Graph Techniques for Matrix Equations
and Eigenvalue Dynamics
JOAKIM ARNLIND
Doctoral Thesis
Stockholm, Sweden 2008
TRITA-MAT-08-MA-02 ISSN 1401-2278
ISRN KTH/MAT/DA 08/02-SE ISBN 978-91-7178-845-0
KTH Matematik SE-100 44 Stockholm SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till oentlig granskning för avläggande av teknologie doktorsexamen i matematik fredagen den 8 februari 2008 kl 14.00 i sal F3, Lindstedtsvägen 26.
c
Joakim Arnlind, 2008
iii
Abstract
One way to construct noncommutative analogues of a Riemannian manifold Σ is to make use of the Toeplitz quantization procedure; i.e., we want to nd maps πn from the Poisson algebra of smooth functions on Σ to a
(noncom-mutative) algebra such that lim n→∞ ˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ 1 i~n[π n(f), πn(g)] − πn({f, g}) ˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ n = 0,
for some sequence ~n → 0. In Paper III and IV, we construct C-algebras
for a continuously deformable class of spheres and tori, and by introducing the directed graph of a representation, we can completely characterize the representation theory of these algebras in terms of the corresponding graphs. It turns out that the irreducible representations are indexed by the periodic orbits and N-strings of an iterated map s : R2→ R2associated to the algebra.
As our construction allows for transitions between spheres and tori (passing through a singular surface), one easily sees how the structure of the matrices changes as the topology changes.
In Paper II, noncommutative analogues of minimal surface and membrane equations are constructed and new solutions are presented some of which correspond to minimal tori embedded in S7.
Paper I is concerned with the problem of nding dierential equations for the eigenvalues of a symmetric N × N matrix satisfyingX = 0. Namely,.. by nding N(N − 1)/2 suitable conserved quantities, the time-evolution of X (with arbitrary initial conditions), is reduced to non-linear equations involving only the eigenvalues of X.
iv
Acknowledgement
I would like to thank my supervisor and friend Jens Hoppe; his knowledge and enthusiasm has been a great source of inspiration during my time as a PhD student. My colleagues at the Dept. of Mathematics also deserve my gratitude for making life at work dynamic and interesting. Furthermore, I would like to thank the Knut and Alice Wallenberg foundation for nancial support. Finally, I wish to thank my family for always supporting me and encouraging me to do the things I want.
Contents
Contents v
1 Graph Techniques for Matrix Equations 1
1.1 The directed graph of a matrix . . . 1 1.2 An example . . . 1 1.3 The dynamical map . . . 3
2 C-algebras 5
2.1 Introduction . . . 5 2.2 Compact surfaces embedded in R3 . . . . 5 2.3 C-Algebras . . . 6
3 Representation theory for a class of C-algebras 11
3.1 Introduction . . . 11 3.2 The C-algebras . . . 11 3.3 Hénon algebras . . . 13
4 Membranes and minimal surfaces 15
4.1 Discrete extremal 3-volumes and minimal surfaces . . . 15 4.2 The double commutator equation . . . 18 4.3 Membranes . . . 20
5 Eigenvalue Dynamics 23
5.1 Making use of conserved quantities . . . 24 5.2 Dierentiating the characteristic equation . . . 26 5.3 Generalization to hermitian matrices . . . 26
Bibliography 29
vi Contents
Paper I
Eigenvalue Dynamics o the Calogero-Moser System (J. Arnlind, J. Hoppe)
Letters in Mathematical Physics 68 (2004), 121129. Paper II
Spinning Membranes
(J. Arnlind, J. Hoppe, S. Theisen) Physics Letters B 599 (2004), 118128. Paper III
Noncommutative Riemann Surfaces
(J. Arnlind, M. Bordemann, L. Hofer, J. Hoppe, H. Shimada) Submitted to Communications in Mathematical Physics. Paper IV
Representation theory of C-algebras for a higher-order class of spheres and tori (J. Arnlind)
Chapter 1
Graph Techniques for
Matrix Equations
1.1 The directed graph of a matrix
Let G = (V, E) be a directed graph with vertex set V = {1, 2, . . . , N} and edge set V ⊆ E × E. For any N × N matrix W we say that G is the digraph of W if it holds that Wij 6= 0 ⇔ (i, j) ∈ E. Thus, the digraph of W encodes the non-zero matrix elements of W . Assume that we are given an equation for W , e.g. f(W ) = 0, is it possible to characterize all graphs that are digraphs of solutions to this equation? In some cases one can obtain such a characterization, and we will study a particular example.
1.2 An example
When studying noncommutative analogues of compact Riemann surfaces the need to nd all representations of so called C-algebras arose ([2]). Hermitian represen-tations could be found by solving the matrix equation
W2W†= αW + βW†W2+ γW W†W, (1.1)
with α, β, γ ∈ R, and it turned out that one can classify all representations in terms of the directed graph of W . Note that for any matrix W satisfying (1.1), the two hermitian matrices D = W W† and ˜D = W†W commute. This is easily seen by multiplying (1.1) from the left with W† and its hermitian conjugate by W from the right. Two commuting hermitian matrices can be simultaneously diagonalized by a unitary matrix, and from now on we assume such a basis to be chosen; we write
D = diag(d1, d1, . . . , dN) ˜
D = diag( ˜d1, ˜d2, . . . , ˜dN). 1
2 ~x s(~x) s2(~x) s2(~x) s3(~x) s3(~x)
Figure 1.1: The digraph of W .
In this notation, equation (1.1) becomes
W D = αW + β ˜DW + γDW (1.2)
together with the associativity condition DW = W ˜D. Written out in components, these equations are
Wij dj− α − β ˜di− γdi= 0 Wij d˜j− di= 0,
and if Wij 6= 0 we write
~xj≡ (dj, ˜dj) = s(~xi) ≡ (α + β ˜di+ γdi, di).
Now, let G = (V, E) be the digraph of W . To every vertex i ∈ V we can assign the pair ~xi = (di, ˜di), and if (i, j) ∈ E we have ~xj= s(~xi), as in Figure 1.1. We call s the dynamical map of the algebra.
Let us de ne a loop to be a directed cycle and a string to be a directed path from a transmitter to a receiver. It is a trivial fact that any nite directed graph must have at least one loop or one string. If the digraph of W has a loop, then sn(~x) = ~x for some integer n > 0 ; i.e. the map s : R2 → R2 must have a periodic orbit of period n. For a transmitter i ∈ V we must have that ˜di= 0 since ˜D = W†W , and for a receiver j ∈ V it follows from D = W W† that d
j= 0. Hence, for a string on n vertices to exist, we need sn−1(d1, 0) = (0, ˜dn) for some d1, ˜dn > 0.
In fact, it turns out that all matrix solutions can be reduced to solutions that are direct sums of such strings and loops, thus giving a classi cation in terms of graphs. One proves the following (see [2]):
Theorem 1.1. Let W be a solution of (1.1). Then W is unitarily equivalent to a matrix whose directed graph is such that every connected component is either a string or a loop.
In fact, up to local injectivity, these methods extend to a larger class of equations (see [1]):
Graph Techniques for Matrix Equations and Eigenvalue Dynamics 3
Theorem 1.2. Let W be a locally injective solution of the equation
W2W† = αW +Xn k=1
βk(W†W )kW +Xn k=1
γk(W W†)kW (1.3)
with α, βk, γk∈ R. Then W is unitarily equivalent to a matrix whose directed graph is such that every connected component is either a string or a loop.
1.3 The dynamical map
For the equation in Theorem 1.2, the same techniques are used; i.e. D and ˜D can be simultaneously diagonalized and there is a dynamical map s such that ~xj = s(~xi) if (i, j) ∈ E. However, this map will no longer be a ne but of the following form
s : x y → α + n X k=1 βkyk+ γkxk x ,
which makes the problem of nding periodic orbits much more involved. The con-verse of Theorem 1.2 is in principle also true, i.e.
• any periodic orbit of s, contained in R2
+= {(x, y) ∈ R2 : x > 0, y > 0}, and • any ~x = (d1, 0) such that d1 > 0 and sk(~x) ∈ R2+ for k = 1, . . . , n − 2 and
sn−1(~x) = (0, ˜d
n) with ˜dn> 0, give rise to matrix solutions.
