Bertrand Banos

Complex solutions of Monge-Ampère equations

We describe a method to reduce partial differential equations of Monge-Ampère type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Ampère equations and the Plebanski equations.

Introduction

A Monge-Ampère equation is a partial differential equation which is non linear in a very specific way: its nonlinearity is the determinant one. In two variables, Monge-Ampère equations are

A ∂ 2 f ∂q 2 1 + 2B ∂ 2 f ∂q 1 ∂q 2 + C ∂ 2 f ∂q 2 2 + D ∂ 2 f ∂q 2 1 • ∂ 2 f ∂q 2 2 -( ∂ 2 f ∂q 1 ∂q 2 ) 2 + E = 0 ,
where coefficients A, B, C and D are smooth functions on jet space J1 R 2 . An important subfamily is the family of "symplectic" Monge-Ampère equations, when coefficients are assumed to be smooth functions on cotangent space T * R 2 . Contact geometry is then replaced by symplectic geometry. In this work, we will only consider Monge-Ampère with constant coefficients which fall in this symplectic subfamily. In higher dimensions, a Monge-Ampère equation is a linear combination of the minors of the hessian matrix Hess(f ) = ∂ 2 f ∂q j ∂q k j,k=1...n .

As examples in dimension 4, we will cite the famous special lagrangian equation described by Harvey and Lawson, or Plebanski equations and Grant equation obtained by reduction of Yang-Mills equations and Einstein equations:

                       ∂ 2 f ∂x 1 ∂x 3 ∂ 2 f ∂x 2 ∂x 4 - ∂ 2 f ∂x 1 ∂x 4 ∂ 2 f ∂x 2 ∂x 3 = 1 (Plebanski I equation) ∂ 2 f ∂x 2 1 ∂ 2 ∂x 2 2 - ∂ 2 f ∂x 1 ∂x 2 2 + ∂ 2 f ∂x 2 ∂x 4 - ∂ 2 f ∂x 1 ∂x 3 = 0 (Plebanski II equation) ∂ 2 f ∂x 2 1 + ∂ 2 f ∂x 1 ∂x 2 ∂ 2 f ∂x 3 ∂x 4 - ∂ 2 f ∂x 1 ∂x 4 ∂ 2 f ∂x 2 ∂x 3 = 0 (Grant equation).
After the seminal paper of Lychagin ([25]), geometry of Monge-Ampère equations is quite well understood in 2 and 3 variables ( [26], [1], [2], [3]) but dimension 4 remains mysterious. This is actually the first dimension in which the space of equivalent classes (modulo a change of independent and dependent variables) is a real moduli space and discrete classification is not possible anymore. We propose here a method to reduce these 4 dimensional equations to complex equations in two variables.

In the first section, we recall the Lychagin correspondence between Monge-Ampère equations and effective forms on the phase space and we describe briefly classification results in dimension 2 and 3. In the second section, we assume that the phase space is endowed with an extra compatible complex structure and we define the notion of complex solution. We explain why complex solutions of a given Monge-Ampère equation depend only of its bieffective part. In the third section we use this method to construct explicit complex solutions of the special lagrangian equation, the real Monge-Ampère equations and the Plebanski equations.

1 Monge-Ampère operators and differential forms 1.1 An example: the special lagrangian equation

Let Ω = i 2 dz 1 ∧ dz 1 + . . . + dz n ∧ dz n be the canonical Kähler form on C n and α = dz 1 ∧ . . . ∧ dz n be the complex volume form. A special lagrangian submanifold is a real n-submanifold L which is lagrangian with respect to Ω and which satisfies the special condition Im(α)| L = 0.

These submanifolds, introduced by Harvey and Lawson in their famous article Calibrated Geometries ( [12]) are minimal submanifolds of C n , and more generally of Calabi-Yau manifolds, transverse in some sense to complex submanifolds. They have been extensively studied after the construction proposed by Strominger, Yau and Zaslow ( [27]) of mirror partners of Calabi-Yau manifolds based upon an hypothetic special lagrangian fibration.

Some examples have been given by many people. We can cite for example Harvey and Lawson ( [12]), Joyce ([18], [19], [20], [21] [22]) and Bryant ([6], [7], [8]).

For every smooth function f on R n , the graph

L f = (q + i ∂f ∂q ), q ∈ R n
is a lagrangian submanifold of C n . The special lagrangian condition becomes then a differential equation on f , called the special lagrangian equation:

• n = 2: ∆f = 0

• n = 3: ∆f -hess f = 0

• n = 4: ∆f -hess 1 f -hess 2 f -hess 3 f -hess 4 f = 0 with ∆f the Laplace operator, hess f the determinant of the hessian matrix and hess i f the (i, i)-minor of the hessian matrix. The Monge-Ampère operators theory developed by Lychagin ([25]) generalizes this correspondence between "calibrated" lagrangian submanifolds of R 2n and Monge-Ampère equations on R n .

The Monge-Ampère operators theory

Let M a n-dimensional manifold and T * M its cotangent bundle endowed with the symplectic canonical form Ω ∈ Ω 2 (T * M ). Denote by q = (q 1 , . . . , q n ) a coordinates system on M and (q, p) the corresponding Darboux coordinates system on T * M such that Ω = dq 1 ∧ dp 1 + . . . + dq n ∧ dp n .

Let ω ∈ Ω n (T * M ) be a n -differential form on the 2n dimensional manifold

T * M . The Monge-Ampère operator ∆ ω : C ∞ (M ) → Ω n (M ) is defined by ∆ ω (f ) = (df ) * (ω)
where df : M → T * M is the differential of the smooth function f . a) A regular solution of the MA equation ∆ ω = 0 is a smooth function f on M such that ∆ ω (f ) = 0. b) A generalized solution is a lagrangian submanifold L of (T * M, Ω) on which vanishes the form ω: Ω| L = 0 and ω| L = 0.

Note that a lagrangian submanifold of T * R n which projects isomorphically on R n is the graph of an exact form df : R n → T * R n . Hence, a generalized solution can be thought of as a smooth patching of local regular solutions. c) Two MA equations ∆ ω1 = 0 and ∆ ω2 = 0 are said (locally) equivalent if there exists a (local) symplectomorphism F : (T * M, Ω) → (T * M, Ω) such that

F * (ω 1 ) = ω 2 .
The symplectomorphism F transforms a generalized solution of ∆ ω2 = 0 into a generalized solution of ∆ ω1 = 0 but regular solutions are not preserved.

For any (n -2)-form θ, the equations ∆ ω = 0 and ∆ ω+θ∧Ω = 0 have the same solutions. We need also to introduce effective n-forms, which satisfy Ω ∧ ω = 0 and the so-called Hodge-Lepage-Lychagin theorem ( [25]) establishes a one-to-one correspondence between effective forms and MA operators:

Theorem (Hodge-Lepage-Lychagin). a) Every n-form ω can be uniquely decomposed ω = ω 0 + ω 1 ∧ Ω with ω 0 effective b) Two effective n-forms which vanish on the same lagrangian subspaces are proportional.

Example. Consider the 2-dimensional equation

hess f = 1.
The corresponding effective form is ω = dp 1 ∧ dp 2 -dq 1 ∧ dq 2 which is transformed into θ = dp 1 ∧ dq 2 + dq 1 ∧ dp 2 by the partial Legendre transformation

Φ : (q 1 , q 2 , p 1 , p 2 ) → (q 1 , p 2 , p 1 , -q 2 )
Hence, hess f = 1 is equivalent to the Laplace equation ∆f = 0. Choose then any harmonic function, for example f (q 1 , q 2 ) = e q1 cos(q 2 ). We obtain a generalized solution L = (q 1 , -e q1 sin(q 2 ), e q1 cos(q 2 ), -q 2 ); (q 1 , q 2 ) ∈ R 2 .

which is, on an open subset, the graph of the non trivial regular solution

u(t 1 , t 2 ) = t 2 arcsin(t 2 e -t1 ) + e 2t1 -t 2 2 .
We know from Jörgens theorem that this regular solution can not be defined on the whole plane.

1.3 Classification results for n = 2 and n = 3

In 1874, Sophus Lie raised the question of linearization of Monge-Ampère equations: when a given equation is equivalent to a linear one ? In the formalism of Monge-Ampère operators, this problem turns into a problem of the Geometric Invariant Theory. Studying action of the symplectic group on effective forms, a complete classification of equivalence classes of Monge-Ampère equations (with constant coefficients) has been obtained in [26] for n = 2 and in [26] and [2] for n = 3.

For n = 2, every MAE with constant coefficient is linearizable (table 1).

∆ ω = 0 ω pf (ω) Geometry ∆f = 0 dq 1 ∧ dp 2 -dq 2 ∧ dp 1 1 A 2 ω = -1 f = 0 dq 1 ∧ dp 2 + dq 2 ∧ dp 1 -1 A 2 ω = 1 ∂ 2 f ∂q 2 1 = 0 dq 1 ∧ dp 2 0 A 2 ω = 0 Table 1: Classification of 2-dimensional MAE
Here, the pfaffian pf(ω) is the scalar defined by ω ∧ ω = pf(ω) Ω ∧ Ω and A ω is the tensor defined by ω(• , •) = Ω(A ω • , •). It is a complex structure for the elliptic equation and a product structure for the hyperbolic equation. It is explained in [3] how this unifying geometry coïncides, in the particular dimension n = 2, with the famous generalized complex geometry introduced by Hitchin ( [15]) and Gualtieri ([11]).

For n = 3, there are three non linear Monge-Ampère equation: the real one, the special lagrangian one and the pseudo special lagrangian one (table 2).

In this table, ε(g ω ) is the signature of the Lychagin-Roubstov metric (see [26]) defined by

g ω (X, Y )Ω 3 = ι X (ω) ∧ ι Y (ω) ∧ Ω,
and A ω is the Hitchin tensor ( [26]), defined for effective 3-forms by

g ω (A ω • , •) = Ω(• , •).
It is explained in [1] how these invariants define a geometry of real or complex Calabi-Yau type.

For dimensions 2 and 3, the quotient space of Monge-Ampère equations for the action of symplectic linear group G = Sp(2n, R) is thus a discrete space. More generally, a Monge-Ampère equation corresponds to a conformal class of an effective form ω. The orbit G • ω of this form is isomorphic to the quotient G/G ω where 2 Complex solutions and bieffective forms

∆ ω = 0 ε(g ω ) A ω 1 hess(f ) = 1 (3, 3) A 2 ω = 1 2 ∆f -hess(f ) = 0 (0, 6) A 2 ω = -1 3 f + hess(f ) = 0 (4, 2) A 2 ω = -1 4 ∆f = 0 (0, 3) A 2 ω = 0 5 f = 0 (2, 1) A 2 ω = 0 6 ∆ q2,q3 f = 0 (0, 1) A 2 ω = 0 7 q2,q3 f = 0 (1, 0) A 2 ω = 0 8 ∂ 2 f ∂q 2 1 = 0 (0, 0) A 2 ω = 0

Complex solutions

A complex structure J on manifold T * M is said to be compatible with symplectic form Ω, if Ω J = Ω(J• , •) is a 2-form. The complex 2-form Θ J = Ω -iΩ J is then a complex symplectic form.

In Darboux coordinates, such a compatible complex structure writes as

J = A B C A t with          B t = -B, C t = -C A 2 + BC = -1 AB + BA t = 0 AC + CA t = 0
As in generalized complex geometry, there are two important families:

J = A 0 0 A t with A a complex structure on M and J = 0 θ -θ -1 0
with θ a symplectic form on M Definition. Let ∆ ω = 0 be a MAE on M and let J be a compatible complex structure on T * M . A J-complex solution is a lagrangian and J-complex submanifold L of T * M on which ω vanishes:

Ω| L = 0 ; J L = L ; ω| L = 0
It is worth mentioning that the condition "complex lagrangian" is equivalent to the condition "real bilagrangian" as it is proved in [16].

Proposition (Hitchin). Let (N, Ω 1 + iΩ 2 ) be a complex symplectic manifold of complex dimension 2m. A real 2m-dimensional submanifold is a complex lagrangian submanifold if and only it is lagrangian with respect to Ω 1 and Ω 2 .

Example. Let us identify C 4 with H 2 endowed with the three complex structures I, J and K. We still denote by Ω the I -Kähler form, Ω J = Ω(J• , •) and

Ω K = Ω(K• , •).
Then Ω-iΩ J is a J-complex symplectic form and it is well known that every J-complex lagrangian submanifold is special lagrangian.

In the formalism of Monge-Ampère operators, this can be seen very simply. It is actually straightforward to check that the special lagrangian form writes as

Im(α) = Ω J ∧ Ω K and therefore, if Ω| L = 0 and Ω J | L = 0 then Im(α)| L = 0.

Bieffective forms

Two MAE ∆ ω = 0 and ∆ ω+θ1∧Ω+θ2∧Ω J = 0 have the same J-complex solutions. To understand this complex reduction, we need then to construct the bieffective part of ω. This is the goal of this section.

