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An introductory walk in integrable woods

Gaetan Borot∗

Mini-course, Leibniz Universitat, Hannover

July 7th, 2015

Contents

1 Lecture 1: Geometry of the KP hierarchy 3

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Hirota derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Finite Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Sato Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Wick theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Boson-fermion correspondence . . . . . . . . . . . . . . . . . . . 13

1.7 KP Tau function and Hirota bilinear equation . . . . . . . . . . 15

2 Lecture 2: avatars of the KP hierachy 17

2.1 Interpretations of the tau function . . . . . . . . . . . . . . . . . 17

2.2 Schur functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 N-solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Hirota bilinear difference equation . . . . . . . . . . . . . . . . . 22

2.5 Algebro-geometric solutions . . . . . . . . . . . . . . . . . . . . . 24

2.6 Example: Witten-Kontsevich generating series . . . . . . . . . . 27

3 Lecture 3: classical integrability and Calogero-Moser systems 30

3.1 Classical integrability . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Hamiltonian reduction . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Example: rational Calogero-Moser system . . . . . . . . . . . . . 33

3.4 The KP hierarchy in Lax form . . . . . . . . . . . . . . . . . . . . 37

3.5 Algebro-geometric perspective . . . . . . . . . . . . . . . . . . . 41

3.6 Appendix: Zhdanov-Trubnikov equation . . . . . . . . . . . . . 41

4 Lecture 4: a excursion into quantum integrability 43

4.1 What does it mean to quantize an integrable system ? . . . . . . 43

4.2 Quantum rational Calogero-Moser . . . . . . . . . . . . . . . . . 43

4.3 Quantum trigonometric Calogero-Moser . . . . . . . . . . . . . 47

∗Max-Planck Institut for Mathematics: [email protected]

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mailto:Max-Planck Institut for Mathematics: [email protected]

Contents

These are lectures notes for a 6h mini-course held in Hannover, July 7-8th 2015. I thank Olivia Dumitrescu and Marcus Sperling for organizing thelectures. The courses were directed to master/graduate students without anybackground assumed. My aim was, with a few representative examples, to il-lustrate several notions appearing in the context of integrable systems. I hopethis can serve as a clarification of mind for somebody interested to delve intothe specialized literature, and may have been confused by the high connectiv-ity of the graph of relations between problems in integrability.

The material presented here is fairly standard. Though I do not follow abook in particular, these monographs were useful in the preparation of thiscourse, and I sometimes borrowed some of their arguments or presentation:

• Introduction to classical integrable systems, O. Babelon, D. Bernard andM. Talon, Cambridge Monographs on Mathematical Physics, CUP (2002)

• Solitons: differential equations, symmetries and infinite dimensional algebras,E. Date and M. Jimbo and T. Miwa, Cambridge Tracts in Mathematics,135, CUP (2000)

• Lectures on Calogero-Moser systems, P. Etingof, Zurich (2005), math/0606233

I also recommend the review article:

• Algebraic theory of the KP equations, M. Mulase, Perspectives in Mathe-matical Physics, eds R. Penner and S. T. Yau, Editors, (1994) 151–218.

2

1 Lecture 1: Geometry of the KP hierarchy

1.1 Introduction

We consider families of compatible flows on a manifold, given by non linearordinary or partial differential equation (ODE or PDE). The variable of a flowis called a ”time”. Informally speaking, an ODE or PDE is integrable when onecan produce a large class of solutions. This is for instance the case when thereis a large (maybe infinite dimensional) group acting on the set of solutions,thus allowing the generation of interesting solutions from a ”trivial” solution.A precise definition of an integrable system will be given in Lecture 3. Actu-ally, there exist several notions of ”integrable systems”, and it is not knownwhether they are equivalent, though they have many examples in common.

For the moment, I just want to convey an idea of the geometry underly-ing most of integrable systems: on a set X, we have an action of an infinite-dimensional group G (think of a semi-infinite matrix), which is stabilized by asubgroup G+ (consisting of strict upper-triangular matrices) ; there is a max-imal abelian group T ⊆ G (the diagonal matrices) whose action generateour compatible flows on X, and a Borel subgroup G− (the invertible, lower-triangular matrices) containing T , that in general acts non-trivially on the setof solutions of the evolution equation.

A fundamental example realizing this geometry is the KP hierarchy. ”Hi-erarchy” means that we have a sequence of times (t1, t2, . . .) corresponding to(countably many) compatible flows. Here, the group will be a certain GL(∞),and the set X is an infinite-dimensional Grassmannian. The KP hierarchy israther important, since many integrable systems have been related to (or em-bedded in) the KP hierarchy or its generalisations.

Historically, integrability is born out of the study of two non-linear PDEs.The KdV equation was proposed in 1877 by Boussinesq – and later studiedby Korteweg and de Vries :

−4∂tu + 6u∂xu + ∂3xu = 0 .

It can be derived from hydrodynamics as an approximated equation for theamplitude of a wave propagating in shallow water. It was proposed to accountfor the observation of tidal bores (”mascaret” in French), which are solitarywaves that can propagate (e.g. on a canal) over several kilometers, withoutdeformation. x is the space variable, and t the time variable. The KP equationis a more recent generalisation introduced by Kadomtsev and Petviashvili(1970):

3∂2yu + ∂x

(− 4∂tu + 6u∂xu + ∂3

xu) = 0 .

It can be derived in plasma physics, in hydrodynamics, and in optics, as anapproximated equation satisfied by the difference amplitude (compared tolinear propagation) for a propagation in a non-linear medium along in thex-axis, with small variations of amplitude in the y axis.

3

1. Lecture 1: Geometry of the KP hierarchy

1.2 Hirota derivatives

Let f , g ∈ C∞(R). The Hirota derivatives are bilinear, differential operatorsDk[ f , g], indexed by an integer k ≥ 0, defined by the following generatingseries:

f (x + y)g(x− y) = ∑k≥0

Dk[ f , g](x)yk

k!.

For instance:

D[ f , g] = g2( f /g)′, D2[ f , g] = f ′′g− 2 f ′g′ + f g′′ ,

and:

D2[ f , f ] = 2 f 2(ln f )′′, D4[ f , f ] = 2 f 2(ln f )(4) − 6(ln f )′′ .

If P ∈ C[x], we define P(D) by substituting the monomial xk with the operatorDk. We see that if P is an odd polynomial, then P(D)[ f , f ] = 0. If f andg are functions of many variables (t1, t2, . . .), we can define similarly Hirota

operators ∏j Dkjtj

as the coefficient of ∏j(t′j)kj /k j! in the Taylor expansion of

f (t + t′)g(t− t′) when t′ → 0.

Using the previous formula, one can recast the x-derivative of the KdVequation as:

(1) u = 2 ∂2x ln τ, (−4DtDx + D4

x)[τ, τ] = 0 .

Similarly, the KP equation is equivalent to:

(2) u = 2 ∂2x ln τ, (3D2

y − 4DtDx + D4x)[τ, τ] = 0 .

In this form, Hirota (1974) was able to construct many solutions of the KdVand the KP equation. Let D = (Dt1 , Dt2 , . . .), and P(D) be an even polynomialin Hirota derivatives, such that P(0) = 0, and consider the equation:

(3) P(D)[τ, τ] = 1

First, we observe that τ = 1 is solution. Then, let us look for an approximatedsolution τ = 1+ ε τ1, modulo ε2. We obtain the equation P(∂t1 , ∂t2 , . . .) · τ1 = 0,which is easy to solve: if (κ(a))1≤a≤N are roots of P, then any linear combina-tion:

τ1 =N

∑a=1

ca exp(

∑j≥1

κ(a)j tj

)leads to a solution of (3) modulo ε2. Actually, if N = 1, we obtain an exactsolution to (3). Indeed, the function fκ(t) = exp(κ · t) satisfies fκ(t + t′) fκ(t−t′) = 1, which shows that P(D)[ fκ, fκ] = 0. Due to the quadratic nature of theHirota derivatives, we get

P(D)[1 + c fκ, 1 + c fκ] = P(D)[1, 1] + 2c P(D)[ fκ, 1] + c2 P(D)[ fκ, fκ] .

4

1.3. Finite Grassmannians

The first and last term vanish, and the middle term reduces to P(∂)[ fκ] = 0,which is zero since we chose P(κ) = 0. For the KdV equation (1), we deducethat for any constants ϕ and κ:

τ(x, t) = 1 + e2(κx+κ3t+ϕ) =⇒ u(x, t) =2κ2

cosh2[κx + κ3t + ϕ]

is an exact solution. This solution has first been found by Rayleigh at theend of the 19th century, by a direct guess. It described a wave perturbing themedium within a finite range of order 1/κ, progressing and preserving itsshape at speed −κ−2, and with maximal amplitude 2κ2. These two underlinedproperties justify the name of ”solitary wave” or ”soliton” for this solution,and the non-linearity of the equation is essential for this to happen.

For any N ≥ 2, Hirota found a way to obtain an exact solution, of the form:

(4) τ(t) = ∑A⊆J1,NK

cA(κ) ∏a∈A

fκ(a)(t)

with well-chosen coefficients cA. Sato then understood that these solutions areexpressing the geometry of the Grassmannians, which we will know explain.It is only in § 2.3 that we shall reveal the structure of the solutions (4), whichare called N-solitons.

1.3 Finite Grassmannians

Let V ' Cn be a vector space, e1, . . . , en its canonical basis, and (e∗i )i the dualbasis. If I is a set consisting of i1 < . . . < ik, we denote eI = ei1 ∧ · · · ∧ eik . Thek-Grassmannian of V is:

Grk(V) =

W ⊆ V, dim W = k

=

span(g · e1, . . . , g · ek), g ∈ GL(V)

'

A ∈ Matn×k(C), rank(A) = k

.

It has a natural structure of a complex manifold. Its dimension is k(n− k): thetangent space at W = span(v1, . . . , vk) consists of a choice of vectors (wi)1≤i≤kchosen in a supplement subspace of W in V, so that adding infinitesimally wito vi will change the subspace. The Plucker map:

∆ : Grk(V) −→ P(ΛkV)span(v1, . . . , vk) 7−→ [v1 ∧ · · · ∧ vk]

is an embedding of complex manifolds. If W ∈ Grk(V), we can decompose∆(W) with homogeneous coordinates on the canonical basis:

∆(W) = ∑I⊆J1,nK|I|=k

∆I(W) eI

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and ∆I(W) are called the Plucker coordinates. For k = 1 or (n − 1), ∆ isan isomorphism, almost by definition of the projective space. In general, thedimension of the Grassmannian is much smaller than the dimension n!/k!(n−k)! − 1 of the projective space. Therefore, there are many relations betweenthe Plucker coordinates. The first non-trivial example is Gr2(C

4) which hasdimension 4, while the projective space P(Λ2C4) has dimension 5. A 2-planein C4 is spanned by two vectors, say v and w, and its Plucker coordinates are:

∆12 = v1w2 − v2w1 ∆34 = v3w4 − v3w4 ,∆13 = v1w3 − w1v3 ∆24 = v2w4 − v4w2 ,∆14 = v1w4 − v4w1 ∆23 = v2w3 − v3w2 ,

and we immediately see that they satisfy the relation:

(5) ∆12∆34 − ∆13∆24 + ∆14∆23 = 0 ,

which cuts out Gr2(C4) in P(C6). To characterize in general the image of the

Grassmannian, we rely on:

1.1 lemma. Let X ∈ ΛkV. Denote LX = ` ∈ V∗, ι`(X) = 0

, and WX =

L⊥X = w ∈ V, ∀` ∈ LX , `(w) = 0

. WX is the smallest subspace of V such thatX ∈ ΛkW, and:

(6) WX = ιλ(X), λ ∈ Λk−1V∗ .

Proof. Let w1, . . . , wm be a basis of WX , that we complete with vectors vm+1, . . . , vnso as to form a basis of V. Let us decompose X on this basis:

X = ∑I⊆J1,mK

J⊆Jm+1,nK|I|+|J|=k

XI,J wI ∧ vJ .

For all ` ∈ LX , we have by definition of WX that ι`(wI) = 0, and by defini-tion of LX that `(X) = 0. Therefore, the coefficients XI,J with J 6= ∅ mustvanish, that is X ∈ ΛkWX . Then, if we take the interior product of X witha λ ∈ Λk−1V∗, we necessarily obtain a vector of WX . Conversely, for anyλ ∈ Λk−1V∗, and any ` ∈ LX , we have ι`(ιλ(X)) = ιλ(ι`(X)) = 0, hencewe have proved the equality (6) by double inclusion. Eventually, if W ⊆ V isanother subspace such that X ∈ ΛkW ′, then taking the interior product withan arbitrary λ ∈ Λk−1V shows that WX ⊆ W ′, i.e. WX is indeed the minimalsubspace whose k-exterior product contains X.

X ∈ Λk(V) represents an element in the Grassmannian iff there existsw1, . . . , wk ∈ V such that X = w1 ∧ · · · ∧ wk. This is equivalent to say thatWX is of dimension k, and to express this, we can say that for any w ∈ WX ,w ∧ X = 0. Hence:

X ∈ ∆(Grk(V)) ⇐⇒ ∀λ ∈ Λk−1V∗, ιλ(X) ∧ X = 0 .

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1.3. Finite Grassmannians

Let us write down these relations in the canonical basis. Let J ⊆ J1, nK ofcardinality k− 1. We have:

ιe∗J (X) =n

∑i=1

(−1)k−pos(i,J)∆Jti(X) ei ,

where the sign is determined by the position of i in the set J t i. Therefore, Xrepresents an element of the Grassmannian iff the coefficients in the expres-sion below vanish:

(7) ιe∗J (X) ∧ X = ∑I⊆J1,nK|I|=k+1

(∑i∈I

(−1)k−pos(i,J)−pos(i,I)+1 ∆Jti(X)∆I\i(X))

eI .

It is true (but we will not prove it) that all these relations can be obtained byalgebraic manipulations from the single relation in Gr2(C

4).The group GL(V) acts naturally on P(ΛkV), and this action preserves the

image of the Grassmannian.

