arX

iv:1

211.

3727

v2 [

hep-

th] 7

Jan 2

013

Imperial-TP-AS-2012-01

On (non)integrability of classical strings in p-brane backgrounds

A. Stepanchuk and A.A. Tseytlin

The Blackett Laboratory, Imperial College, London SW7 2AZ, U.K.

Abstract

We investigate the question of possible integrability of classical string motion in curved p-

brane backgrounds. For example, the D3-brane metric interpolates between the flat and the

AdS5 S5 regions in which string propagation is integrable. We find that while the point-like string (geodesic) equations are integrable, the equations describing an extended string in

the complete D3-brane geometry are not. The same conclusion is reached for similar brane

intersection backgrounds interpolating between flat space and AdSk Sk. We consider, inparticular, the case of the NS 5-brane - fundamental string background. To demonstrate non-

integrability we make a special pulsating string ansatz for which the string equations reduce

to an effective one-dimensional system. Expanding near this simple solution leads to a linear

differential equation for small fluctuations that cannot be solved in quadratures, implying non-

integrability of the original set of string equations.

a.stepanchuk11@imperial.ac.ukAlso at Lebedev Institute, Moscow. tseytlin@imperial.ac.uk

1 Introduction

Strings in curved space are described by 2d sigma models with complicated non-linear equations of

motion. This precludes, in most cases, a detailed understanding of their dynamics. Integrability

selects a subclass of models or target space backgrounds for which one may hope to come as close

as possible to the situation in flat space, i.e. to have a complete quantitative description of classical

string motions and then also of the quantum string spectrum.

A prominent maximally symmetric 10-dimensional example is provided by the superstring in the

AdS5 S5 space-time: much progress towards its solution, that was achieved recently [1], is basedon its integrability. One may wonder if integrability can also apply to string motion in closely

related but less symmetric p-brane backgrounds [24], e.g. the D3-brane background [4,5] which is

a 1-parameter (D3-brane charge Q) interpolation between flat space (Q = 0) and the AdS5 S5space (Q =). While string theory is integrable in these two limiting cases, what happens at finiteQ is a priori an open question. This is the question we are going to address in this paper.

Since quantum integrability of a bosonic model or integrability of its superstring counterpart is

essentially implied (leaving aside some exceptional cases or anomalies) by the integrability of the

corresponding classical bosonic sigma model the study of the latter is the first step. Following the

same method as used previously in [6,7] we will demonstrate non-integrability of classical extended

string motion in the D3-brane background (particle, i.e. geodesic, motion is still integrable). We

will also reach similar conclusions for other p-brane backgrounds that interpolate between flat space

and integrable AdSn Sk backgrounds.As there is no general classification of integrable 2d sigma models, let us start with a brief

survey of some known classically integrable bosonic models describing string propagation in curved

backgrounds. A major class of such models have a target space metric GMN of a symmetric space,

including spheres [8] (and other related constant curvature spaces [9]), group spaces (i.e. principal

chiral model) [10] and various other G/H coset models [1012]. Integrability is also preserved by

some anisotropic deformations of the group space [13,14], the 2-sphere [15] and the 3-sphere [1618]

models. Attempts to find non-diagonal generalisations of the principal chiral model by studying

conditions for the existence of the corresponding zero curvature (Lax) representation were made

in [19, 20] and also in [21].

Allowing for the parity-odd antisymmetric tensor (BMN ) coupling leads, in particular, to the

WZW and related gauged WZW models [22, 23]. The generalised principal chiral model with

the WZ term having an arbitrary coefficient is also integrable [2426]. Some other anisotropic

integrable models with 3d target space were constructed in [18] and new examples of integrable

coset-type models with extra WZ coupling were found in [26].

There are also examples of integrable pp-wave models related to (massive) integrable 2d models

in light-cone gauge, see e.g. [2731].

1

Given an integrable model one can generate a family of other integrable models (i.e. find new

integrable target space backgrounds) by performing transformations that preserve the classical 2d

equations of motion, e.g. using 2d abelian duality (T-duality) or non-abelian duality, combined with

field redefinitions (coordinate transformations). Examples include, in particular, various gaugings

or marginal deformations of WZW and related models, see e.g. [3236]; for a more recent example

related to AdS5 S5 see [37].1 For discussions of an interplay between T-duality and integrabilitysee [21, 40].

