Bihamiltonian Structures of Pdes and Frobenius Manifolds

SMR1665/11

Summer School and Conference on Poisson Geometry

(4 -22 July 2005)

Bihamiltonian structures of PDEs and Frobenius manifolds

Boris Dubrovin SISSA Trieste, Italia

Bihamiltonian structures of PDEs and Frobenius manifolds

Lectures at the ICTP Summer School “Poisson Geometry”,

Trieste, July 11-15, 2005

Boris Dubrovin SISSA, Trieste

July 12, 2005

Abstract

These complementary lecture notes are based on the papers [50], [52], [53].

1 Contents

1 Lecture 1 3 1.1 Brief summary of finite-dimensional Poisson geometry ...... 3 1.1.1 Poisson brackets ...... 3 1.1.2 Dirac bracket ...... 4 1.1.3 Hamiltonian vector fields, first integrals ...... 5 1.1.4 Poisson cohomology ...... 5 1.1.5 Formalism of supermanifolds ...... 8 1.2 Bihamiltonian structures ...... 9 1.2.1 Magri chains ...... 9 1.2.2 Symplectic bihamiltonian structures. Recursion operator . . . . 10 1.2.3 Poisson pencils of constant rank. Construction of hierarchies . . 11 1.2.4 Remark about the “method of argument translation” ...... 13 1.2.5 From Poisson pencils of constant rank to commuting hierarchies 14

2 Lecture 2. 14 2.1 Formal loop spaces ...... 14 2.2 Local multivectors and local Poisson brackets ...... 24

3 Lecture 3. Problem of classification of local Poisson brackets 31 3.1 Extended formal loop space ...... 31 3.2 Miura group ...... 35 3.3 (p, q)-brackets on the extended formal loop space ...... 37 3.3.1 Classification of (n, 0)-brackets ...... 41 3.3.2 Classification of (0, n)-brackets ...... 44

2 1 Lecture 1

1.1 Brief summary of ﬁnite-dimensional Poisson geometry

1.1.1 Poisson brackets

Let P be a N-dimensional smooth manifold. A Poisson bracket on P is a structure of a Lie algebra on the ring of functions F := C∞(P )

f, g 7→ {f, g},

{g, f} = −{f, g}, {af + bg, h} = a{f, h} + b{g, h}, a, b ∈ R, f, g, h ∈ F (1.1.1) {{f, g}, h} + {{h, f}, g} + {{g, h}, f} = 0 (1.1.2) satisfying the Leibnitz rule

{fg, h} = f{g, h} + g{f, h} for arbitrary three functions f, g, h ∈ F. In a system of local coordinates x1, . . . , xN the Poisson bracket reads ∂f ∂g {f, g} = πij(x) (1.1.3) ∂xi ∂xj (summation over repeated indices will be assumed) where the bivector πij(x) = −πji(x) = {xi, xj} satisﬁes the following system of equations equivalent to the Jacobi identity (1.1.2)

∂πij ∂πki ∂πjk {{xi, xj}, xk} + {{xk, xi}, xj} + {{xj, xk}, xi} ≡ πsk + πsj + πsi = 0 ∂xs ∂xs ∂xs (1.1.4) for any i, j, k. Such a bivector satisfying (1.1.4) is called Poisson structure on P . Clearly any bivector constant in some coordinate system is a Poisson structure. Vice versa [98], locally all solutions to (1.1.4) of the constant rank 2n = rk(πij) can be reduced, by a change of coordinates, to the following normal form

π¯ 0 π = (1.1.5) 0 0 with a constant nondegenerate antisymmetric 2n × 2n matrixπ ¯ =π ¯ab. That means that locally there exist coordinates y1, . . . , y2n, c1, . . . , ck, 2n + k = N, s.t.

π¯ab = {ya, yb} and {f, cj} = 0, j = 1, . . . k (1.1.6) for an arbitrary function f.

3 ij −1 For the case 2n = N the inverse matrix (πij(x)) = (π (x)) deﬁnes on P a symplectic structure X i j n Ω = πij(x)dx ∧ dx , Ω 6= 0. i

For 2n < N one obtains on P a structure of symplectic foliation P = ∪c0 Pc0 , c0 = 1 k (c0, . . . c0), of the codimension k = N − 2n

1 1 k k Pc0 := {x | c (x) = c0, . . . , c (x) = c0}. (1.1.7)

The independent functions c1(x), . . . , ck(x) deﬁned in (1.1.6) are called Casimir functions, or simply Casimirs of the Poisson structure. Every leaf Pc0 is a symplectic ∞ ∞ manifold, and the restriction map C (P ) → C (Pc0 ) is a homomorphism of Lie algebras.

Example 1.1.1 Let g be n-dimensional Lie algebra. The Lie - Poisson bracket on the dual space P = g∗ reads i j ij k {x , x } = ck x . (1.1.8) ij Here ck are the structure constants of the Lie algebra. The Casimirs of this bracket are functions on g∗ invariant with respect to the co-adjoint action of the associated Lie group G. The symplectic leaves coincide with the orbits of the coadjoint action with the Berezin - Kirillov - Kostant symplectic structure on them.

1.1.2 Dirac bracket

An arbitrary foliation P = ∪φ0 Pφ0 of a codimension m represented locally in the form

1 1 m m Pφ0 = {x | φ (x) = φ0, . . . , φ (x) = φ0 } will be called cosymplectic if the m × m matrix {φa, φb} does not degenerate on the leaves. In this situation a new Poisson structure { , }D can be deﬁned on P s.t. the a functions φ (x) are Casimirs of { , }D. This is the Dirac bracket given explicitly by the formula X a a b −1 b {f, g}D = {f, g} − {f, φ }{φ , φ } {φ , g}. (1.1.9) a,b It can be restricted in an obvious way to produce a Poisson structure on every leaf. The restriction map ∞ ∞ (C (P ), { , }) → (C (Pφ0 ), { , }D) is a homomorphism of Lie algebras.

4 1.1.3 Hamiltonian vector ﬁelds, ﬁrst integrals

A Poisson bracket deﬁnes an (anti)homomorphism F → V ect(P )

H 7→ XH := {·,H}, (1.1.10)

[XH1 ,XH2 ] = −X{H1,H2}.

XH is called Hamiltonian vector ﬁeld. The corresponding dynamical system ∂H x˙ i = πij(x) (1.1.11) ∂xj is called Hamiltonian system with the Hamiltonian H(x). It is a symmetry of the Poisson bracket

LieXH { , } = 0. (1.1.12) Any function F commuting with the Hamiltonian {F,H} = 0 is a ﬁrst integral of the Hamiltonian system (1.1.11). The Hamiltonian vector ﬁelds XH , XF commute.

1.1.4 Poisson cohomology

The notion of Poisson cohomology of (P, π) was introduced by Lichnerowicz [98]. We need to use the Schouten - Nijenhuis bracket. Denote Λk = H0(P, ΛkTP ) the space of multivectors on P . The Schouten - Nijenhuis bracket is a bilinear pairing a, b 7→ [a, b], Λk × Λl → Λk+l−1 uniquely determined by the properties of supersymmetry [b, a] = (−1)kl[a, b], a ∈ Λk, b ∈ Λl (1.1.13) the graded Leibnitz rule [c, a ∧ b] = [c, a] ∧ b + (−1)lk+ka ∧ [c, b], a ∈ Λk, c ∈ Λl (1.1.14) and the conditions [f, g] = 0, f, g ∈ Λ0 = F, ∂f [v, f] = vi , v ∈ Λ1 = V ect(P ), f ∈ Λ0 = F, ∂xi 1 [v1, v2] = commutator of vector ﬁelds for v1, v2 ∈ Λ . In particular for a vector ﬁeld v and a multivector a [v, a] = Lieva.

5 Example 1.1.2 For two bivectors π = (πij) and ρ = (ρij) their Schouten - Nijenhuis bracket is the following trivector ∂πij ∂ρij ∂πki ∂ρki ∂πjk ∂ρjk [π, ρ]ijk = ρsk + πsk + ρsj + πsj + ρsi + πsi. (1.1.15) ∂xs ∂xs ∂xs ∂xs ∂xs ∂xs Observe that the l.h.s. of the Jacobi identity (1.1.4) reads 1 {{xi, xj}, xk} + {{xk, xi}, xj} + {{xj, xk}, xi} = [π, π]ijk. 2 The Schouten - Nijenhuis bracket satisﬁes the graded Jacobi identity [126] (−1)km[[a, b], c] + (−1)lm[[c, a], b] + (−1)kl[[b, c], a] = 0, a ∈ Λk, b ∈ Λl, c ∈ Λm. (1.1.16) It follows that, for a Poisson bivector π the map ∂ :Λk → Λk+1, ∂a = [π, a] (1.1.17) is a diﬀerential, ∂2 = 0. The cohomology of the complex (Λ∗, ∂) is called Poisson cohomology of (P, π). We will denote it

∗ k H (P, π) = ⊕k≥0H (P, π). In particular, H0(P, π) coincides with the ring of Casimirs of the Poisson bracket, H1(P, π) is the quotient of the Lie algebra of inﬁnitesimal symmetries

v ∈ V ect(P ), Lievπ = 0 over the subalgebra of Hamiltonian vector fields, H2(P, {pi) is the quotient of the space of infinitesimal deformations of the Poisson bracket by those obtained by infinitesimal changes of coordinates (i.e., by those of the form Lievπ for a vector field v). (See details in the lecture notes of J.P.Dufour and N.T.Zung.) On a symplectic manifold (P, π) Poisson cohomology coincides with the de Rham one. The isomorphism is established by “lowering the indices”: for a cocycle a = (ai1...ik ) ∈ Λk the k-form

X i1 ik j1...jk ωi1...ik dx ∧ ... ∧ dx , ωi1...ik = πi1j1 . . . πikjk a

i1<...

Lemma 1.1.3 Let π = (πij(x)) be a Poisson structure of a constant rank 2n < N on a sufficiently small ball U. 1). A one-cocycle v = (vi(x)) ∈ H1(U, π) is trivial iff the vector field v is tangent to the leaves of the symplectic foliation (1.1.7). 2). A 2-cocycle f = (f ij(x)) ∈ H2(U, π) is trivial iff f(dc0, dc00) = 0 (1.1.18) for arbitrary two Casimirs of h.

6 Of course, the statement of the lemma can be easily derived from the results of [98]. Nevertheless, we give a proof since we will use similar arguments also in the inﬁnite dimensional situation. Proof 1). For a coboundary v = ∂f and for any Casimir c of π we have

∂vc = {c, f} = 0.

This means that v is tangent to the symplectic leaves (1.1.7). To prove the converse statement let us choose the canonical coordinates x = (y1, . . . , y2n, c1, . . . , ck) reducing the bracket to the constant form (1.1.5). Here c1,..., ck are independent Casimirs (1.1.6). In these coordinates v = (v1, . . . , v2n, 0,..., 0). The 1-form ω = (ω1, . . . , ω2n, 0,..., 0) given by

2n X j ωi = π¯ijv j=1 has the property dω| = 0. Pc0 Therefore a function g locally exists s.t.

2n k X i X a dg = ωidy + φadc i=1 a=1 for some functions φ1,..., φk. This function is the Hamiltonian for the vector ﬁeld v. 2). We will again use the canonical coordinates for π as in the proof of the ﬁrst part. For an exact 2-cocycle ρ = ∂v and arbitrary two functions c0, c00

0 00 0 i 00 0 i 00 ρ(dc , dc ) = −{c , v }∂ic − ∂ic {v , c }.

This is equal to zero if c0 and c00 are Casimirs of the bracket π. To prove the converse statement we first consider, for every a = 1, . . . , k, the vector field w (depending on a) wi = ρia, i = 1,...,N. (1.1.19) From (1.1.18) it follows that w is tangent to the symplectic leaves of π. The cocycle condition aij ai kj ja ki ij 0 = [π, ρ] = ∂kρ π + ∂kρ π = (∂w) (1.1.20) implies ∂w = 0. According to the first part of the lemma, there exists a function qa(x) s.t. w = ∂qa: ia ik a ρ = π ∂kq , a = 1, . . . , k. (1.1.21) Let us now change the cocycle by a coboundary

ρ 7→ ρ + ∂z

7 where the vector ﬁeld z is given by

k X ∂ z = qa . (1.1.22) ∂ca a=1 After such a change, due to (1.1.21), we obtain

ρia = ρai = 0, i = 1,...,N.