•Dodgson condensation. En passant, here is a nice consequence of the Pluckerrelations in Grn(Cn+2). Take A a square invertible matrix of size n, and let Abe a matrix of size (n + 2)× n:

A =

A1 0 · · · 0 00 0 · · · 0 1

.

Let I = J2, n + 2K and J = J1, n− 1K. Then, the sum over indices i ∈ I that donot belong to J consists of 3 terms i = n, n + 1, n + 2, and the Plucker relationgives for ∆I = ∆I(A):

∆1,...,n∆2,...,n−1,n+1,n+2 − ∆1,...,n−1,n+1∆2,...,n,n+2 + ∆1,...,n−1,n+2∆2,...,n+1 = 0 .

And, expanding with respect to the sparse rows n + 1 and n + 2 when theyare present, we find in terms of A[I; J] = (Aij)i/∈I,j/∈J :

∆1,...,n = det A ∆2,...,n−1,n+1,n+2 = (−1)n det A[1, n; 1, n] ,

∆1,...,n−1,n+1 = (−1)n+1 det A[n; 1] ∆2,...,n,n+2 = det A[1; n] ,

∆1,...,n−1,n+2 = det A[n; n] ∆2,...,n+1 = (−1)n+1 det A[1; 1] .

This pretty identity is easy to remember:

= -

Figure 1: The full square denotes the matrix A, and the shaded orange partthe submatrix of which one should take the determinant.

The identity was proved by Desnanot in 1819 for matrices of size ≤ 6, and

7


in the general case by Jacobi in 1831. It is the basis of the ”condensation al-gorithm” proposed by Dodgson (aka Lewis Carroll) in 1866, which is ratherefficient in the computation (by hand or on a computer) of determinants. Itreduces the computation to 2× 2 determinants. It is also the starting point todefine a deformation of the determinant called ”λ-determinant”, which takesFigure 1 as a rule, but with a prefactor of λ in front of the last term. Theλ-determinant has a rich combinatorial structure, related to alternating signmatrices, integrable spin chains, etc.

1.4 Sato Grassmannian

• Definitions. We now would like to describe the analog geometry for infinitedimensional Grassmannians. We shall work with the vector spaces

V∞ =

∑n∈Z

an zn, : an = 0 for n large enough

, V+ = C[z] .

The basis element zn will also be denoted en+1/2. So, es is indexed by s ∈Z + 1/2, and V+ is conveniently described as the span of (es)s>0. The SatoGrassmannian is defined as:

Gr(V∞) = W ⊆ V∞ πW : W → V+ is Fredholm .

πW denotes the projection of W to V+, and being Fredholm means that itskernel and cokernel are finite dimensional. Then, we can define the index:

qW = dim Ker πW − dim[V+/Im πW ]

We would like to have the analog of a Plucker embedding in an infinitedimensional projective space PF , and an action of a group like GL(V∞) thatpreserves the image of the Grassmannian. Some care is need in this construc-tion, because of the infinite dimensional issues.

• The Fock space. We first define F , which is called the Fock space. It is thelinear span of Maya diagrams:

1.2 definition. A Maya diagram is a map c : Z + 1/2 → , • such thatc(s) = for s large negative enough, and c(s) = • for s large positive enough.

Here are examples of Maya diagrams. The • can be thought as ”particles”,and as ”holes”. The vector |0〉 ∈ F where s > 0 are filled with particles, ands < 0 with holes, is called the ”vacuum”, or the Dirac see. We denote |`〉 ∈ F isthe vector where the Dirac sea is translated by −` ∈ Z. One can pass from oneMaya diagram to another, by defining the representation of a Clifford algebraon F . The Clifford algebra in question is the algebra defined by generators(ψs, ψ†

s )s∈Z+1/2 satisfying the anticommutation relations, for r, s ∈ Z + 1/2:

(8) ψr, ψs = ψ†r , ψ†

s = 0, ψr, ψ†s = δr+s,0 .

Its action on F is defined by considering that a Maya diagram is an ”infinite-

8

1.4. Sato Grassmannian

wedge” vector ∧s : c(s)=•es, with increasing indices from left to right, andputting:

ψs = es ∧ ·, ψ†s = ιe∗−s

.

These operations indeed satisfy the anticommutation relations (8) due to theproperties of the wedge and the interior product. For instance, the vector cor-responding to the first Maya diagram in Figure 2 is e−5/2 ∧ e1/2 ∧ e5/2 ∧ e7/2 ∧· · · , and can be obtained as ψ†

−3/2ψ−5/2|0〉.The ψs, ψ†

s with s < 0 are called ”creation operators”, and the ψs, ψ†s with

s > 0 are the ”annihilation” operators. This summarizes their effect on thevacuum: for s < 0, ψs (resp. ψ†

s ) creates a particle at position s (resp. a hole atposition −s) compared to the Dirac sea ; for s > 0, we have

∀s > 0, ψs|0〉 = ψ†s |0〉 = 0 .

Each Maya diagram represents a point in Gr(V∞): if p1 < . . . < pn < 0 arethe positions in Z + 1/2 of the particles, and 0 < q1 < . . . < qm the positionof the holes, then W = span(ep1 , . . . , epn , (es>0)s 6=qi ). The index of W is:

qW = #particles− #holes

added to obtain the Maya diagram from the vacuum |0〉.F is naturally equipped with the scalar product making the basis of Maya

diagrams orthonormal. If |ω〉 is a vector in F , the scalar product with |ω〉 isdenoted 〈ω|.

It is convenient to make generating series for the operators of the Cliffordalgebra:

ψ(ζ) = ∑s∈Z+1/2

ψs ζ−s−1/2, ψ†(ζ) = ∑s∈Z+1/2

ψ†s ζ−s−1/2 .

Notice that the powers of ζ are integers.

12

32

52

72

− 12− 3

2− 5

2

12

32

52

72

− 12− 3

2− 5

2

12

32

52

72

− 12− 3

2

|0〉

|2〉

Figure 2: Examples of Maya diagrams.

• Normal ordering. If P is a non-constant monomial in ψ’s and ψ†’s, we defineits normal ordering :P: to be the monomial where the annihilation operatorsare on the right of the creation operators, multiplied by the sign of the permu-

9


tation that had to be applied to achieve this ordering. The normal ordering isunique, since the ψ’s (resp. the ψ†) anticommute among themselves. We alsotake the convention :1:= 0, and the normal ordering can be extended by lin-earity to any polynomial in the ψ and ψ†. The essential property of the normalordering is that :P: |0〉 = 0.

We will very often meet the normal ordering of the bilinear expression:

:ψrψ†s :=

−ψ†s ψr if r > 0 and s < 0

ψrψ†s otherwise

,

and since ψ and ψ† always anticommute up to a scalar, there exists a scalar csuch that :ψrψ†

s := ψrψ†s − c. Since normal ordered expressions return 0 on the

vacuum, we find:

(9) :ψrψ†s := ψrψ†

s − 〈0|ψrψ†s |0〉 = ψrψ†

s − δr+s,0δs<0 .

• Action of a Lie algebra on F . Let us define:

gl(V∞) =

c + ∑r,s

Xrs : ψ−rψ†s : c ∈ C, Xrs = 0 for |r− s| large enough

.

In other words, (Xrs)r,s∈Z+1/2 is a band matrix with finite width.

1.3 lemma. gl(V∞) is a Lie algebra, and its action on F is well-defined.

Proof. We have to prove that the commutator of two elements in gl(V∞) is well-defined and still in gl(V∞), which is not obvious due to the possibly infinitenumber of non-zero coefficients Xrs. Let us compute the commutator:

γrs,r′s′ =[

:ψ−rψ†s : , :ψ−r′ψ

†s′ :]= [ψ−rψ†

s , ψ−r′ψ†s′ ] .

For this we use the basic identities [a, bc] = [a, b]c + b[a, c] and [ab, c] =ab, c − a, cb, and we only get contributions from anticommutators of aψ with a ψ†:

γrs,r′s′ = ψ−rψ†s′δr′ ,s − ψ−r′ψ

†s δr,s′ .

Let us rewrite it in terms of normal ordered expressions:

γrs,r′s′ =:ψ−rψ†s′ : δr′ ,s− :ψ−r′ψ

†s : δr,s′ + δr′ ,sδr,s′(δr<0 − δs<0) .

Therefore:

∑r,s,r′ ,s′

XrsX′r′s′ γrs,r′s′ = ∑`,k[X, X′]−k` :ψ−kψ†

` : +∑r,s

XrsX′sr(δr<0 − δs<0) .

The commutator of two band matrices of finite width is well-defined, andagain a band matrix of finite width. Besides, the last sum contains only finitelymany non-zero terms, since s and r must have opposite signs while |s− r| isbounded. This shows that gl(V∞) is a Lie algebra. Since a given Maya di-agram differs from |0〉 only by finitely many particles and holes, applying

10

1.4. Sato Grassmannian

∑r,s Xrs :ψ−rψ†s : to this diagram will result in finitely many non-zero terms, so

the action of the Lie algebra on F is well-defined.

We define the following elements of gl(V∞):

∀n ∈ Z, Hn = ∑s∈Z+1/2

:ψ−sψ†s+n: .

From the previous computations, we can deduce the commutation relation ofthese operators:

1.4 lemma. For any n, n′ ∈ Z:

[Hn, Hn′ ] = nδn+n′ ,0, [Hn, ψ(ζ)] = ζnψ(ζ), [Hn, ψ†(ζ)] = −ζnψ†(ζ) .

In particular, (Hn)n>0 forms a commutative subalgebra of gl(V∞). The follow-ing lemma is useful for computations in gl(V∞): it says that an element isdetermined, up to a scalar, by its commutators with (Hk)k∈Z.

1.5 lemma. If X ∈ gl(V∞) such that [Hk, X] = 0 for any k ∈ Z, then X is scalar.

The proof consists in expressing the constraints on the coefficients Xrsgiven by [Hk, X] = 0, and is left as exercise.

• Remark: the group GL(V∞). So far, we have an action of a Lie algebra. Wecan upgrade it to the action of a group by setting:

GL(V∞) =

eX1 · · · eXm , Xi ∈ gl(V∞)

.

Apart from multiplication by a scalar, the action of an element of this group isdefined by expanding the exponentials ; we get an infinite linear combinationof monomials in the Xi. But since each Xi annihilates at least one object, anda given Maya diagram |ω〉 has only finitely many objects (particle or hole)that can be annihilated before returning 0, there will be only finitely manymonomials whose action on |ω〉 do not vanish. The action of GL(V∞) on F istherefore well-defined.

We will not be concerned with topological questions about this infinite-dimensional group, since it is not necessary to describe the geometry of theKP hierarchy. In a sense, the way we use the group does not contain muchmore information that the Lie algebra itself.

• Geometry of the Sato Grassmannian. Remind that we think of elementsof F as linear combinations of infinite wedge products zs1 ∧ zs2 ∧ · · · of ba-sis elements zi ∈ V∞. Therefore, the operator :ψ−rψ†

s : is interpreted, up to asign, as the operator on F induced by the endomorphism of V∞ that sendszr−1/2 to zs−1/2, and all other basis elements zi to 0. Therefore, it shouldpreserve the Sato Grassmannian – it will be embodied rigorously in Theo-rem 1.6 below. It contains the commutative algebra generated by the Hn forn > 0, which are the operators on F induced by the endomorphism of V∞that sends zi to zi+n for any i ∈ Z, i.e. the multiplication by zn. We have

11


H0 = ∑s>0 ψsψ†−s − ∑s<0 ψ†

−sψs ; when applied to a Maya diagram, it countsthe number of particles minus the number of holes, i.e. it returns the Mayadiagram multiplied by the index qW of the corresponding point in the SatoGrassmannian.

In the finite-dimensional case, the Grassmannian was the orbit of a singlesubspace under the action of GL(V). This is not true anymore in infinite di-mensions. For instance, the action of GL(V∞) preserves the Fredholm index,while we have vectors |`〉 representing elements in the Sato Grassmannianhaving arbitrary Fredholm indices ` ∈ Z. To simplify, we shall work with thebig cell of the Sato Grassmannian, defined as:

Gr0(V∞) =

W ⊆ V πW : W → V+ isomorphism

.

Since GL(V∞) just induce (certain) change of basis in V∞, the orbit of |0〉 byGL(V∞) is exactly the big cell.

1.6 theorem. |ω〉 ∈ F represents an element of Gr0(V∞) iff

(10) ∑s∈Z+1/2

ψ−s|ω〉 ⊗ ψ†s |ω〉 = 0 .

Proof. We shall only prove the implication. (10) holds for ω = |0〉, since eitherψ−s|0〉 = 0 (when s < 0) or ψ†

s |0〉 = 0 (when s > 0). Let us prove that theaction of the Lie algebra preserves this relation, i.e. for any X ∈ gl(V∞),

(11) ∑s∈Z+1/2

Xψ−s|0〉 ⊗ ψ†s |0〉+ ψ−s|0〉 ⊗ Xψ†

s |0〉 = 0 .

It is clear when X is scalar, and for X =:ψ−kψ†` :, we have X|0〉 = 0 since X is in

normal order. Thus, we can replace Xψ−s with its commutator and similarlyin the second term. Using the anticommutation relations (8), the left hand sideof (11) then reads:

∑s

δ`,sψ−k|0〉 ⊗ ψ†s − ψ−s|0〉 ⊗ δk,sψ†

` |0〉 ,

which indeed vanishes. This implies that (10) holds for any ω representing anelement of the big cell.

We see that these Plucker relations are the analog of the Plucker relations(7) for the finite Grassmannians, but now they contain infinitely many termscorresponding to addition/removal of basis elements es ∈ Z+ 1/2. The coeffi-cients in the basis of Maya diagrams of |ω〉 representing a point in the big cell,are the analog of the Plucker coordinates. We will now describe a convenientway to encode all of them at the same time.

Quite often, we will say that a vector |ω〉 is in the image of Sato Grassman-nian, while stricto sensu we mean a projective vector.

12

1.5. Wick theorem

1.5 Wick theorem

The following result is extremely useful to deal with Plucker coordinates, andcan be proved e.g. by induction on the number of ψ and ψ† involved:

1.7 theorem. We have:

〈0|ψa1 · · ·ψam ψ†b1· · ·ψ†

bn|0〉 =

det1≤i,j≤n〈0|ψai ψ

†bj|0〉 if n = m

0 otherwise.