Given a particular string sigma model, to prove its integrability one is to find a zero curvature

representation implying its equations of motion. There is no general method for accomplishing this

and the full set of necessary conditions for the existence of a Lax pair is not known. For example,

the presence of a non-abelian isometry group is not required.2 Among the necessary conditions

should, of course, be that any consistent truncation of the 2d equations of motion to a 1d set of

equations has to represent an integrable mechanical system. For example, rigid string motion on a

sphere or AdS space are described by an integrable Neumann-Rosochatius model [41].

In particular, the point-like string, i.e. geodesic, motion should be integrable. The geodesic

motion in a general class of higher-dimensional (rotating) black hole backgrounds is known to be

integrable [42].3 It is thus not surprising that geodesics are also integrable in p-brane backgrounds

that are discussed below.4

To disprove integrability of a given sigma model it is therefore sufficient to show that there exists

at least one 1d truncation of its equations of motion that is not an integrable system of ordinary

second-order differential equations. This can be done by demonstrating that the corresponding

motion is chaotic, i.e. by showing that the equations for small variations around solutions in phase

space cannot be solved in quadratures [48]. We shall review this method in section 2 (and Appendix

A) and illustrate it on a simple example of the non-integrable two-particle system with the potential

V (x1, x2) =12x

21x

22.

This approach was recently used to show that string motion in AdS5T p,q, AdS5Y p,q [6,7] andconfining supergravity backgrounds [49] is not integrable.5 Here we shall follow a similar method

1One may also find new models by taking orbifolds of known integrable models (see, e.g., [38]), though divisionover discrete subgroups may also lead to a breakdown of integrability (cf. [39]).

2Indeed, generic sigma models obtained from gauged WZW models by solving for the 2d gauge field do not haveisometries (or have only abelian isometries) and yet, following from a gauged WZW action, they must be integrable.

3Also, a particular stationary string motion in AdS-Kerr-NUT backgrounds is integrable as well [43].4In addition, if the general string motion is integrable, then motion of a subclass of strings with large momentum

along a null geodesic or, equivalently, string motion in a pp-wave background which is a Penrose limit of a givencurved background should of course be integrable as well. For the p-brane backgrounds that we are going to considerstring motion in the pp-wave backgrounds representing Penrose limits turn out to be solvable (see e.g. [4447]) asin the light-cone gauge description the string moves in a quadratic potential. However, since the coefficient of thispotential happens to depend on in general, the corresponding models do not appear to be integrable.

5Spaces like T 1,1 and Y p,q are special in that they have isometries, preserve supersymmetry and have integrable

2

to study integrability of strings in curved backgrounds that interpolate between two integrable limits

flat space and AdSn Sm T k.The main cases include the D3-brane, the D5-D1 background and the four D3-brane intersection

that interpolate between flat space and AdS5S5, AdS3S3T 4 and AdS2S2T 6 respectively.While geodesic motion in these backgrounds is integrable, making a special ansatz for string motion

we will find that the resulting 1d system of equations is not integrable by applying the variational

non-integrability technique for Hamiltonian systems as used in [7] (section 3).

In section 4 we shall study the NS5-F1 background and show that integrability is absent for

generic values of the two charges Q1 and Q5, but it is restored in the limit when Q5 .In Appendix A we also give a brief summary of the ideas from differential Galois theory on

which the non-integrability techniques for Hamiltonian systems are based on. In Appendix B we

shall demonstrate non-integrability of string propagation in the pp-wave geometry T-dual to the

fundamental string background [30, 51].

2 On the non-integrability of classical Hamiltonian systems

Integrability of classical Hamiltonian systems is closely related to the behaviour of variations around

phase space curves. Based on this observation, ref. [52] obtained necessary conditions for the

existence of additional functionally independent integrals of motion. These conditions are given

in terms of the monodromy group properties of the equations for small variations around phase

space curves. Using differential Galois theory, refs. [48, 53, 54] improved these results by showing

that integrability implies that the identity component of the differential Galois group of variational

equations normal to an integrable plane of solutions must be abelian (see Appendix A for some

details).

This necessary integrability condition allows one to show non-integrability from the properties of

the Galois group. However, determining the Galois group can be difficult and therefore one usually

takes a slightly different route in analysing the normal variational equation (NVE).

One considers a special class of solutions of the NVE which are functions of exponentials, loga-

rithms, algebraic expressions of the independent variables and their integrals and which are known

as Liouvillian solutions [55]. The existence of such solutions is equivalent to the condition that the

identity component G0 of the Galois group is solvable (see, for example, theorem 25 in [56]). This

in turn implies that if the NVE does not admit Liouvillian solutions then G0 is non-solvable (and

thus non-abelian) leading to the conclusion of non-integrability.

