the Lie algebroids (2.7),(2.8). For general Hamiltonian algebroids the Jacobi identity takes the form

f_{i}^{jk}(x)f_{n}^{im}(x) + {h_{e^{m}}, f_{n}^{jk}(x)} + E_{[ln]}^{j,k,m}h_{ε^{l}} + c.p.(j, k, m) = 0,  (3.26)

where E_{[ln]}^{j,k,m} is an antisymmetric tensor in [ln]. This structure arises in the Hamiltonian systems with the first class constraints and leads to the open algebra of arbitrary rank.^{4}

We have constructed a special Hamiltonian algebroid A_{R}^{H} with the brackets (2.5), the base R equipped with the canonical form (3.19) and the anchor (2.13), (3.21). In this case the last term in (3.26) is absent and we come to the anomalous Jacobi identity similar to (2.12)

f_{i}^{jk}(x)f_{n}^{im}(x) + {h_{m}, f_{n}^{jk}(x)} = 0,  (3.27)

where h_{m} is defined by (3.22). This identity plays the crucial role in the BRST construction.

3.2 BRST construction for Hamiltonian algebroids

Here we consider the BRST construction for general Hamiltonian algebroids. Let A^{H∗} be the dual bundle. Its sections η ∈ Γ(A_{R}^{H∗}) are the odd fields called the ghosts. Let

h_{e^{j}} = \frac{1}{2π} ∮ ⟨η_{j}|F (x)⟩,

where {η_{j}} is a basis in Γ(A^{H∗}) and F (x) = 0 are the moment constraints, generating the canonical algebroid action on R. Introduce another type of odd variables (the ghost momenta) P^{j} ∈ Γ(A_{R}^{H}), j = 1, 2, … dual to the ghosts η_{k}, k = 1, 2, . . .. We attribute the ghost number one to the ghost fields gh(η) = 1, minus one to the ghost momenta gh(P) = −1 and gh(x) = 0 for x ∈ R. Introducing In addition to the non-degenerate Poisson structure on R we introduce the Poisson brackets

{η_{j}, P^{k}} = δ_{j}^{k}, {η^{j}, x} = {P_{k}, x} = 0.  (3.28)

Thus all fields are incorporated in the graded Poisson superalgebra

BFV = (Γ(∧^{•}(A_{R}^{H∗} ⊕ A_{R}^{H})) ⊗ C^{∞}(R). = Γ(∧^{•}A_{R}^{H∗}) ⊗ Γ(∧^{•}A_{R}^{H}) ⊗ C^{∞}(R).

(the Batalin-Fradkin-Vilkovitsky (BFV) algebra).

There exists a nilpotent operator on Q on BFV (Q^{2} = 0, gh(Q) = 1) (the BRST operator) transforming BFV into the BRST complex. The cohomology of BFV complex give rise to the structure of the classical reduced phase space R^{red}. Namely H^{0}(Q) is identified with the classical observables.

Represent the action of Q as the Poisson brackets:

Qψ = {ψ, Ω}, ψ, Ω ∈ BFV.

Due to the Jacobi identity for the Poisson brackets the nilpotency of Q is equivalent to

{Ω, Ω} = 0.

Since Ω is odd, the brackets are symmetric. For a generic Hamiltonian systems Ω has the form of infinite series in the ghost P^{k} expansion

Ω = h_{η} + \frac{1}{4π} ∮ ⟨⌊η, η′⌋|P⟩ + ..., (h_{η} = ∮ ⟨η|F ⟩).

The order of P in Ω is called the rank of the BRST operator Q. For generic Hamiltonian algebroids with the anomalous Jacobi identity (3.26) Q can have the infinite rank.

If A is a Lie algebra bundle defined along with its canonical action on R then Q has the rank one or less. In this case the BRST operator Q is the extension of the Cartan-Eilenberg operator giving rise to the cohomology of A with coefficients in C^{∞}(R) and the first two terms in the previous expression provide the nilpotency of Q.

The distinguish property of A_{R}^{H} is that Ω has the rank one, though the Jacobi identity has additional terms (3.27) in compare with the Lie algebras^{2}

Ω = \frac{1}{2π} ∮ ⟨η|F ⟩ + \frac{1}{4π} ∮ ⟨⌊η, η′⌋|P⟩  (3.29)

4 W_{3}-gravity

4.1 Lie algebroid over SL(3, ℂ)-opers

In our approach the starting point to the W_{3}-gravity is the space infinite dimensional Poisson space of the third order differential operators with smooth coefficients on a Riemann curve Σ_{n} (the SL(3, ℂ)-opers). The curve Σ_{n} play the role of the space part of the W_{3}-gravity, while the time is just ℝ. Locally SL(3, ℂ)-opers are defined as

∂^{3} − T ∂ − W : Ω^{(−1,0)}(Σ_{g}) → Ω^{(2,0)}(Σ_{g}), (∂ = ∂_{z}),  (4.1)

where Ω^{(a,b)}(Σ_{g}) is the space of section of smooth (a, b) forms on Σ_{g}. In fact, they can be defined globally starting from SL(3, ℂ) bundles over Σ_{g}. The connection on the SL(3, ℂ) bundle has the special form corresponding to (4.1)

∇ = ∂ − ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ W & T & 0 \end{matrix} ) .  (4.2)

The SL(N, ℂ) opers were introduced in Ref. 11. The general case was considered in Ref. 12.

The space of SL(3, ℂ)-opers plays the role of the Poisson manifold M = M_{3} = {W, T }, where the brackets are defined as

{T (z), T (w)} = (−2κ∂^{3} + 2T (z)∂ + ∂T (z)) δ(z − w),  (4.3)
{T (z), W (w)} = (−κ∂^{4} + T (z)∂^{2} + 3W (z)∂ + ∂W (z)) δ(z − w), (4.4)
{W (z), W (w)} =  (4.5)

+ (\frac{2}{3}k∂^{5} − \frac{4}{3}T (z)∂^{3} − 2∂T (z)∂^{2} + (\frac{2}{3}(2 − κ)T (z)^{2} − 2∂^{2}T (z) + 2∂W (z)) ∂

+ (∂^{2}W (z) − \frac{2}{3}∂^{3}T (z) + \frac{2}{3}(2 − κ)T (z)∂T (z))) δ(z − w).