Let V be a complex symplectic space of real dimension 4m endowed with a complex symplectic form Θ = Ω 1 + iΩ 2 . Denote by Λ k (V * ) the space of real k-forms on V and Λ p,q (V * ) the space of (p, q)-complex forms, such that

Λ k (V * ) ⊗ C = p+q=k Λ p,q (V * )
Let us introduce for j = 1, 2 the operators ⊤ j and ⊥ j defined by

     ⊤ j θ = θ ∧ Ω j ⊥ j θ = ι XΩ j (θ) with X Ωj the unique bivector satisfying Ω j (X Ωj ) = 1
They have the following properties ( [25])

a) ⊥ j : Λ k (V * ) → Λ k-2 (V * ) is into for k ≥ 2m + 1 b) ⊤ j : Λ k (V * ) → Λ k+2 (V * ) is into for k ≤ 2m -1 c) [⊥ j , ⊤ j ](θ) = (2m -k)θ for θ ∈ Λ k (V * ). A k-form θ is said to be Ω j -effective if ⊥ j θ = 0. For k = 2m, this is equivalent to ⊤ j θ = 0. Let H = [⊥ 1 , ⊤ 1 ] = [⊥ 2 , ⊤ 2 ] and M = [⊥ 2 , ⊤ 1 ]
. We get then the complete list of so-called Lichnerowicz operators which satysfy the following (see [5]):

Proposition (Verbitsky -Bonan). [⊥ 1 , ⊤ 1 ] = H [⊥ 2 , ⊤ 2 ] = H [⊥ 1 , ⊤ 2 ] = -M [⊥ 2 , ⊤ 1 ] = M [⊥ 1 , ⊥ 2 ] = 0 [⊤ 1 , ⊤ 2 ] = 0 [⊥ 1 , H] = -2⊥ 1 [⊥ 2 , H] = -2⊥ 2 [⊤ 1 , H] = 2⊤ 1 [⊤ 2 , H] = 2⊤ 2 [⊥ 1 , M ] = -2⊥ 2 [⊥ 2 , M ] = 2⊥ 1 [⊤ 1 , M ] = -2⊤ 2 [⊤ 2 , M ] = 2⊤ 1 [H, M ] = 0
We obtain then a representation of the Lie algebra sl(2,

C)⊗sl(2, C) on Λ * (V * )⊗C, defining      E 1 = 1 2 (⊥ 1 + i⊥ 2 ) F 1 = 1 2 (⊤ 1 -i⊤ 2 ) H 1 = 1 2 (H + iM )      E 2 = 1 2 (⊥ 1 -i⊥ 2 ) F 2 = 1 2 (⊤ 1 + i⊤ 2 ) H 2 = 1
2 (H -iM ) Representation theory of Lie algebras gives us the existence and uniqueness of bieffective part of a 2m-form.

Theorem 1 ([4]). Every 2m-form ω ∈ Λ 2m (V * ) can be decomposed into a sum ω = ω 0 + ω 1 ∧ Ω 1 + ω 2 ∧ Ω 2 with ω 0 bieffective, that is ω 0 ∧ Ω 1 = 0 = ω 0 ∧ Ω 2 . Moreover the bieffective part ω 0 is unique.
Proof. From Weyl's theorem, we know that Λ * (V * )⊗C decomposes as a unique direct sum of irreducible subspaces. Let W such an irreducible subspace. Since H 1 and H 2 commute, they admit a common eigenvector x ∈ W . But E p 1 E q 2 x is also a common eigenvector. Their exist then p and q such that z = E p 1 E q 2 x is a primitive vector, that is E 1 z = E 2 z = 0. Therefore, W = Gz and every vector w in W writes as

w = j,k a jk F j 1 F k 2 z
We deduce that every 2m-form can be uniquely decomposed into a finite sum

ω = j,k (α jk + iβ jk ) ∧ (Ω 1 + iΩ 2 ) j ∧ (Ω 1 -iΩ 2 ) k
with α jk and β jk primitive. Noting now that 2m-primitive forms are the bieffective forms we obtain the result.

We give now an explicit formula for 4m = 8, which can be easily implemented on a computer.

Proposition. In dimension 8, the bieffective par ω 0 of a 4-form ω is

ω 0 = θ - 1 4 ⊤ 2 ⊥ 2 θ + ⊤ 1 ⊥ 1 θ - 1 4 M (M θ -⊤ 1 ⊥ 2 θ + ⊤ 2 ⊥ 1 θ)
where

θ = ω - (3⊥ 2 1 ω -⊥ 2 2 ω) 64 Ω 2 1 - ⊥ 1 ⊥ 2 ω 8 - (3⊥ 2 2 ω -⊥ 2 1 ω) 64 Ω 2
Proof. We know that

ω = ω 0 + ω 1 ∧ Ω 1 + ω 2 ∧ Ω 2 + ω 11 Ω 1 ∧ Ω 1 + ω 12 Ω 1 ∧ Ω 2 + ω 22 Ω 2 ∧ Ω 2 ,
with ω 0 , ω 1 and ω 2 primitive. Using Verbiski-Bonan relations, we obtain

⊥ 1 ω = 2ω 1 -M ω 2 + (6ω 11 + 2ω 22 )Ω 1 + 2ω 12 Ω 1 and then ⊥ 2 1 ω = 24ω 11 + 8ω 22 ⊥ 2 ⊥ 1 ω = 8ω 12 .
Starting from ⊥ 2 ω we obtain also

⊥ 2 2 ω = 8ω 11 + 24ω 22 . Therefore, ω 11 = 3⊥ 2 1 ω -⊥ 2 2 ω 64 , ω 22 = 3⊥ 2 2 ω -⊥ 2 1 ω 64 , ω 12 = ⊥ 1 ⊥ 2 ω 8 . Define now θ = ω 0 + ω 1 ∧ Ω 1 + ω 2 ∧ ω 2 . Since ⊥ 1 θ = 2ω 1 -M ω 2 and ⊥ 2 θ = M ω 1 + 2ω 2 , we deduce that M ⊤ 1 ω 1 = [M, ⊤ 1 ]ω 1 + ⊤ 1 M ω 1 = 2⊤ 2 ω 1 + ⊤ 1 (⊥ 2 θ -2ω 2 ) = 2⊤ 2 ω 1 -2⊤ 1 ω 2 + ⊤ 1 ⊥ 2 θ and similarly M ⊤ 2 ω 2 = 2⊤ 2 ω 1 -2⊤ 1 ω 2 -⊤ 2 ⊥ 1 θ.
and therefore

M θ = M ⊤ 1 ω 1 + M ⊤ 2 ω 2 = 4(⊤ 2 ω 1 -⊤ 1 ω 2 ) + ⊤ 1 ⊥ 2 θ -⊤ 2 ⊥ 1 θ.
Moreover, the computation of

M (⊤ 2 ω 1 -⊤ 1 ω 2 ) gives M (⊤ 2 ω 1 -⊤ 1 ω 2 ) = -4(⊤ 1 ω 1 + ⊤ 2 ω 2 ) + ⊤ 2 ⊥ 2 θ + ⊤ 1 ⊥ 1 θ. Finally, 4(ω 1 ∧ Ω 1 + ω 2 ∧ Ω 2 ) = ⊤ 2 ⊥ 2 θ + ⊤ 1 ⊥ 1 θ - M 4 (M θ -⊤ 1 ⊥ 2 θ + ⊤ 2 ⊥ 1 θ).

Action of the complex symplectic group

For simplicity, we restrict now to complex dimension 4: V is a 4-dimensional complex vector space endowed with a complex symplectic form Θ = Ω 1 + iΩ 2 . The space of real bieffective 4-forms is

Λ 4 BE (V * ) = ω ∈ Λ 4 (V * ), ω ∧ Ω 1 = ω ∧ Ω 2 = 0 .
Let Λ 2,0 0 (V * ) be the 5 -dimensional complex vector space of (2, 0)-complex forms which are effective with respect to Θ = Ω 1 + iΩ 2 :

Λ 2,0 0 (V * ) = θ ∈ Λ 2,0 (V * ), θ ∧ Θ = 0
The exterior product is non degenerate on Λ 2,0 0 (V * ) and it defines a non degenerate symmetric inner product

< θ 1 , θ 2 > Θ 2 = θ 1 ∧ θ 2 .
We are going to identify bieffective forms with hermitian forms on Λ 2,0 0 (V * ).

Definition. The hermitian form Q ω on C 5 = Λ 2,0 0 (V * ) associated with a bieffective 4-form ω is:

Q ω (θ 1 , θ 2 ) (Θ ∧ Θ) 2 = ω ∧ θ 1 ∧ θ 2 .
Theorem 2. The map

Λ 4 BE (V * ) → su(5) ω → Q ω is an isomorphism.
Moreover, the group Sp(4, C)/Z 2 identifies with SO(5, C) and its action on Λ 4 BE (V * ) is the Hermite action of SO(5, C) on su (5).

Proof. We prove first that Λ 4 BE (V * ) = Λ 2,0 0 (V * ) ⊗ Λ 2,0 0 (V * ). Let ω ∈ Λ 4 BE (V *
) and consider its decomposition

ω = ω 40 + ω 31 + ω 22 + ω 13 + ω 04 with ω pq ∈ Λ p,q (V * ). Since ⊥ C : Λ p,q (V * ) → Λ p-2,q (V * ) is into for p ≥ 3 and ⊥ C : Λ p,q (V * ) → Λ p,q-2 (V * ) is into for q ≥ 3, and since ⊥ C ω = ⊥ C ω = 0, we deduce that ω ∈ Λ 2,2 . But Ker ⊥ C : Λ 2,2 → Λ 0,2 = Λ 2,0 0 (V * ) ⊗ Λ 0,2 so Ker ⊥ C ∩ Ker ⊥ C = Λ 2,0 0 (V * ) ⊗ Λ 2,0 0 (V * ).
We deduce that Λ 4 BE (V * ) and su( 5) have same dimension. Since ω → Q ω is injective, this is an isomorphism. Now, the action of Sp(4, C) preserves this symmetric product, with kernel Z 2 . Since dim C (Sp(4, C)) = 10 = dim C (SO(5, C)), we deduce that Sp(4, C)/Z 2 = SO(5, C).

Moreover, we have

Q F * ω (θ 1 , θ 2 )(Θ ∧ Θ) 2 = F * (ω) ∧ θ 1 ∧ θ 2 = ω ∧ (F -1 ) * (θ 1 ) ∧ (F -1 ) * (θ 2 ) so Q F * ω = F -1 t Q ω F -1 .
This action is completely described by Hong in [17]. Let us briefly explain this result. Let Q be a hermitian matrix and define the canonical form of Q as a direct sum of three hermitian matrices:

J(Q) = H P (Q) ⊕ K N (Q) ⊕ K C (Q) ,
which are obtained from Jordan blocks of Q as follows:

a) H P (Q) = H m1 (λ 1 ) ⊕ . . . ⊕ H mp (λ q )
, where all λ i ≥ 0 and λ 2 i are the positive eigenvalues of

QQ t . b) K N (Q) = K 2n1 (µ 1 ) ⊕ . . . ⊕ H 2nr (µ r )
, where all µ i > 0 and -µ 2 i are the negative eigenvalues of

QQ t . c) K C (Q) = L 2k1 (ξ 1 ) ⊕ . . . ⊕ L 2ks (ξ s )
, where ξ 2 i are the non real eigenvalues of QQ t , with

K 2n (µ) = 0 -iH n (µ) iH n (µ) 0 , L 2k (ξ) = 0 H k (ξ) H ⋆ k (ξ) 0 and 2H m (λ) =           0 0 . . . 0 1 2λ 0 . . . 0 1 2λ 1 . . . . . . 1 2λ 1 0 . . . 1 2λ 1 . . . . . . 1 2λ 1 . . . 0 0 2λ 1 0 . . . 0 0           + i           0 1 0 . . . 0 0 -1 0 1 . . . 0 0 . . . -1 0 1 . . . . . . . . . . . . -1 0 1 0 0 . . . 0 -1 0 1 0 0 . . . 0 -1 0          
Theorem (Hong). Let Q be a hermitian matrix. Then there exists F complex orthogonal and ε = (ε 1 , . . . , ε p )

with ε i = ±1 such that F t QF = J ε (Q) with J ε (Q) = H ε P (Q)⊕K N (Q)⊕K C (Q) = ε 1 H m1 (λ 1 )⊕. . .⊕ε p H mp (λ q ) ⊕K N (Q)⊕K C (Q)
It is therefore difficult to give a complete classification of all possible complex reductions. Nevertheless, to characterize the orbit of a bieffective form ω, it is necessary to know

a) the signature ε(Q ω ) of Q ω , b) the spectrum of Q ω Q t ω
. and it will be sufficient for the examples we are interested in.