Let us call the second set of points in R2 a n-string of s. Hence, nding all solu-tions to (1.3) is equivalent to nding all periodic orbits in R2
+ and n-strings of the dynamical map s.
Chapter 2
C-algebras
2.1 Introduction
In dierential geometry, knowledge about the algebra of continuous functions on a manifold can in some cases give more or less complete information about the underlying space. To study geometry via the corresponding algebra of functions is one of the starting points for noncommutative geometry. In this setting it is natural to ask the question if there are noncommutative analogues of certain man-ifolds; in particular, this question becomes important when one wants to quantize a system on a given manifold by promoting functions to operators. When the manifold is equipped with a symplectic form, inducing a Poisson bracket on the algebra of smooth functions, one way of realizing such a correspondence is to map the function algebra onto an algebra of operators, while respecting the Poisson-algebraic structure to some extent. In this chapter we will present a way to obtain noncommutative analogues of compact surfaces of arbitrary genus by introducing C-algebras. Moreover, we will show how this concept is related to the Toeplitz quantization procedure.
2.2 Compact surfaces embedded in R
3Let C(~x) = P (x) + y22+ z2− c2, with P (x) = α g Y k=1 (x2− k2) − c,
and c > 0. For certain choices of α, the inverse image Σ = C−1(0) describes a compact surface of genus g embedded in R3[6]. By C[~x] we will denote the algebra of polynomials in the commuting variables x1= x, x2= y and x3= z, and by C[ ~X] we denote the algebra of polynomials in the noncommuting variables X1 = X, X2= Y and X3= Z. Let us introduce the following class of functions on Σ.
6
De nition 2.1. For Σ = C−1(0) we set Pp(Σ) = C[~x]/hC(~x)i.
Hence, if Σ is a surface then Pp(Σ) is the algebra of functions on Σ that are restrictions of polynomials in R3. Furthermore, the map {·, ·} : C[~x] × C[~x] → C[~x], de ned by
{p, q} = ∇C · ∇p × ∇q,
is a Poisson bracket and it induces a Poisson bracket on Pp(Σ) by restriction. With this bracket, we regard Pp(Σ) as a Poisson algebra.
2.3 C-Algebras
The idea is to de ne a noncommutative algebra A, and a map π : Pp(Σ) → A, such that commutators and Poisson brackets are related; we propose that a noncommu-tative analogue of Σ should be de ned by the relations
[X, Y ] = i~π({x, y}) = i~π(∂zC) [Y, Z] = i~π({y, z}) = i~π(∂xC) [Z, X] = i~π({z, x}) = i~π(∂yC).
(2.1)
for some ~ > 0, regarded as a small parameter. Let us make this statement more precise.
De nition 2.1. Let ˜π : C[~x] → C[ ~X] be a linear map. We call ˜π a symmetriza-tion map if for each monomial xa1xa2· · · xad ∈ C[~x] there exists an integer N, α1, . . . , αN ∈ C and permutations σ1, . . . , σN ∈ Sd such that
(i) ˜π xa1· · · xad =XN i=1 αiXaσi(1)· · · Xaσi(d) (ii) N X i=1 αi= 1.
As an example of a symmetrization map we can take the map S, completely sym-metrizing a polynomial; e.g., S(xy) = 1
2 XY + Y X
. Let us now de ne the corre-sponding noncommutative algebra.
De nition 2.2 (C-algebra). Let C(~x) be a polynomial and let ˜π1, ˜π2, ˜π3 be sym-metrization maps. Furthermore, we let I be the two-sided ideal in C[ ~X] generated by the relations
[X, Y ] = i~˜π1(∂zC) (2.2)
[Y, Z] = i~˜π2(∂xC) (2.3)
[Z, X] = i~˜π3(∂yC). (2.4)
Graph Techniques for Matrix Equations and Eigenvalue Dynamics 7
It is noteworthy that the de nition of C-algebras is purely algebraic and does not depend on whether C−1(0) describes a regular surface or not. Thus, the C-algebra of a singular surface, described as the inverse image of a polynomial, is perfectly well de ned.
How is the commutator of two noncommutative polynomials related to the Pois-son bracket of the corresponding commutative polynomials? It is easy to prove the following results which will lead us to the Toeplitz quantization procedure. Lemma 2.3. Let Xa1Xa2· · · Xad∈ C~(Σ). For every σ ∈ Sd it holds that
Xa1Xa2· · · Xad= Xaσ(1)Xaσ(2)· · · Xaσ(d)+ O(~). Proof. If we write out equations (2.2)(2.4) we get
XY = Y X + i~˜π1(∂zC) Y Z = ZY + i~˜π2(∂xC) ZX = XZ + i~˜π3(∂yC),
which implies that we can rearrange any monomial at the cost of introducing terms proportional to higher powers of ~.
Any map ˜π : C[~x] → C[ ~X] induces a linear map π : Pp(Σ) → C~(Σ), de ned by π(tk) = pr~ ˜π(tk)(and extended to Pp(Σ) by linearity), where pr~is the projec-tion of C[ ~X] onto C~(Σ) and {t1, t2, . . .} is a basis of Pp(Σ). If ˜π is a symmetrization map, we will call π a symmetrization map.
Proposition 2.4. Let p, q ∈ Pp(Σ) and let π : Pp(Σ) → C~(Σ) be a symmetrization map. Then π(pq) = π(p)π(q) + O(~).
Proof. Assume that t1, t2, . . . is a basis for Pp(Σ) and let us write tk= xa1· · · xad
tl= xb1· · · xbe tk· tl= xc1· · · xcd+e. Since π is a symmetrization map we know that
π(tk) =X
i
αiXaσi(1)· · · Xaσi(d)
π(tl) = X i βiXbτi(1)· · · Xbτi(e) π(tktl) = X i γiXcρi(1)· · · Xcρi(d+e) π(tk)π(tl) = X i,j
8
SincePiγi=Pi,jαiβj = 1, we can use Lemma 2.3 to reorder the terms such that π(tk)π(tl) =
X i
γiXcρi(1)· · · Xcρi(d+e)+ O(~) = π tktl+ O(~).
The relations (2.2)(2.4) do not in general imply that [π(p), π(q)] = i~π {p, q},
for all p, q ∈ Pp(Σ), but up to rst order in ~ we have the following result:
Proposition 2.5. Let π : Pp(Σ) → C~(Σ) be a symmetrization map. Then it holds that
[π(p), π(q)] = i~π {p, q}+ O(~2), for all p, q ∈ Pp(Σ).
Proof. Assume that t1, t2, . . . is a basis of Pp(Σ) and write tk= xa1· · · xad
tl= xb1· · · xbe
π(tk) =X
i
αiXaσi(1)· · · Xaσi(d)
π(tl) = X
i
βiXbτi(1)· · · Xbτi(e).
Using Leibniz' rule, we obtain
π {tk, tl}= π X m,n xa1· · · xam−1xb1· · · xbn−1{xam, xbn} xam+1· · · xadxbn+1· · · xbe ! .
On the other hand, we have that [π(tk), π(tl)] = X i,j,m,n αiβj|Xaσi(1)· · · Xaσi(m−1){zXbτj(1)· · · Xbτj(n−1)} Aijmn ×
× [Xaσi(m), Xbτj(n)] Xaσi(m+1)· · · Xaσi(d)Xbτj(n+1)· · · Xbτj(e)
| {z }
Bijmn
=i~ X
i,j,m,n
αiβjAijmn˜π {xaσi(m), xbτj(n)}Bijmn
for some symmetrization map ˜π. By using Lemma 2.3 and Proposition 2.4, we conclude that
Graph Techniques for Matrix Equations and Eigenvalue Dynamics 9
This makes contact with the Toeplitz quantization procedure in the following way. Let ~1, ~2, . . . be a sequence with lim
n→∞~n = 0 and let πn : Pp(Σ) → C~n(Σ) be symmetrization maps. Furthermore, for each ~n, let || · ||n be a norm on C~n(Σ). Then it follows that
lim n→∞ i~n1 [πn(p), πn(q)] − πn({p, q}) n= limn→∞||O(~n)||n= 0, for all p, q ∈ Pp(Σ).