These brackets are defined in terms of coordinates in a neighborhood of a noncontractable contour on Σ_{g}. The parameter κ comes from the central extension of the Lie brackets on the sections of the algebroid A_{3} and we have 2g nonequivalent extensions (see below (4.20) and Remark 4.1). For κ ≠ 2 the Poisson brackets \{W (z), W (w)\} have two quadratic terms T (z)^{2} and \frac{2}{3}T (z)∂T (z)). In what follows we take κ = 1. Note, that z̄ dependence is hidden these relations. In this way the Poisson structure on M_{3} is just the global version of Adler-Gelfand-Dikii brackets.^{13,14}

Define the Lie algebroid A_{3} over M_{3}. Its sections D_{2} = Γ(A_{3}) are the second order differential operators on Σ_{g} without constant terms. On a disk D_{2} can be trivialized and the sections are represented as

ε^{(1)} = ε^{(1)}(z, z̄)\frac{∂}{∂z} , ε^{(2)} = ε^{(2)}(z, z̄)\frac{∂^{2}}{∂z^{2}} ,

ε^{(1)} ∈ D^{1}, ε^{(2)} ∈ D^{2}, D_{2} = D^{1} ⊕ D^{2}.

The second order differential operators do not generate a closed algebra with respect to the standard commutators. Moreover, they cannot be defined invariantly on Riemann curves in contrast with the first order differential operators. We introduce a new brackets that goes around the both disadvantages.

The antisymmetric brackets on D_{2} are defined by means the Poisson brackets (4.3),(4.4),(4.5) according to the general prescription (2.5).

⌊ε_{1}^{(1)}, ε_{2}^{(1)}⌋ = ε_{1}^{(1)}∂ε_{2}^{(1)} − ε_{2}^{(1)}∂ε_{1}^{(1)}.  (4.6)

⌊ε^{(1)}, ε^{(2)}⌋ = \{ \begin{matrix}−ε^{(2)}∂^{2}ε^{(1)}, & ∈ D^{1} \\ −2ε^{(2)}∂ε^{(1)} + ε^{(1)}∂ε^{(2)}, & ∈ D^{2} \end{matrix}  (4.7)

⌊ε_{1}^{(2)}, ε_{2}^{(2)}⌋ = \{ \begin{matrix}\frac{2}{3}[∂(∂^{2} − T)ε_{1}^{(2)}]ε_{2}^{(2)} − \frac{2}{3}[∂(∂^{2} − T)ε_{2}^{(2)}]ε_{1}^{(2)}, & ∈ D^{1} \\ ε_{2}^{(2)}∂^{2}ε_{1}^{(2)} − ε_{1}^{(2)}∂^{2}ε_{2}^{(2)}, & ∈ D^{2} \end{matrix} (4.8)

The brackets (4.6) are the standard Lie brackets of vector fields and therefore D^{1} is the Lie subalgebra of D_{2}. The structure functions in (4.8) depend on the projective connection T.

Now consider the bundle map A_{3} to T M_{3} defined by the anchor δ_{ε^{(j)}} = ⟨ε^{(j)}|π|\frac{δ}{δT} ⟩, or δ_{ε^{(j)}} = ⟨ε^{(j)}|π|\frac{δ}{δW} ⟩

δ_{ε^{(1)}}T = −2∂^{3}ε^{(1)} + 2T ∂ε^{(1)} + ∂T ε^{(1)},  (4.9)
δ_{ε^{(1)}}W = −∂^{4}ε^{(1)} + 3W ∂ε^{(1)} + ∂W ε^{(1)} + T ∂^{2}ε^{(1)},  (4.10)
δ_{ε^{(2)}}T = ∂^{4}ε^{(2)} − T ∂^{2}ε^{(2)} + (3W − 2∂T)∂ε^{(2)} + (2∂W − ∂^{2}T)ε^{(2)}, (4.11)
δ_{ε^{(2)}}W = \frac{2}{3}∂^{5}ε^{(2)} − \frac{4}{3}T ∂^{3}ε^{(2)} − 2∂T ∂^{2}ε^{(2)}  (4.12)

+(\frac{2}{3}T^{2} − 2∂^{2}T + 2∂W)∂ε^{(2)} + (∂^{2}W − \frac{2}{3}∂^{3}T + \frac{2}{3}T ∂T)ε^{(2)}.

The algebroid structure follows from the identity

[δ_{ε_{1}^{(j)}}, δ_{ε_{2}^{(k)}}] = δ_{⌊ε_{1}^{(j)},ε_{2}^{(k)}⌋}, (j, k = 1, 2).

The Jacobi identity (2.12) in A_{3} takes the form

⌊⌊ε_{1}^{(2)}, ε_{2}^{(2)}⌋, ε_{3}^{(2)}⌋^{(1)} − (ε_{1}^{(2)}∂ε_{2}^{(2)} − ε_{2}^{(2)}∂ε_{1}^{(2)})δ_{ε_{3}^{(2)}}T + c.p.(1, 2, 3) = 0, (4.13)
⌊⌊ε_{1}^{(2)}, ε_{2}^{(2)}⌋, ε_{3}^{(1)}⌋^{(1)} + c.p.(1, 2, 3) = (ε_{1}^{(2)}∂ε_{2}^{(2)} − ε_{2}^{(2)}∂ε_{1}^{(2)})δ_{ε_{3}^{(1)}}T. (4.14)

The brackets here correspond to the product of structure functions in the left hand side of (2.12) and the superscript (1) corresponds to the D^{1} component. For the rest brackets the Jacobi identity is the standard one.

The origin of the brackets and the anchor representations follow from the matrix description of SL(3, ℂ)-opers (4.2). Let E_{3} be the principle SL(3, ℂ)- bundle over Σ_{g}. Consider the set G_{3} of automorphisms of the bundle E_{3}

A → f^{−1}∂f − f^{−1}Af  (4.15)

that preserve the SL(3, ℂ)-oper structure

f^{−1}∂f − f^{−1}A(W, T)f = A(W′, T′),  (4.16)

A(W, T) = ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ W & T & 0 \end{matrix} ) .

It is clear that G_{3} is the Lie groupoid over M_{3} = {W, T } with l(f) = (W, T), r(f) = (W′, T′), f →≪ W, T |f|W′, T′ ≫. The left identity map is the SL(3, ℂ) subgroup of G_{3}

P exp (− ∫_{z_{0}}^{z} A(W, T)) · C · P exp (∫_{z_{0}}^{z} A(W, T)) ,

where C is an arbitrary matrix from SL(3, ℂ). The right identity map has the same form with (W, T) replaced by (W′, T′).