3 Some examples in dimension 4

The choice of the complex structure

The crucial point in this method is the choice of the compatible complex structure, which should depend on the initial Monge-Ampère equation. We choose here five simple complex structures, and give a corresponding complex Darboux coordinates system (z 1 , z 2 , u 1 , u 2 ) in which

Θ J = Ω -iΩ J = dz 1 ∧ du 1 + dz 2 ∧ du 2
The initial coordinate systems on T * R 4 is still (q, p) with

Ω = dq 1 ∧ dp 1 + dq 2 ∧ dp 2 + dq 3 ∧ dp 3 + dq 4 ∧ dp 4 .
We define

A =     0 -1 0 0 1 0 0 0 0 0 0 -1 0 0 1 0     ; Ã =     0 -1 0 0 1 0 0 0 0 0 0 1 0 0 -1 0     ; A 2 =     1 -2 0 0 1 -1 0 0 0 0 1 -2 0 0 1 -1     and J = A 0 0 A t z 1 = q 1 + iq 2 u 1 = p 1 -ip 2 z 2 = q 3 + iq 4 u 2 = p 3 -ip 4 K = 0 A A 0 z 1 = q 1 + ip 2 u 1 = iq 2 + p 1 z 2 = q 3 + ip 4 u 2 = iq 4 + p 3 J = Ã 0 0 Ãt z 1 = q 1 + iq 2 u 1 = p 1 -ip 2 z 2 = q 3 -iq 4 u 2 = p 3 + ip 4 K = 0 Ã Ã 0 z 1 = q 1 + ip 2 u 1 = iq 2 + p 1 z 2 = q 3 -ip 4 u 2 = -iq 4 + p 3 J 2 = A 2 0 0 A t 2 z 1 = q 1 + (-1 + i)q 2 u 1 = (1 -i)p -ip 2 z 2 = q 3 + (-1 + i)q 4 u 2 = (1 -i)p -ip 4

Simple equations

Here is a non exhaustive list of simple complex Monge-Ampère equations which will be our model equations. Function φ is a holomorphic function in (z 1 , z 2 ) and we note

φ jk = ∂ 2 φ ∂z j ∂z k . ε(Q ω ) ∆ ω = 0 spectrum(Q ω Q t ω ) |φ 11 | 2 = 0 (0,0,0,0,0) (1,0) |φ 12 | 2 = 0 (1,0,0,0,0) |φ 11 | 2 = 1 (0,0,0,0,0) (1,1) |φ 12 | 2 = 1
(1,0,0,0,0)

|φ 11 | 2 -|φ 22 | 2 = 0 (-1,-1,0,0,0) |φ 11 | 2 + |φ 12 | 2 = 0 (1,0,0,0,0) (2,0) |φ 11 | 2 + |φ 22 | 2 = 0 (1,1,0,0,0) |φ 11 | 2 + |φ 12 | 2 = 1
(1,0,0,0,0) (2,1) φ 12 + φ 12 -|φ 11 | 2 = 0 (0,0,0,0,0) Table 4: Simple complex Monge-Ampère equations

Examples

We study now the special lagrangian equation, the two real Monge Ampère equations hess f = ±1, the two Plebanski equations and the Grant equation. The corresponding effective forms on (T * R 4 , Ω) are:

ω SLAG = Im dq 1 + idp 1 ) ∧ (dq 2 + idp 2 ) ∧ (dq 3 + idp 3 ) ∧ (dq 4 + idp 4 ) ω H+ = dp 1 ∧ dp 2 ∧ dp 3 ∧ dp 4 -dq 1 ∧ dq 2 ∧ dq 3 ∧ dq 4 ω H-= dp 1 ∧ dp 2 ∧ dp 3 ∧ dp 4 + dq 1 ∧ dq 2 ∧ dq 3 ∧ dq 4 ω P I = dq 1 ∧ dq 2 ∧ dp 1 ∧ dp 2 -dq 1 ∧ dq 2 ∧ dq 3 ∧ dq 4 ω P II = dq 1 ∧ dq 2 + dq 3 ∧ dp 2 + dq 1 ∧ dq 2 ∧ dq 4 ∧ dp 1 + dq 3 ∧ dq ∧ dp 1 ∧ dp 2 ω G = dq 2 ∧ dq 3 ∧ dq 4 ∧ dp 1 -dq 1 ∧ dq 3 ∧ dp 1 ∧ dp 3
We compute for the five compatible complex structures defined above the bieffective part of these forms and the signature of the hermitian form associated with.

Equation J K J K J 2 SLAG 0 0 0 0 (1,1) hess(f ) = 1 (1,1) 0 (1,1) 0 (1,1) hess(f ) = -1 (2,0) (3,2) (2,0) (3,2) (2,0) Plebanski I (2,0) (3,2) (1,1) (3,2) (2,0) Plebanski II (2,1) (3,2) (1,0) (3,2) (2,1) Grant (3,2) (3,2) (3,2) (3,2) (3,2)
Table 5: Examples

This method fails for the Grant equation, at least for this choice of complex structures: signature is always (3,2) and the corresponding complex equation is therefore fully non degenerate. We study the other cases in more details.

Special lagrangian equation

As, we have seen the special lagrangian form ω SLAG has no bieffective part for J and for K since

ω SLAG = Ω J ∧ Ω K
This is the same for any complex structure J = F -1 JF with F in SU (4), since

ω SLAG = F * ω SLAG = F * Ω J ∧ F * Ω K = Ω F -1 JF ∧ Ω F -1 KF
This explains why ω SLAG does not have bieffective part also for J and K. This is the reason of the choice of J 2 : we were looking for a simple complex structure which is not in so(8). In the complex Darboux coordinates system, the bieffective part is

ω BE SLAG = 1 8 (1 + 2i)dz 1 ∧ dz 2 ∧ dz 1 ∧ du 2 + (-1 -2i)dz 1 ∧ dz 2 ∧ dz 2 ∧ du 1 + (1 -2i)dz 1 ∧ du 2 ∧ dz 1 ∧ dz 2 + (1 + 2i)dz 1 ∧ du 2 ∧ du 1 ∧ du 2 + (-1 + 2i)dz 1 ∧ du 1 ∧ dz 1 ∧ dz 2 + (-1 -2i)dz 2 ∧ du 1 ∧ du 1 ∧ du 2 + (1 -2i)du 1 ∧ du 2 ∧ dz 1 ∧ du 2 + (-1 + 2i)du 1 ∧ du 2 ∧ dz 2 ∧ du 1
and this is straightforward to check that

ω BE SLAG = √ 5 4 dZ 1 ∧ dU 2 ∧ dZ 1 ∧ dU 2 -dZ 2 ∧ dU 1 ∧ dZ 2 ∧ dU 1 where (Z 1 , Z 2 , U 1 , U 2 )
is the complex Darboux coordinates system

Z 1 = αz 1 + α -1 u 1 i √ 2 ; U 1 = αz 1 -α -1 u 1 i √ 2 Z 2 = αz 2 -α -1 u 2 √ 2 ; U 2 = αz 2 + α -1 u 2 √ 2 α 2 = 1 + 2i √ 5
We obtain then the following result: Proposition 1. Let φ be a holomorphic solution of

∂ 2 φ ∂z 2 1 2 - ∂ 2 φ ∂z 2 2 2 = 0
and let L φ be the submanifold

L φ = (z 1 , z 2 , ∂φ ∂z 1 , ∂φ ∂z 2 ) ⊂ C 4 .
Then F -1 (L φ ) is special lagrangian in (T * R 4 , Ω, ω SLAG ) where F (q, p) = (z, u) with

z 1 = αq 1 + (-1 + i)αq 2 + (1 -i)α -1 p 1 -iα -1 p 2 i √ 2
z 2 = αq 3 + (-1 + i)αq 4 -(1 -i)α -1 p 3 + iα -1 p 4 √ 2
u 1 = αq 1 + (-1 + i)αq 2 -(1 -i)α -1 p 1 + iα -1 p 2 i √ 2
u 2 = αq 3 + (-1 + i)αq 4 + (1 -i)α -1 p 3 -iα -1 p 4 √ 2

Real Monge-Ampère equations

For every holomorphic function

φ = f + ig : C 2 → C, we have hess R f = | hess C φ| 2
so for J, J and J 2 , which come from complex structures on R 4 , the complex reduction of hess f = ±1 are | hess φ| 2 = ±1 which are equivalent to

∂ 2 ψ ∂z 2 1 2 = ± ∂ 2 ψ ∂z 2 2 2
For example, for J, the corresponding symplectomorphism is the partial Legendre transform

G(z 1 , z 2 , u 1 , u 2 ) = (u 1 , z 2 , -z 1 , u 2 )
We obtain then the following results:

Proposition 2. Let φ be a holomorphic solution of

∂ 2 φ ∂z 2 1 2 - ∂ 2 φ ∂z 2 2 2 = 0 Then G -1 (L φ ) is a generalized solution of hess f = 1 with G(q, p) = (p 1 -ip 2 , q 3 + iq 4 , -q 1 -iq 2 , p 3 -ip 4 ) Proposition 3. Let φ a holomorphic function of the form φ(z 1 , z 2 ) = a(z 1 )b(z 2 ). Then G -1 (L φ ) is a generalized solution of hess f = -1.
Moreover, we see in table 5 that ω H+ has no bieffective part for K and K. We get immediately Proposition 4:

Proposition 4. Any complex lagrangian surface in (T * R 4 , Ω, K) or (T * R 4 , Ω, K) is a generalized solution of hess f = 1.
This result is underlying the strong relationship there is between special lagrangian geometry and "real special lagrangian" geometry or "split special lagrangian geometry", as described in [13].

Plebanski equations

Computing the bieffective part of ω P I , we obtain that the J-complex reduction of Plebanski I equation is

|φ 11 | 2 = -1
which is equivalent up the partial Legendre transform G to

|ψ 11 | 2 + |ψ 12 | 2 = 0 Proposition 5. Let φ a holomorphic function of the form φ(z 1 , z 2 ) = az 1 + b(z 2 ). Then G -1 (L φ ) is a generalized solution of Plebanski I equation.
Similarly, the J-complex reduction is

|φ 12 | 2 = 1 Proposition 6. Let φ a holomorphic function on (R 4 , Ã) of the form φ(z 1 , z 2 ) = z 1 z 2 + a(z 1 ) + b(z 2 ).
Then its real part is a regular solution of Plebanski I equation.

Finally, same computations give analog results for Plebanski II equation:

Proposition 7. Let φ a holomorphic function on (R 4 , A) of φ 12 + φ 12 + |φ 11 | 2 = 0
Then its real part is a regular solution of Plebanski II equation.

Proposition 8. Let φ a holomorphic function on (R 4 , Ã) of the form

φ(z 1 , z 2 ) = a(z 2 ) + b(z 2 )z 1 .
Then its real part is a regular solution of Plebanski II equation.

Conclusion

Studying geometry of 4-bieffective forms on R 8 , we have reduced important equations in Physic to simple -but non empty -complex equations. This has been done for arbitraries complex structures. It would be interesting now, for a given equation in four variables, to study all possible compatible complex structures and to parameterize in this way analytical solutions by pair of compatible complex structures on R 8 , and holomorphic functions on C 2 . Note that bieffective part still exists for 2m-forms on R 4m and such a parametrization should also exist for equations with 2m variables.

A more global approach would be also interesting. We have considered only complex structures with constant coefficients but our decomposition theorem remains valid on a complex symplectic manifold. This suggests to understand "Monge-Ampère calibrations" on complex symplectic manifolds as a generalization of the special lagrangian calibration on HyperKähler manifolds.

Introduction

A Monge-Ampère equation is a partial differential equation which is non linear in a very specific way: its nonlinearity is the determinant one. In two variables, Monge-Ampère equations are

A ∂ 2 f ∂q 2 1 + 2B ∂ 2 f ∂q 1 ∂q 2 + C ∂ 2 f ∂q 2 2 + D ∂ 2 f ∂q 2 1 • ∂ 2 f ∂q 2 2 -( ∂ 2 f ∂q 1 ∂q 2 ) 2 + E = 0 ,
where coefficients A, B, C and D are smooth functions on jet space J 1 R 2 . An important subfamily is the family of "symplectic" Monge-Ampère equations, when coefficients are assumed to be smooth functions on cotangent space T * R 2 . Contact geometry is then replaced by symplectic geometry. In this work, we will only consider Monge-Ampère with constant coefficients which fall in this symplectic subfamily. In higher dimensions, a Monge-Ampère equation is a linear combination of the minors of the hessian matrix

Hess(f ) = ∂ 2 f ∂q j ∂q k j,k=1...n .
As examples in dimension 4, we will cite the famous special lagrangian equation described by Harvey and Lawson, or Plebanski equations and Grant equation obtained by reduction of Yang-Mills equations and Einstein equations:

                       ∂ 2 f ∂x 1 ∂x 3 ∂ 2 f ∂x 2 ∂x 4 - ∂ 2 f ∂x 1 ∂x 4 ∂ 2 f ∂x 2 ∂x 3 = 1 (Plebanski I equation) ∂ 2 f ∂x 2 1 ∂ 2 ∂x 2 2 - ∂ 2 f ∂x 1 ∂x 2 2 + ∂ 2 f ∂x 2 ∂x 4 - ∂ 2 f ∂x 1 ∂x 3 = 0 (Plebanski II equation) ∂ 2 f ∂x 2 1 + ∂ 2 f ∂x 1 ∂x 2 ∂ 2 f ∂x 3 ∂x 4 - ∂ 2 f ∂x 1 ∂x 4 ∂ 2 f ∂x 2 ∂x 3 = 0 (Grant equation).
After the seminal paper of Lychagin ([25]), geometry of Monge-Ampère equations is quite well understood in 2 and 3 variables ( [26], [1], [2], [3]) but dimension 4 remains mysterious. This is actually the first dimension in which the space of equivalent classes (modulo a change of independent and dependent variables) is a real moduli space and discrete classification is not possible anymore. We propose here a method to reduce these 4 dimensional equations to complex equations in two variables.

In the first section, we recall the Lychagin correspondence between Monge-Ampère equations and effective forms on the phase space and we describe briefly classification results in dimension 2 and 3. In the second section, we assume that the phase space is endowed with an extra compatible complex structure and we define the notion of complex solution. We explain why complex solutions of a given Monge-Ampère equation depend only of its bieffective part. In the third section we use this method to construct explicit complex solutions of the special lagrangian equation, the real Monge-Ampère equations and the Plebanski equations.

1 Monge-Ampère operators and differential forms 1.1 An example: the special lagrangian equation

Let Ω = i 2 dz 1 ∧ dz 1 + . . . + dz n ∧ dz n be the canonical Kähler form on C n and α = dz 1 ∧ . . . ∧ dz n be the complex volume form. A special lagrangian submanifold is a real n-submanifold L which is lagrangian with respect to Ω and which satisfies the special condition Im(α)| L = 0.

These submanifolds, introduced by Harvey and Lawson in their famous article Calibrated Geometries ( [12]) are minimal submanifolds of C n , and more generally of Calabi-Yau manifolds, transverse in some sense to complex submanifolds. They have been extensively studied after the construction proposed by Strominger, Yau and Zaslow ( [27]) of mirror partners of Calabi-Yau manifolds based upon an hypothetic special lagrangian fibration.

Some examples have been given by many people. We can cite for example Harvey and Lawson ( [12]), Joyce ([18], [19], [20], [21] [22]) and Bryant ([6], [7], [8]).