Chapter 3
Representation theory for a
class of C-algebras
3.1 Introduction
In this chapter we will present a method to classify hermitian representations of a particular class of C-algebras, in which some algebras correspond to spheres and tori. In doing so, we make use of the graph techniques discussed in Chapter 1, which leads to a characterization of irreducible representations in terms of loops and strings. As a subset of these C-algebras we will introduce the Hénon algebras, and present an example for which we can prove that irreducible loop representations of all dimensions exist.
3.2 The C-algebras
For c, α0, α1, . . . , αn ∈ R, we will consider the following polynomial:
C(~x) = c + 12z2+n+1X k=1
αk−1
2k x2+ y2
k, (3.1)
for which the corresponding Poisson bracket relations are {x, y} = z {y, z} = α0x + xXn k=1 αk x2+ y2k {z, x} = α0y + y n X k=1 αk x2+ y2k.
Let us now choose particular symmetrization maps to de ne the corresponding C-algebra. Namely, writing W = X + iY and V = X − iY we choose the following
12 ordering: [X, Y ] = i~Z [Y, Z] = i~α0X + i~2 n X k=1 ˜βkV (V W )k+ (V W )kW+ ˜γkV (W V )k+ (W V )kW [Z, X] = i~α0Y + i~2i n X k=1 ˜βk(V W )kW − V (V W )k+ ˜γk(W V )kW − V (W V )k with ˜βk+ ˜γk = αk for k = 1, 2, . . . , n. The aim is now to nd hermitian matrices X, Y, Z satisfying these relations. One realizes that Z can be eliminated and the two remaining equations are
Y, [X, Y ]= −~2α0X − ~2 2 n X k=1 ˜βkV (V W )k+ (V W )kW + ˜γkV (W V )k+ (W V )kW [X, Y ], X= −~2α0Y −~2 2i n X k=1 ˜βk(V W )kW − V (V W )k + ˜γk(W V )kW − V (W V )k.
These equations can be entirely written in terms of V and W :
W2V = αW +Xn k=1 βk(V W )kW +Xn k=1 γk(W V )kW (3.2) W V2= αV +Xn k=1 βkV (V W )k+Xn k=1 γkV (W V )k, (3.3)
with α = −2~2α0, β1 = −2~2˜β1− 1, γ1 = −2~2˜γ1+ 2 and βk = −2~2˜βk and γk = −2~2˜γk for k ≥ 2. If X, Y are hermitian then W† = V , which makes (3.3) the hermitian conjugate of (3.2). Hence, nding hermitian representations of the C-algebra is equivalent to solving the matrix equation
W2W† = αW +Xn k=1
βk(W†W )kW +Xn k=1
γk(W W†)kW.
From Theorem 1.2 it follows that any (locally injective) representation is unitarily equivalent to a representation that is a direct sum of loops and strings. Moreover, one can prove that loop and string representations are irreducible [1]. Therefore, we can classify all hermitian representations by the periodic orbits and n-strings
Graph Techniques for Matrix Equations and Eigenvalue Dynamics 13
of the dynamical map. In the linear case, i.e. n = 1, the dynamical map is a ne and one can explicitly construct most representations (see [2]). For these algebras the dimension of representations is typically restricted to a multiple of a positive integer. Going to higher order in n, one can nd algebras that allow for irreducible representations of all dimensions.
3.3 Hénon algebras
When constructing C-algebras for inverse images of the polynomials in (3.1), we have the freedom of choosing ˜βk and ˜γk as long as ˜βk + ˜γk = αk. A particular choice, obtained by setting all but the rst ˜βk to zero, gives us the Hénon algebras, de ned by the relations
W2V = αW + bV W2+Xn k=1 γk(W V )kW (3.4) W V2= αV + bV2W +Xn k=1 γkV (W V )k. (3.5)
The name is justi ed by the fact that the dynamical map
s : x y −→ α + by + p(x) x ,
becomes what is usually referred to as a generalized Hénon map. For the Hénon algebras, every representation is locally injective (since s is invertible), which implies that any representation can be decomposed into loops and strings. Moreover, since disjoint periodic orbit of s give rise to inequivalent loop representations, one can prove that there exists a Hénon algebra of second order having irreducible loop representations of all dimensions.
Chapter 4
Membranes and minimal surfaces
4.1 Discrete extremal 3-volumes and minimal surfaces
Consider d time-dependent hermitian matrices {Xi}di=1 satisfying ¨ Xi= − d X j=1 h Xi, Xj, Xj i . (4.1)
These equations arise as discrete versions of the problem of nding 3-manifolds (in higher dimensional Minkowski spaces) whose mean curvature vanishes ([8]), a problem which also plays an important role in physics, especially since [5]. Let us indicate how this correspondence is realized.
Let X denote a at d + 1-dimensional Minkowski space with metric ηµν = (1, −1, . . . , −1), and let xµ be embedding functions of a 3-dimensional manifold Σ in X. We will denote local coordinates on Σ by σ0, σ1, σ2, and the induced metric by
Gαβ= ∂αxµηµν∂βxν for α, β = 0, 1, 2.
By Gαβ we will denote the tensor de ned by GαβGβγ= δαγ. We are now looking for xµ such that the volume functional
S = −TZ √G d3σ,
with G = det Gαβ, is extremal. Varying this functional gives the following Laplace-Beltrami equation 1 √ G∂α √ GGαβ∂ βxµ = 0 for µ = 0, . . . , d. (4.2)
Let us introduce a metric with restricted indices
grs= −∂rxµηµν∂sxν for r, s = 1, 2 g = det grs.
16
By choosing a particular local parametrization (cp. [8]), we can set x0= σ0, which implies that grs= d X l=1 ∂rxl∂sxl, and we can take Gαβ to be of the following form
Gαβ= G00 0 0 −(grs) . Assuming xµ satis es (4.2), the µ = 0 component gives
∂0 r
g G00 = 0,
since x0 = σ0. Hence, there exists a function (resp. density, s.b.) ρ such that G00= g/ρ2and ∂0ρ = 0. Furthermore, since Gαβ is a metric, ρ must transform as a density under reparametrizations. The remaining components of equation (4.2) give ∂2 0xl=1ρ∂r g ρgrs∂sxl , (4.3)
for l = 1, . . . , d. For smooth functions f, h de ne {f, h} = 1
ρεrs∂rf∂sh, where εrsis a totally antisymmetric tensor with ε12= 1. This is a well de ned Poisson bracket on the surface if ρ transforms as a density under local reparametrizations. Let us now show how we can write (4.3) in terms of this Poisson bracket.
Proposition 4.1. Let grs = d X j=1 ∂rxj∂sxj, grs de ned by X2 s=1 grsgst = δr t and g = det(grs). Setting {f, h} = 1 ρεrs∂rf∂sh gives 1 ρ∂r g ρgrs∂sxi =Xd j=1 n xi, xj , xjo.
Proof. By using the simple fact that ggrs= εruεsvg
uv we can write 1 ρ∂r 1 ρggrs∂sxi = 1ρ∂r 1 ρεruεsvguv∂sxi . Moreover, using that guv= ∂uxj∂vxj gives
1 ρ∂r 1 ρggrs∂sxi =ρ1∂r 1 ρεruεsv∂uxj∂vxj∂sxi =1ρ∂rεruxi, xj ∂uxj =xi, xj , xj +1 ρ xi, xj εru∂2 ruxj. The second term vanishes since εru is anti-symmetric and ∂2
ruxj is symmetric in the two indices.
Graph Techniques for Matrix Equations and Eigenvalue Dynamics 17
From Proposition 4.1 it follows that an extremal solution to the variational problem ful lls the equation
∂2 0xi= d X j=1 xi, xj , xj . (4.4)
Using the correspondence from Chapter 2, substituting {·, ·} by [·, ·]/i~, we arrive at (after identifying σ0 with time) the discretized equations
~2Xi¨ = −Xd j=1
h
Xi, Xj, Xji. (4.5)
How can one nd solutions to these equations? The simplest Ansatz would be to separate time- and matrix-dependence by letting
Xi(t) = x(t)Mi (4.6)
where x(t) is a scalar function and {Mi}d
i=1 are hermitian time-independent matri-ces. With this Ansatz, (4.1) is satis ed, provided
d X j=1 h Mi, Mj, Mj i = λMi (4.7) .. x(t) = −λx(t)3. (4.8)
For positive λ, the solution of (4.8) oscillates around x = 0 and hence, all Xi(t) will simultaneously collapse to zero at some time. This is not a desirable feature and we would like to nd solutions that do not shrink to zero simultaneously.