The local version of (4.16) takes the form

∂X − [( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ W & T & 0 \end{matrix} ) , X] = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ δW & δT & 0 \end{matrix} ) .  (4.17)

It is the sixth order linear differential system for the matrix elements of the traceless matrix X. The matrix elements x_{j,k} ∈ Ω^{(j−k,0)}(Σ_{g}) depend on two arbitrary fields x_{23} = ε^{(1)}, x_{13} = ε^{(2)}. The solution takes the form

X(ε^{(1)}, ε^{(2)}) = ( \begin{matrix}x_{11} & x_{12} & ε^{(2)} \\ x_{21} & x_{22} & ε^{(1)} \\ x_{31} & x_{32} & x_{33} \end{matrix} ) ,  (4.18)

x_{11} = \frac{2}{3}(∂^{2} − T)ε^{(2)} − ∂ε^{(1)}, x_{12} = ε^{(1)} − ∂ε^{(2)},

x_{21} = \frac{2}{3}∂(∂^{2} − T)ε^{(2)} − ∂^{2}ε^{(1)} + W ε^{(2)}, x_{22} = −\frac{1}{3}(∂^{2} − T)ε^{(2)},

x_{31} = \frac{2}{3}∂^{2}(∂^{2} − T)ε^{(2)} − ∂^{3}ε^{(1)} + ∂(W ε^{(2)}) + W ε^{(1)},

x_{32} = \frac{1}{3}∂(∂^{2} − T)ε^{(2)} − ∂^{2}ε^{(1)} + W ε^{(2)} + T ε^{(1)},

x_{33} = −\frac{1}{3}(∂^{2} − T)ε^{(2)} + ∂ε^{(1)}.

The matrix elements of the commutator [X_{1}, X_{2}]_{13}, [X_{1}, X_{2}]_{23} give rise to the brackets (4.6), (4.7), (4.8). Simultaneously, from (4.17) one obtain the anchor action (4.9)-(4.12).

There is a nontrivial cocycle corresponding to H^{1}(A_{3}) with two components

c_{α}(ε_{1}^{(j)}, ε_{2}^{(k)}) = ∮_{γ_{α}} λ(ε_{1}^{(j)}, ε_{2}^{(k)}), (j, k = 1, 2),  (4.19)

The cocycle allows to shift the anchor action

δ̂_{ε^{(j)}}f(W, T) = ∫_{Σ_{g}}⟨δ_{ε^{(j)}}W |\frac{δf}{δW} ⟩ + ∫_{Σ_{g}}⟨δ_{ε^{(j)}}T |\frac{δf}{δT} ⟩ + c^{(j)}, (j = 1, 2).

There exists the 2g central extensions c_{α} of the algebra D_{2}, provided by the nontrivial cocycles from H^{2}(A_{3}, M_{3}). They are the non-contractible contour integrals γ_{α}

c_{α}(ε_{1}^{(j)}, ε_{2}^{(k)}) = ∮_{γ_{α}} λ(ε_{1}^{(j)}, ε_{2}^{(k)}), (j, k = 1, 2),  (4.20)

where

λ(ε_{1}^{(1)}, ε_{2}^{(1)}) = ε_{1}^{(1)}∂^{3}ε_{2}^{(′1)}, λ(ε_{1}^{(1)}, ε_{2}^{(2)}) = ε_{1}^{(1)}∂^{4}ε_{2}^{(2)},
λ(ε_{1}^{(2)}, ε_{2}^{(2)}) =

\frac{2}{3} (−(∂^{2} − T)ε_{1}^{(2)}∂(∂^{2} − T)ε_{2}^{(2)} + ∂^{2}ε_{1}^{(2)}∂(∂^{2} − T)ε_{2}^{(2)} + ∂^{2}ε_{2}^{(2)}∂(∂^{2} − T)ε_{1}^{(2)}) .

It can be proved that sc^{j} = 0 (2.17) and that c^{j} is not exact. These cocycles allow to construct the central extensions of Â_{3}:

⌊(ε_{1}^{(j)}, ∑_{α} k_{α}^{(j)}), (ε_{2}^{(m)}, ∑_{α} k_{α}^{(m)})⌋_{c.e.} = (⌊ε_{1}^{(j)}, ε_{2}^{(m)}⌋, ∑_{α} c_{α}(ε_{1}^{(j)}, ε_{2}^{(m)})).

Remark 4.1 As in the case of the Lie algebras the Poisson structure (4.3), (4.4) and (4.5) can be read off from the Lie brackets (4.6), (4.7), (4.8) along with their central extensions (4.20) according with the Lie brackets for H_{1}, H_{2} ∈ C^{∞}(M_{3})

{H_{1}, H_{2}} = ∮_{γ_{α}}⟨⌊δH_{1}, δH_{2}⌋X⟩ + k ∮_{γ_{α}} λ(δH_{1}, δH_{2}),

where X = W, T and δH = δH/δX ∈ D_{2}.

4.2 Phase space of W_{3}-gravity and BRST operator

The dual fields to T and W are the Beltrami differentials μ ∈ Ω^{(−1,1)}(Σ_{g}) and the differentials ρ ∈ Ω^{(−2,1)}(Σ_{g}). Along with T and W they generate the space R_{3}. It is the classical phase space for the W_{3}-gravity.^{7,8,9} The symplectic form on R_{3} has the canonical form

ω = ∫_{Σ_{g}} DT ∧ Dμ + DW ∧ Dρ.  (4.21)

According with the general theory the anchor (4.9)-(4.12) can be lifted from M_{3} to R_{3}. This lift is nontrivial owing to the cocycle (4.19). It follows from (3.21) that the anchor action on μ and ρ takes the form

δ_{ε^{(1)}}μ = −¯∂ε^{(1)} − μ∂ε^{(1)} + ∂με^{(1)} − ρ∂^{2}ε^{(1)},  (4.22)
δ_{ε^{(1)}}ρ = −2ρ∂ε^{(1)} + ∂ρε^{(1)},  (4.23)
δ_{ε^{(2)}}μ = ∂^{2}με^{(2)} − \frac{2}{3} [(∂(∂^{2} − T)ρ)ε^{(2)} − (∂(∂^{2} − T)ε^{(2)})ρ] ,  (4.24)
δ_{ε^{(2)}}ρ = −¯∂ε^{(2)} + (ρ∂^{2}ε^{(2)} − ∂^{2}ρε^{(2)}) + 2∂με^{(2)} − μ∂ε^{(2)}.  (4.25)

In this way we have defined the Hamiltonian Lie algebroid over R_{3} with the section D_{2}, the brackets (4.6),(4.7),(4.8), and the anchor (4.9)-(4.12), (4.22)- (4.25).