For every smooth function f on R n , the graph

L f = (q + i ∂f ∂q ), q ∈ R n
is a lagrangian submanifold of C n . The special lagrangian condition becomes then a differential equation on f , called the special lagrangian equation:

• n = 2: ∆f = 0

• n = 3: ∆f -hess f = 0 • n = 4: ∆f -hess 1 f -hess 2 f -hess 3 f -hess 4 f = 0
with ∆f the Laplace operator, hess f the determinant of the hessian matrix and hess i f the (i, i)-minor of the hessian matrix. The Monge-Ampère operators theory developed by Lychagin ([25]) generalizes this correspondence between "calibrated" lagrangian submanifolds of R 2n and Monge-Ampère equations on R n .

The Monge-Ampère operators theory

Let M a n-dimensional manifold and T * M its cotangent bundle endowed with the symplectic canonical form Ω ∈ Ω 2 (T * M ). Denote by q = (q 1 , . . . , q n ) a coordinates system on M and (q, p) the corresponding Darboux coordinates system on T * M such that Ω = dq 1 ∧ dp 1 + . . . + dq n ∧ dp n .

Let ω ∈ Ω n (T * M ) be a n -differential form on the 2n dimensional manifold T * M . The Monge-Ampère operator ∆ ω :

C ∞ (M ) → Ω n (M ) is defined by ∆ ω (f ) = (df ) * (ω)
where df : M → T * M is the differential of the smooth function f . a) A regular solution of the MA equation ∆ ω = 0 is a smooth function f on M such that ∆ ω (f ) = 0. b) A generalized solution is a lagrangian submanifold L of (T * M, Ω) on which vanishes the form ω: Ω| L = 0 and ω| L = 0.

Note that a lagrangian submanifold of T * R n which projects isomorphically on R n is the graph of an exact form df : R n → T * R n . Hence, a generalized solution can be thought of as a smooth patching of local regular solutions. c) Two MA equations ∆ ω1 = 0 and ∆ ω2 = 0 are said (locally) equivalent if there exists a (local) symplectomorphism F : (T * M, Ω) → (T * M, Ω) such that

F * (ω 1 ) = ω 2 .
The symplectomorphism F transforms a generalized solution of ∆ ω2 = 0 into a generalized solution of ∆ ω1 = 0 but regular solutions are not preserved.

For any (n -2)-form θ, the equations ∆ ω = 0 and ∆ ω+θ∧Ω = 0 have the same solutions. We need also to introduce effective n-forms, which satisfy Ω ∧ ω = 0 and the so-called Hodge-Lepage-Lychagin theorem ( [25]) establishes a one-to-one correspondence between effective forms and MA operators:

Theorem (Hodge-Lepage-Lychagin). a) Every n-form ω can be uniquely decomposed ω = ω 0 + ω 1 ∧ Ω with ω 0 effective b) Two effective n-forms which vanish on the same lagrangian subspaces are proportional.

Example. Consider the 2-dimensional equation

hess f = 1.
The corresponding effective form is ω = dp 1 ∧ dp 2 -dq 1 ∧ dq 2 which is transformed into θ = dp 1 ∧ dq 2 + dq 1 ∧ dp 2 by the partial Legendre transformation

Φ : (q 1 , q 2 , p 1 , p 2 ) → (q 1 , p 2 , p 1 , -q 2 )
Hence, hess f = 1 is equivalent to the Laplace equation ∆f = 0. Choose then any harmonic function, for example f (q 1 , q 2 ) = e q1 cos(q 2 ). We obtain a generalized solution L = (q 1 , -e q1 sin(q 2 ), e q1 cos(q 2 ), -q 2 ); (q 1 , q 2 ) ∈ R 2 .

which is, on an open subset, the graph of the non trivial regular solution

u(t 1 , t 2 ) = t 2 arcsin(t 2 e -t1 ) + e 2t1 -t 2 2 .
We know from Jörgens theorem that this regular solution can not be defined on the whole plane.

Classification results for n = 2 and n = 3

In 1874, Sophus Lie raised the question of linearization of Monge-Ampère equations: when a given equation is equivalent to a linear one ? In the formalism of Monge-Ampère operators, this problem turns into a problem of the Geometric Invariant Theory. Studying action of the symplectic group on effective forms, a complete classification of equivalence classes of Monge-Ampère equations (with constant coefficients) has been obtained in [26] for n = 2 and in [26] and [2] for n = 3.

For n = 2, every MAE with constant coefficient is linearizable (table 1).

∆ ω = 0 ω pf (ω) Geometry ∆f = 0 dq 1 ∧ dp 2 -dq 2 ∧ dp 1 1 A 2 ω = -1 f = 0 dq 1 ∧ dp 2 + dq 2 ∧ dp 1 -1 A 2 ω = 1 ∂ 2 f ∂q 2 1 = 0 dq 1 ∧ dp 2 0 A 2 ω = 0 Table 1: Classification of 2-dimensional MAE
Here, the pfaffian pf(ω) is the scalar defined by ω ∧ ω = pf(ω) Ω ∧ Ω and A ω is the tensor defined by ω(• , •) = Ω(A ω • , •). It is a complex structure for the elliptic equation and a product structure for the hyperbolic equation. It is explained in [3] how this unifying geometry coïncides, in the particular dimension n = 2, with the famous generalized complex geometry introduced by Hitchin ( [15]) and Gualtieri ([11]).

For n = 3, there are three non linear Monge-Ampère equation: the real one, the special lagrangian one and the pseudo special lagrangian one (table 2).

In this table, ε(g ω ) is the signature of the Lychagin-Rubtsov metric (see [26]) defined by

g ω (X, Y )Ω 3 = ι X (ω) ∧ ι Y (ω) ∧ Ω,
and A ω is the Hitchin tensor ( [26]), defined for effective 3-forms by

g ω (A ω • , •) = Ω(• , •).
It is explained in [1] how these invariants define a geometry of real or complex Calabi-Yau type.

For dimensions 2 and 3, the quotient space of Monge-Ampère equations for the action of symplectic linear group G = Sp(2n, R) is thus a discrete space. More generally, a Monge-Ampère equation corresponds to a conformal class of an effective form ω. The orbit G • ω of this form is isomorphic to the quotient G/G ω where

∆ ω = 0 ε(g ω ) A ω 1 hess(f ) = 1 (3, 3) A 2 ω = 1 2 ∆f -hess(f ) = 0 (0, 6) A 2 ω = -1 3 f + hess(f ) = 0 (4, 2) A 2 ω = -1 4 ∆f = 0 (0, 3) A 2 ω = 0 5 f = 0 (2, 1) A 2 ω = 0 6 ∆ q2,q3 f = 0 (0, 1) A 2 ω = 0 7 q2,q3 f = 0 (1, 0) A 2 ω = 0 8 ∂ 2 f ∂q 2 1 = 0 (0, 0) A 2 ω = 0 Table 2: Classification of 3-dimensional MAE G ω = {F ∈ G, F * ω = ω}
is the stabilizer. For n = 4, the dimension of the group G = Sp(8, R) is 36, and dimension of the effective 4-forms space is 42. Dimensions of some stabilizers have been computed in [9] (table 3)

Equation Stabilizer's dimension SLAG 15 hess(f ) = 1 15 Plebanski I 13 Plebanski II 14 Linear ≥ 16
Table 3: Stabilizers of some 4-dimensional MAE Moreover, generic effective forms have trivial stabilizer as it is explained in [26]. Hence, we need between 6 and 21 parameters to describe the quotient space around a Monge-Ampère equation.

Complex solutions and bieffective forms 2.1 Complex solutions

A complex structure J on manifold T * M is said to be compatible with symplectic form Ω, if

Ω J = Ω(J• , •) is a 2-form. The complex 2-form Θ J = Ω -iΩ J is then a complex symplectic form.
In Darboux coordinates, such a compatible complex structure writes as

J = A B C A t with          B t = -B, C t = -C A 2 + BC = -1 AB + BA t = 0 AC + CA t = 0
As in generalized complex geometry, there are two important families:

J = A 0 0 A t with A a complex structure on M and J = 0 θ -θ -1 0
with θ a symplectic form on M Definition. Let ∆ ω = 0 be a MAE on M and let J be a compatible complex structure on T * M . A J-complex solution is a lagrangian and J-complex submanifold L of T * M on which ω vanishes:

Ω| L = 0 ; J L = L ; ω| L = 0
It is worth mentioning that the condition "complex lagrangian" is equivalent to the condition "real bilagrangian" as it is proved in [16].

Proposition (Hitchin). Let (N, Ω 1 + iΩ 2 ) be a complex symplectic manifold of complex dimension 2m. A real 2m-dimensional submanifold is a complex lagrangian submanifold if and only it is lagrangian with respect to Ω 1 and Ω 2 .

Example. Let us identify C 4 with H 2 endowed with the three complex structures I, J and K. We still denote by Ω the I -Kähler form, Ω J = Ω(J• , •) and

Ω K = Ω(K• , •).
Then Ω-iΩ J is a J-complex symplectic form and it is well known that every J-complex lagrangian submanifold is special lagrangian.

In the formalism of Monge-Ampère operators, this can be seen very simply. It is actually straightforward to check that the special lagrangian form writes as

Im(α) = Ω J ∧ Ω K and therefore, if Ω| L = 0 and Ω J | L = 0 then Im(α)| L = 0.

Bieffective forms

Two MAE ∆ ω = 0 and ∆ ω+θ1∧Ω+θ2∧Ω J = 0 have the same J-complex solutions. To understand this complex reduction, we need then to construct the bieffective part of ω. This is the goal of this section.

Let V be a complex symplectic space of real dimension 4m endowed with a complex symplectic form Θ = Ω 1 + iΩ 2 . Denote by Λ k (V * ) the space of real k-forms on V and Λ p,q (V * ) the space of (p, q)-complex forms, such that

Λ k (V * ) ⊗ C = p+q=k Λ p,q (V * )
Let us introduce for j = 1, 2 the operators ⊤ j and ⊥ j defined by

     ⊤ j θ = θ ∧ Ω j ⊥ j θ = ι XΩ j (θ) with X Ωj the unique bivector satisfying Ω j (X Ωj ) = 1
They have the following properties ( [25])

a) ⊥ j : Λ k (V * ) → Λ k-2 (V * ) is into for k ≥ 2m + 1 b) ⊤ j : Λ k (V * ) → Λ k+2 (V * ) is into for k ≤ 2m -1 c) [⊥ j , ⊤ j ](θ) = (2m -k)θ for θ ∈ Λ k (V * ).
A k-form θ is said to be Ω j -effective if ⊥ j θ = 0. For k = 2m, this is equivalent to

⊤ j θ = 0. Let H = [⊥ 1 , ⊤ 1 ] = [⊥ 2 , ⊤ 2 ] and M = [⊥ 2 , ⊤ 1 ]
. We get then the complete list of so-called Lichnerowicz operators which satysfy the following (see [5]):

Proposition (Verbitskii -Bonan). [⊥ 1 , ⊤ 1 ] = H [⊥ 2 , ⊤ 2 ] = H [⊥ 1 , ⊤ 2 ] = -M [⊥ 2 , ⊤ 1 ] = M [⊥ 1 , ⊥ 2 ] = 0 [⊤ 1 , ⊤ 2 ] = 0 [⊥ 1 , H] = -2⊥ 1 [⊥ 2 , H] = -2⊥ 2 [⊤ 1 , H] = 2⊤ 1 [⊤ 2 , H] = 2⊤ 2 [⊥ 1 , M ] = -2⊥ 2 [⊥ 2 , M ] = 2⊥ 1 [⊤ 1 , M ] = -2⊤ 2 [⊤ 2 , M ] = 2⊤ 1 [H, M ] = 0
We obtain then a representation of the Lie algebra sl(2,

C)⊗sl(2, C) on Λ * (V * )⊗C, defining      E 1 = 1 2 (⊥ 1 + i⊥ 2 ) F 1 = 1 2 (⊤ 1 -i⊤ 2 ) H 1 = 1 2 (H + iM )      E 2 = 1 2 (⊥ 1 -i⊥ 2 ) F 2 = 1 2 (⊤ 1 + i⊤ 2 ) H 2 = 1
2 (H -iM ) Representation theory of Lie algebras gives us the existence and uniqueness of bieffective part of a 2m-form.

Theorem 1 ([4]

). Every 2m-form ω ∈ Λ 2m (V * ) can be decomposed into a sum

ω = ω 0 + ω 1 ∧ Ω 1 + ω 2 ∧ Ω 2
with ω 0 bieffective, that is ω 0 ∧ Ω 1 = 0 = ω 0 ∧ Ω 2 . Moreover the bieffective part ω 0 is unique.

Proof. From Weyl's theorem, we know that Λ * (V * )⊗C decomposes as a unique direct sum of irreducible subspaces. Let W such an irreducible subspace. Since H 1 and H 2 commute, they admit a common eigenvector x ∈ W . But E p 1 E q 2 x is also a common eigenvector. Their exist then p and q such that z = E p 1 E q 2 x is a primitive vector, that is E 1 z = E 2 z = 0. Therefore, W = Gz and every vector w in W writes as

w = j,k a jk F j 1 F k 2 z
We deduce that every 2m-form can be uniquely decomposed into a finite sum

ω = j,k (α jk + iβ jk ) ∧ (Ω 1 + iΩ 2 ) j ∧ (Ω 1 -iΩ 2 ) k
with α jk and β jk primitive. Noting now that 2m-primitive forms are the bieffective forms we obtain the result.