The rest of the proof repeats the arguments of the ﬁrst part. The 2-form

2n X kl ωij = π¯ikπ¯ljρ i, j=1 is closed along the symplectic leaves. Hence there exists a 1-form φ = (φi) s.t.

ω = dφ +ω ˜ where every monomial inω ˜ contains at least one dca for some a. Therefore

ρ = ∂u for the vector ﬁeld

2n i X ik i u = π¯ φk, i = 1,..., 2n, u = 0, i > 2n. k=1 The lemma is proved.

1.1.5 Formalism of supermanifolds

Consider the (N|N) supermanifold N = Π T ∗P . Coordinates on N:

1 N j i i j x , . . . , x , θ1, . . . , θN , x x = x x , θjθi = −θiθj

Bivector 1 ∂ ∂ 1 π = πij(x) ∧ 7→ πij(x)θ θ =:π ˆ 2 ∂xi ∂xj 2 i j (a superfunction on N = Π T ∗P ). (Super)Poisson bracket on N

∂P ∂Q |P | ∂P ∂Q {P,Q} = i + (−1) i (1.1.23) ∂θi ∂x ∂x ∂θi |P | = parity of the superfunction |P |. Claim: Jacobi identity for π ⇔ {π,ˆ πˆ} = 0

8 Proof ∂πij {π,ˆ πˆ} = πskθ θ θ k ∂xs i j 1 ∂πij ∂πki ∂πjk = πsk + πsj + πsi θ θ θ 3 ∂xs ∂xs ∂xs i j k

More generally (see the lecture notes of P.Cartier), for multivectors a ∈ Λk, b ∈ Λl, 1 1 aˆ = ai1...ik θ . . . θ , ˆb = bj1...jl θ . . . θ k! i1 ik l! j1 jl the super-Poisson bracket {a,ˆ ˆb} = [a, b]ˆ (1.1.24) where [a, b] = Schouten - Nijenhuis bracket of multivectors a and b.

1.2 Bihamiltonian structures

1.2.1 Magri chains

Deﬁnition. A bihamiltonan structure on the manifold P is a 2-dimensional linear subspace in the space of Poisson structures on P .

Choosing two nonproportional Poisson structures π1 and π2 in the subspace we obtain that the linear combination

a1π1 + a2π2 (1.2.25) with arbitrary constant coeﬃcients a1, a2 is again a Poisson bracket. This reformulation is usually referred to as the compatibility condition of the two Poisson brackets. It is spelled out as vanishing of the Schouten - Nijenhuis bracket

[π1, π2] = 0. (1.2.26)

An importance of bihamiltonian structures for recursive constructions of integrable systems was discovered by F.Magri [107] in the analysis of the so-called Lenard scheme (see in [67]) of constructing the KdV integrals. The basic idea of these constructions is given by the following simple

Lemma 1.2.4 Let H0, H1, . . . , be a sequence of functions on P satisfying the recursion relation { .,Hp+1}1 = { .,Hp}2, p = 0, 1,... (1.2.27) Then {Hp,Hq}1 = {Hp,Hq}2 = 0, p, q = 0, 1,...

9 For convenience of the reader we reproduce the proof of the lemma. Let p < q and q − p = 2m for some m > 0. Using the recursion and antisymmetry of the brackets we obtain

{Hp,Hq}1 = {Hp,Hq−1}2 = −{Hq−1,Hp}2 = −{Hq−1,Hp+1}1 = {Hp+1,Hq−1}1.

Iterating we arrive at

{Hp,Hq}1 = ... = {Hp+m,Hq−m}1 = 0 since p + m = q − m. Doing similarly in the case q − p = 2m + 1 we obtain

{Hp,Hq}1 = ... = {Hn,Hn+1}1 = {Hn,Hn}2 = 0 where n = p + m = q − m − 1. The commutativity {Hp,Hq}2 = 0 easily follows from the recursion. The Lemma is proved.

1.2.2 Symplectic bihamiltonian structures. Recursion operator

We have not used yet the compatibility condition of the two brackets. It turns out to be crucial in constructing the Hamiltonians satisfying the recursion relation (1.2.27). There are two essentially different realizations of the recursive procedure. The first one applies to the case when the bihamiltonian structure is symplectic , i.e. N = 2n and the Poisson structures of the affine line (1.2.25) do not degenerate for generic a1, a2. Without loss of generality one may assume nondegeneracy of π1. The recursion operator R : TP → TP is defined by −1 R := π2 · π1 . (1.2.28)

Here we identify the tensor π2 with the linear map

∗ π2 : T P → TP

−1 and π1 with −1 ∗ π1 : TP → T P. The main recursion relation (1.2.27) can be rewritten in the form

∗ dHp+1 = R dHp, p = 0, 1,... (1.2.29) where R∗ : T ∗P → T ∗P is the adjoint operator.

10 Theorem 1.2.5 [107, 106] The Hamiltonians 1 H := tr Rp+1, p ≥ 0 p p + 1 satisfy the recursion (1.2.29).

Clearly there are at most n independent of these commuting functions. We say that the bihamiltonian symplectic structure is generic if exactly n of these functions are independent. Let us denote λi = λi(x) the eigenvalues of the recursion operator. Since the characteristic polynomial of R is a perfect square n Y 2 det (R − λ) = (λ − λi) . i=1 only n of these eigenvalues can be distinct, say, λ1 = λ1(x), . . . , λn = λn(x). For generic bihamiltonian symplectic structure these are independent functions on P 3 x.

Theorem 1.2.6 [107, 108, 106] Let π1,2 be a generic symplectic bihamiltonian structure. Then 1) All the commuting Hamiltonians n 1 1 X H = tr Rp+1 = λp+1(x), p = 0, 1, . . . , n − 1 p p + 1 p + 1 i i=1 generate completely integrable systems on P .

2) The eigenvalues λi(x) can be included in a coordinate system λ1, µ1, . . . , λn, µn in order to reduce the two Poisson structures to a block diagonal form where the i-th block in π1 and in π2 reads, respectively 0 1 0 λ , i , i = 1, . . . , n. − 1 0 − λi 0

The last formula gives the normal form of a generic symplectic bihamiltonian structure. Therefore all such structures are equivalent w.r.t. the group of local diﬀeomor- phisms.

1.2.3 Poisson pencils of constant rank. Construction of hierarchies

Let us now consider the degenerate situation. We assume that the Poisson structure (1.2.25) has constant rank for generic a1 and a2. Without loss of generality we may assume that k = corank π1 ≡ corank(π1 + π2) (1.2.30) for an arbitrary suﬃciently small . Let us ﬁrst prove the following useful property of bihamiltonian structures of the constant rank.

11 Lemma 1.2.7 Let the bihamiltonian structure satisfy (1.2.30). Then the Casimirs of π1 commute w.r.t. π2.

Proof Let 2m be the rank of π1. We first reduce the matrix of this bracket to the canonical constant block diagonal form. Denote (πab) the matrix of the second Poisson bracket in these coordinates. Let us now choose two integers i, j such that 2m < i < j ≤ N = 2m + k and form a (2m + 1) × (2m + 1) minor of the matrix π1 + π2 by adding i-th column and j-th row to the principal 2m × 2m minor standing in the first 2m columns and first 2m rows. The condition (1.2.30) is equivalent to vanishing of the determinants of all these minors. It is easy to see that the determinant in question is equal to − πij + O(2). Therefore πij = 0 for all pairs (i, j) greater than 2m. The lemma is proved.

Corollary 1.2.8 For a compatible pair of Poisson brackets of the constant rank (π2 −λπ1) = rankπ1, λ → ∞, 2 π2 ∈ H (P, π1) is a trivial cocycle.

Proof What π2 is a cocycle w.r.t. the Poisson cohomology of (P, π1) follows from (1.2.26). To prove triviality use commutativity of the Casimirs of the ﬁrst Poisson bracket and also Lemma 1.1.3.

A bihamiltonian structure with a marked line λ π1 is called Poisson pencil. Choos- ing another Poisson bracket π2 of the pencil one can represent the brackets of the bihamiltonian structure in the form

πλ := π2 − λπ1. (1.2.31)

The representation (1.2.31) is well-deﬁned up to an aﬃne change of the parameter λ

λ 7→ a λ + b.

In the case of Poisson pencils of constant rank the corank of πλ equals k for λ → ∞. The recursive construction of the commuting ﬂows in this case is given by the following simple statement (cf. [107, 70, 132]).

Theorem 1.2.9 Under the assumption (1.2.30) the coeﬃcients of the Taylor expansion cα(x) cα(x) cα(x, λ) = cα (x) + 0 + 1 + . . . , λ → ∞ (1.2.32) −1 λ λ2 α of the Casimirs c (x, λ), α = 1, . . . , k of the Poisson bracket πλ commute with respect to both the Poisson brackets

α β {cp , cq }1,2 = 0, α, β = 1, . . . , k, p, q ≥ −1.

12 Proof Spelling out the deﬁnition of the Casimirs

α { . , c }λ = 0 for the coeﬃcients of the expansion (1.2.32) we must have ﬁrst that

α { . , c−1}1 = 0. (1.2.33)

That is, the leading coeﬃcients of the Taylor expansions are Casimirs of { , }1. For the subsequent coeﬃcients we get the recursive relations

α α { . , cp+1}1 = { . , cp }2, p = −1, 0, 1,... (1.2.34) From (1.2.34) and Theorem 1 it follows that

α α {cp , cq }1,2 = 0, p, q ≥ −1.

α β The commutativity {cp , cq }1,2 = 0 for α 6= β easily follows from the same recursion trick and from commutativity of the Casimirs

α β {c−1, c−1}2 = 0 (1.2.35) proved in Lemma 1.2.7. The theorem is proved.

1.2.4 Remark about the “method of argument translation”

Example 1.2.10 According to Corollary 1.2.8 there exists a vector ﬁeld Z such that

LieZ π1 = π2.

We say, following [11] that the bihamiltonian structure is exact if the vector ﬁeld Z can be chosen in such a way that

2 (LieZ ) π1 = 0. (1.2.36)

For an exact bihamiltonian structure the generating functions (1.2.32) of the commuting α Hamiltonians cp (x) have the form 1 1 cα(x; λ) = exp (−Z/λ) cα (x) = cα (x) − ∂ cα (x) + ∂2 cα (x) ... (1.2.37) −1 −1 λ Z −1 λ2 Z −1 for every α = 1, . . . , k.

This formula can be easily proved by choosing a system of local coordinates x1, . . . , xN on the phase space P such that the vector ﬁeld Z corresponds to the shift along x1. In these coordinates the tensor of the ﬁrst Poisson bracket depends linearly on x1 and 1 the second Poisson bracket is x -independent. The Poisson pencil πλ is obtained from 1 1 π1 by the shift x 7→ x − 1/λ and by multiplication by − λ.

13 Conversely, if, in a given coordinate system, π1 depends linearly on one of the coordinates and π2 does not depend on this coordinate then the bihamiltonian structure is exact. In particular this trick can be applied to the standard linear Lie - Poisson structures on the dual spaces to Lie algebras (cf [?]). In this case it was called in [118] the method of argument translation. All our bihamiltonian structures on the loop spaces to be studied below will be exact. However, at the moment we do not see their Lie algebraic origin.