For Hamiltonian systems the NVE is a second order linear homogeneous differential equation and

geodesics (in fact, super-integrable with more conserved quantities than degrees of freedom [50]). Yet, the corre-sponding 2d sigma models fail to be integrable.

3

for such equations with rational coefficients Liouvillian solutions can be determined by the Kovacic

algorithm [57] which fails if and only if no such solutions exist. Therefore, whenever no Liouvillian

solutions of the NVE are found by the Kovacic algorithm one can deduce non-integrability of a

Hamiltonian system.

An important subtlety here is that we first need to algebrize the NVE, i.e. rewrite it as a

differential equation with rational coefficients. This can be done by means of a Hamiltonian change

of the independent variable and it was shown in [58] that the identity component of the Galois

group is preserved under this procedure.

In summary, if the algebrized NVE is not solvable in terms of Liouvillian functions or, equivalently,

if the component of the quadratic fluctuation operator normal to an integrable subsystem does not

admit Liouvillian zero modes, the original system is non-integrable. This is consistent with the usual

definition of integrability in the sense of Liouville which, for Hamiltonian systems, implies that the

equations of motion should be solvable in quadratures, i.e. in terms of Liouvillian functions.

The main steps to prove non-integrability of a Hamiltonian system are thus:

Choose an invariant plane of solutions. Obtain the NVEs, i.e. the variational equations normal to the invariant plane. Check that the algebrized NVEs have no Liouvillian solutions using the Kovacic algorithm.This approach was recently used in [7,49] to show the non-integrability of string motion in several

curved backgrounds and we shall also follow it here.

Let us first explain what the algebraization procedure of the last step means. A generic NVE is

a second order differential equation of the form

+ q(t) + r(t) = 0 . (2.1)

A change of the variable t x(t) is called Hamiltonian if x(t) is a solution of the Hamiltoniansystem H(p, x) = p

2

2 + V (x) where V (x) is some potential. Then x satisfies the first integral

equation 12 x2 + V (x) = h, implying that x (and also x) is a function of x only. Changing the

variable t x(t) in (2.1) we obtain (here ddx)

+( xx2

+q(t(x))

x

) +

r(t(x))

x2 = 0 . (2.2)

We should now require that the coefficients in this equation are rational functions of x.

Let us consider the following simple example of a Hamiltonian system with the canonical variables

4

(xi, pi) (i = 1, 2) [59]

H =1

2(p21 + p

22) +

1

2x21x

22. (2.3)

A choice of an invariant plane, referred to above, is P = {(x1, x2, p1, p2) : x2 = p2 = 0} along whichthe solution curves are x1() = + const, p1 = , =

2H. The system along the plane P with

the coordinates (x1, p1) is integrable since the only required constant of motion is provided by the

Hamiltonian.

To determine the direction normal to the solution curves of this integrable subsystem we can

use the Hamiltonian vector field along the curves Xx1 = , Xp1 = 0, Xx2 = 0, Xp2 = 0. Thus

the normal direction is along x2 and p2 and expanding around x2 = 0 with x2 we obtain thefollowing normal variational equation (NVE)

() = x21() (), x1() , (2.4)

which is solved by the parabolic cylinder functions D1/2((i 1)) and D1/2((i + 1)). Theseare not Liouvillian functions and this implies the non-integrability of the original Hamiltonian (2.3).

To illustrate the previously discussed algebraization procedure let us consider the Hamiltonian

H =1

2(p21 + p

22) +

1

8(x21 x22)2 (2.5)

which is equivalent to (2.3) by the trivial canonical transformation

x1 x1 x22

, x2 x1 + x22

, p1 p1 p22

, p2 p1 + p22

. (2.6)

Restricting to the invariant plane P = {(x1, x2, p1, p2) : x2 = p2 = 0} we obtain an integrablesubsystem

x1 = p1, p1 = 12x31. (2.7)

The Hamiltonian vector field on this solution plane is Xp1 = 12 x31, Xp2 = 0, Xx1 = x1, Xx2 = 0.The NVE is now obtained by expanding the Hamiltonian equations for the coordinates normal to

the invariant plane, i.e. x2 and p2, along (2.7). Denoting x2 = we thus obtain

=1

2x21(t) , x1(t) = 2

3/4h1/4 sn((h

2

)1/4t,1

)(2.8)

where x1 is a solution of (2.7) representing the curves in P parametrised by the Hamiltonian h > 0.

5

We can algebrize this NVE by changing the variable t x = x1(t). This leads to ( = ddx)

2x3

8h x4 2x

2

8h x4 = 0 . (2.9)

Further changing the variable = (8h x4)1/4 one obtains the NVE in the normal form

+8hx2 + 2x6

(8h x4)2 = 0. (2.10)

This equation is solved in terms of hypergeometric functions, i.e. it has no Liouvillian solutions.