We give now an explicit formula for 4m = 8, which can be easily implemented on a computer. This action is completely described by Hong in [17]. Let us briefly explain this result. Let Q be a hermitian matrix and define the canonical form of Q as a direct sum of three hermitian matrices:

J(Q) = H P (Q) ⊕ K N (Q) ⊕ K C (Q) ,
which are obtained from Jordan blocks of Q as follows: a) H P (Q) = H m1 (λ 1 ) ⊕ . . . ⊕ H mp (λ q ), where all λ i ≥ 0 and λ 2 i are the positive eigenvalues of

QQ t . b) K N (Q) = K 2n1 (µ 1 ) ⊕ . . . ⊕ H 2nr (µ r )
, where all µ i > 0 and -µ 2 i are the negative eigenvalues of QQ t . c) K C (Q) = L 2k1 (ξ 1 ) ⊕ . . . ⊕ L 2ks (ξ s ), where ξ 2 i are the non real eigenvalues of QQ t , with

K 2n (µ) = 0 -iH n (µ) iH n (µ) 0 , L 2k (ξ) = 0 H k (ξ) H ⋆ k (ξ) 0 and 2H m (λ) =           0 0 . . . 0 1 2λ 0 . . . 0 1 2λ 1 . . . . . . 1 2λ 1 0 . . . 1 2λ 1 . . . . . . 1 2λ 1 . . . 0 0 2λ 1 0 . . . 0 0           + i           0 1 0 . . . 0 0 -1 0 1 . . . 0 0 . . . -1 0 1 . . . . . . . . . . . . -1 0 1 0 0 . . . 0 -1 0 1 0 0 . . . 0 -1 0          
Theorem (Hong). Let Q be a hermitian matrix. Then there exists F complex orthogonal and ε = (ε 1 , . . . , ε p )

with ε i = ±1 such that F t QF = J ε (Q) with J ε (Q) = H ε P (Q)⊕K N (Q)⊕K C (Q) = ε 1 H m1 (λ 1 )⊕. . .⊕ε p H mp (λ q ) ⊕K N (Q)⊕K C (Q)
It is therefore difficult to give a complete classification of all possible complex reductions. Nevertheless, to characterize the orbit of a bieffective form ω, it is necessary to know

a) the signature ε(Q ω ) of Q ω , b) the spectrum of Q ω Q t ω
. and it will be sufficient for the examples we are interested in.

The complex lagrangian grassmannian

Denote by Gr ω the set of all complex lagrangian planes of the complex symplectic space (V, Θ) on which vanishes the bieffective form ω.

The complex isomorphism Θ : V → Λ 1,0 (V * ) transforms a complex basis of such a plane L into a decomposable effective (2, 0) forms θ L ∈ Λ 2,0 0 (V * ). Note that θ L is decomposable if and only if θ L ∧ θ L = 0.

Moreover, the condition ω| L = 0 is equivalent to the condition ω ∧ θ L ∧ θ L = 0.

Proposition. The grassmannian Gr ω identifies with the real algebraic subvariety of P 4 (C) = P (Λ 2,0 0 ) defined by θ ∧ θ = 0 and Q ω (θ) = 0.

Remark. This grassmannian could be empty, for example if Q ω is positive-definite. It could happen therefore that the Monge-Ampère equation ∆ ω = 0 has no generalized complex solution.

3 Some examples in dimension 4

3.1 The choice of the complex structure

The crucial point in this method is the choice of the compatible complex structure, which should depend on the initial Monge-Ampère equation. We choose here five simple complex structures, and give a corresponding complex Darboux coordinates system (z 1 , z 2 , u 1 , u 2 ) in which

Θ J = Ω -iΩ J = dz 1 ∧ du 1 + dz 2 ∧ du 2
The initial coordinate systems on T * R 4 is still (q, p) with

Ω = dq 1 ∧ dp 1 + dq 2 ∧ dp 2 + dq 3 ∧ dp 3 + dq 4 ∧ dp 4 .
We define

A =     0 -1 0 0 1 0 0 0 0 0 0 -1 0 0 1 0     ; Ã =     0 -1 0 0 1 0 0 0 0 0 0 1 0 0 -1 0     ; A 2 =     1 -2 0 0 1 -1 0 0 0 0 1 -2 0 0 1 -1     and J = A 0 0 A t z 1 = q 1 + iq 2 u 1 = p 1 -ip 2 z 2 = q 3 + iq 4 u 2 = p 3 -ip 4 K = 0 A A 0 z 1 = q 1 + ip 2 u 1 = iq 2 + p 1 z 2 = q 3 + ip 4 u 2 = iq 4 + p 3 J = Ã 0 0 Ãt z 1 = q 1 + iq 2 u 1 = p 1 -ip 2 z 2 = q 3 -iq 4 u 2 = p 3 + ip 4 K = 0 Ã Ã 0 z 1 = q 1 + ip 2 u 1 = iq 2 + p 1 z 2 = q 3 -ip 4 u 2 = -iq 4 + p 3 J 2 = A 2 0 0 A t 2 z 1 = q 1 + (-1 + i)q 2 u 1 = (1 -i)p 1 -ip 2 z 2 = q 3 + (-1 + i)q 4 u 2 = (1 -i)p 3 -ip 4

Examples

We study now the special lagrangian equation, the two real Monge Ampère equations hess f = ±1, the two Plebanski equations and the Grant equation. The corresponding effective forms on (T * R 4 , Ω) are:

ω SLAG = Im dq 1 + idp 1 ) ∧ (dq 2 + idp 2 ) ∧ (dq 3 + idp 3 ) ∧ (dq 4 + idp 4 ) ω H+ = dp 1 ∧ dp 2 ∧ dp 3 ∧ dp 4 -dq 1 ∧ dq 2 ∧ dq 3 ∧ dq 4 ω H-= dp 1 ∧ dp 2 ∧ dp 3 ∧ dp 4 + dq 1 ∧ dq 2 ∧ dq 3 ∧ dq 4 ω P I = dq 1 ∧ dq 2 ∧ dp 1 ∧ dp 2 -dq 1 ∧ dq 2 ∧ dq 3 ∧ dq 4 ω P II = dq 1 ∧ dq 2 + dq 3 ∧ dp 2 + dq 1 ∧ dq 2 ∧ dq 4 ∧ dp 1 + dq 3 ∧ dq 4 ∧ dp 1 ∧ dp 2 ω G = dq 2 ∧ dq 3 ∧ dq 4 ∧ dp 1 -dq 1 ∧ dq 3 ∧ dp 1 ∧ dp 3
We compute for the five compatible complex structures defined above the bieffective part of these forms and the signature of the hermitian form associated with. These invariants are given in table 5 and have to be compared with invariants for simple complex equations given in table 4.

In table 4, φ is a holomorphic function in (z 1 , z 2 ) and we note

φ jk = ∂ 2 φ ∂z j ∂z k . ε(Q ω ) ∆ ω = 0 spectrum(Q ω Q t ω ) |φ 11 | 2 = 0 (0,0,0,0,0) (1,0) |φ 12 | 2 = 0 (1,0,0,0,0) |φ 11 | 2 = 1 (0,0,0,0,0) (1,1) |φ 12 | 2 = 1 (1,0,0,0,0) |φ 11 | 2 -|φ 22 | 2 = 0 (-1,-1,0,0,0) |φ 11 | 2 + |φ 12 | 2 = 0 (1,0,0,0,0) (2,0) |φ 11 | 2 + |φ 22 | 2 = 0 (1,1,0,0,0) |φ 11 | 2 + |φ 12 | 2 = 1
(1,0,0,0,0) (2,1) φ 12 + φ 12 -|φ 11 | 2 = 0 (0,0,0,0,0) Table 4: Simple complex Monge-Ampère equations in complex dimension 2

Equation J K J K J 2 SLAG 0 0 0 0 (1,1) hess(f ) = 1 (1,1) 0 (1,1) 0 (1,1) hess(f ) = -1 (2,0) (3,2) (2,0) (3,2) (2,0) Plebanski I (2,0) (3,2) (1,1) (3,2) (2,0) Plebanski II (2,1) (3,2) (1,0) (3,2) (2,1) Grant (3,2) (3,2) (3,2) (3,2) (3,2)
Table 5: Invariants for some Monge-Ampère equations in dimension 4

We see in table 5 that this method fails for the Grant equation, at least for this choice of complex structures: signature is always (3,2) and the corresponding complex equation is therefore fully non degenerate. We study the other cases in more details.

Special lagrangian equation

As, we have seen the special lagrangian form ω SLAG has zero bieffective part for J and for K since ω SLAG = Ω J ∧ Ω K This is the best situation, since every complex lagrangian submanifold is solution. We note that this is the same for J and K and more generally for any complex structure J = F -1 JF with F in SU (4), since

ω SLAG = F * ω SLAG = F * Ω J ∧ F * Ω K = Ω F -1 JF ∧ Ω F -1 KF
Nevertheless, it does not give new solutions: we already know that the action of F ∈ SU (n) transforms a special lagrangian submanifold into an other special lagrangian submanifold. This is the reason of the choice of J 2 : we were looking for a simple complex structure which is not in so( 8) in order to construct other examples of solutions.

In the complex Darboux coordinates system, the bieffective part is where (Z 1 , Z 2 , U 1 , U 2 ) is the complex Darboux coordinates system

ω BE SLAG = 1 8 (1 + 2i)dz 1 ∧ dz 2 ∧ dz 1 ∧ du 2 + (-1 -2i)dz 1 ∧ dz 2 ∧ dz 2 ∧ du 1 + (1 -2i)dz 1 ∧ du 2 ∧
Z 1 = αz 1 + α -1 u 1 i √ 2 ; U 1 = αz 1 -α -1 u 1 i √ 2 Z 2 = αz 2 -α -1 u 2 √ 2 ; U 2 = αz 2 + α -1 u 2 √ 2 α 2 = 1 + 2i √ 5
We obtain then the following result: Then F -1 (L φ ) is special lagrangian in (T * R 4 , Ω, ω SLAG ) where F (q, p) = (z, u) with

z 1 = αq 1 + (-1 + i)αq 2 + (1 -i)α -1 p 1 -iα -1 p 2 i √ 2
z 2 = αq 3 + (-1 + i)αq 4 -(1 -i)α -1 p 3 + iα -1 p 4 √ 2
u 1 = αq 1 + (-1 + i)αq 2 -(1 -i)α -1 p 1 + iα -1 p 2 i √ 2
u 2 = αq 3 + (-1 + i)αq 4 + (1 -i)α -1 p 3 -iα -1 p 4 √ 2

Real Monge-Ampère equations

For every holomorphic function φ = f + ig : C 2 → C, we have hess R f = | hess C φ| 2 so for J, J and J 2 , which come from complex structures on R 4 , the complex reduction of hess f = ±1 are | hess φ| 2 = ±1 which are equivalent to

∂ 2 ψ ∂z 2 1 2 = ± ∂ 2 ψ ∂z 2 2 2
For example, for J, the corresponding symplectomorphism is the partial Legendre transform G(z 1 , z 2 , u 1 , u 2 ) = (u 1 , z 2 , -z 1 , u 2 )

We obtain then the following results:

Proposition 2. Let φ be a holomorphic solution of

∂ 2 φ ∂z 2 1 2 - ∂ 2 φ ∂z 2 2 2 = 0
Then G -1 (L φ ) is a generalized solution of hess f = 1 with G(q, p) = (p 1 -ip 2 , q 3 + iq 4 , -q 1 -iq 2 , p 3 -ip 4 ) Proposition 3. Let φ a holomorphic function of the form φ(z 1 , z 2 ) = a(z 1 )b(z 2 ). Then G -1 (L φ ) is a generalized solution of hess f = -1.

Moreover, we see in table 5 that ω H+ has no bieffective part for K and K. We get immediately Proposition 4: This result is underlying the strong relationship there is between special lagrangian geometry and "real special lagrangian" geometry or "split special lagrangian geometry", as described in [13].

Plebanski equations

Computing the bieffective part of ω P I , we obtain that the J-complex reduction of Plebanski I equation is Then its real part is a regular solution of Plebanski I equation. Then its real part is a regular solution of Plebanski II equation.

Conclusion

Studying geometry of 4-bieffective forms on R 8 , we have reduced important equations in Physic to simple -but non empty -complex equations. This has been done for arbitraries complex structures. It would be interesting now, for a given equation in four variables, to study all possible compatible complex structures and to parameterize in this way analytical solutions by pair of compatible complex structures on R 8 , and holomorphic functions on C 2 . Note that bieffective part still exists for 2m-forms on R 4m and such a parametrization should also exist for equations with 2m variables.

A more global approach would be also interesting. We have considered only complex structures with constant coefficients but our decomposition theorem remains valid on a complex symplectic manifold. This suggests to understand "Monge-Ampère calibrations" on complex symplectic manifolds as a generalization of the special lagrangian calibration on HyperKähler manifolds.

Proposition 1 .

 1 Let φ be a holomorphic solution of let L φ be the submanifoldL φ = (z 1 , z 2 , ∂φ ∂z 1 , ∂φ ∂z 2 ) ⊂ C 4 .

Proposition 4 .

 4 Any complex lagrangian surface in (T * R 4 , Ω, K) or (T * R 4 , Ω, K) is a generalized solution of hess f = 1.

|φ 11 | 2 = - 1 whichProposition 5 . 1 Proposition 6 .

 21516 is equivalent up the partial Legendre transform G to|ψ 11 | 2 + |ψ 12 | 2 = 0 Let φ a holomorphic function of the form φ(z 1 , z 2 ) = az 1 + b(z 2 ). Then G -1 (L φ ) is a generalized solution of Plebanski I equation.Similarly, the J-complex reduction is|φ 12 | 2 = Let φ a holomorphic function on (R 4 , Ã) of the form φ(z 1 , z 2 ) = z 1 z 2 + a(z 1 ) + b(z 2 ).

Finally, same computationsProposition 7 .Proposition 8 .

 78 give analog results for Plebanski II equation: Let φ a holomorphic function on (R 4 , A) of φ 12 + φ 12 + |φ 11 | 2 = 0 Then its real part is a regular solution of Plebanski II equation. Let φ a holomorphic function on (R 4 , Ã) of the form φ(z 1 , z 2 ) = a(z 2 ) + b(z 2 )z 1 .

