Odd symplectic + Odd Poisson Geometry as a Generalization of Symplectic (Poisson) Geometry to the Supercase

A symplectic structure on a manifold M is defined by a non-degenerate closed two-form ω. In a vicinity of an arbitrary point one can consider coordinates (x1, . . . , x2n) such that ω = ∑ni=1 dxidxi+n. Such coordinates are called Darboux coordinates. To a symplectic structure corresponds a non-degenerate Poisson structure { , }. In Darboux coordinates {xi,xj} = 0 if |i−j| ≠ n and {xi,xi+n} = −{xi+n,xi} = 1. The condition of closedness of the two-form ω corresponds to the Jacobi identity {f,{g,h}} + {g,{h,f}} + {h,{f,g}} = 0

for the Poisson bracket. If a symplectic or Poisson structure is given, then every function f defines a vector field (the Hamiltonian vector field) Df such that Dfg = {f,g} = −ω(Df,Dg).

A Poisson structure can be defined independently of a symplectic structure. In general it can be degenerate, i.e., there exist non-constant functions f such that Df = 0. In the case when a Poisson structure is non-degenerate (corresponds to a symplectic structure), the map from T∗M to T M defined by the relation f → Df is an isomorphism.

One can straightforwardly generalize these constructions to the supercase and consider symplectic and Poisson structures (even or odd) on supermanifolds. An even (odd) symplectic structure on a supermanifold is defined by an even (odd) non-degenerate closed two-form. In the same way as the existence of a symplectic structure on an ordinary manifold implies that the manifold is even-dimensional (by the non-degeneracy condition for the form ω), the existence of an even or odd symplectic structure on a supermanifold implies that the dimension of the supermanifold is equal either to (2p.q) for an even structure or to (m.m) for an odd structure. Darboux coordinates exist in both cases. For an even structure, the two-form in Darboux coordinates

zA = (x1,…, x2p1,…, θq) has the form ∑i=1p dxi dxp+i + ∑a=1q εaaa,

where εa = ±1. For an odd structure, the two-form in Darboux coordinates zA = (x1,…,xm1,…,θm) has the form ∑i=1m dxii.

The non-degenerate odd Poisson bracket corresponding to an odd symplectic structure has the following appearance in Darboux coordinates: {xi, xj} = 0, {θij} = 0 for all i,j and {xij} = −{θj,xi} = δji. Thus for arbitrary two functions f, g

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where we denote by p(f) the parity of a function f (p(xi) = 0, p(θj) = 1). Similarly one can write down the formulae for the non-degenerate even Poisson structure corresponding to an even symplectic structure.

A Poisson structure (odd or even) can be defined on a supermanifold independently of a symplectic structure as a bilinear operation on functions (bracket) satisfying the following relations taken as axioms:

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where ε is the parity of the bracket (ε = 0 for an even Poisson structure and ε = 1 for an odd one). The correspondence between functions and Hamiltonian vector fields is defined in the same way as on ordinary manifolds: Dfg = {f, g}. Notice a possible parity shift: p(Df) = p(f) + ε. Every Hamiltonian vector field Df defines an infinitesimal transformation preserving the Poisson structure (and the corresponding symplectic structure in the case of a non-degenerate Poisson bracket).

Notice that even or odd Poisson structures on an arbitrary supermanifold can be obtained as “derived” brackets from the canonical symplectic structure on the cotangent bundle, in the following way.

Let M be a supermanifold and T∗M be its cotangent bundle. By changing parity of coordinates in the fibres of T∗M we arrive at the supermanifold ΠT ∗M. If zA are arbitrary coordinates on the supermanifold M, then we denote by (zA,pB) the corresponding coordinates on the supermanifold T∗M and by (zA,z∗B) the corresponding coordinates on ΠT∗M: p(zA) = p(pA) = p(z∗A) + 1. If (zA) are another coordinates on M, zA = zA(z′), then the coordinates z∗A transform in the same way as the coordinates pA (and as the partial derivatives ∂/∂zA):

pA = ∂zB(z′)/∂zA pB and z∗A = ∂zB(z′)/∂zA z∗B

One can consider the canonical non-degenerate even Poisson structure { , }0 (the canonical even symplectic structure) on T∗M defined by the relations {zA,zB}0 = {pC,pD}0 = 0, {zA,pB}0 = δBA, and, respectively, the canonical non-degenerate odd Poisson structure { , }1 (the canonical odd symplectic structure) on ΠT∗M defined by the relations {zA,zB}0 = {z∗C,z∗D}0 = 0, {zA,z∗B}0 = δAB.

Now consider Hamiltonians on T∗M or on ΠT∗M that are quadratic in coordinates of the fibres. An arbitrary odd quadratic Hamiltonian on T∗M (an arbitrary even quadratic Hamiltonian on ΠT∗M):

S(z,p) = SABpApB (p(S) = 1) or S(z,z∗) = SABz∗Az∗B (p(S) = 0) —– (1)

satisfying the condition that the canonical Poisson bracket of this Hamiltonian with itself vanishes:

{S,S}0 = 0 or {S,S}1 = 0 —– (2)

defines an odd Poisson structure (an even Poisson structure) on M by the formula

{f,g}Sε+1 = {f,{S,g}ε}ε —–(3)

The Hamiltonian S which generates an odd (even) Poisson structure on M via the canonical even (odd) Poisson structure on T∗M (ΠT∗M) can be called the master Hamiltonian. The bracket is a “derived bracket”. The Jacobi identity for it is equivalent to the vanishing of the canonical Poisson bracket for the master Hamiltonian. One can see that an arbitrary Poisson structure on a supermanifold can be obtained as a derived bracket.

What happens if we change the parity of the master Hamiltonian in (3)? The answer is the following. If S is an even quadratic Hamiltonian on T∗M (an odd quadratic Hamiltonian on ΠT∗M), then the condition of vanishing of the canonical even Poisson bracket { , }0 (the canonical odd Poisson bracket { , }1) becomes empty (it is obeyed automatically) and the relation (3) defines an even Riemannian metric (an odd Riemannian metric) on M.

Formally, odd symplectic (and odd Poisson) geometry is a generalization of symplectic (Poisson) geometry to the supercase. However, there are unexpected analogies between the constructions in odd symplectic geometry and in Riemannian geometry. The construction of derived brackets could explain close relations between odd Poisson structures in supermathematics and the Riemannian geometry.

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