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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Symplectic Manifolds

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The canonical example of the n-symplectic manifold is that of the frame bundle, so the question is whether this formalism can be generalized to other principal bundles, and distinguished from the quantization arising from symplectic geometry on the prototype manifold, the bundle of linear frames, a good place to motivate the formalism.

Let us start with an n-dimensional manifold M, and let π : LM → M be the space of linear frames over a base manifold M, the set of pairs (m,ek), where m ∈ M and {ek},k = 1,···,n is a linear frame at m. This gives LM dimension n(n + 1), with GL(n,R) as the structure group acting freely on the right. We define local coordinates on LM in terms of those on the manifold M – for a chart on M with coordinates {xi}, let

qi(m,ek) = xi ◦ π(m,ek) = xi(m)

πji(m,ek) = ej ∂/∂xj

where {ej} denotes the coframe dual to {ej}. These coordinates are analogous to those on the cotangent bundle, except, instead of a single momentum coordinate, we now have a momentum frame. We want to place some kind of structure on LM, which is the prototype of n-symplectic geometry that is similar to symplectic geometry of the cotangent bundle T∗M. The structure equation for symplectic geometry

df= _| X dθ

gives Hamilton’s equations for the phase space of a particle, where θ is the canonical symplectic 2-form. There is a naturally defined Rn-valued 1-form on LM, the soldering form, given by

θ(X) ≡ u−1[π∗(X)] ∀X ∈ TuLM

where the point u = (m,ek) ∈ LM gives the isomorphism u : Rn → Tπ(u)M by ξiri → ξiei, where {ri} is the standard basis of Rn. The Rn-valued 2-form dθ can be shown to be non-degenerate, that is,

X _| dθ = 0 ⇔ X = 0

where we mean that each component of X dθ is identically zero. Finally, since there is also a structure group on LM, there are also group transformation properties. Let ρ be the standard representation of GL(n, R) on Rn. Then it can be shown that the pullback of dθ under right translation by g ∈ GL (n,R) is Rg dθ = ρ(g−1) · dθ.

Thus, we have an Rn-valued generalization of symplectic geometry, which motivates the following definition.

Let P be a principal fiber bundle with structure group G over an m-dimensional manifold M . Let ρ : G → GL(n, R) be a linear representation of G. An n-symplectic structure on P is a Rn-valued 2-form ω on P that is (i) closed and non-degenerate, in the sense that

X _| ω = 0 ⇔ X = 0

for a vector field X on P, and (ii) ω is equivariant, such that under the right action of G, Rg ω = ρ(g−1) · ω. The pair (P, ω) is called an n-symplectic manifold.

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Here, we have modeled n-symplectic geometry after the frame bundle by defining the general n-symplectic manifold as a principal bundle. There is no reason, however, to limit ourselves to this, since we can let P be any manifold with a group action defined on it. One example of this would be to look at the action of the conformal group on R4. Since this group is locally isomorphic to O(2, 4), which is not a subgroup of GL(4, R), then forming a O(2,4) bundle over R4 cannot be thought of as simply a reduction of the frame bundle.

From Vector Spaces to Categories. Part 6.

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We began by thinking of categories as “posets with extra arrows”. This analogy gives excellent intuition for the general facts about adjoint functors. However, our intuition from posets is insufficient to actually prove anything about adjoint functors.

To complete the proofs we will switch to a new analogy between categories and vector spaces. Let V be a vector space over a field K and let V ∗ be the dual space consisting of K-linear functions V → K. Now consider any K-bilinear function ⟨−,−⟩ ∶ V × V → K. We say that the function ⟨−,−⟩ is non-degenerate in both coordinates if we have

⟨u1,v⟩ = ⟨u2,v⟩ ∀ v ∈ V ⇒ u1 = u2, ⟨u,v1⟩ = ⟨u,v2⟩ ∀ u ∈ V ⇒ v1 = v2

We say that two K-linear operators L ∶ V ⇄ V ∶ R define an adjunction with respect to ⟨−, −⟩ if, ∀ vectors u,v ∈ V, we have

⟨u, R(v)⟩ = ⟨L(u), v⟩

Uniqueness of Adjoint Operators. Let L ⊣ R be an adjoint pair of operators with respect to a non-degenerate bilinear function ⟨−, −⟩ ∶ V × V → K. Then each of L and R determines

the other uniquely.

Proof: To show that R determines L, suppose that L′ ⊣ R is another adjoint pair. Thus, ∀ vectors u,v ∈ V we have

⟨L(u), v⟩ = ⟨u, R(v)⟩ = ⟨L′(u), v⟩

Now consider any vector u ∈ V. The non-degeneracy of ⟨−, −⟩ tells us that

⟨L(u), v⟩ = ⟨L′(u), v⟩ ∀ v ∈ V ⇒ L(u) = L′(u)

and since this is true ∀ u ∈ V we conclude that L = L′

RAPL for Operators:

Suppose that the function ⟨−, −⟩ ∶ V × V → K is non-degenerate and continuous. Now let T ∶ V → V be any linear operator. If T has a left or a right adjoint, then T is continuous.

Proof:

Suppose that T ∶ V → V has a left adjoint L ⊣ T, and suppose that the sequence of vectors vi ∈ V has a limit limivi ∈ V. Furthermore, suppose that the limit limiT(vi) ∈ V exists. Then for each u ∈ V, the continuity of ⟨−, −⟩ in the second coordinate tells us that

⟨u, T (limivi)⟩ = ⟨L(u), limivi

= limi⟨L(u), vi

= limi⟨u,T(vi)⟩

= ⟨u, limiT (vi)⟩

Since this is true for all u ∈ V, non-degeneracy gives

T (limivi) = limiT (vi)

The theorem can be made rigorous if we work with topological vector spaces. If (V, ∥ − ∥) is a normed (real or complex) vector space, then an operator T ∶ V → V is bounded if and only if it is continuous. Furthermore, if (V,⟨−,−⟩) is a Hilbert space then an operator T ∶ V → V having an adjoint is necessarily bounded, hence continuous. Many theorems of category have direct analogues in functional analysis. After all, Grothendieck began as a functional analyst.

We can summarize these two results as follows. Let ⟨−,−⟩ ∶ V ×V → K be a K-bilinear function. Then for each vector v ∈ V we have two elements of the dual space Hv, Hv ∈ V defined by

Hv ∶= ⟨v,−⟩ ∶ V → K,

Hv ∶= ⟨−,v⟩ ∶ V → K

The mappings v ↦ Hv and v ↦ Hv thus define two K-linear functions from V to V : H(−) ∶V → V and H(−) ∶ V → V

Furthermore, if the function is ⟨−,−⟩ is non-degenerate and continuous then the functions H(−), H(−) ∶ V → V are both injective and continuous.

the hom bifunctor

HomC(−,−) ∶ Cop × C → Set behaves like a “non-degenerate and continuous bilinear function”……