1.2.5 From Poisson pencils of constant rank to commuting hierarchies

The construction of Theorem 1.2.9 for a bihamiltonian structure of the constant corank k produces k chains of pairwise commuting bihamiltonian flows dx = {x, cα} = {x, cα } , α = 1, . . . , k, p = 0, 1, 2,... (1.2.38) dtα,p p 1 p−1 2 α The chains are labeled by the Casimirs c−1 of the first Poisson bracket. The level p in each chain corresponds to the number of iterations of the recursive procedure (we will keep using this expression although the recursion operator is not defined in the degenerate case). All the family of commuting flows organized by the above recursion procedure is called the hierarchy determined by the bihamiltonian structure. The hierarchy structure of the constructed family of commuting flows depends non- trivially on the choice of { , }1 in the Poisson pencil (1.2.25). On the contrary, a different choice of the second Poisson bracket in the pencil produces a triangular linear transformation of the commuting Hamiltonians, i.e., to the Hamiltonians of the level p it will be added a linear combination of the Hamiltonians of the lower levels. In the finite dimensional case we are discussing now all the chains of the hierarchy will be finite. In other words, the generating functions (1.2.32) of the commuting flows will become polynomials after multiplication by a suitable power of λ (the degrees of an appropriate system of these polynomials correspond to the type of the bihamiltonian structure [132]). A simple necessary and sufficient condition of complete integrability of the flows of the hierarchy was found by A.Brailov and A.Bolsinov (see in [7]). The problem of normal forms of degenerate bihamiltonian structures has been studied by I.M.Gelfand and I.Zakharevich [70], [71] for the case of the corank 1 and by I.Zakharevich [159] and A.Panasyuk [132] for higher coranks.

2 Lecture 2.

2.1 Formal loop spaces

Let M be a n-dimensional smooth manifold. Our aim is to describe an appropriate class of Poisson brackets on the loop space L(M) = {S1 → M}.

14 In our definitions we will treat L(M) formally in the spirit of formal variational calculus of [20, 17]. We define the formal loop space L(M) in terms of ring of functions on it. We also describe calculus of differential forms and vector fields on the formal loop space. In the next section we will also deal with multivectors on the formal loop space. Let U ⊂ M be a chart on M with the coordinates u1, . . . , un. Denote A = A(U) the space of polynomials in the independent variables ui,s, i = 1, . . . , n, s = 1, 2,...

X i1,s1 im,sm f(x; u; ux, uxx,...) := fi1s1;...;imsm (x; u)u . . . u (2.1.1) m≥0

1 with the coeﬃcients fi1s1;...;imsm (x; u) being smooth functions on S × M. Such an expression will be called diﬀerential polynomial. We will often use an alternative notation for the independent variables

i i,1 i i,2 ux = u , uxx = u ,...

Observe that polynomiality w.r.t. u = (u1, . . . , un) is not assumed.

The operator ∂x is deﬁned as follows ∂f ∂f ∂f ∂ f = + ui,1 + ... + ui,s+1 + .... (2.1.2) x ∂x ∂ui ∂ui,s We will often use the notation (k) k f := ∂xf. (2.1.3) The following identities will be useful ∂ ∂ ∂ = ∂ (2.1.4) ∂ui x x ∂ui ∂ ∂ ∂ ∂ = ∂ + (2.1.5) ∂ui,s x x ∂ui,s ∂ui,s−1 ∂ ∂ (here ∂ui,0 := ∂ui ). We deﬁne the space A0,0 = A/R, A0,1 = A0,0 dx, the operator d : A0,0 → A0,1, df := ∂xf dx (2.1.6) and the quotient Λ0 = A0,1/dA0,0. (2.1.7) 1 The elements of the space Λ0 will be written as integrals over the circle S Z ¯ f := f(x; u; ux, uxx,...)dx ∈ Λ0 (2.1.8)

We will use below the following simple statement.

15 Lemma 2.1.1 If R fg dx = 0 for an arbitrary g ∈ A then f ∈ A is equal to zero.

The expressions (2.1.8) are also called local functionals with the density f. The space of local functionals is the main building block of the “space of functions” on the formal loop space. The full ring F = F(L(U)) of functions on the formal loop space by deﬁnition coincides with the suitably completed symmetric tensor algebra of Λ0

ˆ2 ˆ3 F = R ⊕ Λ0 ⊕ S Λ0 ⊕ S Λ0⊕ˆ ... (2.1.9)

ˆk Elements of the (completed) k-th symmetric power S Λ0 will be written as multiple i integrals of diﬀerential polynomials of k copies of the variables that we denote u (x1), i i i ..., u (xk), ux(x1), . . . , ux(xk) etc. Z ˆk f(x1, . . . , xk; u(x1), . . . , u(xk); ux(x1), . . . , ux(xk),...)dx1 . . . dxk ∈ S Λ0 (2.1.10)

(they are also called k-local functionals, cf. [153]). The short exact sequence

d π 0 → A0,0 → A0,1 → Λ0 → 0

(π is the projection) is included in the variational bicomplex x x x    δ δ δ d π 0 → A2,0 → A2,1 → Λ2 → 0 x x x    δ δ δ d π 0 → A1,0 → A1,1 → Λ1 → 0 x x x    δ δ δ d π 0 → A0,0 → A0,1 → Λ0 → 0 x x x    δ δ δ 0 0 0

Here Ak,l are elements of the total degree k + l in the Grassman algebra with the generators δui,s, i = 1, . . . , n, s = 0, 1, 2,... (observe the diﬀerence in the range of the second index of ui,s and δui,s) and dx with the coeﬃcients in A having the degree l in dx. We will often identify δui,0 = δui.

For example, every k-form ω ∈ Ak,0 is a ﬁnite sum 1 ω = ω δui1,s1 ∧ ... ∧ δuik,sk (2.1.11) k! i1s1;...;iksk where the coeﬃcients ωi1s1;...;iksk ∈ A are assumed to be antisymmetric w.r.t. permutations of pairs ip, sp ↔ iq, sq.

16 The exterior differential in the Grassman algebra is decomposed into a sum d + δ. The horizontal differential d : Ak,0 → Ak,1 is defined by dω = dx ∧ ∂xω (2.1.12) where the derivation ∂x,

∂x(ω1 ∧ ω2) = ∂xω1 ∧ ω2 + ω1 ∧ ∂xω2 is given by (2.1.2) on the coeﬃcients of the diﬀerential form and by

i,s i,s+1 ∂xδu = δu .

The elements of the quotient Λk = Ak,1/dAk,0 will be called (local) k-forms on the loop space. k-forms will also be written by integrals Z dx ∧ ω ∈ Λk, ω ∈ Ak,0.

Example 2.1.2 Any one-form has a unique representative Z i φ = dx ∧ φiδu (2.1.13)

(use integration by parts).

More generally, for every k-form ω written as in (2.1.11)

dx ∧ ω ∼ dx ∧ ω˜ (mod d(Ak,0)) where 1 ω˜ = ω˜ δui1 ∧ δui2,s2 ∧ ... ∧ δuik,sk (2.1.14) (k − 1)! i1;i2s2;...;iksk

ω˜i1;i2s2;...;iksk 1 X X s (s−r −...−r ) = (−1)s ω 2 k i1s;i2,s2−r2;...;i ,s −r k r2 . . . rk k k k s≥r2+...+rk 0 ≤ rl ≤ sl 2 ≤ l ≤ k here s s! = (2.1.15) r2 . . . rk r2! . . . rk!(s − r2 − ... − rk)!

17 stands for the multinomial coeﬃcients. The coeﬃcientsω ˜i1;i2s2;...;iksk will be called reduced components of ω. They are antisymmetric w.r.t. pairs i2, s2,..., ik, sk but with the permutation of i1 and i2 they behave as

ω˜i2;i1s2;...;iksk X X t (t−s −t −...−t ) = (−1)t+1 ω˜ 2 3 k i1;i2s2;i3,s3−t3;...;i ,s −t s2 t3 . . . tk k k k t≥s2+t3+...+tk 0 ≤ tl ≤ sl 3 ≤ l ≤ k (2.1.16)

We now deﬁne vertical arrows of the bicomplex. For a monomial

ω = f δui1,s1 ∧ ... ∧ δuik,sk put X ∂f δω = δuj,t ∧ δui1,s1 ∧ ... ∧ δuik,sk , (2.1.17) ∂uj,t t≥0 where we denote ∂ ∂ := . ∂uj,0 ∂uj

This gives vertical diﬀerential δ : Ak,0 → Ak+1,0,

δ2 = 0.

The map δ on Ak,1 is deﬁned by essentially same formula, δdx = 0. Anticommutativity

δd = −dδ justiﬁes action of δ on the quotient Λk.

Example 2.1.3 On Λ0 the diﬀerential δ acts as follows ! Z Z X ∂f δ f dx = dx ∧ (−1)s∂s δui (2.1.18) x ∂ui,s s (the Euler - Lagrange diﬀerential). We will use the notation

δf¯ X ∂f := (−1)s∂s (2.1.19) δui(x) x ∂ui,s s for the components of the 1-form, f¯ = R f dx.

Theorem 2.1.4 [[17]] For M = ball both arrows and columns of the variational bicomplex are exact.

18 Example 2.1.5 A necessary and suﬃcient condition for

δf¯ = 0, i = 1, . . . , n. δui(x) is the existence of a diﬀerential polynomial g = g(x; u; ux; ...) such that f = ∂xg.

Let us now consider the space Λ1 of vector ﬁelds on the formal loop space. These will be formal inﬁnite sums ∂ X ∂ ξ = ξ0 + ξi,k , ξi,k ∈ A (2.1.20) ∂x ∂ui,k k≥0 where we denote ∂ ∂ := . ∂ui,0 ∂ui ¯ R The derivative of a functional f = f(x; u; ux,...)dx ∈ Λ0 along ξ reads

Z ∂f X ∂f ξ f¯ := ξ0 + ξi,k dx. ∂x ∂ui,k The Lie bracket of two vector ﬁelds is deﬁned by

∂η0 ∂ξ0 ∂ [ξ, η] = (ξ0η0 − η0ξ0 + ξj,t − ηj,t ) x x ∂uj,t ∂uj,t ∂x X ∂ηi,s ∂ξi,s ∂ηi,s ∂ξi,s ∂ + ξ0 − η0 + ξj,t − ηj,t (2.1.21) ∂x ∂x ∂uj,t ∂uj,t ∂ui,s s≥0

Evolutionary vector ﬁelds a are deﬁned by the conditions of vanishing of the ∂/∂x- component and the commutativity

[∂x, a] = 0

They are parameterized by n-tuples a1,..., an of elements of A as follows

X ∂ a = ∂sai . (2.1.22) x ∂ui,s s≥0

The corresponding system of evolutionary PDEs reads

i i ut = a (x; u; ux, uxx,...), i = 1, . . . , n. (2.1.23) In particular, an evolutionary vector ﬁeld a is called translation invariant if the coeﬃ- cients ai do not depend explicitly on x,

∂ai = 0, i = 1, . . . , n. ∂x

19 The contraction iξω of a k-form ω ∈ Ak,0 given by (2.1.11) and a vector field ξ is a (k − 1)-form defined by 1 i ω = ξj,tω δui1,s1 ∧ ... ∧ δuik−1,sk−1 . (2.1.24) ξ (k − 1)! jt;i1s1;...;ik−1sk−1 As usual iξiη = −iηiξ for two vector fields ξ, η. For a form ω ∈ Ak,1 the contraction iξω ∈ Ak−1,1 is defined by essentially same formula provided the vector field ξ contains no ∂/∂x-term. It is an easy exercise to check, using (2.1.5), that for an evolutionary vector field a

dia + iad = 0. (2.1.25) It readily follows that contraction with evolutionary vector ﬁelds is well-deﬁned on the quotient ia :Λk → Λk−1. A more strong statement holds true

Lemma 2.1.6 Let ω ∈ Ak,1. It belongs to d(Ak,0) iﬀ iaω ∈ d(Ak−1,0) for an arbitrary evolutionary vector ﬁeld a.

Proof We use induction in k. For k = 1 we can choose a representative of the class of ω ∧ dx ∈ Λ1 with the 1-form ω given by (2.1.13). The contraction reads Z i ia(dx ∧ ω) = − dx a ωi.