Actually, the interesting equation is (4.7), which will also arise from a more sophisticated Ansatz [4], giving non-collapsing solutions. Instead of (4.6), let
Xi= x(t) RM−→i:= x(t)Rij(t)Mj (sum over j) (4.9) where R := eAϕ(t)is an orthogonal d×d matrix with A2= −1,M := (M1, . . . , Md)−→ and ϕ(t) satisfying x2˙ϕ = L (=const). Inserting this Ansatz into (4.1) yields
1 2 .
x2+λ4x4+ L2
2x2 = const., (4.10)
together with (4.7). The centrifugal term L2
2x2 then prevents x(t) from passing through x = 0, provided L 6= 0. Let us now turn to a problem where equation (4.7) also arises.
Minimal surfaces in Sd−1can be found by varying the integral
Z √
18
where g = det ∂r−→m · ∂s−→m, with respect to λ and −→m ∈ Rd. One obtains the equations
1
√g∂r√g grs∂smi = −2mi (4.11)
−
→m2= 1. (4.12)
Using Proposition 4.1 with ρ = √g, one can rewrite (4.11) as d
X j=1 n
{mi, mj}, mjo= −2mi. (4.13)
Again, using the correspondence between Poisson brackets and commutators one obtains d X j=1 h Mi, Mj, Mji= λMi.
4.2 The double commutator equation
At rst glance, d X j=1 h Mi, Mj, Mji= λMi. (4.7)looks like a very speci c equation, but its solutions do in fact incorporate Lie algebras, Cliord algebras and symmetric spaces (through Lie triple systems, see below). Note that λ is not really a free parameter, since by rescaling Mi → ˜Mi :=
1 √
λMi, solutions of (4.7) are always mapped to those of d
X j=1
h ˜
Mi, ˜Mj, ˜Mji= ˜Mi. (4.14)
As a rst example, consider the following real form of a semi-simple Lie algebra L of dimension d. Start with the Cartan-Weyl basis {hαi, eαi, e−αi} (where {αi}Ni=1 are the positive roots of L) and construct a new basis
~g = {gi} := {h1, . . . , hr, e+α1, e−α1, . . . , e+αN, e−αN} where e+ αi := 1 √ 2 eαi+ e−αi e− αi := i √ 2 eαi− e−αi
Graph Techniques for Matrix Equations and Eigenvalue Dynamics 19
and the hj's are chosen such that K(hi, hj) = δij (K denotes the Killing form). One easily checks that (with the standard convention K(eα, e−α)=1)
K(gi, gl) = δil ∀ gi, gl
In any such ortho-normal basis, the structure constants are antisymmetric in all three indices, and therefore
D X j=1 h gi, gj, gj i = XD j,k,l=1 fk ijfkjl gl= D X j,k,l=1 fikjfk ljgl= D X l=1 K(gi, gl)gl= D X l=1 δilgl= gi.
Thus, by taking representations of a Lie algebra in this basis, we can generate solutions to (4.14) of arbitrary (matrix) dimension.
Note that the set of elements {gi} need not form a Lie algebra, since they only have to close with respect to the double commutator. Therefore, one might as well consider subsets of Lie algebras ful lling (4.7). A set of elements, closed under the double commutator, together with certain other conditions, is called a Lie triple system and naturally makes up the basis for nding solutions to (4.7).
Let us now introduce another set of elements, from which one can construct solu-tions. De ne, for arbitrary odd N > 1, N2 independent N × N matrices
U(N) m := N 4πω 1 2m1m2gm1hm2, (4.15) where ω := e4πi N , m := (m1, m2) and (g)ij = ωi−1δij (h)ij = δi,j−1,
and the indices are counted modulo N. These matrices provide a basis for the Lie algebra gl(N, C) and one can easily derive the following two identities
h U(N) m , Un(N) i = −iN2π sin 2π N m × n Um(N)+n (4.16) h U(N) m , Un(N) , U−n(N)i=4πN22sin22π N m × n U(N) m (4.17)
where m × n := m1n2− m2n1. In view of (4.17), one might guess how to construct solutions of (4.7). As an example, it is straightforward to verify that
− → M = Um+ U−m 2 , Um− U−m 2i , Um0+ U−m0 2 , Um0− U−m0 2i , Un+ U−n 2 , Un− U−n 2i , Un0+ U−n0 2 , Un0− U−n0 2i , (4.18)
20 with m = m1 m2 n = n1 n2 m0 =−m2 m1 n0=−n2 n1 ,
is a solution of (4.7) provided m2 = n2. Note that the set of elements in M, as−→ given by (4.18), does not form a Lie algebra, since the set is not closed under the commutator.
We note that forM as in (4.18),−→ −→M2 =N 2π
2
1. For the Lie algebra solution {gi}, note that
~g2=Xd j=1
g2 j
is the second order Casimir of the algebra, since K(gi, gj) = 1. In any irreducible representation, it will be proportional to the identity matrix, by Schur's lemma.
4.3 Membranes
In the Hamiltonian formulation of a theory of a massless relativistic surface, one nds that in the light cone gauge, the equations of motion
.. xi= d X j=1 {{xi, xj}, xj} (4.19)
are supplemented by the constraint (cp. [7, 8]) d
X j=1
{xi,x.i} = 0.
In the context where these equations originally arose, it was necessary to nd nite approximations in order to address quantum dynamics, and therefore, the presented method to replace functions by nite dimensional matrices was developed ([8]). Nowadays, this correspondence between functions and N → ∞ limits of matrices, connecting (4.19) with (4.1), is viewed as a part of non-commutative geometry. From (4.19) one obtains the matrix equations
¨ Xi= −Xd j=1 h Xi, Xj, Xji (4.1) d X j=1 Xi, ˙Xi= 0. (4.20)
Graph Techniques for Matrix Equations and Eigenvalue Dynamics 21
In terms of the Ansatz (4.9), the constraint reads d
X j=1
Mj, AM−→j= 0. (4.21)
Now, there are several ways to ful ll this constraint. Given a solution {M0
i} of (4.7) with d0= d/2 we set−→M = (M0
1, . . . , Md00, 0, . . . , 0) and take A to be the matrix such that AM = (0, . . . , 0, −M−→ 0
1, . . . , −Md00). Then A2= −1 and every term in (4.21) is zero. Note that we could equally well choose−→M = (M−→0,M−→0).
A second way is to have nonzero terms in the sum, and see to it that they add up to zero. As an example, this can be done for su(2N +1) by assigning appropriate signs to the elements of ~g. For bosonic membranes, however, d is smaller or equal to 9, and the only interesting Lie algebra that solves (4.7) with the constraint (4.21) (apart from su(2)) is the real form su(3) of A2.
A third way is to nd solutions such that the elements of−→M commute in pairs, e.g. M2i, M2i+1= 0. Then, by an appropriate choice of A, one can satisfy (4.21). These solutions have the advantage that we don't need to double the matrices or add zeroes, making them, in some sense, irreducible. This is e.g. the case forM−→ de ned as in (4.18).
Chapter 5
Eigenvalue Dynamics
Consider a time-dependent real symmetric N × N matrix X(t). Assume that X(t) satis es the dierential equation ..
X = 0 (5.1)
which has the simple solution
X(t) = X(0) + ˙X(0)t. (5.2)
For given initial conditions X(0) and ˙X(0), one can diagonalize X(t) to obtain the motion of the eigenvalues, i.e.
Q(t) = R−1(t)X(t)R(t) (5.3)
where Q = diag q1(t), . . . , qn(t)and R(t) is an orthogonal matrix.