The scalar products in the determinant have already been computed in(71). The result can be compactly encoded in a generating series:

〈0|ψ(p1) · · ·ψ(pn)ψ†(q1) · · ·ψ†(qn)|0〉 = det

1≤i,j≤n

( 1pi − qj

)(12)

=∆(p)∆(q)

∆(p; q).(13)

where we have introduced:

∆(p) = ∏i<j

(pi − pj), ∆(p; q) = ∏i,j(pi − qj)

for any uples p and q of complex variables. The last equality in (13) is theevaluation of the Cauchy determinant. The formula (12) makes sense for |pi| >|qj| for any i, j. We will very often use the identity:

(14) exp(

∑k>0

zk1zk

2k

)=

11− z1z2

,

which follows immediately from the Taylor expansion of − ln(1− z) at z = 0.

1.6 Boson-fermion correspondence

The Fock space admits a simpler description, in terms of functions of infinitelymany variables. Let t = (t1, t2, t3, . . .), and denote:

H(t) = ∑k>0

tk Hk

If z is a complex variable, we denote [z] the sequence (zk/k)k≥1. We considerthe linear map:

T : F −→ C[[ζ±1, t1, t2, t3, . . .]]|ω〉 7−→ ∑`∈Z ζ`〈`|eH(t)|ω〉 .

1.8 theorem. T is an isomorphism, and we have:

(15) T(

Hk|ω〉)=

∂tk T

(|ω〉) if k > 0

−kt−k T(|ω〉)

if k < 0,

13


and:

T(ψ(z)|ω〉

)= exp

(∑k>0

tk zk)

exp(− ∑

k>0

z−k

k∂tk

)ζ exp(zζ∂ζ) T|ω〉 ,

T(ψ†(z)|ω〉

)= exp

(− ∑

k>0tkzk

)exp

(∑k>0

z−k

k∂tk

)ζ−1 exp(−zζ∂ζ) T|ω〉 .

Proof. We do not justify here that T is an isomorphism. Since (Hk)k>0 is acommutative family, formula (15) is obvious for k > 0. For k < 0, we observethat 〈`|Hk = 0 for any ` ∈ Z. This can be checked directly on the basis, butwe can also remark that Hk is pulling a particle by k < 0 when it can, while|`〉 is a Maya diagram that can never obtained by pulling a particle to the left(since the holes are located to the left of the particles), hence the coefficientof a vector of the form Hk|ω〉 on the basis element |`〉 can only be zero. So,we just need to commute Hk to the left. This can be done by noticing that, forany entire function f , the commutation relation [H−k, Hk] = −k implies that[ f (H−k), Hk] = −k f ′(H−k), and here with f (x) = ext−k we obtain:

T(

Hk|ω〉)= −kt−kT

(|ω〉)

.

Now, let us compute:

T(ψ(z)ω〉

)= ∑

`∈Z

ζ` 〈`|eH(t)ψ(z)|ω〉 .

We use the commutation relation (Lemma 1.4) to move eH(t) to the right ofψ(z), and then formula (15) to obtain:

T(ψ(z)|ω〉

)= exp

(∑k>0

tkzk)

∑`∈Z

ζ` 〈`|ψ(z)eH(t)|ω〉

.

The final formula we want to prove for the left-hand side is equivalent to, forall ` ∈ Z:

(16) 〈`|ψ(z) = z`−1〈`− 1| exp(− ∑

k>0

z−k

kHk

).

Since the same strategy works for all `, let us do the proof for ` = 0. It isenough to prove (16) when applied to any basis element of F . That is, weneed to show that:

〈0|ψ(z)ψ(p2) · · ·ψ(pn)ψ†(q1) · · ·ψ†(qn)|0〉

= z−1〈0|ψ−1/2e−H([z−1]) ψ(p2) · · ·ψ(pn)ψ†(q1) · · ·ψ†(qn)|0〉 .

According to Wick theorem, the left-hand side of (17) is equal to:

(17)∆(p)∆(q)

∆(p; q)∏n

i=2(z− pi)

∏ni=1(z− qi)

.

14

1.7. KP Tau function and Hirota bilinear equation

To compute the right-hand side of (17), let us commute the e−H([z−1]) to theright, and use that eH(t)|0〉 = |0〉. We find:

z−1 exp(

∑k>0

[ n

∑i=2

pki z−k

k−

n

∑i=1

qki z−k

k

])×˛

dw2iπw

〈0|ψ(w)ψ(p2) · · ·ψ(pn)ψ†(q1) · · ·ψ†(qn)|0〉 ,(18)

and in virtue of the identity (14), we get:

1z

∏ni=2(1− pi/z)

∏ni=1(1− qi/z)

˛dw

2iπw∏n

i=2(w− pi)

∏ni=1(w− qi)

∆(p)∆(q)∆(p; q)

,

which is indeed equal to (17). The proof for ψ† is similar.

1.7 KP Tau function and Hirota bilinear equation

Notice that if |ω〉 represents an element in the big cell of Sato Grassmannian,since eH(t) preserves the index, T

(|ω〉)

only contains the power ζ0, so is iden-tified with an element of C[[t1, t2, t3, . . .]].

1.9 definition. A tau function of the KP hierarchy is an element τ(t) ∈C[[t1, t2, t3, . . .]] in the image via T of an element W of the big cell of the SatoGrassmannian. It is a tau function of the KdV hierarchy if moreover z2W ⊆W.

As such, a tau function is defined up to a multiplicative constant. Noticethat the tau function of the KdV hierarchy only depends on the ”odd times”t2j+1. Unless precised, we will use the name ”tau functions” for those of the KPhierarchy. The Plucker relations translate into the Hirota bilinear relations:

1.10 corollary. τ ∈ C[[t1, t2, . . .]] is a tau function iff the following equation holds,order by order in Taylor expansion when u = (u1, u2, . . .)→ 0:

(19)˛

dz exp(

2 ∑k>0

zk uk

)exp

(− ∑

k>0

z−k

k∂vk

)τ(v + u)τ(v− u) = 0 .

Proof. The Plucker relation (10) can be written as:˛

dz ψ(z)|ω〉 ⊗ ψ†(z)|ω〉 = 0 .

Let us apply T ⊗ T, i.e. apply eH(t) ⊗ eH(t′) and take the scalar product with〈1| ⊗ 〈−1|. We then use Theorem 1.8 to find:

˛dz2iπ

exp(

∑k>0

(tk − t′k)zk)

exp(

∑k>0

z−k

k(∂t′k− ∂tk )

)τ(t)τ(t′) = 0 .

Setting (t, t′) = (v + u, v− u) brings the result in desired form.

These relations are equivalent to a collection of non linear PDEs satisfied

15


by the tau functions. To see that, it is convenient to introduce the polynomialspn ∈ C[[t]] such that:

(20) exp(

∑k>0

tk zk)= ∑

n≥0pn(t) zn .

For instance:

p0 = 1, p1 = t1, p2 = t2 +t212

, p3 = t3 + t2t1 +t316

, . . .

Then, using the Hirota derivative notations with D = (Dt1 , Dt2 , . . .), and thepartial derivatives ∂ = (∂t1 , ∂t2 /2, ∂t3 /3, . . .), (19) can be recast:

∑n≥0

pn(u) pn+1(−∂) exp(

∑j>0

uj Dtj

)[τ, τ] = 0 .

For instance, collecting the coefficients of uk1, we get:

[u1] −Dt2 [τ, τ] = 0[u2

1]16(− 4 Dt3 + D3

t1

)[τ, τ] = 0

[u31]

118(

D4t1+ 3D2

t2− 4Dt1 Dt3 + 3D2

t1Dt2 − 6Dt4

)[τ, τ] = 0

We however remind that any odd polynomial of D applied to [τ, τ] vanishes.Thus, the two first equations are trivially satisfied, and the first non-trivialequation is:

(D4t1+ 3D2

t2− 4Dt1 Dt3)[τ, τ] = 0 .

We recognize the KP equation in (2). More generally, the coefficients of mono-mials in u generate a sequence of non-linear PDEs, called the KP hierarchy. Byconstruction, they are all compatible. For this reason, we say that these PDEsare integrable.

16

2 Lecture 2: avatars of the KP hierachy

In this lecture, we will describe several examples of tau functions, and seeseveral equivalent forms of the KP hierarchy.

2.1 Interpretations of the tau function

Let W be an element of the big cell of Sato Grassmannian, and |ωW〉 the cor-responding vector in F . We also denote W(t) the element of the big cell cor-responding to the vector exp(H(t))|ωW〉, and τW(t) the tau function attachedto W.• τW(t) is the coefficient of |0〉 in the decomposition of exp(H(t))|ωW〉 onMaya diagrams.• Since the projection πW : W → V+ is an isomorphism, let us denote fn(z) =π−1

W (zn). Then, we know that W = span( f0(z), f1(z), . . .), and

|ωW〉 = f0(z) ∧ f1(z) ∧ · · ·

Since Hk is the operator induced by the multiplication by zk, we have a similardescription of the subspace W(t), with the basis elements:

fn(z, t) = fn(z) exp(

∑k>0

tkzk)

.

The coefficient of |0〉 = z0 ∧ z1 ∧ · · · in exp(H(t))|ωW〉 is thus given by theFredholm determinant of the change of basis:

τW(t) = Det[πW : W → V+] = detm,n≥0

[ ˛ dz2iπ

z−(m+1) fn(z, t)]

.

• A particular case occur when fn(z) = zn for n > N. In this case, the semi-infinite matrix is block-upper triangular, and the second diagonal block (rowsand columns from N + 1 to ∞) is the identity. Therefore, the determinanttruncates and we have:

(21) τW(t) = det0≤m,n≤N

[ ˛ dz2iπ

z−(m+1) fn(z, t)]

.

Let us denote:F(−N)

n (t) =˛

dz2iπ

z−(N+1) fn(z, t) .

We observe that:˛

dz2iπ

z−(m+1) fn(z, t) = ∂(N−m)t1

F(−N)n (t) .

Hence, the (N + 1)× (N + 1) determinant is actually a Wronskien:

τW(t) = Wr[F(−N)

N−1 (t), . . . , F(−N)0 (t)

],

17

2. Lecture 2: avatars of the KP hierachy

where the derivatives are always taken with respect to the variable t1, alsowritten x.• Let us decompose |ωW〉 = ∑λ cW,λ |λ〉 on the basis of Maya diagrams λ ofindex 0, and let χ|λ〉(t) the tau function for such a diagram. We then have:

τW(t) = ∑λ

cW,λ χλ(t) .

In the next paragraph, we will give several expressions for χλ, in particularthat relates it to the Schur functions.

2.2 Schur functions

• Partitions and Maya diagrams. A partition λ is a finite decreasing sequenceof positive integers ; its length is denoted `(λ). λT denotes the transpositionof λ, it is a partition whose i-th part is the number of parts of λ that are ≥ i. Inparticular, `(λT) = λ1. Very often, we will complete a partition by infinitelymany zeroes – these zeroes are not counted in the length.

To any partition λ, one can associate a Maya diagram |λ〉 as follows. Let λbe a partition, and let us rotate it as in Figure 3. We denote `(λ) the length ofthe partition. The particles are located at the abscissa under the steps up:

(22) pi =

i− λi − 1/2 1 ≤ i ≤ `(λ)i− 1/2 i > `(λi)

,

and the holes are located at the abscissa under the steps down:

(23) qi =

λT

i + 1/2− i if 1 ≤ i ≤ λ11/2− i if i > λ1

.

Here is a second way to describe |λ〉. Let a1 > . . . > an be the arm lengths, andl1 > . . . > lm be the leg lengths of λ (see Figure 4). Then, the particles withnegative abscissa have coordinates −ai + 1/2, while the holes with positiveabscissa have coordinates li − 1/2, therefore:

(24) |λ〉 = (−1)∑i(li−1)ψ†−a1+1/2 · · ·ψ†

−ar+1/2ψ−lr′+1/2 · · ·ψ−l1+1/2|0〉 ,

where the prefactor compensates the signs introduced by the creation of holesin the Dirac sea. This is actually a bijection between Maya diagrams and par-titions, the index being r - r′. They represent elements of the big cell of theGrassmannian iff r = r′.

• Tau function. Let us denote χλ(t) = T(|λ〉), and we will focus on partitions

with r = r′. It is a good exercise for the reader to derive the analog of theformulas below for T

(|λ〉)

with index ` = r− r′ 6= 0.

2.1 theorem.

(25) χλ(t) = det1≤i,j≤r

qai ,lj(t) ,

18

2.2. Schur functions

where:

qa,l(t) = (−1)l−1a+l−1

∑k=a

pk(t)pa+l−(k+1)(−t) .

In the two next formulas, we use the Miwa substitution:

tk =N

∑i=1

xki

k

for x = (x1, x2, . . . , xN) a finite set of variables, and denote:

(26) sλ(x) = χλ(t) .

Let us remind the definition of the complete homogeneous polynomials (hm)m≥0:

∑m≥0

hm(x) zm =N

∏i=1

11− xiz

.

In particular, we deduce from (20) that:

pn(t) = hn(t).

2.2 theorem. sλ defined by (26) coincide with the Schur polynomial:

(27) sλ(x) =det1≤i,j≤N

[x

λj−j+Ni

]det1≤i,j≤N

[x−j+N

i] .

It satisfies the Giambelli formula:

(28) sλ(x) = det1≤i,j≤N

[hλi−i+j(x)

].

Proof. We compute from (24):

〈0|eH(t)|λ〉 = (−1)∑i(li−1)〈0|eH(t)ψ†−a1+1/2 · · ·ψ†

−ar+1/2ψ−lr′+1/2 · · ·ψ−l1+1/2|0〉 .

We can commute the eH(t) to the right with Lemma 1.4, that can be rewritten:

eH(t)ψs =

˛dz2iπ

zs−1/2 exp(

∑k>0

tk zk)

ψ(z)

= ∑k≥0

pk(t)ψs+k ,

and similarly:eH(t)ψ†

s = ∑k≥0

pk(−t)ψ†s+k .