There are two Hamiltonians, defining by the anchor and by the cocycle

h^{(1)} = ∫_{Σ_{g}}(μδ_{ε^{(1)}}T + ρδ_{ε^{(1)}}W) + c^{(1)}, h^{(2)} = ∫_{Σ_{g}}(μδ_{ε^{(2)}}T + ρδ_{ε^{(2)}}W) + c^{(2)}.

After the integration by part they take the form

h^{(1)} = ∫_{Σ_{g,n}} ε^{(1)}F^{(1)},

h^{(2)} = ∫_{Σ_{g,n}} ε^{(2)}F^{(2)},

where F^{(1)} ∈ Ω^{(2,1)}(Σ_{g,n}), F^{2} ∈ Ω^{(3,1)}(Σ_{g,n})

F^{(1)} = −¯∂T − ∂^{4}ρ + T ∂^{2}ρ − (3W − 2∂T)∂ρ−  (4.26)
−(2∂W − ∂^{2}T)ρ + 2∂^{3}μ − 2T ∂μ − ∂T μ,

F^{(2)} = −¯∂W − \frac{2}{5}∂^{5}ρ + \frac{4}{3}T ∂^{3}ρ + 2∂T ∂^{2}ρ(−\frac{2}{3}T^{2} + 2∂^{2}T − 2∂W)∂ρ+ (4.27)

+(−∂^{2}W + \frac{2}{3}∂^{3}T − \frac{2}{3}T ∂T)ρ + ∂^{4}μ − 3W ∂μ − ∂W μ − T ∂^{2}μ.

They carry out the moment map

m = (m^{(1)} = F^{(1)}, m^{(2)} = F^{(2)}) : R_{3} → Γ^{∗}(A_{3}).

We assume that m^{(1)} = 0, m^{(2)} = 0. Then the coadjoint action of D_{2} preserve m = (0, 0). The moduli space W_{3} of the W_{3}-gravity ( W_{3}-geometry) is the symplectic quotient with respect to the groupoid G_{3} action

W_{3} = R_{3}//G_{3} = {F^{1} = 0, F^{2} = 0}/G_{3}.

The space W_{3} is finite-dimensional. It has dimension

dim W_{3} = 16(g − 1).

The prequantization of W_{3} can be realized in the space of sections of a linear bundle L over the space of orbits M̃_{3} ∼ M_{3}/G_{3}. The sections are functionals Ψ(T, W) on M_{3} satisfying the following conditions

δ̂_{ε^{(j)}}Ψ(T, W) := ⟨δ_{ε^{(j)}}W |\frac{δΨ}{δW} ⟩ + ⟨δ_{ε^{(j)}}T |\frac{δΨ}{δT}⟩ + c^{(j)}Ψ = 0, (j = 1, 2).

Presumably, the bundle L can be identified with the determinant bundle det(∂^{3} − T ∂ − W).

The moment equations F^{(1)} = 0, F^{(2)} = 0 are the consistency conditions for the linear system

\{ \begin{matrix}(∂^{3} − T ∂ − W)ψ(z, z̄) = 0, \\ (¯∂ + (μ − ∂ρ)∂ + ρ∂^{2} + \frac{2}{3}(∂^{2} − T)ρ − ∂μ) ψ(z, z̄) = 0, \end{matrix}  (4.28)

where ψ(z, z̄) ∈ Ω^{(−1,0)}(Σ_{g}). The last equation represents the deformation of the antiholomorphic operator ¯∂ by the first order operator μ∂ and by the second order differential operator ρ∂^{2}. The left hand side is the exact form of the deformed operator when it acts on Ω^{(−1,0)}(Σ_{g}). This deformation cannot be supported by the structure of a Lie algebra and one leaves with the Hamiltonian algebroid symmetries.

Instead of the symplectic reduction one can apply the BRST construction. Then cohomology of the moduli space W_{3} are isomorphic to H^{j}(Q). To construct the BRST complex we introduce the ghosts fields η^{(1)}, η^{(2)} and their momenta P^{(1)}, P^{(2)}. Then it follows from the general construction that for

Ω = ∑_{j=1,2} h^{(j)}(η^{(j)}) + \frac{1}{2} ∑_{j,k,l=1,2} ∫_{Σ_{g,n}}(⌊η^{(j)}, η^{(k)}⌋P^{(l)}).

The operator QF = {F, Ω} is nilpotent and defines the BRST cohomology in the complex

⋀^{•}(D_{2} ⊕ D_{2}^{∗}) ⊗ C^{∞}(R_{3}).

4.3 Chern-Simons description

Here we follow the approach proposed in Ref. 8. Consider the Chern-Simons functional on Σ_{g} ⊕ R^{+}

S = ∫_{Σ_{g}⊕R} tr (AdA + \frac{2}{3}A^{3}) , A = (A, Ā, A_{t}).

In the hamiltonian picture the components A, Ā are elements of the phase space R_{SL(3,ℂ)} with the symplectic form

∫_{Σ_{g}} DA ∧ DĀ,

while A_{t} is the Lagrange multiplier for the first class constraints

F (A, Ā) := ¯∂A − ∂Ā + [Ā, A] = 0.  (4.29)

The phase space R_{3} can be derived from the phase space of the Chern-Simons theory R_{SL(3,ℂ)}. The flatness condition (4.29) generates the gauge transformations

A → f^{−1}∂f + f^{−1}Af, Ā → f^{−1} ¯∂f + f^{−1}Āf, f ∈ G_{SL(3,ℂ)}  (4.30)

The result of the gauge fixing with respect to the whole gauge group G_{SL(3,ℂ)} is the moduli space M_{3}^{flat} of the flat SL(3, ℂ) bundles over Σ_{g}.