Using Lemma 2.1.1 we obtain ωi = 0 for all i. Let us assume validity of the lemma for any (k − 1)-form. We will prove that the condition iaω∧dx = 0 ∈ Λk−1 implies vanishing of all the reduced components (2.1.14). By induction the above condition is equivalent to

ib2 . . . ibk iaω ∧ dx ∈ d(A0,0) for arbitrary evolutionary vector ﬁelds b2,..., bk. Integrating by parts we rewrite the last line in the form Z i a φi dx = 0 (2.1.26) where s2 i2 sk ik φi = k ω˜i;i2s2;...;iksk ∂x b2 . . . ∂x bk . i i From (2.1.26) it follows that φi = 0 for all i. Since b2,..., bk are arbitrary diﬀerential polynomials, this impliesω ˜i;i2s2;...;iksk = 0. That means that the form ω∧dx is equivalent to zero, modulo d(Ak,0). The lemma is proved.

Corollary 2.1.7 A form ω ∈ Ak,1 belongs to d(Ak,0) iﬀ

ia1 . . . iak ω ∈ d(A) for arbitrary evolutionary vector ﬁelds a1,..., ak.

20 R Example 2.1.8 For the one-form ω = δ f dx the contraction iaω reads Z δf¯ i ω = ai dx ∈ Λ . a δui(x) 0 This is the time derivative of the functional f¯ = R f dx w.r.t. the evolutionary system (2.1.23).

i Example 2.1.9 For a one-form ω = ωiδu ∧ dx ∈ Λ1 the condition of closedness

δω = 0 ∈ Λ2 reads ∂ωi X t ∂ωj = (−1)t ∂t−s (2.1.27) ∂uj,s s x ∂ui,t t≥s for any i, j = 1, . . . , n, s = 0, 1,.... This is the classical Volterra’s criterion [153] for the system of ODEs

ω1(x; u; ux, uxx,...) = 0, . . . , ωn(x; u; ux, uxx,...) = 0 to be locally representable in the Euler - Lagrange form δf¯ ω = , i = 1, . . . , n i δui(x) (use exactness of the variational bicomplex).

Example 2.1.10 For a 2-form 1 ω = ω δui,s ∧ δuj,t, 2 is;jt the contraction with two evolutionary vector ﬁelds a and b can be represented in the form Z i s j iaibω = −2 a ω˜i;js∂xb dx (2.1.28) where s 1 X X t ω˜ = (−1)t ∂t−s+rω i;js 2 s − r x is;jr r=0 t≥s−r are the reduced components (2.1.14). The tilde will be omitted in the subsequent formulae. According to this we will often represent 2-forms in the reduced form

i j,s dx ∧ ω = ωi;jsdx ∧ δu ∧ δu . (2.1.29) The reduced coeﬃcients must satisfy the antisymmetry conditions (2.1.16). They are spelled out as follows X t ω = (−1)t+1 ∂t−sω (2.1.30) i;js s x j;it t≥s (integrate by parts in (2.1.28) and use arbitrariness of a and b).

21 Example 2.1.11 A 2-form dx ∧ ω = δdx ∧ φ for

i φ = φiδu has the reduced representative (2.1.29) with

! 1 ∂φi X t ∂φj ω = + (−1)t+1 ∂t−s . (2.1.31) i;js 2 ∂uj,s s x ∂ui,t t≥s

Example 2.1.12 A 2-form (2.1.29) is closed, δω = 0, iﬀ

t+s m−s t ! X X X X m ∂ωj;k,t−r ∂ωi;j,s ∂ωi;k,t + (−1)m ∂m−r−s + − = 0 r s x ∂ui,m ∂uk,t ∂uj,s m=s r=0 m≥t+s+1 r=0 (2.1.32) for any i, j, k = 1, . . . n, s = 0, 1, 2,....

Proof By deﬁnition

X ∂ωi;js δ(ω) = δui ∧ δuj,s ∧ δuk,l ∧ dx. ∂uk,l So δω = 0 means that for any three evolutionary vector ﬁelds

X ∂ X ∂ X ∂ a = (ai)(s) , b = (bi)(s) , c = (ci)(s) ∂ui,s ∂ui,s ∂ui,s the contraction iaibicδ(dx ∧ ω) ∈ d(A0,0), i,e, Z ∂ω i;j,s ai (bk)(l) (cj)(s) − ai (bj)(s) (ck)(l) − (ak)(l) bi (cj)(s) + (ak)(l) (bj)(s) ci ∂uk,l +(aj)(s) bi (ck)(l) − (aj)(s) (bk)(l) ci dx = 0 (2.1.33)

Using integration by parts we get Z ∂ω i;j,s ai (bk)(l) (cj)(s) − ai (bj)(s) (ck)(l) ∂uk,l X m ∂ωi;j,s + (−1)m+1 ∂m−l−r ak (bi)(l) (cj)(s+r) − ak (bj)(s+l) (ci)(r) l r x ∂uk,m X m ∂ωi;j,m + (−1)m ∂m−s−r aj (bi)(s) (ck)(l+r) − aj (bk)(l+s) (ci)(r) dx s r x ∂uk,l = 0

The above identity is equivalent to ∂ω ∂ω i;j,s − i;k,l ∂uk,l ∂uj,s

22 l+s m−l s ! X X X X m ∂ωk;j,s−r + + (−1)m+1 ∂m−l−r l r x ∂ui,m m=l r=0 m≥l+s+1 r=0 l+s m−s l ! X X X X m ∂ωj;k,l−r + + (−1)m ∂m−s−r s r x ∂ui,m m=s r=0 m≥l+s+1 r=0 l+s m−l s ! X X X X m ∂ωk;i,m + + (−1)m ∂m−l−r l r x ∂uj,s−r m=l r=0 m≥l+s+1 r=0 l+s m−s l ! X X X X m ∂ωj;i,m + + (−1)m+1 ∂m−s−r = 0 r s x ∂uk,l−r m=s r=0 m≥l+s+1 r=0 Now by using the antisymmetry condition (2.1.30) we see that the third term in the above sum is equal to the fourth term, and by using the identity (2.1.5) and the antisymmetry condition (2.1.30) again we see that the last two terms equal to the second and ﬁrst term respectively. Thus we arrive at the proof of (2.1.32).

Remark 2.1.13 The equations (2.1.32) were derived by Dorfman in the theory of the so-called local symplectic structures [28], see also the book [29].

Corollary 2.1.14 Any solution to (2.1.32) satisfying (2.1.30) can be locally represented in the form (2.1.31).

This follows from exactness of the variational bicomplex. We will briefly outline necessary points of the global picture of functionals, differential forms and vector fields on the formal loop space L(M) for a general smooth manifold M (i.e., not only for a ball). The corresponding objects must be defined for any chart U ⊂ M as it was explained above. On the intersections U ∩ V they must satisfy certain consistency conditions. For example, functionals in the charts U, V with 1 n 1 n the coordinates u ,..., u and v ,..., v are defined by densities fU (x; u; ux,...) and fV (x; v; vx,...) s.t. ∂v f (x; v(u); u ,...)dx = f (x; u; u ,...)dx (mod Im d). V ∂u x U x

Such objects comprise the space Λ0(M). As above, we obtain the ring of functions on the formal loop space taking the tensor algebra of Λ0(M). One-forms in the charts U, U V V are described by their reduced components ωi and ωa s.t., on U ∩ V transform as components of a covector ∂v ∂ui ωV (x; v(u); u ,...) = ωU (x; u; u ,...) a ∂u x i x ∂va etc. The components of evolutionary vector ﬁelds transform like vectors ∂v ∂vk ak (x; v(u); u ,...) = ai (x; u; u ,...). (2.1.34) V ∂u x ∂ui U x

23 The contraction iadx ∧ ω of a 1-form with an evolutionary vector ﬁeld is well-deﬁned as an element of Λ0(M). The global theory of the variational bicomplex was developed in [146], [152].

2.2 Local multivectors and local Poisson brackets

We first define more general, i.e. non-local k-vectors as elements of (Λ1)∧k. They will be written as infinite sums of expressions of the form 1 α = αi1s1;...iksk (x , . . . , x ; u(x ), . . . , u(x ); u (x ), . . . , u (x ),...) k! 1 k 1 k x 1 x k ∂ ∂ × i ,s ∧ ... ∧ i ,s (2.2.1) ∂u 1 1 (x1) ∂u k k (xk) (in this subsection we will consider only multivectors not containing ∂/∂x). The coefficients must satisfy the antisymmetry condition w.r.t. simultaneous permutations

ip, sp, xp ↔ iq, sq, xq.

The exterior algebra structure on multivectors is introduced in a usual way: the product of a k-vector α by a l-vector β is a (k + l)-vector

i1s1;...;iksk;ik+1sk+1;...;ik+lsk+l (α ∧ β) (x1, . . . , xk+l; u(x1), . . . , u(xk+l); ...) 1 X = sgnσ αiσ(1)sσ(1);...;iσ(k)sσ(k) (x , . . . , x ; u(x ), . . . , u(x ); ...) k!l! σ(1) σ(k) σ(1) σ(k) σ∈Sk+l

iσ(k+1)sσ(k+1);...;iσ(k+l)sσ(k+l) ×β (xσ(k+1), . . . , xσ(k+l); u(xσ(k+1)), . . . , u(xσ(k+l)); ...) (2.2.2)

Example 2.2.1 Lie derivative of a k-vector α (2.2.1) along a vector ﬁeld (2.1.20) reads

i1s1;...;iksk Lieξα = k ∂ ∂ X 0 i1s1;...;iksk jp,tp i1s1;...;iksk ξ (xp; u(xp); ...) α + ξ (xp; ...) α ∂x ∂ujp,tp (x ) p=1 p p k ip,sp X ∂ξ (xp; ...) − αi1s1;...;ip−1sp−1;jptp;...;iksk . (2.2.3) ∂ujp,tp (x ) p=1 p

Here we assume that ξ0 does not depend on uj,t.

Deﬁnition.A k-vector α is called translation invariant if

Lie∂x α = 0

24 and also the sum of partial derivatives of the coeﬃcients αi1s1;...iksk i1s1;...iksk = α (x1, . . . , xk; u(x1), . . . , u(xk); ux(x1), . . . , ux(xk) equals zero ∂ ∂ + ... + αi1s1;...iksk = 0. ∂x1 ∂xk

Lemma 2.2.2 Every translation invariant k-vector α has coeﬃcients of the form

i1s1;...;iksk α (x1, . . . , xk; u(x1), . . . , u(xk); ...) = ∂s1 . . . ∂sk Ai1...ik (x , . . . , x ; u(x ), . . . u(x ); ...) (2.2.4) x1 xk 1 k 1 k

i1...ik where the diﬀerential polynomials A (x1, . . . , xk; u(x1), . . . , u(xk); ...) are antisymmetric w.r.t. simultaneous permutations

ip, xp ↔ iq, xq and also they satisfy

i1...ik i1...ik A (x1 + t, . . . , xk + t; u(x1), . . . , u(xk); ...) = A (x1, . . . , xk; u(x1), . . . , u(xk); ...) for any t.

i1...ik The functions A (x1,... ; xk; u(x1), . . . , u(xk); ...) will be called components of the translation invariant k-vector α. Translation invariant multivectors form a graded Lie subalgebra of the full graded Lie algebra of multivectors closed w.r.t. Schouten - Nijenhuis bracket.