One could ask: What dierential equations do the qi(t)'s satisfy? (Of course, no matter how complicated those equations are, they are solved by diagonalizing X(t).) In the following, we will try to nd these equations. If we let
X = RQR−1 ˙X = RLR−1 we get L = ˙Q + [M, Q] .. X = RL + [M, L]. R−1
where M := R−1˙R. The equationX = 0, is therefore equivalent to the (Lax-pair).. equation
.
L = [L, M]. (5.4)
24
Written out in detail, (5.4) is equivalent to ..
qi= −2X k6=i
MikMki(qi− qk) (5.5)
˙Mij(qi− qj) + 2Mij( ˙qi− ˙qj) =X k
MikMkj(qi+ qj− 2qk) (i 6= j). (5.6)
We observe that the equations for the qi's are highly coupled to the matrix elements of R (through M). In principle, one could, from the initial conditions, calculate R(t) and hence M(t), and insert the result into the above equations. This would give dierential equations for qi(t), that involve t explicitly. However, for general N, this is not a realistic procedure, and we would like to nd other ways to eliminate the Mij's from (5.5), such that the resulting equations depend only on qi's and their derivatives (and not explicitly on time).
For a hermitian matrix X(t), it is well known (see e.g. [9]) that for (very special) initial conditions, satisfying
[X(0), ˙X(0)]jk= i(1 − δjk), eq (5.6) can be identically satis ed, by choosing
Mj6=k= i qj(t) − qk(t)2 Mjj = iX k6=j 1 qj(t) − qk(t)2
and that (5.5) yields the Calogero-Moser equations of motion ..
qi= 2X k6=i
1
(qi− qk)3. (5.7)
Paper I ([3]) is concerned with the problem of nding the dierential equations for the eigenvalues of a real symmetric matrix, satisfying X = 0, for arbitrary.. initial conditions. In the following sections we take two dierent approaches to the problem.
5.1 Making use of conserved quantities
Our goal is to eliminate the Mij's in (5.5), to obtain dierential equations involving only the eigenvalues. Let us start by introducing a more convenient set of variables,
F := [L, Q]
Graph Techniques for Matrix Equations and Eigenvalue Dynamics 25
Equations (5.5) and (5.6) become
.. qi= 2X k6=i f2 ik (qi− qk)3. (5.8) ˙fij = − X k6=i,j fikfkj (q 1 i− qk)2− 1 (qk− qj)2 ! (5.9)
There exists a large set of conserved quantities related to this system. As an example, consider the quantity Tr LFLF. Since the trace is invariant under con-jugation, dierentiating with respect to t gives
d
dtTr LFLF
=dtd Tr L[L, Q]L[L, Q]= dtd Tr ˙X[ ˙X, X] ˙X[ ˙X, X]= 0
since X = 0. Thus, Tr LFLF.. = const = Tr L(0)F(0)L(0)F(0). Now we can solve for one of the fij's and insert the expression into (5.8). If we nd enough conserved quantities, we could (at least in principle) eliminate all the fij's. The resulting dierential equations would contain the values of the conserved quantities used, which are dependent on the initial conditions.
Can we nd suitable conserved quantities to use? We would like the degree (as functions of the fij's) to be as low as possible, in order not to bring too much trouble when trying to solve for them. Since the fij's appear quadratically in (5.8), it would be very convenient to nd su ciently many (independent) conserved quantities being linear in f2
ij.
To accomplish this, we rst note that the following traces, which are not con-served quantities, only contain f2
ij
Eα,β:= Tr QαFQβF.
For 0 ≤ α ≤ β ≤ N − 2, this gives N(N−1)2 linearly independent equations linear in f2
ij. The crucial point is that one can construct conserved quantities Jα,βcontaining the Eα,β, Tr Qkand a number of conserved quantities; i.e. Jα,βdepends on fijonly through the Eα,β.
After the elimination, (5.8) reads ..
qi= 2X k6=i
Aik[q1, . . . , qN; c1, . . . , cγ]
(qi− qk)3 . (5.10)
Since c1, . . . , cγ are dependent on qi(0), ˙qi(0) and fij(0), not all choices of constants in (5.10) will describe the original eigenvalue dynamics. Given qi(0), ˙qi(0) and fij(0), one should rst calculate the values of c1, . . . , cγ and then consider the dierential equation (5.10) for those values c1, . . . , cγ.
26
5.2 Dierentiating the characteristic equation
Since the determinant is invariant under conjugation, we havedet X(t) − λ1= det Q(t) − λ1= (q1− λ)(q2− λ) · · · (qN − λ). On the other hand
det X(t) − λ1= det X(0) + t ˙X(0) − λ1, hence
det X(0) + t ˙X(0) − λ1= (q1− λ)(q2− λ) · · · (qN − λ).
The order of t, on the left hand side, is equal to the rank k of ˙X(0). Thus, by dierentiating this equation, with respect to t, k times and comparing both sides with respect to dierent powers of λ, we get N k-th order dierential equations (not explicitly dependent on t) for the eigenvalues.
This method is more straightforward than the previous one, but the drawback of this procedure is that when k > 2, one gets dierential equations for the eigenvalues, of degree higher than 2.
5.3 Generalization to hermitian matrices
So far, we have considered real symmetric matrices, but we can easily generalize the result to general hermitian matrices. We write, for a complex hermitian matrix X,
X = UQU−1 (5.11)
where Q is diagonal and U is a unitary matrix. While R, in equation (5.3), is de-termined up to discrete transformations, U is only dede-termined up to multiplication by an arbitrary diagonal matrix D, satisfying D†D = 1. The de nition
M := U−1˙U
makes M an anti-hermitian matrix. While, in the real case, the diagonal elements of the (real antisymmetric) matrix M vanish, in the complex case they are purely imaginary. Consider a transformation exploring the freedom in U
U → ˜U := UD. This induces a transformation of M
Graph Techniques for Matrix Equations and Eigenvalue Dynamics 27
Since D†D = 1, the o-diagonal elements of M will only be multiplied by a phase under this transformation. Clearly
|Mij|2= | ˜Mij|2. (i 6= j)
Writing D = diag eiϕ1(t), . . . , eiϕN(t)gives, for the diagonal elements of M Mkk→ ˜Mkk= Mkk+ i ˙ϕk.
If one likes, one could eliminate these extra (compared to the real symmetric case) degrees of freedom by imposing that
˙ϕk(t) = i Mkk(t) =⇒ Mkk(t) ≡ 0
In any case, these degrees of freedom will not enter in the equations we are interested in. The matrix F := [L, Q] is anti-hermitian but still purely o-diagonal, and since
| ˜fjk|2= |(D†FD)jk|2= |fjk|2 equation (5.5) becomes .. qi= 2X k6=i |fik|2 (qi− qk)3, (5.12)
for all choices of D. Furthermore, the quantities Tr QαFQβF, will again be linear in |fik|2 and we can use the same technique, as for the real case, to eliminate the |fik|2's in (5.12).
Bibliography
[1] J. Arnlind. Representation theory of C-algebras for a higher-order class of spheres and tori. arXiv:0711.2943.
[2] J. Arnlind, M. Bordemann, L. Hofer, J. Hoppe, H. Shimada. Noncommutative Riemann Surfaces. arXiv:0711.2588.
[3] J. Arnlind, J. Hoppe. Eigenvalue Dynamics o the Calogero-Moser System. Letters in Mathematical Physics 68 (2004), 121129.
[4] J. Arnlind, J. Hoppe, S. Theisen. Spinning Membranes. Physics Letters B 599 (2004), 118128.
[5] T. Banks, W. Fischler, S.H. Shenker, L. Susskind M-theory as a matrix model: A conjecture. Phys.Rev. D55 (1997) 5112-5128.
[6] L. Hofer. Surfaces de Riemann compactes. Master thesis. Univ. Mulhouse, 2002.
[7] J. Hoppe Membranes and matrix models. IHES/P/02/47, hep-th/0206192. [8] J. Hoppe Quantum theory of a massless relativistic surface and a two
dimen-sional bound state problem. PhD Thesis (MIT, 1982). http://www.aei.mpg.de/∼hoppe/.
[9] A.M. Perelomov Integrable systems of classical mechanics and Lie algebras, vol 1. Birkhäuser Verlag, 1990.