Let P be the maximal parabolic subgroup of SL(3, ℂ) of the form

P = ( \begin{matrix}∗ & ∗ & 0 \\ ∗ & ∗ & 0 \\ ∗ & ∗ & ∗ \end{matrix} ) ,

and G_{P} be the corresponding gauge group. We partly fix the gauge with respect to G_{P}. A generic connection ∇ can be gauge transformed by f ∈ G_{P} to the form (4.2).

The form of Ā can be read off from (4.29)

Ā = ( \begin{matrix}a_{11} & a_{12} & −ρ \\ a_{21} & a_{22} & −μ \\ a_{31} & a_{32} & a_{33} \end{matrix} )  (4.31)

a_{11} = −\frac{2}{3}(∂^{2} − T)ρ + ∂μ, a_{12} = −μ + ∂ρ,

a_{21} = −\frac{2}{3}∂(∂^{2} − T)ρ + ∂^{2}μ − W ρ, a_{22} = \frac{1}{3}(∂^{2} − T)ρ,

a_{31} = −\frac{2}{3}∂^{2}(∂^{2} − T)ρ + ∂^{3}μ − ∂(W ρ) − W μ,

a_{32} = −\frac{1}{3}∂(∂^{2} − T)ρ + ∂^{2}μ − W ρ − T μ, a_{33} = \frac{1}{3}(∂^{2} − T)ρ − ∂μ.

The flatness (4.29) for the special choice A (4.2) and Ā (4.31) gives rise to the moment constraints F^{(2)} = 0, F^{(1)} = 0 (4.26),(4.27). Namely, one has F (A, Ā)|_{(3,1)} = F^{(2)} (4.26), F (A, Ā)|_{(2,1)} = F^{(1)} (4.27), while the other matrix elements of F (A, Ā) vanish identically. In this way, we come to the matrix description of the moduli space W_{3}.

The groupoid action on A, Ā plays the role of the rest gauge transformations that complete the G_{P} action to the G_{SL(3,ℂ)} action. The algebroid symmetry arises in this theory as a result of the partial gauge fixing by G_{P}. Thus we come to the following diagram

R_{SL(3,ℂ)}

|  ↘ G_{P}
G_{SL(3,ℂ)} | R_{3}
↓  ↓ Γ(G_{3})
M_{SL(3,ℂ)}^{flat}  W_{3}

The tangent space to M_{SL(3,ℂ)}^{flat} at the point (A = 0, Ā = 0) coincides with the tangent space to W_{3} at the point (W = 0, T = 0, μ = 0, ρ = 0). Their dimension is 16(g − 1). But their global structure is different and the diagram cannot be closed by the horizontal isomorphisms. The interrelations between M_{SL(3,ℂ)}^{flat} and W_{3} were analyzed by Hitchin.^{25} 5 Sklyanin algebra

5.1 Hamiltonian algebroid for Sklyanin algebra

We start from the Poisson bivector defined on a ℂ^{4} with generators (S_{0}, S⃗ = (S_{α}), α = 1, 2, 3) ^{d}

{S_{α}, S_{0}} = 2J_{βγ}S_{β}S_{γ}, {S_{α}, S_{β}} = −2ϵ_{αβγ}S_{0}S_{γ},  (5.1)

where J_{βγ} = ℘_{β} − ℘_{γ}, ℘_{α} = e_{α}, ^{e} and

e_{1} = ℘ (\frac{1}{2}) , e_{2} = ℘ (\frac{1 + τ}{2} ) , e_{3} = ℘ (\frac{τ}{2} ) ,  (5.2)

and ℘ = ℘(u; τ) is the Weierstrass function. In this way the algebra is parameterized by the modular parameter τ. The Jacobi identity is provided by the identity e_{1} + e_{2} + e_{3} = 0. It is the Sklyanin algebra S. ^{17} It has two Casimirs

K_{0} = ∑ S_{α}^{2} , K_{1} = S_{0}^{2} + ∑ ℘_{α}S_{α}^{2}.  (5.3)

The Lie algebroid brackets (2.5) for (ε_{0}, ε⃗) ∈ T^{∗}S take the form

⌊ε_{β}, ε_{0}⌋_{α} = 2J_{βα}S_{β}ε_{α},
⌊ε_{α}, ε_{β}⌋_{0} = −2ϵ_{αβγ}S_{γ}ε_{α}ε_{β},  (5.4)
⌊ε_{α}, ε_{β}⌋_{γ} = −2ϵ_{αβγ}S_{0}ε_{α}ε_{β}.

It is convenient to arrange the Poisson brackets on S in the form of the classical Yang-Baxter equation. Introduce three functions φ_{α}(z) depending on a formal (spectral) parameter z living on the elliptic curve Σ_{τ} = ℂ/(ℤ + τℤ)

φ_{1}(z) = ϕ (\frac{1}{2}, z) , φ_{2}(z) = −e^{πiτ} ϕ (\frac{1 + τ}{2} , z) , φ_{3}(z) = e^{πiτ} ϕ (\frac{τ}{2} , z) , (5.5)

where

ϕ(u, z) = \frac{ϑ(u + z)ϑ′(0)}{ϑ(u)ϑ(z)},

and

ϑ(z|τ) = q^{\frac{1}{8}} ∑_{n∈Z}(−1)^{n}e^{πi(n(n+1)τ+2nz)}

d In this section to reconcile with the accepted notations of the Sklyanin algebra we do not comply with the positions of the upper and lower indices.

e We use the standard notations of the elliptic functions.^{26}

is the odd theta-function. Consider the Lax operator

L(z) = S_{0}Id + iS(z),  (5.6)

where

S(z) = ∑ S_{α}φ_{α}(z)σ_{α}

and σ_{α} are the Hermitian sigma matrices σ_{α}σ_{β} = −iϵ_{αβγ}σ_{γ}. Then (5.1) is equivalent to the matrix relation

{L_{1}(z), L_{2}(w)} = [r(z − w), L_{1} ⊗ L_{2}],  (5.7)

where r is the classical SL(2, ℂ) elliptic r-matrix

r(z) = φ_{α}(z)σ_{α} ⊗ σ_{α}.  (5.8)

Note that

det(L^{2}) = K_{0}℘(z) − K_{1}.  (5.9)

5.2 Sklyanin algebra from canonical brackets

Let E_{2} be the principle GL(2, ℂ)-bundle over Σ_{τ} with deg(E_{2}) = 1. Consider the group element g ∈ Ω^{(0)}(Σ_{τ} , GL(2, ℂ))

g(z, z̄) = g_{0}(z, z̄) + ic ∑_{α} g_{α}(z, z̄)σ_{α},

where c is a relativistic parameter. In what follows for brevity we omit the z̄ dependence. To be a section of the degree one bundle the functions g_{α} should satisfy the quasi-periodicity conditions

g(z + 1) = σ_{3}g(z)σ_{3}^{−1}, g(z + τ) = Λg(z)Λ^{−1},  (5.10)

Λ = Λ(z; τ) = −σ_{1} exp −π(\frac{1}{2}τ + z).

