ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Pages 119–120 (November 4, 1997) S 1079-6762(97)00034-6

QUANTIZATION OF POISSON STRUCTURES ON R^{2}

DMITRY TAMARKIN

(Communicated by Alexandre Kirillov)

Abstract. An ‘isomorphism’ between the ‘moduli space’ of star products on R^{2} and the ‘moduli space’ of all formal Poisson structures on R^{2} is established.

The problem of quantization of Poisson structures has been posed in [1]. It is well known that any Poisson structure on a two-dimensional manifold is quantizable. In this paper we establish an ‘isomorphism’ between the ‘moduli space’ of star products on R^{2} and the ‘moduli space’ of all formal Poisson structures on R^{2} by construction of a map from Poisson structures to star products. Certainly, this isomorphism follows from the Kontsevich formality conjecture [2]. Most likely, our map can be used as a first step in constructing an L_{∞}-quasiisomorphism in the formality conjecture for R^{2}. The author would like to thank Boris Tsygan and Paul Bressler for the attention and helpful suggestions.

The set of all star-products S is acted upon by the group D ×^{.} DiffeoR^{2}, where D is the group of operators of the form 1 + hD_{1} + h^{2}D_{2} + ⋯ with D_{k} to be arbitrary differential operators. The set of all formal Poisson structures P consists of formal series in h with bivector fields as the coefficients. Formal Poisson structures are acted upon by the group Diffeo R^{2} ×^{.} exp(hVect[[h]]), where Vect is the Lie algebra of vector fields on R^{2}. These actions define equivalence relations. We want to have a pair of maps f_{1} : S → P and f_{2} : P → S such that

f_{1} ∘ f_{2}(x) ∼ x, f_{2} ∘ f_{1}(x) ∼ x,

x ∼ y → f_{1,2}x ∼ f_{1,2}y. (1)

By a map from S we mean a differential expression in terms of the coefficients of the bidifferential operators corresponding to the star products. Maps from P are defined similarly.

We can replace S by a subspace. Let P, Q be a nondegenerate pair of (real) polarizations of R^{2}. Define a subset S_{P,Q} of S in the following way: m ∈ S_{P,Q} iff m(f, g) = fg if f is constant along P or g is constant along Q.

Proposition 1. Let x, y be a nondegenerate coordinate system on R^{2} such that x is constant along Q and y is constant along P. Then there exists a unique map S → D : m ↦ U(m) = 1 + hV (m) such that

\begin{matrix}1) & m_{P,Q}(m) = U^{−1}(m(Uf, Ug)) ∈ S_{P,Q}, \\ 2) & Ux = x, Uy = y, U1 = 1. \end{matrix} (2)

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

Received by the editors September 2, 1997.

1991 Mathematics Subject Classification. Primary 81Sxx.

U is uniquely defined by the condition U(x^{∗m} ∗ y^{∗n}) = x^{m}y^{n} (where star denotes the star product m).

We denote by m_{P,Q} : S → S_{P,Q} the map which sends m to m_{P,Q}(m). Further, x, y will mean the same as in Proposition 1. Thus, it is enough to find maps p_{1} : S_{P,Q} → P and p_{2} : P → S_{P,Q} with the same properties as f_{1}, f_{2} have. Indeed, put

f_{2} = i ∘ p_{2}, f_{1} = p_{1} ∘ m_{P,Q} (3)

(here i : S_{P,Q} → S is the inclusion).

The following theorem gives an explicit construction for p_{2} which appears to be a bijective map so that we can put p_{1} = p_{2}^{−1}. Denote by C_{P} (resp. C_{Q}) the space of functions, constant along Q (resp. P). Denote by V_{P} (resp. V_{Q}) the space of vector fields preserving the polarizations and tangent to P (resp. Q). Denote by D_{P} the subalgebra of the algebra of differential operators consisting of operators D such that D(C_{Q}) ⊂ C_{Q} and D(fg) = fD(g) if f ∈ C_{P}. Denote by D_{Q} the same algebra where P and Q are interchanged. In the coordinates x, y we have C_{P} = {f(x)}, V_{P} = {f(x)∂_{x}}, D_{P} = ∑ f_{i}(x)∂_{x}^{i} and the same things with P replaced by Q and x replaced by y. Denote by \overline{D_{P}} (resp. \overline{D_{Q}}) the subring of D_{P} (resp. D_{Q}) consisting of operators which annihilate constant functions.

Note that the space of bivector fields is isomorphic to V_{P} ⊗_{R} V_{Q}. Let D_{P,k} be the space of maps V p^{⊗k} → \overline{D_{P}} (which are differential operators in terms of the coefficients).

Theorem 1. a) There exists a unique sequence c_{k} ∈ D_{P,k} ⊗ D_{Q,k}, k = 0, 1, 2, …, c_{k} = ∑_{i} a_{k}^{i} ⊗ b_{k}^{i}, c_{0}(X, Y) = 1 ⊗ 1, such that for any bivector field Ψ = ∑_{i} X_{i} ∧ Y_{i}, X_{i} ∈ V_{P} , Y_{i} ∈ V_{Q}, the formula

\begin{aligned}m(Ψ, P, Q, f, g) & = fg + ∑_{k,i_{1},... ,i_{k+1}} h^{k+1}L_{X_{i_{1}}} \{a_{k}^{n}(X_{i_{2}}, X_{i_{3}}, …, X_{i_{k+1}})f\} \\ & × L_{Y_{i_{1}}} \{b_{k}^{n}(Y_{i_{2}}, Y_{i_{3}}, …, Y_{i_{k+1}})g\} \\ & = ∑_{k} m_{k}(f, g). \end{aligned} (4)

gives a star-product.

b) Put p_{2} : P → S_{P,Q} : Ψ → m(Ψ, P, Q, ·, ·). Put p_{1} = p_{2}^{−1}. Then p_{1} and p_{2} provide an isomorphism of P and S by (3).

References

[1] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, Deformation theory and quantization, I, II, Ann. Phys. 11 (1978), 61–151. MR 58:14737a, MR 58:14737b

[2] M. Kontsevich, Formality conjecture, preprint, to appear in Proc. of Summer School on Deformation Quantization in Ascona.

Department of Mathematics, Pennsylvania State University, 218 McAllister Building, University Park, PA 16802

   E-mail address: tamarkin@math.psu.edu