19


Since eH(t) leaves |0〉 invariant, we obtain using Wick theorem:

〈0|eH(t)|λ〉 = (−1)∑i(bi−1) ∑k1,...,kr≥0k′1,...,k′r′≥0

r

∏i=1

pki(t)

r′

∏i=1

pk′i(−t)

〈0|ψ†−a1+1/2+k1

· · ·ψ†−ar+1/2+kr

ψ−br+1/2+k′r · · ·ψ−b1+1/2+k′1|0〉

= (−1)∑i(bi−1) det1≤i,j≤r

[∑

k,k′≥0pk(t)pk′(−t)δ−ai−bi+k+k′+1,0 δ−ai+1/2+k>0

]

= (−1)∑i(bi−1) det1≤i,j≤r

[ bj−1

∑l=0

pai+l(t)pbj−1−l(−t)]

,

which yields formula (25).

For Theorem 2.2, we rather start from the fact that |λ〉 is represented bythe infinite wedge f0(z) ∧ f1(z) ∧ · · · with fi(z) = zpi−1−1/2, and pi are theposition of the particles given in (22). By the remark of § 2.1, the tau functionis a determinant of size `(λ)× `(λ):

χλ(t) = det1≤m,n≤`(λ)

[ ˛ dz2iπ

z−m+pn−1/2 exp(

∑k>0

tkzk)]

= det1≤m,n≤`(λ)

[ ˛ dz2iπ

z−m+pn−1/2N

∏i=1

11− zxi

]= det

1≤i,j≤`(λ)

[∑k≥0

˛dz2iπ

z−m+pn−1/2+k hk(x)]

= det1≤i,j≤`(λ)

[hm−pn+1/2(x)

],

which is Giambelli formula (27). To derive (28), we evaluate the third line ofthe previous equation in a different way, by expanding the determinant:

χλ(t) = ∑σ∈SN

ε(σ)

˛ N

∏i=1

dzi2iπ

z−σ(i)+pi−1/2i

N

∏i,j=1

11− zixj

.

We recognize the evaluation of the Cauchy determinant:

χλ(t) = ∑σ∈SN

ε(σ)

˛ N

∏i=1

dzi2iπ

z−σ(i)+pi−1/2i ∆−1(z)∆−1(x) det

1≤i,j≤N

( 11− zixj

)

=

˛ N

∏i=1

dzi zpi−1/2i

2iπdet1≤i,j≤N [z

−ji ]

∆(z)∆(x)det

1≤i,j≤N

( 11− zixj

)=

1∆(x)

˛ N

∏i=1

dzi zpi−1/2−Ni2iπ

det1≤i,j≤N

( 11− zixj

)

20

2.2. Schur functions

=1

∆(x)det

1≤i,j≤N

[ ˛dz2iπ

zpi−1/2−N

1− zxj

]

=det1≤i,j≤N [x

N−pi−1/2j ]

∆(x).

In the process, we have used the evaluation of the Vandermonde determinant,in the form:

det1≤i,j≤n

z−ji =

N

∏i=1

z−Ni ∆(z) .

We conclude by reminding that pi = i− λi − 1/2.

λ1 λ2

λ3

p1 p2 q1p3 q2q3q5 q4 p4 p5 p6q6

λT1

λT2

Figure 3: Coding of a partition by a Maya diagram.

λ1a1 = 5 a2 = 3

λ2

λ3

l1 = 5

l2 = 2

Figure 4: Arm lengths (a1, a2, . . .) and leg lengths (b1, b2, . . .) of a partition.

21


2.3 N-solitons

Let κ1 < . . . < κN be generic real parameters, and A ∈ MatN×k(R). Weconsider the element g of GL(V∞) induced by the change of basis on V∞:

(29)

zj 7−→ ∑Ni=1 Aij exp(zλi) 0 ≤ j ≤ k− 1

zj 7−→ zj otherwise.

It follows from (21) and expansion of the sum in (29) out of the determinantthat:

2.3 lemma. We have a KP tau function

(30) τg|0〉(t) =1

∏k−1m=1 m!

∑

I⊆J1,NK|I|=k

∆I(A)∆(κI) ∏i∈I

exp(

∑`>0

κ`i t`)

.

These are the N-soliton solutions promised in § 1.2. They are parametrizedby a full rank N × k matrix A, and we see that the coefficients of the linearcombination are not arbitrary, they must satisfy Plucker relations. Remind thatthe relation between the tau function and the solution of the KP equation isu = 2∂2

x ln τ with x = t1. One may wonder when is (30) a smooth solution tothe KP equation. Since the κ’s are ordered, ∆(κI) > 0. A sufficient conditionfor u to be smooth is then: ∆I(A) > 0 for all subsets I. It turns out the converseis true:

2.4 theorem (Kodama-Williams, 2011). If u(t1, t2, t3, 0, 0, . . .) defined by (30) isa smooth solution of the KP equation (2) for all times t1, t2, t3, then ∆I(A) > 0 forall subsets I of J1, NK.

The set of full rank N× k matrices A such that ∆I(A) > 0 for all I is calledthe positive Grassmannian. It has a rich combinatorial structure, and is alsorelated to cluster algebras.

2.4 Hirota bilinear difference equation

We follow the argument of Gekhtman and Kasman (2005). If V1 and V2 arevector spaces in which two Grassmannians G1 and G2 are embedded, andf : V1 → V2, we say that f is Grassmannian-preserving (GP) if f (G1) = G2.GP maps allow to transport Plucker relations. When V1 is an exterior productof a vector space V0, the morphisms induced by change of basis in V0, and bytaking the dual V0 → V∗0 , are GP. In particular, we can find GP maps betweenthe (big cell of) Sato Grassmannian and the finite Grassmannians. This givesanother characterization of KP tau functions. If (zi)

Ni=1 and I ⊆ J1, NK, remind

the notation ∆(zI) = ∏i,j∈Ii<j

(zj − zi).

22

2.4. Hirota bilinear difference equation

2.5 theorem. If τ is KP tau function, then for any integers 1 ≤ k ≤ N:

(31) ∑I⊆J1,NK|I|=k

∆(κI) τ(

t + ∑i∈I

[κi])

eI

is in the image of Grk(CN) in P(ΛkCN).

2.6 corollary. If τ is a KP tau function, it must satisfy the Hirota bilinear differenceequation:

τ(t + [z1] + [z2]− [z3]− [z4]) · τ(t)z12z34

z13z24z14z23

=τ(t + [z1]− [z3]) · τ(t + [z2]− [z4])

z13z24− τ(t + [z1]− [z4]) · τ(t + [z2]− [z3])

z14z23

where zij = zi − zj.

Proof. We construct a GP map L : F → ΛkCN , for which ∆(zI)τ(t + ∑i∈I [zi]

)are the Plucker coordinates in the target. Actually, L is obtain by composingthe following GP maps:

• KP flow eH(t) : F → F . A element W of the big cell is mapped to avector

(32) |ωW(t)〉 = ∑I

cI(t) eI .

where eI is the semi-infinite wedge product of zi with i ∈ I.

• Projection on (zi−k)i≥0: W is necessarily mapped to a linear combinationof zi1 ∧ zi2 ∧ · · · with I = i1, i2, . . . of the form −k,−k + 1, . . . \ J forsome subset J of cardinality k.

• Dual in this image: W is mapped to a linear combination of zj1−k ∧· · · zjk−k.

• A change of basis zi−k 7−→ ∑Nj=1 κi

j zj−k.

• Projection on (zi−k)N−1i=0 .

The final result takes the form:

(33) L(|W〉

)= ∑

J⊆J0,N−1K|J|=k

(∑

I⊆J0,N−1KcIk [J](t)

[det

(i,j)∈(I,J)κi

j])

eJ ,

where Ik = −k,−k + 1, . . . \ i − k, i ∈ I and cI are the coefficients ap-pearing in (32). By construction, it must be in the image of Grk(C

N) under the

23


Plucker embedding. Now, we can also compute from (32) using Theorem 2.2:

τW

(t + ∑

j∈J[κj])= ∑

I⊆J0,N−1K|I|=k

cIk (t) sλ[I](κJ) ,

where λ[I] is the partition in which the holes are located at I− k, and sλ is theSchur function:

sλ[I](κJ) =det(i,j)∈(I,J) κi

j

∆(κJ).

The result follows by comparing with (33). The corollary is a specialization toGr2(C

4) and shifting t → t− [z3]− [z4] since then, (32) is just the expressionof the Plucker relation (5).

The converse also holds, i.e. the Hirota bilinear difference equation (32)characterizes KP tau functions, but we shall not prove it.

2.5 Algebro-geometric solutions

We shall only sketch the notion of algebro-geometric solutions, and give a hintwhy Riemann surfaces play a prominent role in integrable systems.

• Riemann surfaces and Schottky problem. Let Σ be a compact Riemannsurface of genus g > 0. Then, H1(Σ, Z) ' Z2g, and it admits a basis (αj, β j)

gj=1

such that:

(34) ∀i, j ∈ J1, gK, αi ∩ αj = βi ∩ β j = 0, αi ∩ β j = δij .

On the other hand, the space of holomorphic forms over Σ has dimension g.The linear forms ω 7→ ¸αi

ω defines a basis of its dual, and thus determines aunique basis h1, . . . , hg such that:

∀i ∈ J1, gK,˛

αi

hj = δij .

Then, we can introduce the Riemann matrix of periods:

τij =

˛βi

hj

It is a fundamental result of Riemann that τ is symmetric and Im τ > 0. If oneperforms a change of basis:(

α′

β′

)=

(d cb a

)(αβ

),

which respect the pairing (34), this imposes that the matrix of change of basisis in Sp2g(Z). The new basis of holomorphic 1-forms is:

h′ = (cτ + d)−1h

24

2.5. Algebro-geometric solutions

and τ becomes:τ′ = (cτ + d)−1(aτ + b) .

Modulo this action of Sp2g(Z), the matrix of periods only depends on thecomplex structure of Σ.

Let Hg be the space of complex symmetric matrices with definite positiveimaginary part: it is called the ”Siegel upper-half plane”. One immediatelynotices, that dimHg = g(g + 1)/2 is in general larger than dimMg = 3g−3+ δg,1 of the moduli space of complex structures on Σ. These dimensions areequal only for g = 1, 2, 3, and differ starting from g ≥ 4:

dimH4 = 10, dim M4 = 9 .

The locus of Hg consisting of Riemann matrices of periods is highly non trivial,and characterizing it is called the Schottky problem.

• Theta functions. If τ ∈ Hg, we can construct the Siegel theta function:

θ(w|τ) = ∑m∈Zg

exp[iπm · τ ·m + 2iπw ·m

],

which is a function of w ∈ Cg. It is a pseudo-periodic function:

(35) ∀n, n′ ∈ Zg, θ(w+n′+τ ·n) = exp[− 2iπn · (w+τ ·n/2)

]θ(w|τ) .

In particular, we deduce that if n and n′ are integer vectors such that n · n′ isodd, then:

θ(

c =n′ + τ · n

2

∣∣∣τ) = 0 .

We say that c is a half-integer odd characteristics.Thanks to (35), ratios of theta functions can be used to produce meromor-

phic functions on Σ, or meromorphic sections of line bundles of Σ. Actually,the set of isomorphism classes of line bundles over Σ is parametrized by theJacobian of the curve:

Jac(Σ) = Cg/(Z⊕ τZg)

We do not fully justify this fact, but just remark that if w ∈ Cg, we can considerthe function:

fw,o : z 7→ θ(w + u(z)− u(o) + c)θ(u(z)− u(o) + c)

defined for z in the universal cover of Σ (here o is an arbitrary, reference pointon Σ). When z is carried along the cycle n′ · α + n · β, we have:

fw,o(z) −→ exp[−2iπn ·w] fw,o(z) .

Besides, two w differing by an element of the lattice Zg ⊕ τZg will give thesame phase shift. It is therefore not surprising that Jac(Σ) should classifymorphisms π1(Σ)→ C∗, i.e. line bundles over Σ.

• Riemann theta functions . . . When τ is the matrix of periods of a compactRiemann surface Σ, one can show (see e.g. Mumford’s Tata lectures on Theta)

25


that:

λc(z) =g

∑i=1

∂iθ(c) hi(z) ,

are holomorphic forms with double zeroes, therefore one can define the primeform:

(36) E(z1, z2) =θ(u(z1)− u(z2)c)√

dλc(z1)dλc(z2)

, u(z1) =

ˆ zh .

Here h = (h1, . . . , hg) is the vector of holomorphic forms. (36) is defined as asection of a line bundle on Σ× Σ: it is single-valued for z1, z2 on the universalcover of Σ. One can also show that there exists c such that E is not identicallyzero1. Such a c is called a non-singular, and E is independent of the choiceof such a c. Notice that E(z1, z2) has a simple zero when z1 = z2. The thetafunction attached to a Riemann surface satisfies Fay identity:

2.7 theorem (Fay, 1970). Let Σ be a compact Riemann surface of genus g > 0, c bea non-singular half-integer odd characteristics, and τ a Riemann matrix of period ofΣ. For any w ∈ Cg, we have:

θ(w + c + u(z1) + u(z2)− u(z3)− u(z4)|τ)θ(w + c|τ) E(z1, z2)E(z3, z4)

E(z1, z3)E(z1, z4)E(z2, z3)E(z2, z4)

=θ(w + c + u(z1)− u(z3)|τ) θ(w + c + u(z2)− u(z4)|τ)

E(z1, z3)E(z2, z4)

− θ(w + c + u(z1)− u(z4)|τ) θ(w + c + u(z2)− u(z3)|τ)E(z1, z4)E(z2, z3)

.

Fay identity is quite non trivial. Its proof consists of showing that despiteall appearances, as a function of z1, the difference D(z1, z2, z3, z4) of the leftand right-hand side has no pole when zi = zj, i 6= j. Therefore, it is theholomorphic section of a line bundle over Σ4. Algebro-geometric argumentsthen allow to conclude that it must be constant, and specializing to coincidingpoints, a computation in local coordinate shows that this constant must be 0.