The holomorphic structure on E_{2} is defined by the operator

dĀ = k ¯∂ + Ā : Ω^{(j,k)}(Σ_{τ} , sl(2, ℂ)) → Ω^{(j,k+1)}(Σ_{τ} , sl(2, ℂ)),

Ā = A_{0} + i ∑_{α} Ā_{α}σ_{α}.

Here we have introduced the central charge k. The moduli space M of stable holomorphic bundles is the factor space

M = {k ¯∂ + Ā}/G,

where G = {f ∈ Ω^{(0)}(Σ_{τ} , GL(2, ℂ))} is the gauge group

Ā → f^{−1}k ¯∂f + f^{−1}Āf.  (5.11)

There exists the canonical symplectic structure on the pair of fields R′ = {g, Ā}

ω′ = ∫_{Σ_{τ}} tr (D(Āg^{−1}) ∧ Dg) + \frac{k}{2} ∫_{Σ_{τ}} (g^{−1}Dg ∧ ¯∂(g^{−1}Dg)) .  (5.12)

The canonical class of Σ_{τ} is incorporated in the integrals. We fix it in what follows.

If c → 0 then in the first order in c we come to the symplectic structure on the cotangent bundle to the holomorphic structures on E_{2}.^{27} The element ∑_{α} g_{α}σ_{α} being multiply on the canonical class plays the role of the Higgs field in the Hitchin construction.

The form generates the following Poisson brackets in the space of smooth functionals over R′

{H_{1}(Ā, g), H_{2}(Ā, g)} =

∫_{Σ_{τ}} tr ([\frac{δH_{1}(Ā, g)}{δĀ}, \frac{δH_{2}(Ā, g)}{δĀ}] Ā) + k ∫_{Σ_{τ}} tr (\frac{δH_{1}(Ā, g)}{δĀ} ¯∂\frac{δH_{2}(Ā, g)}{δĀ})

+ ∫_{Σ_{τ}} tr (\frac{δH_{2}(Ā, g)}{δĀ} \frac{δH_{1}(Ā, g)}{δg}g) − ∫_{Σ_{τ}} tr (\frac{δH_{1}(Ā, g)}{δĀ} \frac{δH_{2}(Ā, g)}{δg}g) .

In particular,

\{Ā_{α}(z), Ā_{β}(w)\} = ϵ_{αβγ} Ā_{γ}(z)δ^{(2)}(z − w) + \frac{k}{2} δ_{α,β} ¯∂δ^{(2)}(z − w),

\{Ā_{0}(z), Ā_{0}(w)\} = \frac{k}{2} δ_{α,β} ¯∂δ^{(2)}(z − w), (5.13)

\{Ā_{0}(z), g_{α}(w)\} = \frac{1}{2}g_{α}(z)δ^{(2)}(z − w),

\{Ā_{0}(z), g_{0}(w)\} = \frac{1}{2}g_{0}(z)δ^{(2)}(z − w),

\{Ā_{α}(z), g_{β}(w)\} = −\frac{1}{2}ϵ_{αβγ}g_{γ}(z)δ^{(2)}(z − w) − \frac{1}{2}δ_{α,β}g_{0}(z)δ^{(2)}(z − w),

\{Ā_{α}(z), g_{0}(w)\} = \frac{1}{2}g_{α}(z)δ^{(2)}(z − w), (5.14)
{g_{α}, g_{β}} = {g_{0}, g_{β}} = 0,  (5.15)

where δ^{(2)}(z − w) is the functional f(0) = ∫_{Σ_{τ}} f(w)δ^{(2)}(z − w).

Consider the canonical transformation of the fields

Ā_{0} → Ā_{0} + c(z, z̄), Ā_{α} → Ā_{α}, g → g,

where c(z, z̄) ∈ Ω^{(0,1)}(Σ_{τ}). The corresponding moment map is the functional

log det g(z) = log(g_{0}^{2} + ∑_{α} g_{α}^{2}).  (5.16)

Let fix the gauge as Ā_{0} = 0. The form ω′ being pushed down on the reduced phase space

R′′ = {g, Ā | Ā_{0} = 0, g_{0}^{2} + ∑_{α} g_{α}^{2} = const}  (5.17)

is non-degenerate. The Poisson structure on R′′ has the form (5.13),(5.14),(5.15) where Ā_{0} = 0. The determinant

det g(z) = g_{0}^{2} + ∑_{α} g_{α}^{2}  (5.18)

is the Casimir functional with respect to the new brackets.

To complete the model we add some finite degrees of freedom. Consider the cotangent bundle T^{∗}SL(2, ℂ) represented by T ∈ sl(2, ℂ), h ∈ SL(2, ℂ) with the canonical form

ω_{0} = tr (D(Th^{−1}) ∧ Dh) .  (5.19)

The whole upstairs phase space is

R_{2} = R′′ ∪ T SL(2, ℂ) ∼ {dĀ, g, T, h}

with the symplectic structure

ω = ω′ + ∫_{Σ_{τ}} ω_{0}δ^{(2)}(z, z̄).  (5.20)

The form ω is invariant under the gauge transformation (5.11) accompanied by

g → f^{−1}gf, (f ∈ G),  (5.21)
T → f_{0}^{−1}Tf_{0}, h → f_{0}^{−1}hf_{0}, f_{0} = f(0),

These transformations generate the moment map

μ : R_{2} → Lie^{∗}(G)μ = −Ā + gĀg^{−1} − k ¯∂gg^{−1} − Tδ^{2}(z).