Letters in Mathematical Physics 68: 121–129, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands. 121
Eigenvalue-Dynamics
off the Calogero–Moser System
JOAKIM ARNLIND and JENS HOPPEDepartment of Mathematics, Royal Institute of Technology, Stockholm, Sweden. e-mail: {joakim.arnlind, hoppe}@math.kth.se
(Received: 2 October 2003; revised version: 5 April 2004)
Abstract. By finding N(N − 1)/2 suitable conserved quantities, free motions of real
symmetric N× N matrices X(t), with arbitrary initial conditions, are reduced to nonlinear equations involving only the eigenvalues of X – in contrast to the rational Calogero-Moser system, for which [X(0), ˙X(0)] has to be purely imaginary, of rank one.
Mathematics Subject Classification (2000). 70H06.
Key words. eigenvalue dynamics, symmetric matrix, integrable systems.
1. Introduction
Among finite-dimensional integrable models, the rational Calogero–Moser [4, 11] system has played a prominent role. It consists of a finite number of particles mov-ing, in one dimension, under the influence of a repulsive 2-body force proportional to the cube of the distance(s). One of its (many) nice features is that, classically, the positions of the N particles, as functions of time, can be interpreted as the eigenvalues of a Hermitian N× N matrix, X(t), satisfying the simplest possible second-order equation, X..= 0.
This identification, however, requires the initial conditions for X(t) to be very special (namely such that [X(0), ˙X(0)] is a purely imaginary matrix whose off-diag-onal elements are all equal to each other). How do the eigenvalues of X evolve for other choices of initial conditions? In this Letter, we will give the answer, for arbi-trary real-symmetric matrices X.
So, let X(t) be a real symmetric time-dependent N× N matrix satisfying
..
X= 0. Our aim is to derive differential equations describing the time-evolution of (just) the eigenvalues of X. Writing
X(t)= R(t)Q(t)R−1(t), ˙X(t) =R(t)L(t)R−1(t) (1)
with Q diagonal, R∈ SO(N) one finds that
122 JOAKIM ARNLIND AND JENS HOPPE with M= R−1˙R (= −MT) ∈ so(N), and
..
X= 0 becomes equivalent to the coupled nonlinear equations .. qi= −2 k=i MikMki(qi− qk), (3) ˙ Mij(qi− qj) + 2Mij( ˙qi− ˙qj) = k MikMkj(qi+ qj− 2qk) (i = j). (4)
Whereas, for complex (anti-Hermitian) M, it is consistent to set Mjk= iδjk l=j 1 (qj− ql)2− i 1− δjk (qj− qk)2, (5)
corresponding to having chosen initial conditions satisfying [X(0), ˙X(0)]jk =
i(1 − δjk), yielding the famous Calogero–Moser [4,11] equations of motion, ..
qj= 2
k=j
1
(qj− qk)3, (6)
the elimination of M for other initial conditions, in particular real X, appears to be unknown (despite of the fact that, as we gradually discovered, Equations (3)/(4), and related ones, were studied in quite a variety of contexts [3, 8–10, 12, 13, 15, 16]; after the submission of our article (resp [1]), an interesting paper [7] appeared in which a choice of initial conditions is pointed out which allows for the simple reduction Mij= −˙qi˙qj/(qi− qj)).
Motivated by (5), resp. the structure of the left hand side of (4), let us rewrite (3)/(4) in terms of fjk:= Mjk(qj− qk)2=: Mjkqjk2 = [L, Q]jk= [[M, Q], Q]jk: .. qi= 2 k=i f2 ik (qi− qk)3, (7) ˙ fij= − k=i,j fikfkj 1 q2 ik − 1 q2 kj . (8)
As will be shown below, we can eliminate the f2
ik by finding (N (N− 1))
2 linear(-ly independent) equations for them, involving only q, and constants of motion. Before presenting this elimination, let us note that, while X.. = R ˙L+ [M, L]R−1= 0 is equivalent to
˙L =[L,M] (9)
implying the time-independence of
Qk:=1kTr Lk, (10)
EIGENVALUE-DYNAMICS OFF THE CALOGERO–MOSER SYSTEM 123 it is not difficult to see that L(n):= [Ln, Q], for any n ∈ N, also satisfies (9); hence
d dt Tr L+ λ1F+ λ2L(2)+ · · · + λN−1L(N−1) L(λ) k = 0, (11)
due to ˙L(λ) = [L(λ), M], where the Lax matrix L(λ), containing (N − 1) spectral parameters, λ1, λ2, . . . , λN−1, is defined in (11).
Another way to construct conserved quantities is to note that
TrQα1Fβ1Qα2Fβ2· · · QαlFβl= (−) βrTrXα1Kβ1Xα2Kβ2· · · XαlKβl, (12) with X(t)= X(0) + t ˙X(0), and K := [X, ˙X] constant, is a polynomial in t of degree α = lr=1αr, and can be expressed as a polynomial of the Tr Qk, k= 1, 2, . . . , α (with coefficients that are constants of motion).
2. Eliminating the
f
ij2As our aim is to eliminate the N(N− 1)/2 variables fij (appearing quadratically in (7)) it is natural to look at the quantities
Eαβ:= Tr QαFQβF = − i<j qiαqjβ+ qjαqiβ f2 ij= i<j Tij(αβ)fij2. (13)
Each of them gives rise to a conserved quantity
Jαβ= TrQαFQβFTr Lα+β− − Tr α r=1 Qα−rLQr−1FQβF+ β s=1 QαFQβ−sLQs−1F· Tr Lα+β−1Q+ · · · + +(−)α+βTr LαFLβF Tr Qα+β (14)
being of the form α+βk=0(−)kLα+β−kRk, with
Lα+β= Eα,β, Rα+β=(α + β)!1 Tr Qα+β,
Rk−1:= ˙Rk, Lk−1:= ˙Lk. (15)
Note that, in (14), Tr QαFQβF is not the only term involving F. However, it turns out that we can use the same technique (as for Jα,β) to eliminate these traces in terms of the conserved quantities Cαβ:= Tr LαFLβF by a recursive procedure. To get a feeling for how this works, let us work out the elimination of Tr LFQ2F. By using the same kind of construction as in (15), we can produce the three conserved quantities
Tr QL Tr L− Tr L2Tr Q= const., (16)
Tr LFQLF Tr L− Tr LFL2F Tr Q= const., (17)
Tr LFQ2F Tr L2−Tr LFQLF+ Tr LFLQFTr QL+
124 JOAKIM ARNLIND AND JENS HOPPE From these equations we can express Tr LFQ2F as a function of qi’s and the con-served quantities Tr L, Tr L2 and C12= Tr LFL2F. For traces of higher order in Q, we simply continue this process. Thus, from each Jαβ we obtain a linear equation in fij2 involving qi’s and a number of conserved quantities C1, . . . , Cγ.
As Eαβ= Eβα, and F= [L, Q], and the Cayley–Hamilton identity expressing QN as a polynomial in the Qk<N, we will focus on the set α β{Eαβ}N−2α,β=0 (since for α or β N − 1, Eαβ will contain QN). Expressing Eαβ in terms of the variables sk:= Tr Qk, and a certain number of conserved quantities, C1, C2, . . . , Cγ, taking values c1, c2, . . . , cγ, yielding
Eαβ= eαβ(s1, s3, . . . ; c1, . . . , cγ),
one gets N(N− 1)2 linear equations for the fi<j2 :
− i<j qα iqjβ+ qiβqjα f2 ij= eαβ(s1, s3, . . . ; c1, . . . , cγ), α, β = 0, . . . , N − 2; (19) thus obtaining explicit expressions
f2 i<j= ij =Fij[q ] (= Fji), (20) = − detqiαqjβ+ qiβqjα ,
ij= determinant of T with the (ij)th column replaced by
the N(N − 1) 2 -dimensional ‘vector’α β{eαβ} N−2 α,β=0. Using (20), (7) becomes .. qi= 2 k=i Fik[q; c1, . . . , cγ] (qi− qk)3 . (21)
As the constants of motion C1, . . . , Cγ are functions of the fij as well as q, ˙q, not all choices qi(0), ˙qi(0), c1, . . . , cγ will describe the original eigenvalue dynamics. Rather, for given qi(0), ˙qi(0), fij(0) (resp. X(0), ˙X(0) – with R(0) = 1) one should
first calculate the numerical values for c1, . . . , cγ – which (when inserted into (21)) then consistently present the second-order equations for the eigenvalues of X.