• . . . and KP hierarchy. We remark the similarity between the Fay identityand the Hirota bilinear difference equation (Corollary 2.6) satisfied by a taufunction of KP. Actually, the results of Krichever show that one can build a KPtau function2 out of the theta function, so that the times are coordinates onthe vector space of meromorphic forms on Σ. Writing down the exact formulafor the tau function would require more notations, so we shall not present it.But let us say that, a fortiori, we obtain a solution of the KP equation:

(37) u(x, t2, t3) = 2∂2x ln θ(xv1 + t2v2 + t3v3 + c|τ) + C ,

1This is not clear a priori, since the zero locus of θ(·|τ) is a complex dimension g− 1 manifold,while zi only vary over a complex dimension 1 manifold.

2Strictly speaking, we have a solution of a ”multicomponent KP”, where there is a collectionof times (tk)k>0 attached to each point in the divisor.

26

2.6. Example: Witten-Kontsevich generating series

where vi are the β-cycle integral of a certain meromorphic 1-form on Σ with apole of order i, and C a constant (with respect to the time) expressed in termsof this 1-form.

2.6 Example: Witten-Kontsevich generating series

While studying 2d quantum gravity, Witten conjectured – and Kontsevichproved – that a suitable generating series of intersection numbers on the mod-uli space of curves is a tau function of the KdV hierarchy.

• Background. Let Mg,n be the moduli space of Riemann surfaces of genusg, with n punctures. For 2g − 2 + n > 0, this is a (non compact) complexorbifold, of dimension 3g− 3 + n. It admits a compactification (named afterDeligne and Mumford)Mg,n, which roughly speaking is obtained by allowingRiemann surfaces pinched at several points (called ”nodal points”), so that theconnected components of the complement of the nodal points are Riemannsurfaces whose genus h and number of punctures k satisfy the inequality 2h−2 + k > 0. This condition ensures that the automorphism group of a nodalsurface is finite, so that Mg,n is also a complex orbifold. In H•(Mg,n), thereexist a subring of cohomology classes that can be constructed geometrically,called the tautological ring. The intersection numbers that we are concernedwith are the integrals of these tautological classes overMg,n.

A point inMg,n is the data (Σ, p1, . . . , pn) of a Riemann surface Σ of genusg with n labeled points on Σ. We can associate to it T∗pi

Σ, and this forms acomplex line bundle Li overMg,n. Its Chern class is denoted ψi = c1(Li).

The intersection numbers are compactly encoded in the Witten-Kontsevichpartition function:

Z(t) = exp(

∑n≥1g≥0

1n! ∑

d1,...,dn≥0d1+···+dn=3g−3+n

ˆMg,n

ψ∧d11 ∧ · · · ∧ ψ∧dn

n

n

∏i=1

(2di + 1)!! t2di+1

).

For instance, we have:ˆM0,3

1 = 1,ˆM1,1

ψ1 =1

24,

where the first identity just follows from the fact thatM0,3 = pt, while thesecond one comes from a computation inM1,1 'H/PSL2(Z). In general, thecoefficients are rather non trivial, and it is desirable to have at least a recursiveway to compute them.

2.8 theorem. Z(t) satisfies the Virasoro constraints:

(38) ∀n ≥ 0, Kn · Z[(t2k+1 + δk,1)k

]= 0 ,

with:Kn =

32

H2n+1 +14 ∑

k∈Z

H2k+1H2n−k−3 +δn,1

16.

27


and we remind that Hk acts as multiplication by −kt−k if k < 0, and ∂tk if k > 0.

2.9 theorem. Z(t) is a KdV tau function.

Kontsevich (1991) proved the Virasoro constraints using a combinatorialdecomposition of Mg,n, that allowed him to write Z(t) as a sum over fat-graphs ; the Virasoro constraints are then consequence of the properties of thesum over fatgraphs and the weights assigned to them. Actually, this sum overfatgraphs can be written has a matrix integral, and generalizing the construc-tion of N-solitons for KP, he could prove also Theorem 2.9.

• Constructing the tau function. Let us split C[[t1, t3, . . .]] in graded pieces byassigning the degree di to the variable t2di+1. It is not complicated to see that

either of these theorems, together with the initial condition Z() = exp(

t316 +

t324 + · · · ) where · · · are terms of degree > 1, determines a unique formalseries Z(t). Therefore, if we find a KdV tau function that satisfies (38) and theinitial conditions, we have found Z(t).

Here, we will construct this KdV tau function, following the article Geomet-ric interpretation of the partition function of 2d gravity, V. Kac and A. Schwarz,Nucl. Phys. B 257 3-4, 329–333 (1991).

First, we claim that:

Kn = ∑s∈Z

s + n2

:ψ−sψ†s+2n: +cte .

This is proved, using Lemma 1.4, by showing that the commutators with(H`)`∈Z of both sides give the same result, hence the equality according toLemma 1.5. We can now read the action of this operator in the basis (zi)i∈Z,and show that it is induced on F by the operator on V∞:

D =3z2

+12z

ddz− 1

4z2 .

Therefore, we would have constructed our tau function if we can show:

2.10 theorem. There exists a unique element W of the big cell of Sato Grassmannian,such that z2W ⊆W and DW ⊆W.

Proof. First consider the uniqueness. If W is in the big cell, there exists aunique ϕ ∈ W of the form ϕ = 1 + ∑i>0 ϕi z−i. Since Dn ϕ = zn + O(zn−1),(Dn ϕ)n≥0 forms a basis of W. This means that the data of ϕ determines W. Inparticular, we should be able to express z2 ϕ ∈W on this basis. Since we have:

D2 =1

4z2d2

dz2 +12

(3− 1

z3

) ddz

+9z2

4+

516z4 ,

we deduce that D2 ϕ− 94 z2 ϕ belongs to W but is o(1), so:

(39) D2 ϕ =94

z2 ϕ .

28

2.6. Example: Witten-Kontsevich generating series

This gives the differential equation:(1

4z2d2

dz2 +12

(3− 1

z3

) ddz

+5

16z4

)ϕ = 0 ,

and inserting ϕ = 1 + ∑i>0 ϕiz−i, we identify the coefficient of z−(i+4) andobtain:

i(i + 1)4

ϕi +i2

ϕi −3(i + 3)

2ϕi+3 +

516

ϕi = 0 ,

that is:ϕi+3 =

ϕi6

(i2 + 3i +

54

).

Since ϕ−1 = ϕ−2 = 0, we deduce that ϕ3i−1 = ϕ3i−2 = 0, and with the initialcondition ϕ0 = 1, we can solve:

(40) ϕ3i =1

72i

i−1

∏j=0

(6j + 5)(6j + 1)3(j + 1)

=1

216i(6i + 3)!!

i!(2i + 1)!!,

or equivalently:

ϕ3i =1

2i+1π

Γ(i + 5/6)Γ(i + 1/6)Γ(i + 1)

.

Therefore, ϕ is unique, and so must be W. Conversely, if we take ϕ determinedby (40), then Dz2n ϕ = z2n+1 + O(z2n), therefore the span of (z2nDϕ, z2n ϕ)n≥0defines an element W of the big cell. It is obviously stable by multiplicationby z2. And since ϕ satisfies (39) by construction, we also have DW ⊆ W. Thisproves the existence.

By construction, the tau function of W is a solution to Kn · τW(t) = 0 forany n ≥ 0. It can be expressed:

τW(t) = detm,n≥0

[ ˛dz

2iπzm+1 Dn ϕ(z) exp

(∑k≥0

t2k+1 z2k+1)]

.

Note that the differential equation (39) can be simplified by introducingthe new function:

y(x) =exp(− 2

3 x3/2)

x1/4 ϕ[(2/3)1/3x1/2] .

Computation shows that ∂2xy = xy, i.e. y is solution to the Airy differential

equation, and (40) gives the coefficients of the asymptotic expansion of theAiry function at x → ∞.

29

3. Lecture 3: classical integrability and Calogero-Moser systems

3 Lecture 3: classical integrability and Calogero-Moser

systems

3.1 Classical integrability

• Poisson and symplectic manifolds. A Poisson algebra is a commutativealgebra A over K = R or C, equipped with a Poisson bracket ·, · : A⊗A → A which is C-linear, satisfying:

• Antisymmetry: f , g = −g, f .

• Leibniz rule: f , gh = f , gh + g f , h.

• Jacobi identity: f , g, h+ cyclic = 0.

The Poisson bracket is non-degenerate if for any f ∈ A, Ker f , · = C. APoisson manifold is a smooth manifold M such that C∞(M) is equipped withthe structure of a Poisson algebra. Then, there must be a Poisson bivectorΠ ∈ Γ(M, Λ2TM) such that:

f , g = Π(d f ⊗ dg)

The Poisson bracket induces a morphism of Lie algebras:

C∞(M) −→ Γ(TM)f 7−→ X f = f , · ,

as can be checked using local coordinates. Functions H ∈ C∞(M) are called”Hamiltonians”, and by the morphism above, they determine a vector fieldX f , hence a flow (φt)t on M, such that, for any g ∈ C∞(M):

dg(φt(x))dt

= f , g(φt(x))

We remark that H is a conserved quantity along the flow it generates.A symplectic manifold is a smooth manifold equipped with a non-degenerate,

closed 2-form ω. Then, it is a Poisson manifold, with bracket defined in termsof local coordinates (xi)

Ni=1, seen as functions on an open set of M:

xi, xj = Π(dxi ⊗ dxj) = Ω−1ij , Ω = (ω(∂xi , ∂xj))1≤i,j≤N .

Conversely, any Poisson manifold with non-degenerate bracket is a symplecticmanifold: the formula above defines ω on coordinate vector fields in terms ofthe Poisson bivector, and the Jacobi identity implies that dω = 0. Remark thata symplectic manifold must be even dimensional.

The fundamental example of symplectic manifold is T∗CN , with coordi-nates (q1, . . . , qN , p1, . . . , pN) and symplectic form:

ω =N

∑i=1

dpi ∧ dqi .

30

3.1. Classical integrability

If H is a Hamiltonian, it defines the flow on T∗CN by the Hamilton-Jacobiequations:

∀i ∈ J1, NK,dqidt

=∂H∂pi

,dpidt

= −∂H∂qi

.

• Coadjoint orbits. The second important example of symplectic manifoldis provided by coadjoint orbits. Let G be a connected, finite-dimensional Liegroup, and g its Lie algebra. G has a left action on g by conjugation (the adjointaction), as well as on g∗ (the coadjoint action):

∀x ∈ g, ∀` ∈ g∗, Adg(x) = gxg−1, Ad∗g(`)[x] = `(g−1xg) .

g∗ has a natural structure of Poisson manifold when equipped with the Kostant-Kirillov bracket. To define it, we identify the differential of a smooth functionh on g∗ at a point ` ∈ g∗ with an element (dh)` ∈ g, since g∗∗ is canonicallyisomorphic to g. The bracket is:

h1, h2` = `([dh1, dh2])

where [·, ·] is the bracket of the Lie algebra.

3.1 lemma. The kernel of the Kostant-Kirillov bracket consists of the Ad∗ invariantfunctions.

Proof. If x ∈ g, the flow (Ad∗exp(tx))t corresponds to the vector field [·, x]. h ∈C∞(g∗) is in the kernel iff for any ` ∈ g∗ we have `([dh, ·]) = 0, which exactlymeans that h is invariant under the coadjoint action.

A coadjoint orbit is a submanifold O ⊆ g∗ which is invariant under thecoadjoint action of G. As a consequence, the restriction to O of an Ad∗-invariant function on g∗ is a constant function on O. This implies that therestriction of the Kostant-Kirillov bracket on O is non-degenerate, and there-fore O is a symplectic manifold. A side consequence is that coadjoint orbitsmust be even dimensional.

• Liouville integrability. Let M be a symplectic manifold of dimension N.

3.2 definition. A Hamiltonian H on M is (Liouville) integrable if there existsHamiltonians H1 = H, H2, . . . , HN such that Hi, Hj = 0 for all i, j ∈ J1, NK,and (dH1, . . . , dHN) is linearly independent at any point of M.

By extension, we say that (H1, . . . , HN) forms an integrable system on M.When it is the case, H1, . . . , HN are conserved quantities along the flow gen-erated by H1. More than that, if we call ti the flow defined by Hi, all the Hjare conserved quantities along these flows, which are therefore all compatible.Then, the values (Hi, ti)

Ni=1 define local coordinates on M, such that:

ω =N

∑i=1

dHi ∧ dti ,

31


and the flows induce a linear motion in these coordinates. We can equivalentlydescribe an integrable system on M as a fibration on M, whose fibers areabelian varieties.

If H is integrable, the trajectories of the flows can be obtained in finitelymany steps by computing primitives. It is thus much simpler to solve than anon-linear ODE. Being integrable is a strong property, which is not true for ageneric hamiltonian. Historically, integrable system were found in mechanicsby clever guesses of conserved quantities. It is not obvious to prove that ahamiltonian is not integrable: one may wonder if one has not been yet smartenough to find enough independent conserved quantities. There is however anecessary condition, that has been used e.g. by Audin to exhibit non-integrablesystems:

3.3 theorem (Morales-Ramis (1999)). If M is a compact, complex manifold and(H1, . . . , HN) are meromorphic functions that form an integrable system on M, then,for any solution ϕ of the evolution equations, the linearization around ϕ of the evolu-tion equations is an ODE such that the connected component of id in its differentialGalois group is abelian.

3.2 Hamiltonian reduction

Here is a method that often lead to the construction of integrable systems. LetM be an even dimensional Poisson manifold, with maybe degenerate Poissonbracket. Assume that a connected Lie group G acts on M by Poisson automor-phisms: in general, this means that the Poisson bracket is degenerate. Assumethat we are given a moment map µ : M → g∗. By definition, this is a mapsuch that, if we define µ∗ : g → C∞(M) by µ∗(a)x = µ(x) · a, then for anya ∈ g, the hamiltonian µ∗(a) generates the flow (exp(ta))t induced by theaction of G.

Let O ⊆ g∗ be a coadjoint orbit of G, and assume that µ∗ is a submersionat any point of µ−1(O). So, µ−1(O) is a submanifold of M. G still acts onµ−1(O), and let us assume this action is free and properly discontinuous.Then, µ−1(O)/G is a manifold, of dimension:

dim[µ−1(O)/G] = dim M− 2 dim G + dimO .