Example 2.2.3 The Lie derivative of a bivector α with the components Aij(x−y; u(x), u(y); ...) along a translation invariant vector ﬁeld a with the components i a (u; ux,...) has the components ∂Aij ∂Aij Lie αij = ∂t ak(u(x); ...) + ∂t ak(u(y); ...) a x ∂uk,t(x) y ∂uk,t(y) ∂ai(u(x); ...) ∂aj(u(y); ...) − ∂t Akj − ∂t Aik. (2.2.5) ∂uk,t(x) x ∂uk,t(y) y

Example 2.2.4 Let α, β be two translation invariant bivectors with the components ij ij A (x − y; u(x), u(y); ux(x), ux(y),...) and B (x − y; u(x), u(y); ux(x), ux(y); ...) that ij ij we redenote resp. Ax,y and Bx,y for brevity. The Schouten - Nijenhuis bracket [α, β] is a translation invariant trivector with the components ∂Aij ∂Bij ∂Aij ∂Bij [α, β]ijk = x,y ∂sBlk + x,y ∂sAlk + x,y ∂sBlk + x,y ∂sAlk x,y,z ∂ul,s(x) x x,z ∂ul,s(x) x x,z ∂ul,s(y) y y,z ∂ul,s(y) y y,z ∂Aki ∂Bki ∂Aki ∂Bki + z,x ∂sBlj + z,x ∂sAlj + z,x ∂sBlj + z,x ∂sAlj ∂ul,s(z) z z,y ∂ul,s(z) z z,y ∂ul,s(x) x x,y ∂ul,s(x) x x,y ∂Ajk ∂Bjk ∂Ajk ∂Bjk + y,z ∂sBli + y,z ∂sAli + y,z ∂sBli + y,z ∂sAli . (2.2.6) ∂ul,s(y) y y,x ∂ul,s(y) y y,x ∂ul,s(z) z z,x ∂ul,s(z) z z,x

25 For a translation invariant k-vector α and k 1-forms ω1,..., ωk,

j j i,s ω = ωisδu ∧ dx ∈ A1,1, j = 1, . . . , k the contraction

< α, ω1 ∧ ... ∧ ωk > Z 1 X sgn σ σ(1) σ(k) := (−1) ω (x1; u(x1); ...) . . . ω (xk; u(xk); ...) k! i1s1 iksk σ∈Sk

i1s1;...;iksk ⊗ˆ k α (x1, . . . , xk; u(x1), . . . , u(xk); ...)dx1 . . . dxk ∈ Λ0 (2.2.7)

⊗k is well deﬁned on Λ1 .

Example 2.2.5 The value of a translation invariant k-vector α with the components Ai1...ik on the 1-forms δf¯1,..., δf¯k equals

< α, δf¯1 ∧ ... ∧ δf¯k > Z δf¯1 δf¯k i1...ik = i ... i A (x1, . . . , xk; u(x1), . . . , u(xk); ...)dx1 . . . dxk δu 1 (x1) δu k (xk) ⊗ˆ k ∈ Λ0 . (2.2.8)

The transformation law of components of translation invariant multivectors w.r.t. changes of coordinates on the intersection of two coordinate charts (U, u1, . . . , un) and (V, v1, . . . , vn) is analogous to the transformation law of components of multivectors on a ﬁnite-dimensional manifolds: ∂v ∂v Aa1...ak (x , . . . x ; v(u(x )), . . . , v(u(x )); u (x ),..., u (x ),...) V 1 k 1 k ∂u x 1 ∂u x k a1 ak ∂v ∂v i1...ik = (x1) ... (xk)AU (x1, . . . , xk; u(x1), . . . , u(xk); ux(x1), . . . , ux(xk),...). ∂ui1 ∂uik (2.2.9)

We now proceed to the main definition of local multivectors. They are translation invariant multivectors α such that their dependence on x1,..., xk is given by a finite order distribution with the support on the diagonal x1 = x2 = ... = xk X Ai1...ik = Bi1...ik (u(x ); u (x ),...)δ(p2)(x −x )δ(p3)(x −x ) . . . δ(pk)(x −x ). p2,...,pk 1 x 1 1 2 1 3 1 k p2,p3,...,pk≥0 (2.2.10) The coefficients Bi1...ik (u(x ); u (x ),...) are differential polynomials in A not de- p2,...,pk 1 x 1 pending explicitly on x. All the sums at the moment are assumed to be finite. In the next section we will relax this condition. Delta functions and their derivatives and products are defined by the formulae Z Z f(y)δ(x − y)dy = f(x), f(y)δ(p)(x − y)dy = f (p)(x) (2.2.11)

26 Z (p2) (p3) (pk) f(x1, . . . , xk)δ (x1 − x2)δ (x1 − x3) . . . δ (x1 − xk)dx2 . . . dxk

= ∂p2 . . . ∂pk f(x , . . . , x )| . x2 xk 1 k x1=x2=...=xk

Lemma 2.2.6 The value (2.2.7) of a local k-vector α on k 1-forms ω1,..., ωk is given by

< α, ω1 ∧ ... ∧ ωk > Z = Bi1...ik (u; u , u ,...)ω1 (x; u; u ,...)∂p2 ω2 (x; u; u ,...) p2...pk x xx i1 x x i2 x

. . . ∂pk ωk (x; u; u ,...) dx ∈ Λ . (2.2.12) x ik x 0 It gives a well-deﬁned polylinear map

⊗k α :Λ1 → Λ0.

In calculations with local multivectors various simple identities for delta-functions will be useful. All of them are simple consequences of the deﬁnition (2.2.11). First,

p X p f(y)δ(p)(x − y) = f (q)(x)δ(p−q)(x − y). (2.2.13) q q=0 Next,

δ(x1 − x2) . . . δ(x1 − xk) = δ(x2 − x1)δ(x2 − x3) . . . δ(x2 − xk) = ...

= δ(xk − x1) . . . δ(xk − xk−1). (2.2.14)

Differentiating (2.2.14) w.r.t. x1,..., xk we will obtain relations between products of derivatives of delta-functions. We leave as a simple exercise for the reader to prove that the space of local multivectors that we denote ∗ k Λloc = ⊕Λloc is closed w.r.t. the Schouten - Nijenhuis bracket. Warning: this space is not closed w.r.t. the exterior product! Because of this we were to introduce a wider algebra of multivectors to introduce the definition of the Schouten - Nijenhuis bracket according to the rules one uses in the finite dimensional case.

Example 2.2.7 The component of a local bivector $ has the form

ij X ij (s) $ = As (u(x); ux(x),...)δ (x − y). (2.2.15) s≥0

i i The value of the bivector on two 1-forms φ = φiδu ∧ dx and ψ = ψiδu ∧ dx equals Z ij s φiAs ∂xψjdx. (2.2.16)

27 The conditions of antisymmetry of the bivector reads

X t Aji = (−1)t+1 ∂t−sAij. (2.2.17) s s x t t≥s

Proof Let us explain how to prove the antisymmetry condition (2.2.17). We must have

X ji (s) X ij (s) As (u(y); ...)δ (y − x) = − As (u(x); ...)δ (x − y). s

Using δ(s)(y − x) = (−1)sδ(s)(x − y) and (2.2.13) we obtain (2.2.17).

Remark 2.2.8 One can represent the bivector as d Aij(u(x); u (x),... ; )δ(x − y). (2.2.18) x dx Here the diﬀerential operators Aij are

d X ds Aij(x; u(x); u (x),... ; ) = Aij . x dx s dxs s For local multivectors of higher rank the language of diﬀerential operators was used by Olver [129].

Example 2.2.9 The Schouten - Nijenhuis bracket of the bivector

$ = πijδ(x − y), where πij is a constant antisymmetric matrix, with α of the form (2.2.15) reads

 (r) ij ki ! ∂A X q + r + s ∂Aq+r+s [$, α]ijk = t πlk + (−1)q+r+s πlj x,y,z ∂ul,s q r ∂ul,t−q  jk !(r) X q + r + t ∂As−q + (−1)q+r+t πli δ(v)(x − y)δ(s)(x − z). (2.2.19) q r ∂ul,q+r+t 

Proof Substituting into (2.2.6) we obtain

∂Aij(x) [$, α]ijk = t πlkδ(v)(x − y)δ(s)(x − z) x,y,z ∂ul,s(x) ∂Aki(z) ∂Ajk(y) + t πljδ(v)(z − x)δ(s)(z − y) + t πliδ(v)(y − z)δ(s)(y − x). (2.2.20) ∂ul,s(z) ∂ul,s(y)

28 ij jk ki ij jk ki Here At (x), At (y), At (z) stand for At (u(x); ...), At (u(y); ...), At (u(z); ...) resp. Use the identities (2.2.14)

(v) (s) t s δ (z − x)δ (z − y) = (−∂x) (−∂y) [δ(z − x)δ(z − y)]

t X t = (−∂ )t(−∂ )s[δ(x − y)δ(x − z)] = (−1)t δ(s+q)(x − y)δ(t−q)(x − z), x y q q=0 (v) (s) s t δ (y − z)δ (y − x) = (−∂x) (−∂z) [δ(y − z)δ(y − x)] s X s = (−∂ )s(−∂ )t[δ(x − y)δ(x − z)] = (−1)s δ(s−q)(x − y)δ(t+q)(x − z) x z q q=0 and also (2.2.13) to arrive at (2.2.19).

Deﬁnition. A local Poisson structure on the formal loop space is a local bivector 2 $ ∈ Λloc (2.2.15) satisfying [$, $] = 0. Adopting notation common in the physics literature we will represent the Poisson structure in the form

i j X ij (s) {u (x), u (y)} = As (u(x); ux(x), uxx(x),...)δ (x − y). (2.2.21) s ¯ R The Poisson bracket of two local functionals f = f(x; u; ux,...)dx and R g¯ = g(x; u; ux,...)dx can be written in the following equivalent forms (see above the general theory of multivectors) ZZ δf¯ δg¯ {f,¯ g¯} =< $, δf¯∧ δg¯ >= dxdy {ui(x), uj(y)} δui(x) δuj(y) (s) X Z δf¯ δg¯ = dx Aij(u; u , u ,...) ∈ Λ . (2.2.22) δui(x) s x xx δuj(x) 0 s Therefore it is again a local functional. The crucial property of local Poisson brackets is that, the Hamiltonian systems ¯ i i ij s δH u = −i ¯ $ = {u (x), H¯ } = A (u; u , u ,...)∂ (2.2.23) t δH s x xx x δuj(x) with local translation invariant Hamiltonians Z ¯ H = H(u; ux,...)dx are translation invariant evolutionary PDEs (2.1.23). Living in the inﬁnite dimensional loop space we will not impose conditions on the rank of the Poisson bracket. However, in the main examples the corank of the bivector will be ﬁnite.

29 Example 2.2.10 For a constant antisymmetric matrix πij the bivector {ui(x), uj(y)} = πijδ(x − y) (2.2.24) is a local Poisson structure. It is called ultralocal Poisson bracket. This is a symplectic structure on the loop space iﬀ detπij 6= 0. The Hamiltonian evolutionary PDEs read δH¯ ui = πij . t δuj(x) Reducing the nodegenerate matrix πij to the canonical form we arrive at the Hamilto- nian formulation of 1+1 dimensional variational problems ¯ ¯ i δH i δH qt = , p t = − i . δpi(x) δq (x) Example 2.2.11 For a constant symmetric matrix ηij the bivector {ui(x), uj(y)} = ηijδ0(x − y) (2.2.25) is a local Poisson structure. Under the assumption det ηij 6= 0 this Poisson bracket has n independent Casimirs Z Z u¯1 = u1 dx, . . . , u¯n = un dx. (2.2.26)

The annihilator of (2.2.25) is generated by the above Casimirs. The Hamiltonian evolutionary PDEs read δH¯ ui = ηij∂ . (2.2.27) t x δuj(x) We finish this section with spelling out the transformation law of coefficients of local ij ab bivectors imposed by the general formula (2.2.9). If As (u; ux,...) and At (v; vx,...) are the coefficients of a local bivector in two coordinate charts (U, u1, . . . , un) and (V, v1, . . . , vn) resp. then, on U ∩ V one has (s−t) ∂v X s ∂va ∂vb Aab(v; u ,...) = Aij(u; u ,...). (2.2.28) t ∂u x t ∂ui ∂uj s x s≥t Example 2.2.12 Applying the transformation 1 u = v2 (2.2.29) 4 to the bivector 1 {u(x), u(y)} = u(x)δ0(x − y) + u0(x)δ(x − y) (2.2.30) 2 we obtain a constant Poisson bracket of the form (2.2.25) {v(x), v(y)} = δ0(x − y). (2.2.31) Hence (2.2.30) is itself a Poisson structure. This is the Lie - Poisson bracket on the space dual to the Lie algebra of vector fields on the circle [46].