Table 2 :

 2 Classification of 3-dimension MAEG ω = {F ∈ G, F * ω = ω} is the stabilizer. For n = 4, the dimension of the group

	G = Sp(8, R) is 36, and dimension of the effective 4-forms space is 42. Dimensions of
	some stabilizers have been computed in [9] (table 3)
	Equation	Stabilizer's dimension
	SLAG	15
	hess(f ) = 1	15
	Plebanski I	13
	Plebanski II	14
	Linear	≥ 16
	Table 3: Stabilizers of some 4-dimensional MAE
	Hence, generic orbits are 21-dimensional in a 42-dimensional space and 21 param-
	eters are needed to describe the quotient space.

  dz 1 ∧ dz 2 + (1 + 2i)dz 1 ∧ du 2 ∧ du 1 ∧ du 2 + (-1 + 2i)dz 1 ∧ du 1 ∧ dz 1 ∧ dz 2 + (-1 -2i)dz 2 ∧ du 1 ∧ du 1 ∧ du 2 + (1 -2i)du 1 ∧ du 2 ∧ dz 1 ∧ du 2 + (-1 + 2i)du 1 ∧ du 2 ∧ dz 2 ∧ du 1 ∧ dU 2 ∧ dZ 1 ∧ dU 2 -dZ 2 ∧ dU 1 ∧ dZ 2 ∧ dU 1

	and this is straightforward to check that
	ω BE SLAG =	√ 5 4	dZ 1

LMAM, Université de Bretagne Sud, Centre Yves Coppens, Campus de Tohannic BP 573, 56017 VANNES, FRANCE email:bertrand.banos@univ-ubs.fr

Acknowledgements

I am very grateful to Volodya Roubtsov for his helpful comments and suggestions.

Acknowledgements

I am very grateful to Volodya Rubtsov and Jenya Ferapontov for their helpful comments and suggestions.

Proposition. In dimension 8, the bieffective par ω 0 of a 4-form ω is

where

Proof. We know that

with ω 0 , ω 1 and ω 2 primitive. Using Verbitskii-Bonan relations, we obtain

Starting from ⊥ 2 ω we obtain also

Therefore,

and therefore

Action of the complex symplectic group

For simplicity, we restrict now to complex dimension 4: V is a 4-dimensional complex vector space endowed with a complex symplectic form Θ = Ω 1 + iΩ 2 . The space of real bieffective 4-forms is

Let Λ 2,0 0 (V * ) be the 5 -dimensional complex vector space of (2, 0)-complex forms which are effective with respect to Θ = Ω 1 + iΩ 2 :

The exterior product is non degenerate on Λ 2,0 0 (V * ) and it defines a non degenerate symmetric inner product

We are going to identify bieffective forms with hermitian forms on Λ 2,0 0 (V * ). Definition. The hermitian form Q ω on C 5 = Λ 2,0 0 (V * ) associated with a bieffective 4-form ω is:

Theorem 2. The map

Moreover, the group Sp(4, C)/Z 2 identifies with SO(5, C) and its action on Λ 4 BE (V * ) is the Hermite action of SO(5, C) on su (5).

Proof. We prove first that Λ 4 BE (V * ) = Λ 2,0 0 (V * ) ⊗ Λ 2,0 0 (V * ). Let ω ∈ Λ 4 BE (V * ) and consider its decomposition ω = ω 40 + ω 31 + ω 22 + ω 13 + ω 04 with ω pq ∈ Λ p,q (V * ). Since ⊥ C : Λ p,q (V * ) → Λ p-2,q (V * ) is into for p ≥ 3 and ⊥ C : Λ p,q (V * ) → Λ p,q-2 (V * ) is into for q ≥ 3, and since

We deduce that Λ 4 BE (V * ) and su(5) have same dimension. Since ω → Q ω is injective, this is an isomorphism. Now, the action of Sp(4, C) preserves this symmetric product, with kernel Z 2 . Since dim C (Sp(4, C)) = 10 = dim C (SO(5, C)), we deduce that Sp(4, C)/Z 2 = SO(5, C).

Moreover, we have


Un algorithme distribu é d' énum ération des noeuds d'un r éseau et application au calcul des distances entre 2 noeuds quelconques et du diam ètre d'un r éseau Y. Métivier, J.M. Robson, et A. Zemmari Université de Bordeaux -Bordeaux INP, LaBRI, UMR CNRS 5800 351 cours de la Libération, 33405 Talence, France {metivier, robson, zemmari}@labri.fr On considère un réseau où les noeuds (processus) communiquent par passage de messages. On suppose qu'il existe un noeud distingué. Dans ce travail, nous proposons et analysons un algorithme distribué d'énumération des noeuds d'un réseau utilisant des messages de taille O(1) et tel que si deux noeuds u et v ont des numéros consécutifs alors ils sont à distance au plus 3. Nous montrons que l'algorithme se termine en O(n) rondes, où n est le nombre de noeuds du réseau ; la taille des messages étant O(1), la complexité en bits (nombre maximum de bits transitant par un canal de communication) est O(n). Comme application de cette énumération, on présente un algorithme permettant de calculer les plus petites distances entre 2 noeuds quelconques d'un réseau et un algorithme permettant de

The Model and the Problem

We consider the standard message passing model for distributed computing. The communication model consists of a point-to-point communication network described by a connected graph G = (V (G), E(G)) (= (V, E) for short) where the vertices V represent network processes and the edges E represent bidirectional communication channels. Processes communicate by message passing: a process sends a message to another by depositing the message in the corresponding channel. In the sequel, we consider only connected graphs. We assume the system is fully synchronous, namely, all processes start at the same time and time proceeds in synchronised rounds. Time Complexity. A round (cycle) of each process is composed of the following three steps: 1. Send messages to (some of) the neighbours, 2. Receive messages from (some of) the neighbours, 3. Perform some local computation. As usual the time complexity is the number of rounds needed until every vertex has completed its computation. Bit Complexity. By definition, in a bit round each vertex can send/receive at most 1 bit from each of its neighbours. The bit complexity of algorithm A is the number of bit rounds to complete algorithm A.

Network and Processes Knowledge. The network G = (V, E) is anonymous: unique identities are not available to distinguish the processes. We only assume that there is an elected (a distinguished) vertex denoted Leader. We do not assume any global knowledge of the network, not even its size or an upper bound on its size. The processes do not require any position or distance information. Each process knows from which channel it receives or to which channel it sends a message, thus one supposes that the network is represented by a connected graph with a port numbering function.

All Pairs Shortest Paths, Diameter, Girth, Cut-Edge and Cut-Vertex. A walk in a graph G = (V, E) is a finite alternating sequence of vertices and edges, beginning and ending with a vertex and where each edge is incident with the vertices immediately preceding and following it. A trail is a walk in which no edge occurs more than once. A path is a trail in which all vertices are different, except that the initial and final vertices may be the same. A walk with at least 3 edges in which the first and last vertices are the same but all other vertices are distinct is called a cycle.

Definition 1 Let G = (V, E) be a connected graph, let u, v ∈ V .
The distance dist G (u, v) between u and v in G, is the length of a shortest path between u and v in G.

The eccentricity of v is the greatest distance from v to any other vertex.

The diameter of G, denoted D(G), is the maximum distance between any two vertices of G.

The girth of G is the length of a shortest cycle of G.

A cut-vertex is a vertex whose removal increases the number of connected components.

A cut-edge is an edge whose removal increases the number of connected components.

The all pairs shortest paths (APSP for short) problem in G is to compute the length of shortest paths between any pair of vertices in G.

A tree is a connected acyclic graph. A rooted tree is a tree in which one of the vertices is distinguished from the others (called Leader in this work). A spanning-tree of a connected graph

G = (V, E) is a tree T = (V, E ′ ) such that E ′ ⊆ E.

Our Contribution

We present a distributed enumeration algorithm, denoted DEA, which assigns to each vertex of a graph G of size n having a distinguished vertex, denoted Leader, a unique integer of {1, 2, . . ., n} such that the distance between any two vertices having two consecutive numbers is at most 3. This algorithm uses messages of size O(1) and has a time complexity equal to O(n).

The steps of Algorithm DEA are:

1. computation of a Breadth-First-Search (BFS) spanning-tree of G whose root is Leader;

2. enumeration of the vertices with respect to a special traversal of the BFS spanning-tree.

In the following, we explain how the nodes of the BFS spanning-tree are numbered (we skip the BFS construction for the lack of place). This algorithm is denoted Trav and can be described as follows:

We add a loop on each leaf of the BFS spanning-tree and vertices are visited twice in a Depth-First-Search (DFS) traversal (a leaf is visited on arriving and by following the loop).

The traversal Trav is defined by:

• if it is the first visit to a vertex then go to the first child of the vertex if it has a child; if it has no child (i.e., it is a leaf) go from the leaf to itself;

• if it is the second visit to a vertex go to the next sibling of the vertex (two vertices v and w are siblings if they have the same parent), if it has a next sibling; if it has no next sibling go to the parent of the vertex if it has a parent else stop since it is the root of the BFS spanning-tree and the traversal is finished.

Thus, a visit to v is immediately followed by a visit to a child or to a sibling (through the parent) or to the parent of v or to v itself (if v is a leaf). Let v be a vertex, ν

v (resp. ν

(2) v ) denotes the number of vertices visited before the first visit to v (resp. before the second visit to v). We can prove (by induction) that ν

(1) v is even iff the level of v is even and ν

(2) v is even iff the level of v is odd. The number of a vertex v is obtained by computing the number of visited vertices during the tree traversal before the first or the second visit to v. More precisely: the vertex numbered k is the k th visited vertex such that an even number of vertices have been visited before it; it is denoted v k . An example of a run of Trav and the numbering of vertices is given in Fig. 1.

This enumeration enables the initialisation of anonymous waves, with respect to the enumeration order, i.e., the first wave is initialised by the vertex numbered 1, the second wave by the vertex numbered 2, etc. Anonymous waves reach vertices with respect to the enumeration order (i.e., the wave initialised by the vertex numbered i reaches any vertex after the wave initialised by the vertex numbered i -1 and before the wave initialised by the vertex numbered i + 1) and thus are implicitly identified. This fact allows each vertex to compute its distance to any vertex without the computation and the use of the distance itself but by inference from the time; in this way all pairs shortest paths are obtained in time O(n), each message having a constant size so that the bit complexity is O(n). We deduce also a distributed algorithm for graph diameter with a bit complexity and a time complexity equal to O(n). The diameter is obtained by the following steps:

1. Breadth-First-Search Tree Computation initiated by Leader;

2. Numbering of vertices;

3. Calculating distance between Leader and each vertex;

4. Waves initiation and all pairs shortest paths calculation;

5. Centralisation of the maximum distance and broadcast of the diameter.

We can note that only the maximum distance between any two nodes needs to be known, which is independent of node names. Furthermore, Frischknecht et al. proved ([FHW12], Theorem 5.1) that: "For any n ≥ 10 and B ≥ 1 and sufficiently small ε any distributed randomized ε-error algorithm A that computes the exact diameter of a graph requires at least Ω(n/B) time for some n-node graph even when the diameter is at most 5," where B is the size of messages. From this result we deduce that the bit complexity of our algorithm is optimal and the time complexity is also optimal for messages of size O(1) The waves initiated by vertices with respect to the numbering of vertices can be used also for computing girth and for the determination of cut-edges and cut-vertices.

We may wonder whether it is possible to obtain an enumeration of vertices such that the distance between any two vertices having two consecutive numbers is at most 2. We explain now why the answer is negative. It indicates that in some certain sense our enumeration is optimal.

The enumeration of vertices of a connected graph such that two consecutive vertices of the enumeration are at distance at most 3 is also presented in [START_REF] Sekanina | On an algorithm for ordering of graphs[END_REF]; this paper presents a sequential algorithm for computing such an enumeration.

Let G be a graph. A Hamiltonian path in G is a path that includes all the vertices of G.

We recall that the cube of a graph G, denoted G 3 , is the graph with the set of vertices of G in which there is an edge between two vertices u and v if the distance between u and v in G is at most 3. It was noticed by C. Gavoille [Gav14] that the existence of such an enumeration is equivalent to the fact that the cube of a connected graph G contains a Hamiltonian path.

A cycle containing all vertices of G is called a Hamiltonian cycle of G, and G is called a Hamiltonian graph. From our enumeration result we deduce a well known result [START_REF] Chartrand | The cube of every connected graph is 1-hamiltonian[END_REF]:

Theorem 1 If G is a connected graph then G 3 is a Hamiltonian graph.
As for the cube of a graph, the square of a graph G, denoted G 2 , is the graph with the set of vertices of G in which there is an edge between two vertices u and v if the distance between u and v in G is at most 2. The previous theorem is no longer true for the square of a tree as indicated by the next theorem. Let K 1,3 be the tree with one internal vertex and three leaves. Let S(K 1,3 ) be the subdivision of K 1,3 formed by inserting a vertex of degree two on each edge of K 1,3 . A graph G ′ = (V ′ , E ′ ) is called a subgraph of a graph G = (V, E) if V ′ ⊆ V , E ′ ⊆ E and V ′ contains all the endpoints of the edges in E ′ . Regarding the characterisation of trees with Hamiltonian square, Harary and Schwenk [START_REF] Harary | Trees with hamiltonian square[END_REF] proved that: Theorem 2 Let T be a tree with at least 3 vertices. T 2 is a Hamiltonian graph if and only if T does not contain S(K 1,3 ) as a subgraph.