Indeed, for any o ∈ O, µ−1(o) has dimension dim M−dim G, and its quotientby the free action of G has dimension dim M− 2 dim G. Then varying o ∈ Oadds a dimension dimO.

µ−1(O)/G is a Poisson manifold, and its Poisson algebra consists on (therestriction of) the smooth, G-invariant functions on M, modulo the ideal gen-erated by the µ∗(a) for a ∈ g. Notice that, since coadjoint orbits are symplecticmanifolds, µ−1(O)/G is also even-dimensional.

Now, imagine that the bracket is non-degenerate on µ−1(O)/G, i.e. itis a symplectic manifold. Imagine also that we had G-invariant hamiltoni-ans H1, . . . , Hn on M which Poisson commuted. They induce hamiltoniansH1, . . . , Hn on µ−1(O)/G that still Poisson commute. If it turns out that thedimension of µ−1(O)/G is 2n and Hi are independent, we have found an

32

3.3. Example: rational Calogero-Moser system

integrable system. There is a better chance for that than initially since thedimension of the manifold is much smaller.

This procedure is called hamiltonian reduction, and is a rich source ofintegrable systems, despite the fact that it seems to rely on many assumptions.Here we have presented the simplest case, assuming all varieties to be smooth,but there exist generalizations that address the (subtle) issue of singularities.

3.3 Example: rational Calogero-Moser system

The rational Calogero-Moser system is the evolution equation for the positions(qi(t))N

i=1 of N particles:

(41) ∀i ∈ J1, NK, pi =dqidt

,dpidt

= −∑j 6=i

2(qi − qj)3 .

It comes from the Hamiltonian:

(42) HCM =N

∑i=1

p2i −∑

j 6=i

1(qi − qj)2

on T∗CN equipped with its canonical symplectic structure. We will show thatHCM is integrable, and actually construct the other conserved quantities, andthe solution of the flow equations.

• Construction by hamiltonian reduction. Let M = T∗MatN×N(C) with co-ordinates (Q, P), and symplectic form ω(Q, P) = tr(P∧Q). We shall considerthe action of G = PGLN(C) on M by conjugation. A moment map for thisaction is µ(Q, P) = [P, Q], and we would like to apply Hamiltonian reductionon the coadjoint orbit:

O =

T ∈ MatN×N(C), Tr T = 0 and rank(T + 1) = 1

.

3.4 lemma. G acts freely and property discontinuously on µ−1(O).

Proof. It can be easily checked that the action is proper, so we focus on thefirst point. Let (Q, P) ∈ µ−1(O), and assume that A ∈ GLN(C) such that[A, Q] = [A, P] = 0. The first condition means rank([P, Q] + 1) = 1, hence theeigenvalues of [P, Q] are −1 with multiplicity N − 1, and N − 1. The secondcondition implies that any eigenspace E for A is stable under action of Q, Pand [P, Q]. Therefore, we can consider the restriction C of the endomorphism[P, Q] restricted to E. The eigenvalues of C must be a subset of −1, . . . ,−1, N−1. But since it is a commutator, it must be also be traceless. This can onlyhappen if E = 0, or E = CN and all eigenvalues appear. Thus, A must bescalar, i.e. identity in PGLN(C).

3.5 lemma. dim[µ−1(O)/G] = 2N, and more precisely, over the dense open subsetof matrices with pairwise disjoint eigenvalues, any point (Q, P) in µ−1(O)/G has a

33


representative of the form:

Q = diag(q1, . . . , qN), Pij =1

qi − qjfor i 6= j ,

which is unique up to conjugation by a permutation matrix.

Proof. Up to conjugation, for (Q, P) in a dense open subset, we can always finda representative such that Q = diag(q1, . . . , qN). Let us express the conditionto be in µ−1(O): there exists vi, wi ∈ C such that

∀i, j ∈ J1, NK, [P, Q]ij + δij = viwj .

For i = j, we find that viwi = 1. For i 6= j, we find Pij(qj− qi) = vi/vj. If we actfurther by conjugation by a diagonal matrix diag(a1, . . . , aN), Q is unchangedbut P changes so that, now:

Q = diag(q1, . . . , qN), Pij =1

qi − qj.

We can still permute the qi’s, but this is all what is left of the freedom toconjugate simultaneously Q and P by an element of G. Therefore, we havefound that the N eigenvalues of Q, and the diagonal elements Pii providecoordinates on µ−1(O)/G.

It is easy to check that:

3.6 lemma. Let CN∆ = q ∈ CN , ∀i 6= j, qi 6= qj, and U ⊆ µ−1(O)/G such

that Q has simple eigenvalues. We have an isomorphism of symplectic varieties:

T∗CN∆ /SN −→ U

(q, p) 7−→[

Q = diag(q1, . . . , qN) ; Pij =

pi if i = j(qi − qj)

−1 if i 6= j

].

On M, the functions:

(43) Hk = (Tr Pk)Nk=1

provide N G-invariant hamiltonians, which Poisson commute since the entriesof P are Poisson commuting. They descend to give N independent hamilto-nians (Hk)

Nk=1 on µ−1(O)/G, i.e. an integrable system. Using the coordinates

presented in Lemma 3.6, the first two hamiltonians are:

H1 = Tr P =N

∑i=1

pi, H2 =N

∑i=1

p2i −∑

j 6=i

1(qi − qj)2 ,

and we recognize in H2 the Calogero-Moser hamiltonian (42).On M, we see from the symplectic structure ω(Q, P) = tr(P ∧ Q) that

the flows generated by Tr Pk (we denote tk the corresponding time) leave P

34

3.3. Example: rational Calogero-Moser system

constant, and give linear evolutions for the entries of Q:

(44) Q(t) = Q(0) +N

∑k=1

ktkPk−1 .

The induced flow on µ−1(O)/G is just π(Q(t), P) where π : M → M/G isthe projection map. If we force a representative of π(Q(t)) to be of the formdiag(q1(t), . . . , qN(t)), it means that there exists Ψ(t) ∈ PGLN(C) such that:

(45) Q(t) = Ψ(t)−1diag(q1(t), . . . , qN(t))Ψ(t) .

Notice that (45) defines Ψ(t) up to right multiplication by a diagonal matrixthat may depend on t. Then, if we define:

(46) L = Ψ(t)PΨ−1(t), M(k)(t) = ∂tk Ψ(t) ·Ψ−1(t) ,

and differentiate (44) along tk, we find:

∀i, j, k ∈ J1, NK, M(k)ij (qj − qi) + δij∂tk qi =

N

∑k=1

k[Lk−1]ij ,

where now all quantities will depend implicitly on the times, unless precised.This yields:

(47) ∂tk qi = [kLk−1]ii, M(k)ij =

[kLk−1]ijqi − qj

.

The freedom in the definition of Ψ(t) can be fixed by imposing that, if J is thematrix full of 1’s,

M(k) J = 0

which means according to (47) that we choose:

(48) M(k)ii = −∑

j 6=i

[kLk−1]ijqi − qj

.

Other choices are possible ; this one will turn out convenient in § 3.6. Tosummarize:

3.7 lemma. The integrable system generated by (Hk)Nk=1 is equivalent to the equa-

tion:

(49) ∂tk L = [M(k), L] ,

with M(k) given by (47)-(48), and:

(50) Lij =

∂t2 qi if i = j(qi − qj)

−1 if i 6= j.

35


• Remark: integrable systems in Lax form. (49) is a necessary condition forthe existence of a solution Ψ to the system:

(51) ∂tk Ψ = M(k)Ψ, LΨ = zΨ .

In words, we can say that L follows an isospectral evolution under the flowstk – by comparison with (46), we see that Ψ is nothing but the matrix ofeigenvectors of the constant matrix P, multiplied on the left by Ψ(t). Noticethat the compatibility of ∂tk Ψ = M(k)Ψ among themselves demands that:

(52)

∀k, l ∈ J1, NK, [∂tk −M(k), ∂tl −M(l)] = ∂tl M(k)− ∂tk M(l)+[M(k), M(l)] = 0 .

Equations of the form (51) or (52) are called Lax equations: they give non-linear PDEs on the entries of M(k) and/or L, which express the compatibilityof a family of linear ODEs. Most (but not all) of the known integrable sys-tems can be put in Lax form, i.e. one can build matrices L and/or M(k) out ofthe dynamical variables, such that Lax equations are satisfied. The Lax repre-sentation is certainly not unique. But if it can be found, it shows directly theintegrability of the non-linear PDEs, since it follows from (51) that Tr Lk (thesymmetric functions of the eigenvalues of L) are conserved quantities.

When one only has equations (52) without a matrix L following an isospec-tral evolution, we speak of an isomonodromic system, because it expressesthe compatibility of linear ODEs whose solutions have a monodromy inde-pendent of the tk’s. In another language, (52) is a zero-curvature condition forthe connection d− ∑k M(k)dtk on the space of tkN

k=1. Solving – for Ψ – forthe linear ODEs is then closely related to solving a so-called Riemann-Hilbertproblem. Jimbo, Miwa and Ueno have introduced a notion of tau function forisomonodromic systems. Its essential meaning was found by Malgrange, un-der assumptions we do not reproduce here: the tau function τ(t) vanishes att = t0 iff the Riemann-Hilbert problem has no solution for the value t = t0.

• Calogero-Moser and pole dynamics. We present an equivalent formulationof Calogero-Moser system. For simplicity, we focus on the flow (42), but asimilar result can be formulated for all the hamiltonians Hk.

3.8 theorem. The equations:

(53) L(t)Ψ(t, z) = z Ψ(t, z), ∂tΨ(t, z) = M(2)(t) Ψ(t, z)

for a column vector Ψ(x, t) are equivalent to requiring that:

(54) ϕ(x, t) =( N

∑i=1

Ψi(t, z)x− qi(t)

)exp(xz + tz2/2)

36

3.4. The KP hierarchy in Lax form

is solution to the linear PDE:

(55)[2∂t − ∂2

x − u(x, t)]ϕ(x, t, z) = 0, u(x, t) = −

N

∑i=1

2(x− qi(t))2 .

Once stated, the theorem can be proved straightforwardly, by inserting (54)in (55) and identifying the coefficients of the triple, second and simple orderpoles that can appear at x = qi(t), and comparing with (53).

(55) can be considered as a Schrodinger equation with time variable t andspace variable x, in a time dependent potential u(x, t) which has singulari-ties at x = qi(t). If the poles qi(t) were arbitrary, the solution would havemonodromies when x moves around the qi(t). Requiring the existence of aquasirational solution – i.e. a solution have no monodromy around x = qi(t)– of the form (54) imposes the pole qi(t) to satisfy non-linear evolution equa-tions. Here, this equation is precisely the Calogero-Moser evolution (41).

In the remaining of the lecture, we shall explain that Theorem 3.8 allowsto embed the Calogero-Moser system in the KP hierarchy.

3.4 The KP hierarchy in Lax form

Let R be a ring of test functions of one variable x (e.g. smooth functions on acompact of R, or Schwartz functions on R). We introduce the set of pseudo-differential operators:

A =

∑n∈Z

ak(x) ∂kx, ak ∈ R and ak = 0 for k > 0 large enough

.

We can put the structure of a ring on A, by asking that ∂x satisfies the Leibnizrule, and:

∂−1x a = ∑

j≥0(−1)j a(j)(x) ∂

−(j+1)x .

We denote A− the space of elements of A containing only negative powers of∂x.

• From the Grassmannian to A. If W is in the big cell of Sato Grassmannian,we denote:

(56) GWn (t) =

˛dz2iπ

π−1W (zn) exp

(∑k>0

tkzk)

,

and identify the time t1 with x. We have in particular:

∂−mx GW

n (t) =˛

dz2iπzm π−1

W (zn) exp(

∑k>0

tkzk)

.

We remind that the tau function of W reads:

τW(t) = detm,n≥0

[∂−(m+1)x GW

n (t)]

.

37


3.9 definition. We define a pseudodifferential operator D , depending on Wand t, by its action on a test function:

D · f = τ−1W (t) det

m,n≥0

[∂−m

x f∂−m

x GWn (t)

].

We define the wave function as:

(57) Φ(z, t) = D[

exp(

∑k>0

tkzk)]

.

As we can see by expanding the Fredholm determinant with respect to itsfirst line, D takes the form3

(58) D = 1 + ∑i≥1

di(t) ∂−ix .

By the properties of the determinant, we also have:

∀n ≥ 0, D [GWn ] = 0 .

The wave function can be computed in terms of the tau function.

3.10 theorem.

Φ(z, t) =τW(t− [z−1])

τW(t)exp

(∑k>0

tkzk)

.

The proof follows from elementary manipulations with Definition 3.9, and isleft as exercise.

• Action of KP flows. Let us now study the effect of the KP flows on theoperator and the wave function. Note that, for an element:

(59) D = 1 + ∑i≥1

di ∂−ix ,

there exists a unique element:

(60) D−1 = 1 + ∑i≥1

di ∂−ix ,

such that DD−1 = D−1D = 1. The coefficients di are found by substitution:

(61) d1 = −d1, d2 = −d2 + d21, etc.

If P ∈ A, we may decompose it P = P− + P+ with P− ∈ A− and P+ adifferential operator.

3Not all pseudo-differential operators in 1 +A− come from a subspace W ; one can see forinstance that the special form of GW

n in (56) imposes some constraints on the coefficients of D in(58).

38

3.4. The KP hierarchy in Lax form

3.11 theorem. We have:

∂tkD = −[D∂kxD−1]−D , ∂tk Φ = Lk Φ ,(62)

with Lk = [D∂kxD−1]+.

Proof. The second formula is a direct consequence of the first formula and thedefinition (57) of the wave function. To prove the first formula, we will showboth sides agree when we truncate modulo ∂

−(N+1)x , for any N ≥ 0. Let us

write:D = DN∂−N

x + O(∂−(N+1)x ) .

DN = ∂N + · · · is a differential operator of degree N. We need to prove:

(63) ∂tkDN = −DN∂kx + [DN∂k

xD−1N ]+DN .

and then, we can conclude by taking the limit N → ∞. Since both sides of (63)are differential operators of degree N − 1, it is enough to find N independentfunctions that annihilate them. Coming back to the Definition 3.9 of D asan infinite determinant, we see that its truncation DN is computed by thetruncation to a determinant of size (N + 1):

DN · f = τ−1W (t) det

0≤m≤N0≤n≤N−1

[∂−m f

∂−mGn

].