In the case of degree one bundles on elliptic curves the connection Ā can be represented as the pure gauge Ā = f^{−1} ¯∂f. The same gauge transform brings g in the form

g = f^{−1}Lf.  (5.22)

We preserve the same notations for the transformed variables T and h. Then the moment condition μ = 0 takes the form

k ¯∂LL^{−1} = Tδ^{2}(z, z̄).  (5.23)

Solutions of this equation should satisfy the consistency condition

k(Res|_{z=0}L) = T(Res|_{z=0}L),  (5.24)

(see^{18}). Taking into account the quasi-periodicity of g (5.10) we can write the general solution in the form (5.6)

L(z) = S_{0} + i ∑_{α} S_{α}φ_{α}(z)σ_{α}.

Then (5.24) takes the form

−ikS_{α} = ϵ_{αβγ}T_{β}S_{γ},  (5.25)

∑_{α} S_{α}T_{α} = 0.

The last equation is consistent with the system (5.25). To come to nontrivial solutions S⃗ of (5.25) we should assumed that its determinant vanishes. It is happened in the case k = 0, or

k^{2} = ∑_{α} T_{α}^{2}.  (5.26)

This condition restricts the space T^{∗}SL(2, ℂ) to the coadjoint orbit of SL(2, ℂ). Fortunately, we still have at hand the constant gauge transformations preserving the gauge condition Ā = 0. The condition (5.26) is the moment constraint in the symplectic reduction of T^{∗}SL(2, ℂ) generating by the constant matrices. In this way we come to the Lax operator L (5.6), where S_{α} are related to the elements of the coadjoint orbit via the solutions of (5.25).

Finally, we should demonstrate that the brackets (5.13), (5.14), (5.15) being restricted on shell (5.23) after the gauge fixing Ā = 0 lead to the Sklyanin algebra (5.1). Using the Dirac procedure we define

{g(z), g(w)}^{∗}|_{on shell} = − ∫_{Σ_{τ}} ∫_{Σ_{τ}}dvdu{g(z), Ā(v)}C(v, u){g(w), Ā(u)}|_{on shell} .

(5.27)

On shell only the second term in the r.h.s. of (5.13) is essential. In this way C(v, u) is the inverse operator to

k ¯∂: Ω^{(0)}(Σ_{τ} , sl(2, ℂ)) → Ω^{(0,1)}(Σ_{τ} , sl(2, ℂ)),

C(v, u) : η(u) ∈ Ω^{(0,1)}(Σ_{τ} , sl(2, ℂ)) → ∫_{Σ_{τ}} G(v − u)η(v) ∈ Ω^{(0)}(Σ_{τ} , sl(2, ℂ)).

Note, that its action is diagonal

η(u)_{α} → ∫_{Σ_{τ}} G_{α}(v − u)η_{α}(v)

Taking into account the quasi-periodicity (5.10) we find

G_{α} = φ_{α}(v).

Substituting in (5.27) g(z)|_{on shell} = L(z) and C(v − u) we come to the Yang-Baxter equation (5.7) with r(z−w) replaced by \frac{1}{k} r(z−w). In this way, after the symplectic reduction of the upstairs space R_{2} we come to the two-dimensional phase space R^{S} = {S_{0}, S}/(K_{0}, K_{1}). Remark 5.1 The upstairs space R′ = {g, Ā} is similar to the upstairs space for the elliptic Ruijsenaars-Schneider (ERS) system.^{18} The only difference is that for ERS system the bundle E_{2} is trivial (deg(E_{2}) = 0). There exists the symplectic map from the phase space R^{ERS} of the two-body ERS to R^{S}. In fact, it is the same map that works for the two-body elliptic Calogero-Moser (ECM) system - SL(2, ℂ)-elliptic rotator correspondence.^{28} It follows from the statement that ERS and ECM are governed by the same r-matrix.^{29} This symplectic map is a twist of the r-matrices. Note that the Hamiltonian of the elliptic rotator H^{rot} comes from the representation of the Casimir

K_{1} = \frac{1}{2}S_{0} + H^{rot}.

5.3 Symplectic reduction in the Sklyanin algebra

Here reproduce part of results from Ref. 19. To accomplished our general scheme we introduce the phase space

R_{S} = {S_{0}, S⃗; ξ_{0}, ξ⃗},

and the symplectic form (3.19)

ω = Dξ_{0} ∧ DS_{0} + Dξ⃗ ∧ DS⃗.

Note that we put here the central charge k = 0 and therefore the Poisson sigma-model is one-dimensional. The form ω is invariant with respect the algebroid action (2.4), (3.21)

δ_{ε}S_{0} = −2J_{βγ}S_{β}S_{γ}ε_{α}, δ_{ε}S_{α} = −2ϵ_{αβγ}S_{0}S_{γ}ε_{β},
δ_{ε}ξ_{0} = −2ϵ_{αβγ}S_{γ}ξ_{β}ε_{α}
δ_{ε}ξ_{α} = 2J_{αβ}S_{β}ξ_{γ}ε_{0} − 2ϵ_{αβγ}S_{0}ξ_{γ}ε_{β}.

The moment map takes the form

μ_{0} = 2J_{βγ}S_{β}S_{γ}ξ_{α}, μ_{α} = 2ϵ_{αβγ}S_{0}S_{γ}ξ_{β} − 2J_{βγ}S_{β}S_{γ}ξ_{0}.

Assume that

μ_{0} = μ_{2} = μ_{3} = 0, μ_{1} = 2ν.  (5.28)

The Hamiltonian vector field V_{ε} preserving this form of the moment has the form

δ_{ε}S_{1} = 0, δξ_{1} = 0,
δ_{ε}S_{0} = J_{23}S_{2}S_{3}ε, δ_{ε}ξ_{0} = (ξ_{2}S_{3} − ξ_{3}S_{2})ε,
δ_{ε}S_{2} = −S_{0}S_{3}ε, δ_{ε}ξ_{2} = −(S_{0}ξ_{3} + J_{23}S_{3}ξ_{0})ε,  (5.29)
δ_{ε} = S_{0}S_{2}ε, δ_{ε} = (S_{0}ξ_{2} − J_{23}S_{2}ξ_{0})ε.