30 3 Lecture 3. Problem of classiﬁcation of local Pois- son brackets

The last example of the previous section is the simplest issue of the problem of reduction of local Poisson brackets to the simplest (possibly, to the constant one) form. In this example the reduction to the constant form was achieved by a change of coordinates in the target space M (M was one-dimensional). We give now another well-known example: to transform the bivector (the Magri bracket for the KdV equation) 1 {u(x), u(y)} = u(x)δ0(x − y) + u0(x)δ(x − y) − δ000(x − y) (3.0.1) 2 to the constant form (2.2.31) one is to use the celebrated Miura transformation [120] 1 u = v2 + v0. (3.0.2) 4 Our strategy will be to classify local Poisson brackets on the loop space L(M) (the target space M will be a ball in this section) with respect to the action of the group of Miura-type transformations. The problem of reduction of certain classes of Poisson brackets to a canonical form by coordinate transformations was first investigated in [44] for the Poisson brackets of hydrodynamic type and in [45] for the so-called differential geometric Poisson brackets (see also [46] and the references therein). Some results regarding reduction of the local Poisson brackets to the canonical form by using Miura - Bäcklund transformations were obtained in[2], [69], [130], [134] (in the latter non translation invariant Poisson brackets were studied). We want to classify local Poisson brackets w.r.t. general Miura type transformations of the form i i i u → u˜ = F (u; ux, uxx,...). (3.0.3) The problem is that these transformations do not form a group. The main trouble is with inverting such a transformation. E.g., to invert the Miura transformation one is to solve Riccati equation (3.0.2) w.r.t. v. To resolve this problem we will extend the class of Miura-type transformations. Simultaneously we will also be extending the class of local functionals, vector fields, and Poisson brackets.

3.1 Extended formal loop space

Let us introduce gradation on the ring A of diﬀerential polynomials putting

degui,k = k, k ≥ 1, degf(x; u) = 0. (3.1.4)

We extend the gradation onto the spaces Ak,l of diﬀerential forms by degdx = −1, degδui,s = s, s ≥ 0.

31 The diﬀerentials d and δ preserve the gradation. Introduce a formal indeterminate of the degree deg = −1. Let us deﬁne a subcomplex ˆ −1 Ak,l ⊂ Ak,l ⊗ C[[], ] collecting all the elements of the total degree k − l. The corresponding subcomplex ˆ −1 Λk ⊂ Λk ⊗ C[[], ] spanned by the elements of the total degree k − 1. In particular, the space of local ˆ functionals Λ0 consists of integrals of the form Z ¯ f = f(u; ux, uxx,... ; )dx,

∞ X k (k) f(u; ux, uxx,...) = fk(u; ux, . . . , u ), fk ∈ A, degfk = k. (3.1.5) k=0 We will still call such a series differential polynomials when it will not cause confusions. ˆ Taking the tensor algebra of Λ0 we obtain the ring of functionals on the extended formal loop space that we will denote Lˆ(M). The vertical differential δ of the bicomplex must be renormalized 1 δ 7→ δˆ = δ. ˆ ˆ It follows from Theorem 2.1.4 the bicomplex (Ak,l, d, δ) is exact The gradation on the vector and multivector fields is defined by ∂ ∂ deg = 1, deg = −s, s ≥ 0. ∂x ∂ui,s

Observe that ∂x increases degrees by one:

deg∂xf = degf + 1. The space Λˆ 1 of vector ﬁelds on Lˆ(M) is obtained by collecting all the elements in Λ1⊗C[[], −1] of the total degree 1. In particular, the translation invariant evolutionary vector ﬁelds are ∞ i X k−1 i (k) i i a = ak(u; ux, . . . , u ), ak ∈ A, degak = k. (3.1.6) k=0 The corresponding evolutionary system of PDEs reads

i −1 i i i 2 ut = a0(u) + a1(u; ux) + a2(u; ux, uxx) + O( ) i i j a1(u; ux) = vj(u)ux, 1 ai (u; u , u ) = bi (u)uj + ci (u)uj uk (3.1.7) 2 x xx j xx 2 jk x x

32 etc. Proceeding in a similar way we introduce the subspace

k k −1 Λˆ ⊂ Λ ⊗ C[[], ] of k-vectors of the total degree k.

Lemma 3.1.1 The Schouten - Nijenhuis bracket gives a well-deﬁned map

[ , ]: Λˆ k × Λˆ l → Λˆ k+l−1.

There is an important subtlety with the grading of the local multivectors. Indeed, a local k-vector is a map ⊗k Λ1 → Λ0 not ⊗k ⊗ˆ k Λ1 → Λ0 (cf. the formulae (2.2.7), (2.2.10) and (2.2.12)). Such a map does not respect the grading. So we must assign a nonzero degree to delta-function

degδ(x − y) = 1, degδ(s)(x − y) = s + 1 (3.1.8) ˆ Therefore a general local bivector in Λloc will be represented by an inﬁnite sum

∞ X {ui(x), uj(y)} = k{ui(x), uj(y)}[k] (3.1.9) k=−1 k+1 i j [k] X ij (s) (k−s+1) {u (x), u (y)} = Ak,s(u; ux, . . . , u )δ (x − y), s=0 ij ij Ak,s ∈ A, degAk,s = s, s = 0, 1, . . . , k + 1.

More explicitly, the ﬁrst three terms in the expansion (3.1.9) read

{ui(x), uj(y)}[−1] = πij(u(x))δ(x − y) (3.1.10) i j [0] ij 0 ij k {u (x), u (y)} = g (u(x))δ (x − y) + Γk (u(x))uxδ(x − y) (3.1.11) i j [1] ij 00 ij k 0 {u (x), u (y)} = a (u(x))δ (x − y) + bk (u(x))uxδ (x − y) 1 +[cij(u(x))uk + dij (u(x))ukul ]δ(x − y) (3.1.12) k xx 2 kl x x

ij ij ij ij ij ij ij where π , g (u), Γ (u), a (u), bk (u), ck (u), dkl(u) are some functions on the manifold M.

33 Remark 3.1.2 Our rules of introducing the gradation can be memorized using the following simple trick. Do a rescaling of the independent variable x,

x 7→ x. (3.1.13)

The x-derivatives ui,k = dkui/dxk will change

ui,k 7→ kui,k. (3.1.14)

We also have dx 7→ −1dx. (3.1.15) The delta-function, according to the deﬁnition (2.2.11) must be rescaled as

δ(x) 7→ δ(x). (3.1.16)

In other words, “delta-function” is not a function but a density. Simultaneously with the rescaling we will also redeﬁne the integrals Z Z k . dx1 . . . dxk 7→ . dx1 . . . dxk.

After such a rescaling we expand all the formulae of the previous two sections in a power series in to arrive at our grading conventions.

Example 3.1.3 Rescaling (3.1.13) the KdV equation ut = uux + uxxx one obtains

2 ut = (uux + uxxx).

One usually introduces slow time variable t 7→ t to recast the last equation into the form 2 ut = uux + uxxx. (3.1.17) This is the small dispersion expansion of the KdV equation. The smooth solutions of (3.1.17) describe solutions to KdV slow varying in space and time.

Remark 3.1.4 Besides the rescaling procedure, one can arrive at the above series (3.1.5), (3.1.6), (3.1.9) considering the continous limits of diﬀerential-diﬀerence systems. E.g., for the well-known example of Toda lattice

un+1 un u˙ n = vn − vn−1, v˙n = e − e , n ∈ Z (3.1.18) the continuous limit un = u(n) = u(x), vn = v(n) = v(x), t 7→ t gives an evolutionary system of the form (3.1.7) with an inﬁnite series in the r.h.s. 1 1 u = (v(x) − v(x − )) = [v0 − v00 + O(2)], t 2 1 1 v = eu(x+) − eu(x) = [(eu)0 + (eu)00 + O(2)]. (3.1.19) t 2

34 Replacing in the Hamiltonian structure

{um, un} = {vm, vn} = 0, {um, vn} = δmn − δm,n+1

−1 −1 the Kronecker symbols δmn by δ(x−y), δm,n+1 by δ(x−y−), we obtain a Poisson bracket of the (3.1.9) form

{u(x), u(y)} = {v(x), v(y)} = 0, 1 {u(x), v(y)} = [δ(x − y) − δ(x − y − )] = δ0(x − y) − δ00(x − y) + O(2). 2 (3.1.20)

The Hamiltonian of the “interpolated” Toda lattice (3.1.19) is a local functional of the form Z 1 H = v2 + eu dx. (3.1.21) 2

3.2 Miura group

The next deﬁnition will be that to extend the class of Miura-type transformations (3.0.3). Let us consider the transformations

∞ i i X k i (k) u 7→ u˜ = Fk(u; ux, . . . , u ), i = 1, . . . , n k=0 i i Fk ∈ A, deg Fk = k, (3.2.22) ∂F i(u) det 0 6= 0. ∂uj

Lemma 3.2.5 The transformations of the form (3.2.22) form a group. The Lie algebra ˆ 1 of the group is isomorphic to the subalgebra Λev of all translation invariant evolutionary vector ﬁleds in Λˆ 1 with the Lie bracket operation.

Example 3.2.6 Let us invert the clasical Miura transformation 1 u = v2 + v0 4 using successive approximations. Rewriting the equation in the form

√ √ v0 √ u0 v = 2 u − v0 = 2 u − √ + O(2) = 2 u − + O(2) u u we obtain ﬁrst two terms of the solution v = F (u; u0,... ; ).

35 Remark 3.2.7 This way of solving the Riccati equation is essentially equivalent to the classical WKB method of solving the related linear second order ODE

1 y0 2y00 = u y, v = 4 . 4 y Substituting the above series solution to Riccati we obtain the WKB asymptotic solution to the second order ODE with the small parameter → 0 1 Z √ y = u−1/4 exp udx (1 + O()) . 2

Definition. The group G of all the transformations of the form (3.2.22) is called Miura group. The Miura group G looks to be a natural candidate for the role of the group of “local diffeomorphisms” of the extended formal loop space Lˆ(M) (recall that at the moment M is a ball). G contains the group of diffeomorphisms Diff(M) of the manifold M as a subgroup. It coincides with the semidirect product of Diff(M) and the pro-unipotent subgroup G0 consisting of Miura-type transformations close to identity, 1 ui 7→ ui + Ai (u)uj + 2 Bi(u)uj + Ci (u)uj uk + ... (3.2.23) j x j xx 2 jk x x

The product in the group G0 reads

i i i Aj = A1j + A2j i i i i k Bj = B1j + B2j + A2kA1j 1 Ci = C i + C i + ∂ A i A s + ∂ A i A s + A i ∂ A s + A i ∂ A s jk 1jk 2jk 2 s 2j 1k s 2k 1j 2s k 1j 2s j 1k ... (3.2.24)

The following simple statements immediately follow from Lemma 3.2.5.

Lemma 3.2.8 The class of local functionals (3.1.5), evolutionary PDEs (3.1.7), and local translation invariant multivectors (see the formula (3.1.9) for the bivectors) on the extended formal loop space Lˆ(M) is invariant w.r.t. the action of the Miura group.

Lemma 3.2.9 An arbitrary vector ﬁeld a of the form (3.1.6) with a0 6= 0 can be reduced, by a transformation of the Miura group, to a constan form

a0 = const, ai = 0 for i > 0.

36 This is an infinite dimensional analogue of the theorem of “rectifying of a vector field”. Proof By using the theorem of “rectifying of a vector field” on a finite dimensional 1 n manifold, we can reduce the vector field (a0, . . . , a0 ) to a constant one by performing a Miura transformation of the form (3.2.22) with Fk = 0, k ≥ 1. We prove the lemma by induction. Let us assume the vector field a to be of the form

i i k i (k) k+1 a = a0 + ε ak(u, ux, . . . , u ) + O(ε )

i (k) with deg ak = k. Since a0 6= 0, we can ﬁnd diﬀerential polynomials Fk(u, . . . , u ) of degree k such that ∂F i al k + ai = 0. 0 ∂ul k Then the Miura transformation

i i k i (k) u¯ = u + ε Fk(u, . . . , u ) reduces the vector ﬁeld a to the form

i i k+1 a = a0 + O(ε ).