Equation (8) on the other hand, via 2fijf˙ij= ˙Fij= ˙q∇Fij, yields the first-order equations ∇Fij[q1, . . . , qN]· ˙q = −2 k fikfkjfji 1 q2 ik − 1 q2 jk , (22)
which, together with other equations, like (1, 1, . . . , 1)· ˙q=p =const., could be used to obtain
EIGENVALUE-DYNAMICS OFF THE CALOGERO–MOSER SYSTEM 125 (implying ¨q=Pj∇jP, resp. (21)) as describing the motion of the N eigenvalues of a
real symmetric N× N matrix satisfying X..= 0.
An alternative method to derive differential equations for just the eigenvalues (of a, in fact general Hermitian, matrix X(t) satisfying X.. = 0) is to differentiate detX(t)− λ1= detX(0)− λ1 + t ˙X(0) k times (with respect to t), where k is the rank of ˙X(0). This yields N equations that are linear in the kth derivatives, qj(k), and may be used to express them in terms of lower derivatives, and a number of constants.
3. The Case N=3
To illustrate the general formulas, let us work out the N= 3 case in detail: with fij=: k εijkfk, αi:= j,kεijk/qik2, (3)(4) read
.. q1= 2 f2 3 q3 12 −f22 q3 31 , q..2= 2 f2 1 q3 23 −f32 q3 12 , q..3= 2 f2 2 q3 31 −f12 q3 23 , (24) ˙ f1= α1f2f3, f˙2= α2f3f1, f˙3= α3f1f2. (25) In order to eliminate the three f ’s, we consider the three quantities
E00= Tr F2, E
10= Tr QF2, E11= Tr QFQF, (26)
appearing in the three conserved quantities J00= Tr F2(= − f 2) = const.,
J10=Tr QF2Tr L− Tr LF2Tr Q,
J11=Tr QFQFTr L2− 2Tr QFLFTr QL+Tr LFLFTr Q2. (27) Denoting Tr Q by q, the conserved quantity Tr LF2 by C
10 (and Tr LFLF by C11), Tr L by p (the total momentum which we assume to be nonzero) and 12Tr L2 by H (the ‘Energy’; ‘Hamiltonian’ aspects are treated separately [2]), as well as elim-inating Tr QFLF, Tr QL (cp. [15]) and Tr Q2 via the conserved quantities
M1,0:=
Tr LFQFTr L−Tr LFLFTr Q, (28)
K1,1:=Tr QLTr L− Tr L2Tr Q, (29)
C := Tr Q2Tr L2−Tr QLTr QL, (30)
(and, finally, assigning to all appearing constants of motion specific values – which we denote by the corresponding small letters) we obtain the desired expressions
e00= j00, e10= 1 p j10+ qc10 ,
126 JOAKIM ARNLIND AND JENS HOPPE e11= 1 2h j11− c11q2 +hp12m10+ qc11 k11+ 2hq =j11 2h + 1 hp2 hc11q2+ 2hm10q + m10k11− c11 4h cp2+ k2 11 , (31)
for the right-hand side e(q, c1, . . . , c6) of the linear system
− q2+ q2 3q3+ q2 1q1+ q2 2 2q2q3 2q3q1 2q1q2 f 2 1 f2 2 f2 3 = ee0010 e11 . (32)
Assuming (as we did all along) that all eigenvalues are different (which implies that ∼i<j(qi− qj)N−2 is nonzero), we can solve (19), resp. (32), obtaining
f2 1= h1[q] q31q12 = F1 [q], f2 2= h2[q] q12q23= F2 [q], f2 3= h3[q] q23q31= F3 [q], hj[q]= αq2+ γ q + ε − µqj− νqqj− λqj2, (33) where α = c11 2p2, γ = m10 p2 , µ = j10 p , ν =c10 p , λ = − 1 2j00, ε =j11p 2+ 2m 10k11−c2h11 cp2+ k2 11 4hp2 (34)
are the values c1, . . . , c6 assigned to the corresponding six conserved quantities C1, . . . , C6.
Unlike (5) (satisfying (4) identically), (33) will yield equations involving q, the values c1, . . . , c6, and ˙q (linearly), cf. (25):
∇Fj· ˙q = 2αjf1f2f3 ∇F1= 1 q31q12 (2αq −νq1+ γ ) 11 1 −(µ+2λq1+ νq) 10 0 − −h1 q31 −10 1 − h1 q12 −11 0 (+cycl.) (35)
EIGENVALUE-DYNAMICS OFF THE CALOGERO–MOSER SYSTEM 127 In order to eliminate ˙q explicitly, it is easiest to use
q11 q12 q13 α1∇F2−α2∇F1 ˙q = p 1 p k1,1+ 2H q 0 . (36)
The method mentioned at the end of Section 2, on the other hand, for ˙X(0) of rank 2 (e.g.), yields
q2+ q1 3q3+ q1 1q1+ q1 2 q2q3 q3q1 q1q2 .. q1 .. q2 .. q3 = µ1− 2 0 ˙q1˙q2+ ˙q2˙q3+ ˙q3˙q1 µ2− 2q1˙q2˙q3+ ˙q1q2˙q3+ ˙q1˙q2q3 with µ1= 2c11c22− |c12|2+ c22c33− |c23|2+ c33c11− |c31|2 and µ2= 2q1(0)(c22c33− |c23|2) + q2(0)(c33c11− |c31|2) + q3(0)(c11c22− |c12|2) where cij:= ˙X(0)ij.
Addendum: ‘The Goldfish’
After the submission of our Letter, a very interesting paper [7] appeared, in which it is pointed out that upon substituting
fij(t) = −(qi− qj)˙qi˙qj, (37)
the ‘Euler–Calogero–Moser equations (7), (8) reduce to the equations
..
qi= 2
j=i
˙qi˙qj
qi− qj (38)
previously studied by Calogero [5], Ruijsenaars and Schneider [14], as well as many others. Calogero mentions that these ‘goldfish’ [6] equations correspond to
X(t) = a√1+ c1t √c1c2t · · · √c1cNt c1c2t a2+ c2t · · · √c2cNt .. . ... ... ... √c 1cNt √c2cNt · · · aN+ cNt , (39) where ai= qi(0), ci= ˙qi(0).
128 JOAKIM ARNLIND AND JENS HOPPE So, while in the time development of the eigenvalues of a Hermitian matrix it is
[X(t), ˙X(t)]= ig 0 1· · · 1 1 0· · · 1 .. . ... ... ... 1 1· · · 0
that gives the reduction to the celebrated Calogero–Moser equations of motion, it is now ˙X itself, √c icj = S c20· · · 0 0 0· · · 0 .. . ... ... ... 0 0· · · 0 ST,
that is of rank 1, allowing (37). One should perhaps note that the phase-space functions gij(q, p) := 2(qi − qj)√pipj, quite amazingly, satisfy ordinary so(N )
Poisson-bracket relations, thus ‘explaining’ (37) from a Hamiltonian point of view. What about first-order equations (which our approach predicts)?
Due to det
X(t) − λ1= detS−1X(0)S − λ1 + t diagc2, 0, . . . , 0 being linear in t (for the choice (39)), its differentiation with respect to t leads to N conserved quantities, b, linear in the velocities ˙qj(t), resp. a matrix equation
A[q] · ˙q = b, (40)
which due to det A being (proportional to) i<j(qi− qj) (= 0, for noncoinciding
eigenvalues) can be inverted to the desired first-order equations, ˙q= A−1[q]b. For N= 3, e.g., they read
˙q1=q−1 12q31 q2 1b1− q1b2+ b3 , ˙q2=q−1 23q12 q2 2b1− q2b2+ b3 , ˙q3=q−1 31q23 q2 3b1− q3b2+ b3 , (41)
where b1=c1+c2+c3=p, b2= i=jaicj, and b3=(c1a2a3+c2a3a1+c3a1a2). It is straightforward to verify that solutions of (41) solve (38) (and knowing the relation of (38) to (39), vice versa), thus providing a first-order formulation of the goldfish.