Therefore, DNGn = 0 for n ∈ J0, N− 1K. Using this property, and coming backto the definition (56) of Gn, we also compute:

(∂tkDN)Gn = −DN∂tk Gn = −DN∂kxGn .

However, DN∂kx is not a differential operator of degree N − 1. But we can

add the vanishing term [DN∂kxD−1N ]+DNGn to make it so. As we argued, this

justifies (63), and finishes the proof.

Since Φ is a solution, the equations (62) must be compatible:

∀l, k ≥ 1, ∂tl Lk − ∂tkLl + [Lk, Ll ] = 0 .

Therefore, we have put the KP equations in Lax form. For instance, we cancompute with (59)-(61):

(64) L1 = ∂x, L2 = [D∂2xD−1]+ = ∂2

x − 2(∂xd1), etc.

Plugging the solution given in Theorem 3.10 for Φ(z, t) in the equation ∂tk Φ =LkΦ gives an expression of the coefficients of Lk in terms of (derivatives) ofthe tau function. For instance:

3.12 lemma. ∂xd1 = ∂2x ln τ.

39


Proof. Let us expand the numerator in Theorem 3.10 in powers of z−1:

Φ(z, t) =

1− z−1∂x ln τ +z−2

2

(∂2xτ

τ− ∂t2 ln τ

)+ o(z−2)

exp

(∑k>0

tkzk)

.

Now, we compute:

∂t2 Φ(z, t) =

z2 +12

(∂2xτ

τ− ∂t2 ln τ

)+ o(1)

exp

(∑k>0

tkzk)

and:

∂2xΦ(z, t) =

z2 +

12

(∂2xτ

τ− ∂t2 ln τ

)− 2∂x(∂x ln τ) + o(1)

exp

(∑k>0

tkzk)

.

This entails the result by comparison with (64).

• Question. Deduce from this formalism conserved quantities for the KdVand KP hierarchy, in the form:

ˆdxPk(u, ∂xu, ∂2

xu, . . .)

with Pk a well-chosen polynomial.

• Polynomial tau functions and Calogero-Moser dynamics. Assume that wehave a KP tau function of the form:

(65) τ(t) =N

∏i=1

(x− qi(t)), t = (t2, t3, . . .) .

Then, the wave function is:

ϕ(t, z) =N

∏i=1

x− z−1 − qi(t− ˜[z])x− qi(t)

exp(

∑k>0

tkzk)

=(

1 +N

∑i=1

ϕi(t, z)x− qi(t)

)exp

(∑k>0

tkzk)

,

where ˜[z] = (zk/k)k≥2. And, it must be solution to [∂tk −Lk]Φ = 0 for k ≥ 2.This implies non-linear evolution equation for the qi(t). With k = 2, we haveaccording to Lemma 3.12 with our special form (65) of the tau function:

L2 = ∂2x − u(x, t), u(x, t) = −

N

∑i=1

2(x− qi(t)))2 .

If we identify t2 = t/2, we can prove directly that, in terms of the matricesL and M = M(2) defined in (50) and (48) in terms of the qi’s, the equation

40

3.5. Algebro-geometric perspective

L2 ϕ = 0 is equivalent to:

LΦ = −zΦ− J, ∂tΦ = M(2)Φ ,

where Φ is the column vector with components ϕi(t, z). Were the constantcolumn vector J – full of 1’s – absent, this would be the equivalent form of theCalogero-Moser system found in (53). But, if Φpart is a solution of LΦpart =−zΦpart − J (this equation can have a solution when z is an eigenvalue of L),then using that M(2) · J = 0, one can check that:

Ψ = Φ− Φpart

is exactly solution to the Calogero-Moser system (53).

3.5 Algebro-geometric perspective

Novikov’s conjecture roughly stated that τ is a matrix of period iff (37) satisfythe KP equation, i.e. gives a solution of the Schottky problem in terms of theKP equation. It was later proved by Shiota (1986), building on earlier workof Krichever, as well as Mulase. Actually, we have the following result, statedinformally:

3.13 theorem (Mulase, 1984). Any finite-dimensional orbit of the KP flow is iso-morphic to the Jacobian of a curve.

We quote the author of the theorem for a very concise explanation:

”Why do algebraic curves have something to do with the KP equations ? [...] Firstof all, the KP equations govern all possible isospectral deformations of an arbitrarylinear ordinary differential operator. On the other hand, every ordinary differentialoperator defines a unique algebraic curve as a set of eigenvalues with resolved mul-tiplicity. The eigenspace of the operator defines a vector bundle on this curve. Sinceisospectral deformations preserve the eigenvalues, they should correspond to defor-mations of vector bundles on the curve. In particular, these deformations generateJacobian varieties.”

3.6 Appendix: Zhdanov-Trubnikov equation

We give here another example of non-linear, non-local PDE, which admits ra-tional solutions. The Zhdanov-Trubnikov equation modelizes the propagationof a flame:

(66) ∂tφ− ν∂2xφ +H[∂xφ] +

12(∂xφ

)2+

c2(H[∂xφ]

)2= 0 .

Here, φ(x, t) is the altitude of the interface between the fresh and the burntmedium, above the abscissa x, at time t ; c is a parameter. The first two termsdescribe a diffusion that smoothes out the interface. The operator H is theHilbert transform:

H[ 1

x− iq

]=−isgn(q)

x− iq

41


In Fourier space, the Hilbert transform acts like multiplication by the signfunction. It is a non-local operator, and occurs typically because a perturba-tion of the interface has a finite range effect below in the medium below theinterface. (66) was derived by the named authors (the case c = 0 was pre-viously obtained by Michelson-Sivashinsky), assuming a small amplitude ofvariation and slow evolution of the interface, and a not too large relative den-sity of the burnt/fresh medium.

If q ∈ R, we have:

H[ 1

x− iq

]= − i sgn(q)

x− iq.

Therefore, searching for (real-valued) solutions of (66) of the form:

φ(x, t) = γN

∑k=1

ln[(x− iqk)(x + iqk)

],

one indeed finds one provided:

γ = − 2ν

1− c,

and the location of the poles satisfy the evolution equation:

dqkdt

+ 1− ν

(∑l 6=k

1qk − ql

+1 + c1− c

N

∑l=1

1qk + ql

)= 0 .

It would be interesting to know, for combustion-theoretic problems, whetherthis equation is ”integrable”, since one would like to produce a large class ofphysical solutions, and study their stability (it is necessary to have some ex-plicit formula for a solution in order to be able to study its stability !).

42

4 Lecture 4: a excursion into quantum integrability

4.1 What does it mean to quantize an integrable system ?

If A is a Poisson algebra, a quantization A of A is an associative algebra Asuch that:

• There is an isomorphism of vector spaces σ : A[[h]]→ A.

• For any f , g ∈ A, considered as elements of A[[h]], we have σ( f ) · σ(g) =σ( f g) + O(h), and [σ( f ), σ(g)] = hσ( f , g) + O(h2).

If M is a Poisson manifold of dimension 2n and A = C∞(M), a quantizationof an integrable system H1, . . . , Hn is a commutative subalgebra H of a quan-tization A of A, generated by H1, . . . , Hn such that σ(Hi) = Hi + O(h). Wethen speak of a quantum integrable system on M. In order to understand A,one can look at its representations on an algebra of operators. If we call ρ sucha representation, the commutativity of H tells us that the equations:

∀i ∈ J1, NK, ρ(Hi)ψ = λiψ

have a common solution. Solving a quantum integrable system means beingable, for any ”interesting” representation ρ, to find explicitly the eigenvaluesλi and the common eigenvectors ψ of H1, . . . , Hn. This is in general a difficultproblem, for which many techniques have been developed in the last 40 years:Bethe Ansazte, quantum separation of variables (for which we shall see anexample in the next paragraph), etc.

It is often possible to find faithful representations of A in terms of dif-ferential operators on M. The simplest example of quantization is T∗CN withits coordinates (q, p), where we actually restrict A to be the space of functionswhich are smooth in q but polynomials in p. Then, A = C∞(CN)[h∂q1 , . . . , h∂qN ]is a quantization of A.

We stress that, although there exists general results about the existenceof quantizations, one cannot ask for uniqueness, and apart from cotangentspaces of vector spaces, quantizations are hardly canonical. Finding ”nice”quantizations is a major problem underlying many aspects of quantum fieldtheories.

4.2 Quantum rational Calogero-Moser

We will show that the operator:

(67) HCM =N

∑i=1

∂2qi−∑

j 6=i

c(c + 1)(qi − qj)2

can be fit in a commutative algebra (Hi)Ni=1 which quantizes the Calogero-

Moser hamiltonians of (Hk)Nk=1 of (43).

• Dunkl operators. It is more transparent at this stage to work in the moregeneral framework a euclidean vector space V, together with a finite reflection

43

4. Lecture 4: a excursion into quantum integrability

group W. Let S be the set of all reflections in W. For each s ∈ S, we choose(arbitrarily) a vector vs ∈ V with norm 2 and such that s(vs) = −vs. Then, thereflection can be written:

s(x) = x− (vs|x)vs

We also fix the data of a function c : S→ R which is W-invariant. In the caseof W = SN , S consists of reflections with respect to the hyperplanes (ei − ej)

⊥

for i < j, on which W acts transitively, thus c is just a constant. We introducesome basic operations on smooth functions on RN :

• If ϕ is an endomorphism of V, we denote we denote [ϕ f ](x) = f (ϕ(x)).

• If ` ∈ V∗, we denote ( ˆ f )(x) = `(x) f (x).

• If a ∈ V, we denote (∂a f )(x) = dx f · a.

Let RN∆ be the complement of all reflection hyperplanes in RN . Following

Dunkl, we introduce the differential operators acting on smooth functions onRN

∆ .

4.1 definition. For any a ∈ V, let:

Da = ∂a −∑s∈S

cs(vs|a)(vs|•)

(1− s) .

4.2 theorem. Let w ∈W, a, b ∈ V and ` ∈ V∗. We have:

(i) wDaw−1 = Dw(a).

(ii) [Da, ˆ] = `(a)−∑s∈S cs(vs|a) `(vs)s.

(iii) Da applied to a polynomial function is a polynomial function.

(iv) [Da, Db] = 0.

Proof. (i) is obvious since c is W-invariant. For (ii), we compute:

[Da, ˆ] = `(a) + ∑s∈S

cs(vs|a)(vs|•)

[s, ˆ] ,

and:

[s ˆ − ˆ s] f (x) = (`(s(x))− `(x)) f (s(x)) = −`(vs) (vs|x) s f (x) ,

hence the formula. For (iii), we remark that polynomial functions are linearcombinations of ˆ1 · · · ˆn · 1 for some `i ∈ V∗. In the expression Da ˆ1 · · · ˆn · 1,we can commute Da to the right using (ii), and when Da finally hits 1, ityields 0. In the process, only products of ˆ i appear, i.e. Da leaves stable the setof polynomial functions. (iv) can be proved by direct computation, but thereis a more clever argument. First, by a density argument it is enough to prove

44

4.2. Quantum rational Calogero-Moser

it for polynomial functions. Since [Da, Db] · 1 = 0, by the previous remark, it isenough to show that [[Da, Db], ˆ] = 0 for any ` ∈ V∗. This is indeed the case:

[[Da, Db], ˆ] = [[Da, ˆ], Db]− [Da, [Db, ˆ]]

= ∑s∈S

cs`(vs)(vs|b) [s, Da]− (vs|a) [s, Db]

,(68)

and we have sDa − Da s = (Ds(a) − Da)s = −(vs|a) Dvs s, so (68) yields 0 byantisymmetry in a and b.

• Restriction to W-invariant polynomials. Let C[V]W be the set of W-invariantpolynomial functions on V, e1, . . . , en an orthonormal basis of V, and (e∗i )

Ni=1

the dual basis. We deduce from Lemma 4.2 that, when P ranges over C[V]W ,the operators P(∑n

i=1 Dei ei) are well-defined, act on C[V]W , and form a com-mutative algebra, denoted J . Since the scalar product is W-invariant, J al-ways contain the quadratic W-invariant:

J2 =N

∑i=1

D2ei

=N

∑i=1

∂2

ei−∑

s∈S

(cs

(vs|ei)

(vs|•)(∂ei − ∂s(ei)

) +cs (vs|ei)

2

(vs|•)2

)+ ∑

s,s′∈S

cscs′(vs|ei)(vs′ |ei)

(vs|•)(vs′ |•))

=N

∑i=1

(∂2

ei+ ∑

s∈Scs

(vs|ei)2

(vs|•)∂vs

)+ ∑

s∈S

2cs(cs + 1)(vs|•)2 + ∑

s 6=s′cscs′

(vs|vs′)

(vs|•)(vs′ |•).

Notice that the factors (1 − s) eventually disappear since we are applyingthese operators to W-invariant functions. We claim that the last term vanishes.Indeed, if we multiply it by ∏s∈S(vs|•), we find:

(69) ∑s 6=s′

cscs′ ∏s′′ 6=s,s′

(vs′′ |•) (vs|vs′) ,

and this is a polynomial, W-anti-invariant of degree ≤ |S| − 2. But the smallestpossible degree for a polynomial anti-invariant is |S|. Therefore (69) must be0, and using ∑N

i=1(vs|ei)2 = (vs|vs) = 2, we arrive to:

J2 =N

∑i=1

∂2ei−∑

s∈S

2cs

(vs|x)∂vs .

J2 contains first order derivatives. We can always get rid of them by conjugat-ing it with the multiplication by a suitable function (this is sometimes calleda ”gauge transformation”).

4.3 lemma. Let ∆(x) = ∏s∈S |(vs|x)|cs . We have:

H2 = ∆−1 J2∆ =N

∑i=1

∂2ei−∑

s∈S

2cs(cs + 1)(vs|•)2 .

45


Proof. We have:

∆−1 J2∆ =N

∑i=1

∂2ei−∑

s∈S

2cs

(vs|x)∂vs

+N

∑i=1

(∂e2i

ln ∆) + (∂ei ln ∆)2 + 2(∂ei ln ∆)∂ei −∑s∈S

2cs

(vs|x)(∂vs ln ∆) .