3.3 (p, q)-brackets on the extended formal loop space

Let us write explicitly down the transformation law of the coeﬃcients of a local Poisson bracket w.r.t. transformations from the Miura group (cf. [129]). Let Akl be the diﬀerential operator of the Poisson bracket given in (2.2.18). In the new “coordinates” u˜i of the form (3.2.22) the Poisson bracket will be given by the operator

˜ij ∗i kl j A = L kA Ll (3.3.25)

i ∗i where the matrix-valued operator Lk and the adjoint one L k are given by

X ∂u˜i X ∂u˜i Li = (−∂ )s ◦ ,L∗i = ∂s. k x ∂uk,s k ∂uk,s x s s

Main Problem. To describe the orbits of the action of the Miura group G on Λˆ 2.

In our opinion this problem is the natural setup of the problem of classiﬁcation of Poisson structures of one-dimensional evolutionary PDEs with respect to Miura-type transformations (they are called also Darboux, or Bianchi, or B¨acklund transformations. Our transformations do not involve a change of the independent variable x since we consider only translation invariant PDEs).

37 Our conjecture is that, for a reasonable class of Poisson brackets to be defined below, the orbits are labelled by certain finite-dimensional geometrical structures on the underlying manifold M. Below we will illustrate this claim describing two orbits being, in a certain sense, generic. The full problem remains open. ˆ 2 We first explain how a local Poisson bracket from Λloc induces certain finite- dimensional geometrical structures on M.

Lemma 3.3.10 The subgroup Diff(M) ⊂ G acts independently on every term { , }[k] ij of the expansion (3.1.9), k ≥ −1. In particular, the leading term Ak,0(u) is a (2, 0)- tensor ﬁeld on M, symmetric/antisymmetric for even/odd k.

Proof This follows from the transformation law (2.2.28). The symmetry/antisymmetry of the coeﬃcients follows from the general antisymmetry condition (2.2.17). The lemma is proved.

The transformations of the Miura group mix the terms of the expansion (3.1.9). However, the following simple statement holds true.

Lemma 3.3.11 After a transformation from the Miura group G the k-th term of the transformed bracket is expressed via { , }[−1], { , }[0],..., { , }[k] for any k ≥ −1.

This follows from the explicit formula (3.3.25).

Lemma 3.3.12 The ﬁrst non-zero term in the expansion (3.1.9) is itself a local Pois- son bracket.

This is obvious.

Corollary 3.3.13 The coeﬃcient πij(u) in (3.1.10) is a Poisson structure on M. This ˆ 2 Poisson structure is invariant w.r.t. the action of G on Λloc.

We obtain a map

ˆ 2 Λloc/G → Poisson structures on M. (3.3.26)

ij Let us assume that the Poisson structure π (u) on M has constant rank p = 2p1. Denote q := n − p the corank of πij(u). Deﬁnition. (3.1.9) is called (p, q)-bracket if the coeﬃcient gij(u) in (3.1.11) does ij ∗ not degenerate on Ker π (u) ⊂ Tu M on an open dense subset in M 3 u.

Example 3.3.14 The ultralocal Poisson bracket (2.2.24) with a non-degenerate matrix πij is a (n, 0)-bracket.

38 Example 3.3.15 The Poisson bracket (2.2.25) with a non-degenerate matrix ηij is a (0, n)-bracket.

ij Example 3.3.16 Let ck be the structure constants of a semisimple n-dimensional Lie algebra g. The Killing form ηij on g deﬁnes a central extension gˆ of g called Kac - Moody Lie algebra [85]. The Lie - Poisson bracket (1.1.8) on the dual space gˆ∗ 1 1 {ui(x), uj(y)} = ηijδ0(x − y) + cijukδ(x − y) (3.3.27) k can be considered as a Poisson bracket of the form (3.1.9) on the loop space Lˆ(g∗). Here is the central charge. This is a (p, q)-bracket with

q = rk g, p = dim g − rk g.

Let a (p, q)-Poisson bracket on Lˆ(M) of the form (3.1.9) - (3.1.11) be given. We will now construct a flat metric on the base of the symplectic foliation of M defined by the finite-dimensional Poisson bracket πij(u). Let us assume that M is a small ball such that the symplectic foliation defines a fibration

M → N, dim N = q. (3.3.28)

Functions on N are Casimirs of the finite-dimensional Poisson bracket πij(u). Define first a symmetric bilinear form ( , )∗ on T ∗N putting ∂f ∂f (df , df )∗ := 1 2 gij(u) (3.3.29) 1 2 ∂ui ∂uj for any two Casimirs of πij(u). By the assumption this bilinear form does not degenerate. Define the non-degenerate symmetric tensor ( , ) on TN by

( , ) := [( , )∗]−1 . (3.3.30)

Theorem 3.3.17 The metric (3.3.30) on TN is well-deﬁned and ﬂat.

Proof The simplest way to prove that the metric (3.3.30) is constant along symplectic leaves and also prove vanishing of the curvature of the metric is the following one. a α Choose local coordinates u = (w , v ) on M, a = 1, . . . , p = 2p1, α = 1, . . . , q, p+q = n, such that w1, . . . , wp are canonical coordinates on the symplectic leaves and v1, . . . , vq is a system of independent Casimirs of πij(u). The v’s can be considered as coordinates on N. In the local coordinates the Poisson bracket (3.1.9) reads 1 {wa(x), wb(y)} = πabδ(x − y) + Aab(u)δ0(x − y) + Aab(u, u )δ(x − y) + O() 00 01 x α a αa 0 αa {v (x), w (y)} = A00 (u)δ (x − y) + A01 (u, ux)δ(x − y) + O() α β αβ 0 αβ a αβ γ {v (x), v (y)} = g (u)δ (x − y) + Γa (u)wx + Γγ (u)vx δ(x − y) + O() (3.3.31)

39 Here πab is a constant antisymmetric nondegenerate matrix, the q × q matrix gαβ(u) coincides with the Gram matrix of the bilinear form (3.3.29) in the basis dv1,..., dvq. Let us consider the following foliation on the loop space Lˆ(M)

a a w (x) ≡ w0 , a = 1, . . . , p (3.3.32)

1 p ab for arbitrary given numbers w0,..., w0. Due to nondegeneracy of π this foliation is cosymplectic. The corresponding Dirac bracket { , }D can be considered as a Poisson bracket on Lˆ(N) since, by deﬁnition

α a a b {v (x), w (y)}D = {w (x), w (y)}D = 0.

Let us show that the Dirac bracket is a (0, q)-bracket on Lˆ(N) with the same leading term α β α β {v (x), v (y)}D = {v (x), v (y)} + O(). Introduce the diﬀerential operator a0b0 d a0b0 2 Π = π − π 0 A (u) + A (u, u ) π 0 + O( ) ab ab a a 00 dx 01 x b b inverse to the operator Aab. Then the Dirac bracket has the form

p α β α β X αa bβ {v (x), v (y)}D = {v (x), v (y)} − A ΠabA δ(x − y). (3.3.33) a,b=1

We obtain a (0, q) Poisson bracket on Lˆ(N) eventually depending on the parameters 1 p w0,..., w0. The leading term

α β [0] αβ 0 αβ γ {v (x), v (y)}D = g (v, w0)δ (x − y) + Γγ (v, w0)vxδ(x − y) is itself a Poisson bracket (the so-called Poisson bracket of hydrodynamic type). Ac- cording to the theory of such brackets [44] one can choose local coordinates vα on N in such a way that αβ αβ g = const, Γγ = 0. This proves the theorem.

An alternative way to prove the Theorem is to write explicitly down the terms of the order −1 in the Jacobi identity

{{vα(x), vβ(y)}, wa(z)} + (cyclic) = 0

α β in order to prove that the leading term in {v (x), v (y)} does not depend on w, wx. Then from the leading term in the Jacobi identity

{{vα(x), vβ(y)}, vγ(z)} + (cyclic) = 0

40 it follows, as in [44], vanishing of the curvature of the metric gαβ(v). We believe that the problem of classiﬁcation of (p, q)-brackets (and also classiﬁcation of pencils of (p, q)-brackets) is very important in the Hamiltonian theory of integrable PDEs. In this paper we will mainly consider (0, n)-brackets leaving the general case for a subsequent publication. To illustrate our technique we will begin with a more simple example of (n, 0)- brackets.

3.3.1 Classiﬁcation of (n, 0)-brackets

Our ﬁrst result is

ˆ 2 Theorem 3.3.18 If M is a ball then all (n, 0) Poisson brackets in Λloc are equivalent w.r.t. the action (3.3.25) of the Miura group G.

Proof First we choose the Darboux coordinates for the symplectic structure on M. The Poisson bracket in question will read

∞ 1 X {ui(x), uj(y)} = πijδ(x − y) + k{ui(x), uj(y)}[k]. (3.3.34) k=0 Next we will try to kill all the terms of the expansion (3.3.34) by transformations of i i the form (3.2.22) with F0(u) = u , i = 1, . . . , n. To this end an appropriate version of Poisson cohomology will be useful. We deﬁne the Poisson cohomology H∗(Lˆ(M), $) ˆ 2 for a Poisson structure $ ∈ Λloc as the cohomology of the complex

ˆ 0 ∂ ˆ 1 ∂ ˆ 2 ∂ 0 → Λloc → Λloc → Λloc → ... (3.3.35) with the diﬀerential ∂β := [$, β]. In the present proof $ = πijδ(x − y). The cohomology is naturally decomposed into the direct sum

k k,m H = ⊕m≥−1H (3.3.36) with respect to monomials in , where Hk,m consists of the cocycles proportional to m. Denote ˜ k k,m H := ⊕m≥0H . (3.3.37)

The following obvious statement holds true.

Lemma 3.3.19 The ﬁrst non-zero term in the expansion (3.3.34) is a 2-cocycle in the Poisson cohomology H˜ 2 of the ultralocal Poisson bracket (2.2.24).

41 So, we will be able to kill this ﬁrst nonzero term of the expansion if we prove that, for ˆ 2 [−1] any 2-cocycle ∈ Λloc of the ultralocal bracket $ := { , } , there exists a vector ﬁeld a of the form (3.1.6) such that the Lie derivative Liea$ gives the cocycle and a|=0 = 0. This will follow from the following general statement about triviality of cohomology of the ultralocal Poisson bracket.

Lemma 3.3.20 For M = ball and the ultralocal Poisson bracket $ (2.2.24) with det(πij) 6= 0 the Poisson cohomology H˜ 1(Lˆ(M), $), H˜ 2(Lˆ(M), $) vanish.

The first proof of the lemma (and of the lemma 3.3.22 below) was obtained by E. Getzler [73] (also triviality of the higher cohomology has been proved). Independently, L. Degiovanni, F. Magri, V. Sciacca obtained another proof [18]. We have decided to present here our own proofs of triviality of cohomologies that closely follow the finite- dimensional case. Our proof will also be useful in the study of bihamiltonian structures below. Let us prove first triviality of H1. Let an evolutionary vector field a with the components a1,..., an be a cocycle. Denote

j ωi = πija

ij where the constant matrix πij is inverse to π . The condition ∂a = Liea$ = 0 reads ∂ωi X t ∂ωj = (−1)t ∂t−s . ∂uj,s s x ∂ui,t t≥s

Using (2.1.27) we conclude that there exists a local functional f¯ = R f dx such that

δf¯ ω = . i δui(x) Therefore the vector ﬁeld is a Hamiltonian one, δf¯ ai = πij . δuj(x)

Let us now proceed to the proof of triviality of H2. The idea is very simple: the bivector ij X ij (s) $ + εα = π δ(x − y) + ε As δ (x − y) s satisﬁes the Jacobi identity

[$ + εα, $ + εα] = 0(mod ε2) iﬀ the inverse matrix is a closed diﬀerential form 1 1 π dx ∧ δui ∧ δuj + ε ω dx ∧ δui ∧ δuj,s (mod ε2) 2 ij 2 i;js

42 where pq ωi;js := πipπjqAs . (3.3.38) Denote 1 ω = ω δui ∧ δuj,s. 2 i;js

From the condition of closedness δ(dx∧ω) = 0 ∈ Λ3 we derive, due to Corollary 2.1.14, i existence of a one-form dx ∧ φ, φ = φiδu such that δ(dx ∧ φ) = dx ∧ ω. The vector ﬁeld a with the components i ij a = π φj gives a solution to the equation [$, a] = α.