In fact for our work, a priori, we only need a Hamiltonian path. In [START_REF] Radoszewski | Hamiltonian paths in the square of a tree[END_REF], it is proved that the square of a tree T has a Hamiltonian path if and only if T is a horsetail. The definition of a horsetail is rather technical thus we do not provide it; in our context the important fact is that the family of trees which are not horsetails is infinite. Finally, it is not possible to obtain an enumeration of vertices such that the distance between any two vertices having two consecutive numbers is at most 2.

  calculer le diamètre d'un réseau en un temps O(n) avec des messages de taille O(1) ce qui donne une complexité en bits linéaire en le nombre de noeuds du réseau. Ces résultats améliorent les résultats obtenus par Peleg et al. (ICALP 2012) et Wattenhofer et al. (SODA 2012) qui présentent des algorithmes pour le calcul des plus petites distances entre deux noeuds quelconques ou le diamètre en temps O(n) avec des messages de taille O(log n). Enfin nous montrons qu'il n'existe pas d'énumération des noeuds telle que deux noeuds ayant des numéros consécutifs sont à distance au plus 2. Une présentation complète de ce travail peut être consultée dans [MRZ16].

Fig. 1 :

 1 Fig. 1: An example of a run of Trav with the associated numbering of the vertices.

  This table summarises the comparison between the complexities of various diameter algorithms and the complexities of the diameter algorithm presented in this work.

		Time	Message size (number of bits)	bit complexity
	Almeida et al. [ABC11]	O(D)	O(n log n)	O(Dn log n)
	Holzer and Wattenhofer [HW12]	O(n)	O(log n)	O(n log n)
	Peleg et al. [PRT12]	O(n)	O(log n)	O(n log n)
	This paper	O(n)	O(1)	O(n)


Abhijit L Chavda

L K Chavda

Quantized gravitational waves and a

come

I. INTRODUCTION

Over the past three decades, compelling theoretical and observational evidence has built up that the visible matter in the universe makes up a very small fraction of the total matter composition of the universe [START_REF] Gelmini | TASI 2014 Lectures: The Hunt for Dark Matter[END_REF]. Galaxies are observed to have dark matter halos which contain far more matter than their visible regions [START_REF] Combes | [END_REF][3][START_REF] Burkert | The structure of dark matter halos. Observation versus theory[END_REF]. Most galaxies are not dominated by dark matter inside their optical disks, which suggests that the dark matter has properties that segregate it from visible matter.

According to the recent observational results obtained from the Planck satellite, the universe is composed of 68.3% dark energy, 26.8% dark matter, and 4.9% visible matter [START_REF] Ade | [END_REF][START_REF] Ade | Planck 2015 results. XIII. Cosmological param-eters[END_REF][START_REF] Ade | Planck 2015 results. XX. Constraints on inflation[END_REF]. Consequently, discovering the nature and identity of dark matter and dark energy has become one of the most challenging problems facing modern cosmology.

Several candidates for dark matter particles have been proposed, which include WIMPs [START_REF] Bernabei | [END_REF], Standard Model neutrinos [9,10], Sterile neutrinos [9,11], Axions [9,12], Supersymmetric particles [9,13,14], dark matter from Little Higgs models [START_REF] Arkani-Hamed | [END_REF][17][18], Kaluza-Klein dark matter [19,20], WIMPZILLAs [START_REF] Kolb | Proceedings of the 2nd International Conference on Dark Matter in Astro and Particle Physics (DARK98)[END_REF], Cryptons [START_REF] Ellis | [END_REF]23], primordial black holes [START_REF] Dolgov | Gravitational waves, baryogenesis, and dark matter from primordial black holes[END_REF][START_REF] Blais | [END_REF][26],

super-heavy X-particles [27][28][29], and Holeums [30], among others. Dark energy, which has repulsive gravitational interactions, is hypothesized to explain observations that indicate the accelerating expansion of the universe. Candidate theories for dark energy include the cosmological constant, which is a constant energy density that permeates space homogeneously, and dynamic scalar fields such as quintessence and moduli, whose energy densities are variable in space and time, among others [31][32][33].

The cold dark matter (CDM) model is the leading theory of dark matter and galaxy formation in the universe [START_REF] Peter | Proceedings of Science (Bash11)[END_REF][START_REF] Garrett | [END_REF]. Computer simulations using this model predict that structures in the universe grow hierarchically through gravitational instability, with the smallest structures forming first, followed by galaxy-sized structures and finally by galactic cluster-sized structures [36]. If dark matter is made up of stable, nonrelativistic, purely gravitating subatomic particles, then inhomogeneities in their distributions such as galactic dark matter halos should cause the formation of large-scale self-gravitating dark matter structures.

Consider one such self-gravitating system composed of a very large number of dark matter particles. As it stabilizes over time, it should assume a spherical shape, and its constituent dark matter particles' gravitational interaction with the system's mass should lead to the formation of a composite bound state made up of all the particles in the system. Thus, over time, we expect dark matter structures to take the form of macroscopic gravitational bound states of large numbers of dark matter particles.

Let us examine the nature of the system. We are dealing with a large, classical-sized system which is composed of a very large number of subatomic particles. It may have a diameter of the order of centimeters, meters, or even hundreds of meters. Since the particles are extremely small, we can safely assume them to never collide, even when their number density is very large. Similarly, because the individual particles' masses are very small compared to the total mass of the system, we can assume the particle-particle gravitational interactions to be negligible compared to the interaction of each particle with the total gravity of the system. Since this is a self-gravitating system, none of the particles has sufficient momentum to escape the system's gravitational potential well. The particles describe orbits around the center of mass of the system. For the sake of simplicity, we assume the particles to orbit the system's center of mass at classical distances. In other words, we assume the number of particles that orbit the system's center of mass at very short, non-classical distances to be negligible.

We must address whether the system is classical or quantum mechanical in nature. For this, we note that in the absence of particle-particle gravitational interactions, each individual particle can be treated as being alone in the gravitational potential of the system. This is similar to the experiment of ultracold neutrons under the effect of the Earth's gravity [37,38], which is a good example of a quantum mechanical gravitational bound state of subatomic particles and a very large mass. Thus, since our system's constituent dark matter particles are of subatomic sizes, it is reasonable to expect their bound state, though macroscopic and classical in scale, to be quantum mechanical in nature.

Since we expect these bound states to form primarily in galactic dark matter halos, we call them DArk Compact Halo Objects (DACHOs).

There have been several studies of Massive Compact Halo Objects (MACHOs) in galactic halos. Microlensing experiments have determined that MACHOs in the mass range 0.6×10 -7 M ⊙ < M < 15M ⊙ are ruled out as the primary occupants of the Milky Way dark matter halo [39]. Therefore in this paper we examine the properties of DACHOs of small sizes, having radii ranging from 1 cm to 10 m in length and masses less than 10 -8 M ⊙ .

Let us consider a spherical system of N dark matter particles of masses µ 1 , µ 2 , µ 3 , ...µ N which are held together by the system's gravity (a self-gravitating system). Let the system's mass distribution be spherically symmetric and its radius be R. Here we take R to be the radius of orbit of the outermost dark matter particle. Let N ≫ 1 and let R be large enough that the system, as a whole, has classical, not quantum dimensions. We assume the particles to have non-relativistic velocities. This is a classical system, and each particle has a classical velocity which is such that the particle's distance from the system's center of mass never exceeds R.

The average mass of the particles that constitute the system is given by

μ = 1 N N i=1 µ i (1)
The total mass of the system is

M tot = N μ (2)
The gravitational potential acting upon a dark matter particle of mass µ located a distance r from the center of the mass M tot is given by

V (r) =            1 2 GM tot µ R 3 r 2 -3 2 GM tot µ R if r ≤ R - GM tot µ r if r ≥ R (3)
where G is the gravitational constant. For the sake of simplicity, consider M tot to be isolated, i.e. assume that there is no dark matter particle (or any other particle) at a distance greater than R from the center of the system M tot . In that case, the potential acting upon the i th particle of the system is

V (r i ) = 1 2 µ i ω 2 r 2 i -V 0 i (4)
where

ω = GM tot R 3 = 4 3 πGρ ( 5
)
where ρ is the system's mass density, and

V 0 i = 3 2 GM tot µ i R (6)
This consists of a constant attractive potential -V 0 i plus a repulsive isotropic harmonic oscillator potential 1 2 µ i ω 2 r 2 i . If we assume the dark matter particles to be distinguishable (µ 1 = µ 2 = . . . = µ N ) and non-interacting (we assume the particle-particle gravitational interactions to be negligible compared to the interaction of each particle with the total gravity of the system), then the Schrödinger equation of the N-particle system

Ĥψ n 1 l 1 ,n 2 l 2 ,...n N l N ( r 1 , r 2 , . . . r N ) = E n 1 l 1 ,n 2 l 2 ,...n N l N ψ n 1 l 1 ,n 2 l 2 ,...n N l N ( r 1 , r 2 , . . . r N ) (7)
separates into N one-particle equations [START_REF] Zettili | Quantum Mechanics: Concepts and Applications[END_REF] -

2 2µ i ∇ 2 i + V i ( r i ) ψ n i l i ( r i ) = E n i l i ψ n i l i ( r i ) (8)
with

E N = N i=1 E n i l i (9)
where E N is the system's total binding energy and

ψ n 1 l 1 ,n 2 l 2 ,...n N l N ( r 1 , r 2 , . . . r N ) = ψ n 1 l 1 ( r 1 ) ψ n 2 l 2 ( r 2 ) . . . ψ n N l N ( r N ) = N i=1 ψ n i l i ( r i ) (10)
The solutions of these equations yield the single-particle energies E n i l i .

III. THE DACHO BOUND STATE

Let ψ ( r, t) be the wave function of a dark matter particle of mass µ located a distance r from the center of the mass M tot of the system. The Hamiltonian of this system therefore writes

H = p 2 2µ + V ( r) (11)
The potential is given by

V ( r) = 1 2 µω 2 r 2 -V 0 0 ≤ | r| ≤ R ( 12
)
where ω is given by eq. ( 5) and

V 0 = 3 2 GM tot µ R (13)
This gives us the time-independent Schrödinger equation for the particle:

∇ 2 + 2µ 2 (E -V ( r)) ψ ( r) = 0 (14)
Assuming a spherically symmetric potential

V ( r) = V (| r|) = V (r) = V (15)
and using separation of variables in the form

ψ (r, θ, φ) = R (r) Y (θ, φ) (16)
we obtain two separated equations

1 R ∂ ∂R r 2 ∂R ∂r - 2µr 2 2 (V -E) = l (l + 1) (17)
and

1 Y sin θ ∂ ∂θ sin θ ∂Y ∂θ + 1 Y sin 2 θ ∂ 2 Y ∂φ 2 = -l (l + 1) (18)
where l (l + 1) is a constant term. The solutions to the angular equation ( 18) are the spherical harmonics:

Y m l (θ, φ) = (2l + 1) 4π (l -m)! (l + m)! P m l (cos θ) e imφ (19)
which are orthonormal:

π θ=0 2π φ=0 Y m l Y m ′ l ′ * sin θdθdφ = δ ll ′ δ mm ′ (20)
We can rewrite the radial function eq. ( 17) using total derivatives since R (r) depends only on r:

1 R d dR r 2 dR dr - 2µr 2 2 (V -E) = l (l + 1) (21)
Making the substitution

u (r) = rR (22)
and simplifying, we obtain

- 2 2µ d 2 u dr 2 + 1 2 µω 2 r 2 + 2 2µ l (l + 1) r 2 u = (E + V 0 ) u (23)
which has the solution

E = ω 2k + l + 3 2 -V 0 (24)
E = ω n + 3 2 -V 0 (25)
Let the average value of n be given by

n = 1 N N i=1 n i (26)
and the average attractive potential V0 be given by

V0 ≡ 1 N N i=1 V 0 i = 3 2 GM tot R μ ( 27
)
Then the system's total energy is given by

E N = N ω n + 3 2 -V0 (28)
The total mass of the bound state is given by

M N = M tot + E N c 2 (29)
which gives us

M N = M tot + N c 2 ω n + 3 2 -V0 (30)
IV. THE GRAVITATIONAL SPECTRUM When a particle in the bound state makes a transition from a higher energy state E 2 to a lower energy state E 1 , it emits quantized gravitational radiation of frequency given by

ν = E 2 -E 1 h = (n 2 -n 1 ) ω 2π ( 31
)
where E 2 and E 1 are both given by eq. ( 25). If the particle's transition from state E 2 to state E 1 causes the system's average principal quantum number to change from n2 to n1 , then the system emits quantized gravitational radiation of frequency given by Several results are presented in Table 1. Since the systems have classical sizes, we have considered very large values of the quantum number n. By varying the particle mass, particle number, and system radius, a wide range of bound state masses M N and gravitational wave frequencies ν, including frequencies in the Advanced LIGO range (∼ 1 to 10 3 Hz) are obtained. The largest DACHO masses M N obtained here are of the order of 10 -11 M ⊙ and are therefore not ruled out by the microlensing studies [39]. It is possible to obtain a much larger range of frequencies than shown here. We find that the system radius R is always greater than the Schwarzschild radius of the mass M N even for values of N approaching N th .

ν = (n 2 -n1 ) Nω 2π (32) μ N R (m) ν (Hz) M N (kg) R S (m) N th 1 MeV 1 ×
Therefore DACHOs are stable, non-black hole bound states.

VII. DISCUSSION AND CONCLUSIONS

We propose a new form of dark matter -DACHOs. If galactic dark matter halos consist of purely gravitating cold dark matter particles, DACHO formation should take place therein.

Since objects in the mass range 0.6 × 10 -7 M ⊙ < M < 15M ⊙ are ruled out as the primary occupants of the Milky Way dark matter halo [39], we propose that DACHOs have masses smaller than 0.6 × 10 -7 M ⊙ . They will therefore be too small to be detected by current microlensing technology.