The reduced phase space

R_{S}^{red} = {μ_{0} = μ_{2} = μ_{3} = 0, μ_{1} = 2ν}/δ_{ε} action (5.29)

is parameterized by the canonical coordinates (v_{1}, v_{2}; u_{1}, u_{2}) such that

ξ_{0} = u_{1}, ξ_{1} = 0, ξ_{2} = 0, ξ_{3} = u_{2},

S_{0} = v_{1}, S_{1} = 0, S_{2} = −\frac{ν}{v_{1}u_{2} − J_{32}v_{2}u^{1}} , S_{3} = v_{2},

and

ω|_{R_{S}^{red}} = ω^{red} = Dv_{1} ∧ Du_{1} + Dv_{2} ∧ Du_{2}.

In these coordinates the Casimirs (5.3) being reduced on R_{S}^{red} take the form

K_{0} = \frac{v_{1}^{2}}{2} + \frac{ν^{2}}{2(v_{1}u_{2} − J_{32}u_{1}v_{2})^{2}} , K_{1} = \frac{1}{2}(v_{1}^{2} + J_{32}v_{2}^{2}).

Since they commute this two-dimensional system is completely integrable. In the trigonometric limit (Imτ → ∞) e_{2} = e_{3} and thereby J_{32} = 0. Then v_{1} = const and K_{0} describes the rational two-body Calogero system with the coupling constant depending on v_{1}. The Sklyanin algebra is still quadratic in this limit. Nevertheless, the result of the symplectic reduction is essentially the same as when the upstairs space has the linear gl(2, ℂ) brackets.^{30} Acknowledgments

I am grateful for hospitality the Max-Planck-Institut für Mathematik (Bonn), where the paper was prepared. I would like to thank A. Odessky for illuminating discussion about the elliptic algebras. The work was partly supported by grants RFBR-00-02-16530, 00-15-96557 for support of scientific schools, and INTAS-99-01782.

References

1. M.S. Marinov and M.V. Terentev, “Dynamics on the group manifolds and the path integral,” Fortschr. Phys. 27, 511 (1979).

2. A. Levin and M. Olshanetsky, “Hamiltonian Algebroid Symmetries in W-gravity and Poisson sigma-model,” hep-th/0010043.

3. I. Batalin, “Quasigroup construction and first class constraints,” J. Math. Phys. 22, 1837 (1981).

4. M. Hennaux and C. Teitelbom, Quantization of Gauge Systems, (Princeton Univ. Press, Princeton, New Jersey, 1992).

5. N. Ikeda, “Two-dimensional gravity and nonlinear gauge theory,” Ann. Phys. 235, 435-464 (1994).

6. P. Schaller and Th. Strobl, “Poisson Structure Induced (Topological) Field Theories,” Mod. Phys. Lett. A 9, 3129-3136 (1994).

7. A. Polyakov, “Gauge transformations and diffeomorphisms,” Int. Jorn. Mod. Phys. A 5, 833-842 (1990).

8. V. Fock, “Towards the geometrical sense of operator expansions for chiral currents and W -algebras,” Preprint ITEP (1990); A. Bilal, V. Fock and Ia. Kogan, “On the origin of W -algebras,”Nucl. Phys. B359, 635–672 (1991).

9. A. Gerasimov, A. Levin and A. Marshakov, “On W -gravity in twodimensions,” Nucl. Phys. B 360, 537-558 (1991).

10. S. Carlip, Quantum gravity in 2+1 dimensions, (Cambridge Univ. press, Cambridge 1998).

11. C. Teleman, “Sur les structures homographiques d’une surface de Riemann,” Comment. Math. Helv. 33 , 206-211 (1959).

12. A. Beilinson and V. Drinfeld, “Opers,” preprint (1993).

13. M. Adler, “On a trace functional for formal pseudodifferential operators and the symplectic structure of the Korteveg-de Vries equations,” Inv. Math. 50, 219-248 (1979).

14. I.M. Gelfand and L.A. Dikii, “A family of Hamiltonian structures related to nonlinear integrable differential equations,” in Collected papers of I.M.Gelfand, Vol 1, Berlin, Heidelberg, New York : Springer (1987).

15. S. Barannikov and M. Kontsevich, “Frobenius manifolds and formality of Lie algebras of polyvector fields,” Internat. Math. Res. Notices No 4, 201-215 (1998).

16. J. Thierry-Mieg, “BRS analysis of Zamolodchikov’s spin 2 and 3 current algebras,” Phys. Lett. B197, 368-372 (1987).

17. E. Sklyanin, “Some algebraic structures connected with the Yang-Baxter equation,” Funct. Anal. Appl. 16, 283 (1982).

18. G. Arutyunov, S. Frolov and P. Medvedev, “Elliptic Ruijsenaars-Schneider model from the cotangent bundle over the two-dimensional current group,” hep=th/9608013.

19. V. Dolgushev, “Sklyanin Bracket and Deformation of the Calogero-Moser System”, Mod. Phys. Lett. A16, 1711 (2001).

20. K. Mackenzie, “Lie Groupoids and Lie Algebroids in Differential Geometry,” London Math. Soc. Lect. Notes series, Cambridge Univ. Press, 124, (1987).

21. M. Karasev and V. Maslov, Nonlinear Poisson Brackets. Geometry and Quantization, (in Russian), (Nauka, 1991).

22. A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, (Berkeley Mathematical Lectures, Amer. Math. Soc., Providence 10, 1999).

23. M. Karasev, “Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets,” Math. USSR Izvestia 28, 497-527 (1987).

24. B. Fuchssteiner, “The Lie algebra structure of degenerate Hamiltonian and bi-Hamiltonian systems” Progr. Theor. Phys. 68, 1082-1104 (1982).

25. N. Hitchin, “Lie groups and Teichmuller theory,” Topology 31, 451-487 (1992).

26. N.I. Akhiezer, “ Elements of the theory of elliptic functions,” AMS, Providence, (1990)

27. N. Hitchin, “Stable bundles and integrable systems”, Duke Math. Jour. 54, 91-114 (1987).

28. A. Levin, M. Olshanetsky and A. Zotov, “Hitchin Systems - symplectic maps and two-dimensional version”, nlin.SI/0110045.

29. Yu. Suris, “Elliptic Ruijsenaars-Schneider and Calogero-Moser hierarchies are governed by the same r-matrix,” solv-int/9603011.

30. M. Olshanetsky and A. Perelomov, “Explicit solution of the Calogero model in the classical case and geodesic flows on symmetric spaces of zero curvature,” Lett. Nuovo. Cim. 16, 333-339 (1976).