If we want the first order derivative terms to disappear, we must choose ∆such that:

∀i ∈ J1, NK, (∂ei ln ∆) = ∑s∈S

cs(vs|ei)

(vs|x).

This is solved by ∆(x) = ∏s∈S |(vs|x)|cs . With this choice, we compute further:

∆−1 J2∆ =N

∑i=1

∂2ei+

N

∑i=1

∑s∈S−cs

(vs|ei)2

(vs|x)2

+ ∑s,s′∈S

cscs′(vs|ei)(vs′ |ei)

(vs|x)(vs′ |ei)− ∑

s,s′∈S2cscs′

(vs|vs′)

(vs|x)(vs′ |x)

.

Using the remark around (69) and simplifying, we find the desired result.

In the case of W = SN , C[V]W is the space of symmetric polynomials in Nvariables, and our commutative algebra of operators is generate by the powersums of the Dunkl operators:

∀k ∈ J1, NK, Hk =N

∑i=1

Dkei

.

We have for instance:

H1 =N

∑i=1

∂ei ,

and H2 coincides with the Calogero-Moser operator (67).

• Quantum separation of variables. The problem of finding the eigenvaluesand common eigenvectors of the commutative algebra J , or equivalent ofH = ∆−1J ∆ reduce to the problem of finding the eigenvalues and commoneigenvectors of the Dunkl operators. We already know that Da · 1, thereforefor any J ∈ J , J · 1 = 0, which means that for any H ∈ H, H · ∆−1 = 0.Therefore, we already have found an eigenvector (the ”ground state”), witheigenvalue 0.

4.4 theorem (Dunkl). There exists a unique isomorphism U : C[V]W → C[V]W

such that U · 1 = 1 and Dei U = U∂ei .

The knowledge of U reduces the solution of the spectral problem to solvinglinear ODEs, instead of linear PDEs: one speaks of a ”quantum separationof variables”. Unfortunately, the explicit form of U for the Calogero-Moser

46

4.3. Quantum trigonometric Calogero-Moser

system is only known for N ≤ 3.

4.3 Quantum trigonometric Calogero-Moser

The spectrum for the rational Calogero-Moser is not discrete, due to the ab-sence of confinement when xi ∈ R. We will study a slightly modified system,for simplicity in the case of W = SN .

• Dunkl operators.

4.5 definition. We introduce the trigonometric Dunkl operator, for i ∈ J1, NK:

Di = zi∂zi − c ∑j 6=i

zmin(i,j)

zi − zj(1− sij) + c

(− i +

N + 12

)(70)

= ziDei − c ∑j<i

sij + cN − 1

2.

It is expressed in terms of trigonometric functions if we write zj = eiθj ,hence their name. This family of operators is still commutative, and at leastwhen c ≤ 0, it has the advantage to be self-adjoint for the scalar product:

(71) 〈 f , g〉 =˛

TN

N

∏m=1

dzm

zm

f (z) g(z−1)

∆(z),

where T is the unit circle, and:

∆(z) = ∏i<j

∣∣(zi/zj)1/2 − (zj/zi)

1/2∣∣2c, T = z ∈ C, |z| = 1 .

The assumption c ≤ 0 ensures that the integral is convergent. In this formula,the real-valued polynomial functions f and g defined on RN are extended topolynomial functions on CN , thus 〈 f , f 〉 ≥ 0.

4.6 theorem. For any i, j ∈ J1, NK, we have:

(i) Di is self-adjoint for (71).

(ii) For any w ∈ SN , wDiw−1 = Dw(i).

(iii) [Di, Dj] = 0.

Note that the dependence in i of the constant in (70) is crucial to have thesymmetry (ii).

Proof. We have:

(72) zi∂zi ln ∆(z) = c ∑j 6=i

(zi/zj)1/2 + (zj/zi)

−1/2

(zi/zj)1/2 − (zj/zi)1/2 = c ∑j 6=i

zi + zj

zi − zj,

47


and notice that:

[Dig](z−1) = −zi∂zi [g(z−1)]− c ∑

j 6=i

z−1min(i,j)

z−1i − z−1

j

[g− sijg](z−1) + cte g(z−1)

= −zi∂zi [g(z−1)] + ∑

j 6=ic

zmax(i,j)

zi − zj[g− sijg](z−1) + cte g(z−1) .(73)

We find by integration by parts:

˛TN

N

∏m=1

dzm

zm

Dzi f (z) · g(z−1)

∆(z)

= cte〈 f , g〉+˛

TN

N

∏m=1

dzm

zm

(∑j 6=i−c

zmin(i,j)

zi − zj

[ f − sij f ](z) · g−1(z)

∆(z)

− f (z) · zi∂zi [g(z−1)]

∆(z)+ c

f (z) g(z−1)

∆(z)∑j 6=i

zi + zj

zi − zj

).

Since ∆ is symmetric, the term the action of sij on f can be transferred to gwith the change of variable x 7→ sij(x):

˛TN

N

∏m=1

dzm

zm

[Di f ](z) · g(z−1)

∆(z)

= cte〈 f , g〉+˛

TN

N

∏m=1

dzm

zm

f (z) · [zi∂zi g](z−1)

∆(z)

+

˛TN

N

∏m=1

dzm

zm

c∆(z)

(∑j 6=i

zmax(i,j)

zi − zjf (z) · g(z−1)−

zmax(i,j)

zi − zjf (z) · [sijg](z−1)

)

=

˛TN

N

∏m=1

dzm

zm

f (z)Dig(z−1)

∆(z),

hence (i). For (ii), it is enough to prove the claim when w is a transpositionsk,k+1. We shall give the details to prove the relation:

(74) si,i+1Di = Di+1 si,i+1 .

The other cases:

si,i−1Di = Di−1 si,i−1, sk,k+1Di = Di sk,k+1, k /∈ i− 1, i

can be proved in a similar way. We use the obvious relation:

(75) sij sjk = sjk ski

48


to compute:

si,i+1Di = zi+1∂zi+1 si,i+1 − c ∑j 6=i

si,i+1zmin(i,j)

zi − zj(1− sij) + c

(− i +

N + 12

)= zi+1∂zi+1 si,i+1 − c ∑

j 6=i

zmin(i+1,si,i+1(j))

zi+1 − zsi,i+1(j)(1− si+1,j)si,i+1 + c

(− i +

N + 12

).

We recognize in the right-hand side Di+1 si,i+1, except for the term j = i + 1 inthe sum, for which we have si,i+1(j) = i. So:

si,i+1Di =(

Di+1 + czi+1

zi+1 − zi(1− si,i+1) + c

)si,i+1 − c

zizi+1 − zi

(1− si,i+1)si,i+1,

and since (1− si,i+1)si,i+1 = −(1− si,i+1), this entails (74).For (iii), let us use (71) and the commutation relations already known

(Theorem 4.2) for Di:

[Di, Dj] = zi[Di, zj]Dj − zj[Dj, zi]Di − c ∑l<j

[ziDi, sl,j] + c ∑k<i

[zjDj, si,k] + c2 ∑k<il<j

[si,k, sj,l ]

= c(zi si,jDj − zj si,jDi) + c− ziDi ∑

k<jsj,k + δi<jzjDj si,j + ∑

k<jk 6=i

ziDi sj,k

+zjDj ∑k<i

si,k − δj<iziDi si,j −∑k<ik 6=j

zjDj sj,k

+c2

∑k=l<min(i,j)

[si,k, sj,k] + ∑k=j<i

l<j

[si,j, sj,l ]

.

Since we have (zi si,jDj − zj si,jDi) = (ziDi − zjDj)si,j, the terms proportional toc cancel. One can also check using repeatedly (75) that the term proportionalto c2 cancel.

• Common basis of eigenvectors. If λ = (λ1, . . . , λN) is a N-uple of non-negative integers, we define λord the sequences of λi’s in decreasing order.

4.7 definition. If λ, µ are N-uples, we say that λ / µ if λord < µord for thelexicographic order, or λord = µord but λ > µ for the lexicographic order.

This gives an order relation, and we can consider the ordered basis Zλ =

∏Ni=1 zλi

i of C[z1, . . . , zN ], and orthogonalize it with respect to the scalar prod-uct (71). The outcome is a new basis (Eλ)λ such that:

Eλ = Zλ + ∑µ/λ

dλ,µ Zµ .

49


4.8 lemma. For any i ∈ J1, NK, there exists coefficients di,λ,µ such that:

DiZλ =

λi + c(i− N + 12

)Zλ + ∑

µ/λ

di,λ,µ Zµ .

Proof. We have zi∂zi Zλ = λiZλ. Let us denote Ki,j =zmin(i,j)zi−zj

(1− si,j). We have:

Ki,j[zλii z

λjj ] =

0 if λi = λj

−∑λj−λi−1m=0 zλi+1+m

i zλj−1−mj if i < j and λi < λj

∑λi−λj−1m=0 zλi+m

i zλj−mj if i < j and λi > λj

−∑λj−λi−1m=0 zλi+m

i zλj−mj if i > j and λi < λj

∑λi−λj−1m=0 zλi−1−m

i zλj+1+mj if i > j and λi > λj

.

So, when we compute for a fixed i the sum −c ∑j 6=i Ki,jZλ, the result is a linearcombination of monomials:

• including Zλ with coefficient −c for the third case, and c in the fourthcase. In total that makes Zλ with a prefactor c

(− ∑j>i 1 + ∑j<i 1

)=

c(2i− (N + 1)). Note that this is twice the opposite of the constant termin Di

• including Zλ′ in which λ′ is obtained from permutation of λi and λj inthe second and fifth case. In the second case, i < j and λi < λj, therefore,λ′ / λ.

• and all the other terms involve Zµ for uples µ differing from λ at the i-thand j-th position only, and such that max(µi, µj) < max(λi, λj), henceµ / λ.

Collecting all the terms entails the result.

In other words, the matrix of Di in the basis of Zλ is upper triangular forthe order /. This implies that the matrix of Di in the basis of Eλ is also uppertriangular, and of the form:

DiEλ =

λi + c(

i− N + 12

)Eλ + ∑

µ/λ

di,λ,µ Eµ .

Since the (Eλ)λ basis is orthogonal, this implies:

4.9 theorem. Eλ is an eigenvector of Di with eigenvalue λi + c(i− N−1

2)

The Eλ are called the non-symmetric Jack polynomials.Now, if P is any symmetric polynomial in N variables, we deduce that the

operators P(D1, . . . , DN) are well-defined, all commute, and on the space ofsymmetric polynomials in N variables, they have eigenvectors:

Esymλ (x) = ∑

σ∈SN

Eλ(σ(x))

50


with eigenvalue:

1N! ∑

σ∈SN

P[

λσ(i) +(

i− N + 12

)N

i=1

].

The Esymλ , up to a normalization, are called the Jack polynomials. In particular:

4.10 theorem. Esymλ forms the basis of eigenvectors of:

Jtrig2 =

N

∑i=1

D2i −

c2N(N2 − 1)12

=N

∑i=1

(zi∂zi )2 − c ∑

i 6=j

zi + zj

zi − zj(zi∂zi − zj∂zj)− 2c ∑

i 6=j

zizj

(zi − zj)2

with eigenvalues ∑Ni=1 λ2

i . In particular, the spectrum is discrete and the minimumeigenvalue is 0, which is reached for the eigenfunction 1.

Proof. (76) is a simple computation, remarking that si,j = id on the space ofsymmetric polynomials. Without the shift by a scalar, the eigenvalue is:

1N! ∑

σ∈SN

N

∑i=1

[λσ(i) + c

(i− N + 1

2

)]2

=N

∑i=1

λ2i + c2

N

∑i=1

(i− N + 1

2

)2+

2cN

( N

∑i=1

λi

) N

∑i=1

(i− N + 1

2

)=

N

∑i=1

λ2i +

c2N(N2 − 1)12

.(76)

• Gauge transformation. As in Lemma 4.3, we can conjugate Jtrig2 to get rid of

the first order derivatives.

4.11 lemma.

Htrig2 = ∆−1 Jtrig

2 ∆ + c2N(N − 1)2 =N

∑i=1

(zi∂zi )2 −∑

i 6=j

4c(c + 1)zizj

(zi − zj)2 .

Proof. Using (72), we find:

Htrig2 =

N

∑i=1

(zi∂zi )2 − c ∑

j 6=i

zi + zj

zi − zj(zi∂zi − zj∂zj)− 2c ∑

j 6=i

zizj

(zi − zj)2

+N

∑i=1

(zi∂2zi

ln ∆) + (zi∂zi ln ∆)2 − c ∑i 6=j

zi + zj

zi − zj(zi∂zi − zj∂zj) ln ∆

51


Htrig2 =

N

∑i=1

(zi∂zi )2 − 2c ∑

j 6=i

zizj

(zi − zj)2 +N

∑i=1

c2 ∑

j,k 6=i

zi + zj

zi − zj

zi + zkzi − zk

−2c ∑j 6=i

zizj

(zi − zj)2 − c2 ∑i 6=j

zi + zj

zi − zj

(∑k 6=i

zi + zkzi − zk

−∑k 6=j

zj + zk

zj − zk

)

=N

∑i=1

(zi∂zi )2 − 4c ∑

j 6=i

zizj

(zi − zj)2

+c2−∑

i 6=j

(zi + zj)2

(zi − zj)2 + ∑i,j,k pairwise

distinct

zi + zj

zi − zj

zj + zk

zj − zk

.

And the last line is equal to:

−c2

∑i 6=j

( 4zizj

(zi − zj)2 + 1)+ ∑

i,j,k pairwisedistinct

1

= −c2(

∑i 6=j

4zizj

(zi − zj)2 + N(N− 1)2)

.

Htrig2 is the trigonometric Calogero-Moser operator. Putting all results to-

gether, we have shown that its eigenvectors are given by the functions ∆(x) ·Esym

λ , with eigenvalues:

N

∑i=1

λ2i + c2N(N − 1)2 .

The constant c2N(N − 1) represents the ground state energy. The other gen-erators of the commuting algebra are given by the conjugation by:

Htrigk = ∆−1

( N

∑i=1

Dki

)∆ , k ∈ J1, NK .

52

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