Although DACHOs cannot be detected by current microlensing technology, detecting their gravitational wave spectrum is possible. Because DACHOs are made up of extremely large numbers of particles, transitions between quantum states should occur constantly, leading to continuous emission of gravitational waves. We therefore predict that galactic halos should be excellent sources of quantized gravitational waves, including those in the Advanced LIGO frequency range.

Our most interesting and significant result is a new quantum effect wherein gravitational bound states of very large numbers of particles can acquire negative masses. DACHOs acquire negative mass if the particle number N exceeds N th . This has a repulsive gravity effect like dark energy. The gravitational potential between a negative mass -M N and a positive mass m is

V = - G (-M N ) m r = + GM N m r (39)
This is a repulsive potential. A negative mass repels ordinary, positive masses and attracts other negative masses. We therefore propose that negative-mass DACHOs, if they exist, should be the source of at least a part of the universe's dark energy. If this is indeed true, it means that dark energy is a matter component that can cluster.

A system with an initial particle number N > N th will form a negative-mass bound state.

A system with an initial particle number N < N th can acquire a negative mass by accreting dark matter particles from the galactic dark matter halo until N > N th .

Consider a DACHO that has a radius of 10 m, an initial particle number N = 10 49 and whose particles have an average mass μ = 1 MeV. From table 1 we see that it has a mass M N = 1.78 × 10 19 and N th = 5.05 × 10 57 . Over time, the DACHO will accrete more dark matter particles from the galactic halo. As its particle number N increases, the DACHO's mass M N will reach a maximum, and then start decreasing as N approaches N th , eventually reaching nearly zero. As the DACHO's mass decreases, the rate at which it accretes dark matter particles will also decrease. Eventually, the DACHO's mass will reach so close to zero that the accretion of a single dark matter particle will cause it to acquire negative mass.

This will take a very long time to happen due to its very small mass. The resultant negativemass DACHO will have a minuscule negative mass. It will immediately be expelled from its galactic halo and will wander until it reaches a position of equilibrium with the positive and negative masses in the universe. Because of its very small negative mass, it will make a very small contribution to the universe's expansion.

On the other hand, a system with an initial particle number N ≫ N th will form a bound state with a large negative mass. We expect such DACHOs with large negative masses to form fairly early in the universe, possibly when the first galaxies and dark matter halos formed, a few 100 million years after the Big Bang. We expect the subsequent conversion of DACHOs into dark energy via accretion of dark matter particles to be a very slow process, possibly taking billions of years to build into a significant repulsive force capable of causing the universe's expansion. There also exists the possibility of the merger of two or more large DACHOs in galactic halos, which will result in the formation of a bound state with a significantly large negative mass. The frequency of such mergers will depend on the size and density of the halo.

The cosmological implications of the existence of negative-mass DACHOs are straightforward. If they exist, they will cause, or, at least, contribute to the expansion of the universe by repelling ordinary masses. They will be undetectable. Over time, as more DACHOs acquire negative mass, the net negative mass of the universe will increase, thereby accelerating the expansion of the universe. This will continue until the total amount of positive mass and negative mass in the universe become roughly equal. Once the negative mass content of the universe exceeds the positive mass content, the universe will start contracting again, possibly leading to the so-called "big crunch". Thus, unlike the cosmological constant -which stays constant, we predict the net repulsive force (dark energy) of negative-mass DACHOs in the universe to first slowly increase over time until equilibrium between positive and negative masses is reached, and then to slowly decrease over time as negative masses begin to predominate over positive masses.

  10 36 1 × 10 -2 1.73 1.78 × 10 6 2.64 × 10 -21 5.05 × 10 54 1 MeV 1 × 10 44 1 1.73 × 10 1 1.78 × 10 14 2.64 × 10 -13 5.05 × 10 56 1 MeV 1 × 10 49 1 × 10 1 1.73 × 10 2 1.78 × 10 19 2.64 × 10 -8 5.05 × 10 57 1 GeV 1 × 10 33 1 × 10 -2 1.73 1.78 × 10 6 2.64 × 10 -21 5.05 × 10 51 1 GeV 1 × 10 41 1 1.73 × 10 1 1.78 × 10 14 2.64 × 10 -13 5.05 × 10 53 1 GeV 1 × 10 46 1 × 10 1 1.73 × 10 2 1.78 × 10 19 2.64 × 10 -8 5.05 × 10 54 1 TeV 1 × 10 30 1 × 10 -2 1.73 1.78 × 10 6 2.64 × 10 -21 5.05 × 10 48 1 TeV 1 × 10 38 1 1.73 × 10 1 1.78 × 10 14 2.64 × 10 -13 5.05 × 10 50 1 TeV 1 × 10 43 1 × 10 1 1.73 × 10 2 1.78 × 10 19 2.64 × 10 -8 5.05 × 10 51 1 PeV 1 × 10 27 1 × 10 -2 1.73 1.78 × 10 6 2.64 × 10 -21 5.05 × 10 45 1 PeV 1 × 10 35 1 1.73 × 10 1 1.78 × 10 14 2.64 × 10 -13 5.05 × 10 47 1 PeV 1 × 10 40 1 × 10 1 1.73 × 10 2 1.78 × 10 19 2.64 × 10 -8 5.05 × 10 48

OF THE TOTAL MASS

From eq. [START_REF] Combes | [END_REF] we see that M tot varies as N and from eq. (28) we see that E N varies as ˜(-N 2 ). This indicates that there is a threshold value of N beyond which the total mass M N will be negative. For the bound state to have mass, we must have M N > 0, which gives us

or, equivalently,

We denote the threshold values of N and V0 respectively as

and

Thus, the bound state has negative mass if

or, equivalently, if V0 > Vth (38) The bound state has negative mass when the average attractive potential acting on the particles of the system is greater than the average of the particles' total energy. This is a new quantum effect, wherein the system's binding energy -E N c 2 exceeds its total mass M tot , causing the system to acquire a negative mass.

VI. BOUND STATE PROPERTIES

Table 1 Gravitational wave frequencies, bound state masses, Schwarzschild radii and N th values for a range of particle masses, particle numbers, and system radii. Here n 2 = R × 10 9 , n 1 = n 2 -1.


Cyrille Suire
email: cyrille.suire@univ-lr.fr

Axel Jean

Jean-Christophe Burie
email: jcburie@univ-lr.fr

User Activity Characterization in a Cultural Heritage Digital Library System

Keywords: H.3.7 [Information Storage and Retrieval]: Digital libraries Information Seeking, User Modeling, User Behavior, Cultural Heritage, Task Models, Humanities

Digital access to large amount of heterogeneous data can create methodological biases regarding the discovery and exploitation of resources, particularly when it comes to Social Sciences. In order to provide relevant adaptivity for social scientists, it is important to fully consider their research practice diversity. To do so, we consider an activity-based approach for researchers' information search behavior. We have also conducted an experiment in a Cultural Heritage use case. The main result shows us that social scientists have the same research behaviors as those observed in exact Sciences.

Paper accepted in JCDL16 Conference DOI: http://dx.doi.org/10.1145/2910896.2925459

CONTEXT AND MOTIVATION

Social scientists and humanists have an increasing number of web platforms to access heterogeneous resources. The new challenge for researchers is the discovery and exploitation of these data. The tools commonly used to access this huge amount of data can create methodological problems [START_REF] Milligan | Illusionary Order: Online Databases, Optical Character Recognition, and Canadian History, 1997-2010[END_REF]. Moreover, we know that the practices of social scientists are very diverse [START_REF] Audenaert | What Humanists Want: How Scholars Use Source Materials[END_REF]. They need to be analyzed and understood to avoid most methodological issues. To investigate this, we present an activity-based approach. Following Marchionini's conceptual framework [START_REF] Marchionini | Exploratory Search: From Finding to Understanding[END_REF], users' research activities can be distinguished in two main groups: lookup tasks and exploratory search tasks. A recent paper by Athukorala et. al. [START_REF] Athukorala | Is exploratory search different? A comparison of information search behavior for exploratory and lookup tasks[END_REF] has shown that it is possible to distinguish the type of task the users are engaged in, when they are using a scientific search engine. The purpose of our work is to determine if this stands in another context -using a digital library platform and experimenting with a different type of public.

HYPOTHESE

As proposal, we want to prove that user behavior evaluation based on activity type is valid even if several points of our approach differ from [START_REF] Athukorala | Is exploratory search different? A comparison of information search behavior for exploratory and lookup tasks[END_REF]. Indeed, our system is significantly different and is built around the following considerations: its design and implementation is closer to a digital library web portal than a search engine like Google Scholar. Considering the work of [START_REF] Agosti | Understanding user requirements and preferences for a digital library Web portal[END_REF] this kind of systems requires a specific evaluation. We implemented precise observers in our system to log users' behavior and compute features in two different ways. First, we focus on the first query iteration. A query is composed of every user events from the first input in the search field to the next one or to the end of the task. Then we focus on the overall task by averaging the values of each feature over all queries from the task.

• Query Length (f1): this feature is defined as the number of terms in the query. • Duration (f2): duration in seconds.

EXPERIMENT

We conducted a first experiment with forty master students who are a the beginning of their specialization in the Cultural Heritage field of study. They have a medium level of topic knowledge and are able to adapt their behavior to the task type. Our experimental corpus must be precise enough to allow participants to perform realistic tasks. It contains 240 documents related to the Cultural Heritage field of study which the students are relatively familiar with. The resources of the corpus are heterogeneous: general works, scientific articles, iconographic documents, multimedia resources and even primary documents.

To carry out our experiment, we chose to make the students perform tasks in each Marchionini's overall category [START_REF] Marchionini | Exploratory Search: From Finding to Understanding[END_REF]. These tasks were designed by a specialist of the Cultural Heritage field of study. We defined the tasks such as follows. Lookup category: Tasks T1 and T2 referred to the fact retrieval subcategory. For these two first tasks we suggested two straightforward questions. Participants had to find the date of a particular event and where an event took place. The search goal was simple and participants were able to decide when they had found the correct information. During the third task (T3), students had to retrieve a precise document using given information. This task belongs to the known item search subcategory. Exploratory category: We have identified one task for Marchionini's learn and investigate group referring in particular to the knowledge acquisition and accretion low level subcategories. The exploratory task (T4) is an important and regular work for Humanities researchers and students. Participants had to identify and write a research problem on the Cultural Heritage topic in one of its aspect covered by our data set.

At the beginning of our experiment, all the participants received a fifteen minutes training on the use of our experimental platform. In order to control technical factors that could affect user behavior, all participants did the experiment under the same conditions. They used the same web browser connected to our experimental platform on a desktop computer and a 22-inch display. We gave participants the same instructions concerning the four required tasks. Users had to select the type of task inside a list and click on the start button to begin each task. At the end of the task, users had to click on the end button and provide the answers and results required by the current task.

RESULTS AND DISCUSSION

For the four tasks, features were computed regarding the first query iteration (fn) and the entire task (f ′ n ). Because the raw data does not follow a normal distribution, we used non parametric methods. We performed a Friedman test on each feature, followed by a pairwise Wilcoxon signed rank test to evaluate the differences between two different tasks. Our results are consistent with the analysis made by [START_REF] Athukorala | Is exploratory search different? A comparison of information search behavior for exploratory and lookup tasks[END_REF]. Indeed, most of the features are fundamentally different when it comes to exploratory tasks. However, we find ourselves in a different context than the study led by Athukorala. Our system is a digital library storing not only scientific articles but more heterogeneous resources. Conducting the experiment with a different public -participants coming from social sciences and Humanities -leads us to the same results as Athukorala who took an interest in Computer Science researchers. Yet, these two types of researchers have very different habits in terms of information search.

Going further than the confirmation of previous results, we made the following additional observations. We can ob- serve that some of the features presented in Table 1, computed on the first query and the overall task behave similarly. Identifying the type of task the user is engaged in as early as possible is key to the system adaptation. To discriminate user's task as soon as possible, features f1, f2, f3 and f5 are significant. When looking at the results in Table 1, it seems that T1 behave differently than T2 and T3 when compared to T4. This is particularly true for f2, f3, f6 and f7, in the case of the first query iteration. We made another statistical test comparing T1 and T2 that confirmed this observation. It seems that when users discover the platform for the first time, some features are clearly impacted. This could be used to evaluate how difficult it is to get familiar with the platform and should be taken into account to efficiently contextualize user behavior traces.

•

  Maximum scroll value (f3): This corresponds to the proportion of the page seen by the user. It reflects the number of items observable in the result list. • Number of clicked items (f4): how many resources were clicked on in the results page. That corresponds to an access to more precise metadata. • Position of those items in the result list (f5): this is a mean of the position of the items selected by the user inside the search engine results page. • Number of viewed documents (f6): how many original documents were actually opened by the user. • Duration dwelling (f7): time the user spent investigating documents outside the results page (in seconds).

Table 1 :

 1 Statistical differences between an Exploratory task (T4) and three different Look-up tasks (T1, T2 and T3). The left-hand side of this table describe the results during the first query iteration, while the right-hand side concerns the overall task. We use a Wilcoxon signed rank test. pvalues with *s are statistically significant, with * for p < 0.05, ** for p < 0.01 and *** for p < 0.001.

		first query (f)	overall task (f ′ )
			T4			T4
		T1	T2	T3	T1	T2	T3
				query length
	p	***	***	***	*	***	***
	Z -4.28 -3.33 -4.16	-2.56 -4.05 -4.08
				duration
	p	0.1	***	***	***	***	***
	Z 1.61 -3.75 -3.78	-4.33 -4.55 -4.53
				max scroll value
	p 0.06	**	**	**	0.18	*
	Z -1.87 -2.97 -3.74	-2.64 -1.34 -2.09
				clicked items position
	p	*	*	0.05	***	***	***
	Z -2.25 -2.31 -1.96	-3.39 -3.92 -3.51