Acknowledgement
J.H. would like to thank M. Bordemann, T. Damour, J. Fr ¨ohlich, G.M. Graf, K. Johansson, E.-J. Kim, M. Kontsevich, E. Langmann, N. Nekrasov, A. Per-elomov, J. Plefka, T. Ratiu, S. Theisen and S. Wojciechowski for discussions, the
EIGENVALUE-DYNAMICS OFF THE CALOGERO–MOSER SYSTEM 129 Albert Einstein Institut, the Institut des Hautes Etudes Scientifiques, and the Insti-tute of Theoretical Physics of ETHZ for hospitality, and F. Calogero for corre-spondence.
References
1. Arnlind, J. and Hoppe, J.: Eigenvalue dynamics off Calogero–Moser, IHES Preprint P/03/41 (July 2003).
2. Bordemann, M. and Hoppe, J.: Hamiltonian Reductions off Calogero–Moser, in prepa-ration.
3. Barucchi, G. and Regge, T.: Conformal properties of a class of exactly solvable N body systems in space dimension one, J. Math. Phys. 18 (1977), 1149.
4. Calogero, F.: Exactly solvable one-dimensional many-body problems, Lett. Nouvo
Ci-mento 13 (1975), 411–416.
5. Calogero, F.: Motion of poles and zeros of special solutions of nonlinear and linear partial differential equations, and related ‘solvable’ many-body problems, Nuovo
Cimen-to 43B (1978), 117–241.
6. Calogero, F.: The ‘neatest’ many-body problem amenable to exact treatment (a ‘gold-fish’?), Physica D 152–153 (2001), 78–84.
7. Calogero, F.: A technique to identify solvable dynamical systems, and a solvable gen-eralization of the goldfish many-body problem, Submitted to J. Math. Phys.
8. Gibbons, J. and Hermsen, T.: A generalization of the Calogero–Moser system,
Physica D 11 (1984), 337–348.
9. Haake, F.: Quantum Signatures of Chaos, Springer-Verlag, Berlin, 2001.
10. Jevicki, A.: Nonperturbative collective field theory, Nuclear Phys. B 376 (1992), 75–98. 11. Moser, J.: Three integrable hamiltonian systems connected with isospectral
deforma-tions, Adv. Math. 16 (1975), 197–220.
12. Nekrasov, N.: Infinite-dimensional algebras, many-body systems and gauge theories, In:
Advances in the Mathematics Science, Moscow Seminar in Mathematical Physics, Amer.
Math. Soc. Tranl. 191, Amer. Math. Soc., Providence 1999, pp. 263–299.
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Lett. 51 (1983), 943–946.
14. Ruijsenaars, S. and Schneider, H.: A new class of integrable systems and its relation to solitons, Ann. Phys. (NY) 170 (1986), 370–405.
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Physics Letters B 599 (2004) 118–128
www.elsevier.com/locate/physletb
Spinning membranes
Joakim Arnlind
a, Jens Hoppe
a, Stefan Theisen
baDepartment of Mathematics, Royal Institute of Technology, 10044 Stockholm, Sweden bAlbert-Einstein-Institut, Am Mühlenberg 1, D-14476 Golm, Germany
Received 27 June 2004; received in revised form 12 August 2004; accepted 13 August 2004 Available online 25 August 2004
Editor: P.V. Landshoff
Abstract
We present new solutions of the classical equations of motion of bosonic (matrix-)membranes. Those relating to minimal surfaces in spheres provide spinning membrane solutions in AdSp× Sq, as well as in flat space–time. Nontrivial reductions of
the BMN matrix model equations are also given.
2004 Published by Elsevier B.V.
1. Introduction
Starting from the premise that ‘membranes are to M-theory what strings are to string theory’ the search for classical solutions of membrane dynamics needs almost no justification. Given the additional fact that promising approaches to M-theory are within the context of matrix mechanics, solutions to its equations of motion are equally relevant. The observation that a discretized formulation of membrane dynamics is matrix mechanics[1]links the two.
In the context of string theory, the study of classical solutions was recently revived in[2](see[3]for a review of further interesting subsequent developments). Relating time-dependent classical solutions of the string sigma-model in an AdS5× S5target space–time to the dual conformal field theory, extends the testable features of the
duality between string theory andN = 4 SYM, i.e., of the AdS/CFT correspondence.
A likely extension of these ideas to M-theory is to consider their motion on maximally supersymmetric back-grounds which, aside from eleven-dimensional Minkowski space, are AdS7× S4and AdS4× S7. The former is
the near-horizon limit of a stack of N coincident M5 branes with12RAdS= RS= lp(π N )1/3and the latter is the
near-horizon limit of a stack of N M2 branes with 2RAdS= RS= lP(32π2N )1/6. The dualities between classical
supergravity on these background and the conformal field theories on the world-volume of the branes which create
E-mail addresses:joakim.arnlind@math.kth.se(J. Arnlind),hoppe@math.kth.se(J. Hoppe),theisen@aei.mpg.de(S. Theisen). 0370-2693/$ – see front matter2004 Published by Elsevier B.V.
J. Arnlind et al. / Physics Letters B 599 (2004) 118–128 119
them has been studied. In particular for the AdS7× S4case, if the duality holds, nontrivial information about the (0, 2) conformal field theory of N interacting tensor multiplets in six dimensions has been obtained, e.g., its
con-formal anomaly has been computed[4,5]. Direct verifications have, however, so far been impossible, mainly due to the lack of knowledge of the interacting (0, 2) theory.
One of the open problems in string theory is its quantization in nontrivial backgrounds, such as AdS5× S5.
An exception is the gravitational plane wave background which is obtained as the Penrose limit of the AdS5× S5
vacuum of type IIB string theory. In this background light-cone quantization leads to a free theory on the world-sheet whose spectrum is easily computed[6]. This opens the way to the duality between string theory and another sector of large-N SYM, which is characterized by large R-charge (∼√N ) and conformal weight (∼√N ). The
extensive activity to which this has led was initiated in[7].
The difficulties related to quantization are much more severe in M-theory where quantization on any back-ground is still elusive. The semiclassical analysis, which in the case of string theory provides valuable nontrivial information about the dual conformal field theory, can, however, be extended to M-theory. While the equations of motion of strings on AdS5× S5reduce, for special symmetric configurations, to classical integrable systems[8,9],
this is not as simple for membranes. Also, the integrable spin-chains which appear in the discussion of the dual gauge theory[10,11], have so far no known analogue in the (0, 2) tensor theory. However, the matrix model of the discrete light cone description of M-theory on plane waves obtained as Penrose limits of AdS4× S7and AdS7× S4
is known[7]and has been studied (see, e.g.,[12]).
In this Letter we present new solutions to bosonic matrix model equations (in Minkowski space, and of the BMN matrix model), as well as make a first step towards the semi-classical analysis of M-theory in AdSp× Sq
backgrounds, where we will find that the equations of motion, upon imposing a suitable ansatz, may be reduced to the equations describing minimal embeddings of 2-surfaces into higher spheres (as well as generalizations thereof).
2. The bosonic matrix model equations
The time evolution of spatially constant SU(N ) gauge fields inR1,d as well as of regularized membranes in
R1,d+1[1]is governed by equations of motion
(1) ¨Xi= − d j=1 [Xi, Xj], Xj
involving d Hermitean traceless N× N time-dependent matrices, with the constraint (‘Gauss law’, respectively, reflecting a residual diffeomorphism invariance in a light cone orthonormal gauge description of relativistic mem-branes) (2) d i=1 [Xi, ˙Xi] = 0.
As shown in[13], solutions of these equations may be found by making the ansatz
(3)
Xi(t)= x(t)Rij(t)Mj,
withR(t) = eAϕ(t )a real, orthogonal d× d matrix and {M
j}dj=1time-independent N× N matrices. Define M:=
(M1, M2, . . . , Md) and requireA2M= − M. Imposing that no component of both M andA M vanishes, restricts
d to be even. By a suitable change of basis one can always castA into the form
(4) A = diag(J, . . . , J ) with J = 0 1 −1 0 ,