To be on the safe side we will now show, by straightforward calculations, that, indeed, the above geometrical arguments work. First, from the antisymmetry condition (2.2.17) for the bivector α it readily follows the antisymmetry condition (2.1.30) for the 2-form ω with the reduced components (3.3.38). Next, we are to verify that from the cocycle condition [$, α] = 0 where the Schouten - Nijenhuis bracket [$, α] is written in (2.2.19), it follows closedness (2.1.32) of dx ∧ ω ∈ Λ2. First we will rewrite the formula for the bracket in a slightly modiﬁed form. Diﬀerentiating the antisymmetry condition

X m Aik = − (−1)m ∂m−sAki s s x m w.r.t. ul,t and using the commutators (2.1.5) we obtain

(r) ik ki ! ∂A X q + r + s ∂Aq+r+s s = − (−1)q+r+s . ∂ul,t q r ∂ul,t−q

So the coeﬃcients of the Schouten - Nijenhuis bracket (2.2.19) can be rewritten as follows " ∂Aij ∂Aik [$, α]ijk = t πlk − s x,y,z ∂ul,s ∂ul,t  jk !(r) X q + r + t ∂As−q + (−1)q+r+t πli δ(t)(x − y)δ(s)(x − z). (3.3.39) q r ∂ul,q+r+t 

The coeffcient of δ(t)(x − y)δ(s)(x − z) must vanish for every t and s. Multiplying this coefficient by πiaπjbπkc we arrive at the condition of closedness (2.1.32) of the 2-form (3.3.38). Using Corollary 2.1.14 we establish existence of differential polynomials φ1, ..., φn representing the 2-form as in (2.1.31). The translation invariant vector field

i ij a = π φj will satisfy ∂a = α. This proves the lemma, and also the theorem.

43 3.3.2 Classiﬁcation of (0, n)-brackets

Let us now consider the Poisson structures in Λˆ 2 with identically vanishing leading term { , }[−1]. As above, the first nonzero term (3.1.11) is itself a Poisson bracket. The leading coefficient gij(u) of it determines a symmetric tensor field on M invariant w.r.t. the action of the Miura group. This gives a map

Λˆ 2/G → symmetric tensors on M.

Theorem 3.3.21 Let M be a ball. Then the only invariant of a (0, n) Poisson bracket in Λˆ 2 with respect to the action of the Miura group is the signature of the quadratic form gij(u).

Proof. The symmetric nondegenerate tensor gij(u) deﬁnes a pseudoriemannian metric

i j ij−1 gij(u)du du , (gij) = g on the manifold M. From the general theory of [44] of the Poisson brackets of the form (3.1.11) it follows that the Riemann curvature of the metric vanishes, and that ij k the coefficient Γk (u) in (3.1.11) is related to the Christoffel coefficients Γij(u) of the Levi-Civita connection for the metric by

ij is j Γk = −g Γsk. Using standard arguments of diﬀerential geometry we deduce that, locally coordinates v1(u), . . . , vn(u) exist such that, in the new coordinates the metric becomes constant

∂vk ∂vl gij(u) = ηkl = const. ∂ui ∂uj The Christoﬀel coeﬃcients in these coordinates vanish. Of course, all constant symmetric matrices ηkl of a given signature are equivalent w.r.t. linear changes of coordinates. We have reduced the proof of the theorem to reducing to the normal form (2.2.25) the Poisson bracket

∞ X {ui(x), uj(y)} = ηijδ0(x − y) + k{ui(x), uj(y)}[k] (3.3.40) k=1

i by the transformations of the form (3.2.22) with F0 =id. As in the proof of Theorem 3.3.18 the latter problem is reduced to proving triviality of the second Poisson cohomology of the Poisson bracket α of the form (2.2.25). Triviality of the cohomology is somewhat surprising from the point of view of ﬁnite-dimensional Poisson geometry. Indeed, as we have seen above this Poisson bracket degenerates. So we will be to also prove that all the cocycles are tangent to the leaves of the symplectic foliation (see the end of Section 1.1).

44 Lemma 3.3.22 For M = ball all the cocycles in H1(Lˆ(M)) and H2(Lˆ(M)) vanishing at = 0 are trivial.

Denote, like in (3.3.37), ˜ k k,m H = ⊕m>0H . (3.3.41) We are to prove that H˜ 1 = H˜ 2 = 0. Let us begin with proving triviality of H˜ 1. Using (2.2.5) we obtain

(t)! X ∂aj ∂ $ = Lie $ij = − ∂ (−1)tηip δ(x − y) a a x ∂up,t t≥0 " (t−r)# X ∂ai X t + 1 ∂aj − ηpj + (−1)t ηip δ(r+1)(x − y). (3.3.42) ∂up,r r + 1 ∂up,t r≥0 t≥r

Since (t) δa¯j X ∂aj = (−1)t δup(x) ∂up,t t≥0 is a diﬀerential polynomial in Λ0 ⊗ C[[]] of the degree 0 vanishing at = 0, from vanishing of the coeﬃcient in front of δ(x − y) we derive that

δa¯j = 0, j, p = 1, . . . , n. δup(x)

Using Example 2.1.5 we derive existence of diﬀerential polynomials bj s.t.

j j a = ∂xb , j = 1, . . . , n.

This is the crucial point in the proof: we have shown that the vector ﬁeld a is tangent to the level surface of the Casimirs (2.2.26). The remaining part of the proof is rather straightforward. Using (2.1.5) and also the Pascal triangle identity

m m m + 1 + = n n − 1 n we rewrite the coeﬃcient of δ(r+1)(x − y) in the form

" (t−r)# ∂ωk X t + 1 ∂ωl ∂ − (−1)t x ∂ul,r r ∂uk,t t≥r

∂ω ∂ω + k + (−1)r l = 0. ∂ul,r−1 ∂uk,r−1 Here i ij −1 ωk = ηlib , (ηij) = (η ) ,

45 the last two terms are not present for r = 0. As above, for r = 0 we derive that

(t) ∂ωk X ∂ωl = (−1)t . ∂ul ∂uk,t t≥0

R i Proceeding by induction in r we prove that the 1-form dx∧ωi δu is closed. Using the Volterra criterion we derive existence of a diﬀerential polynomial f s.t. ω = δ R f dx. Hence δf¯ ai = ηij∂ . x δuj(x) We proved triviality of H˜ 1. Let us proceed to prove the triviality of H˜ 2. The condition ∂α = 0 for α of the form (2.2.15) can be computed similarly to Example 2.2.9. We obtain a system of equations

(r) ij ki ! ∂A X q + r + s ∂Aq+r+s t ηlk + (−1)q+r+s ηlj ∂ul,s−1 q r ∂ul,t−q−1

jk !(r) X q + r + t ∂As−q + (−1)q+r+t ηli = 0 q r ∂ul,q+r+t−1 for any i, j, k, s, t (3.3.43)

(it is understood that the terms with s − 1, t − q − 1 or t + q + r − 1 negative do not appear in the sum). Recall that the crucial point in the proof of triviality of the 2-cocycle is to establish validity of (1.1.18) for the Casimirs (2.2.26) of $. Explicitly, we need to show that Z i j ij α(δu¯ , δu¯ ) = A0 dx = 0 for any i, j. (3.3.44)

We ﬁrst use (3.3.43) for s = t = 0 to prove that

(r) jk ! X ∂A ∂ (−1)r 0 ηli = 0. x ∂ul,r r Hence jk jk A0 = ∂xB for some differential polynomial Bjk. This implies (3.3.44). The rest of the proof is identical to the proof of Lemma 1.1.3. We first construct the vector field z (see the proof of the lemma 1.1.3). To this end we use the equation (3.3.43) for s = 0, t > 0:

(r) (r) ki ! jk ! X q + r ∂Aq+r X t + r ∂A (−1)q+r ηlj + (−1)t+r 0 ηli = 0. r ∂ul,t−q−1 r ∂ul,t+r−1 q,r r

46 Diﬀerentiating the antisymmetry condition

ik X r+1 ki(r) A0 = (−1) Ar w.r.t. ul,t−1 we identify the ﬁrst term of the previous equation with

∂Aik − 0 ηlj. ∂ul,t−1 The resulting equation coincides with the condition ∂ak = 0 of closedness of the 1- cocycle k i ik (a ) = A0 for every k = 1, . . . , n (see (3.3.42) for the explicit form of this condition). Using the ﬁrst part of Lemma we arrive at existence of n diﬀerential polynomials q1,..., qn s.t.

δq¯k Aik = ηis∂ . (3.3.45) 0 x δus(x)

The last step, as in the proof of Lemma 1.1.3, is to change the cocycle α to a cohomological one to obtain a closed 2-cocycle

α 7→ α + ∂z =: α0 for ∂ z = qi . ∂ui 0 ij The new 2-cocycle α will have the same form as above with A0 = 0. Denote

ij gi;js := ηipηjqAs , s ≥ 1.

We will now show existence of diﬀerential polynomials ωi;j0, ωi;j1,. . . s.t.

gi;j1 = ∂xωi;j0,

gi;js = ∂xωi;j,s−1 + ωi;j,s−2 for s ≥ 2. (3.3.46)

From (3.3.43) for s = 1, t = 0 we obtain

(r) jk ! X ∂A ∂ (−1)r 1 = 0. x ∂ul,r r As we already did many times, from the last equation it follows that

(r) jk ! X ∂A (−1)r 1 = 0. ∂ul,r r

47 This shows existence of ωi;j0. Using (3.3.43) for s = 1 and t > 0 we inductively prove existence of the diﬀerential polynomials ωi;j,t−1. Actually, we can obtain X s−l−2 ωi;jl = ∂x gi;js. (3.3.47) s≥l+2

From this it readily follows that the coeﬃcients ωi;js satisfy the antisymmetry conditions (2.1.30). Thus they determine a 2-form ω. Let us prove that the 2-form ω is closed. Denote t+s m−s t ! X X X X m ∂ωj;k,t−r J := + (−1)m ∂m−r−s ijk;st r s x ∂ui,m m=s r=0 m≥t+s+1 r=0 ∂ω ∂ω + i;j,s − i;k,t ∂uk,t ∂uj,s the l.h.s. of the equation (2.1.32) of closedness of a 2-form. Let us show that the coeﬃcient of δ(v)(x − y)δ(s)(x − z) in (3.3.43) is equal to

∂xJijk;t−1,s−1 + Jijk;t−1,s−2 + Jijk;t−2,l−1. (3.3.48) To this end we replace the second sum in (3.3.43) by ∂Aik − s ηlj. ∂ul,t−1

Lowering the indices by means of ηij and using (3.3.46) we obtain (3.3.48). From vanishing of (3.3.48) we inductively deduce that Jijk;st = 0 for all i, j, k = 1, . . . , n and all s, t ≥ 0 (observe that the coefficients Jijk;t0 = Jijk;0s = 0 due to our assumption ij R A0 = 0. This proves that the 2-form ω is closed. So ω = δ dx ∧ φ for some 1-form i φ = φiδu . Introducing the vector field i ik a = η φk we finally obtain, for the original cocycle α, α = ∂(a − z). Theorem is proved.

Example 3.3.23 The Poisson brackets (3.1.20) of the “interpolated” Toda lattice (3.1.19) can be reduced to the canonical form {u(x), w(y)} = δ0(x − y), {u(x), u(y)} = {w(x), w(y)} = 0 by the Miura-type transformation (u, v) 7→ (u, w), ∞ 1 X 1 v0(x) = [w(x + ) − w(x)] = ( ∂ )nw(x). (3.3.49) (n + 1)! x n=0 The inverse transformation reads ∞ −1 X Bn w(x) = ∂ e ∂x − 1 v(x) = ( ∂ )nv(x). (3.3.50) x n! x n=0

Here Bn are Bernoulli numbers.

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