This implies non-integrability of (2.5), in agreement with the previous discussion of of the equivalent

system (2.3).

To summarise, proving non-integrability of a Hamiltonian system can be achieved by demon-

strating that given an integrable subsystem defined by some invariant plane in the phase space, the

corresponding normal variational equations are not integrable in quadratures and thus do not admit

sufficiently many conserved quantities.

3 String motion in p-brane backgrounds

In what follows we shall consider classical bosonic string motion in a curved background. In the

conformal gauge the action is

I = 14

dd

[GMN (Y )aY

MaY N + abBMN (Y )aYMbY

N], (3.1)

and the corresponding equations of motion should be supplemented by the Virasoro constraints

GMN YM Y N = 0 , (3.2)

GMN (YM Y N + YM Y N ) = 0 . (3.3)

In this section we shall consider the case of BMN = 0 and the target space metric given by the

following p-brane ansatz (Q 0)

ds2 = f2(r) dxdx + f2(r)(dr2 + r2 d2k) , (3.4)

f(r) =(1 +

Q

rn

)m, n,m 6= 0 (3.5)

= diag(1,+1, ...,+1), , = 0, ..., p

where d2k is the metric on a k-sphere.

6

For k = 8p, n = 7p > 0 andm = 14 this metric describes the standard single p-brane geometryin D = 10 supergravity. It may be supported by a R-R field strength and dilation backgrounds, but

since they do not couple to the classical bosonic string we shall ignore them here.

Keeping k, n,m, p generic also allows one to describe lower-dimensional backgrounds representing

(up to a flat factor) some special brane intersection geometries. In particular, the metric (3.4) with

nm = 1 describes a one-parameter interpolation [60, 61] between flat space (for Q = 0) and the

AdSp+2 Sk space (for Q).6Special cases include:

(i) n = 4, m = 14 , k = 5, p = 3: D3-brane interpolating between flat space and AdS5 S5;(ii) n = 2, m = 12 , k = 3, p = 1: D5-D1 (or NS5-F1 with non-zero BMN , see next section)

background [62, 63] with Q5 = Q1 = Q interpolating between flat 10d space and AdS3 S3 T 4;(iii) n = 1, m = 1, k = 2, p = 0: four equal-charge D3-brane intersection [64] (or U-duality

related backgrounds, see, e.g. [65]) interpolating between flat 10d space and AdS2 S2 T 6; thismay be viewed as a generalised Bertotti-Robinson geometry.

Below we shall show that the point-like string (i.e. geodesic) motion in this background is

integrable. We shall then demonstrate that extended string motion in the p-brane geometry with

p = 0, .., 6 and in the backgrounds (ii), (iii) is not integrable for generic values of Q, despite

integrability being present in the limits Q = 0 and Q =.

3.1 Complete integrability of geodesic motion

The symmetries of the metric (3.4) are shifts in the x coordinates giving p+1 conserved quantities

and spherical symmetry which gives k/2 conserved commuting angular momenta for even k and

(k+1)/2 for odd k (generators of the Cartan subgroup of SO(k+1)). Thus for k 3 the sphericalsymmetry does not, a priori, provide us with sufficiently many conserved quantities.

Parametrising the k-sphere by k+1 embedding coordinates yi the effective Lagrangian for point-like

string motion is

L = f2(r)xx + f2(r)(r2 + r2~y2) + (~y2 1). (3.6)

The first integrals for x and r are E = f2(r)x and the Hamiltonian respectively. The equation

of motion for yi reads

(f2(r)r2yi) + yi = 0 . (3.7)

6More precisely, the metric (3.4) interpolates between the flat-space region (rn Q) and the AdSp+2Sk region

(rn Q). Moreover, since f(r) =(c+ Q

rn

)mgives a solution for any constant c including c = 0, AdSp+2S

k is alsoan exact solution.

7

Taking the scalar product of this equation with ~y and with ~y and using yiyi = 1, yiyi = 0 we

conclude that

= f2(r)r2~y2 = f2(r)r2m2 , m2 = f4(r)r4~y2 = const . (3.8)

Substituting this expression into (3.7) allows one to eliminate giving

(f2(r)r2yi) +m2f2(r)r2yi = 0 . (3.9)

Multiplying this equation by f2(r)r2yi for fixed i and integrating once we obtain the constants of

motion Ci (no summation over i)

f4(r)r4(yi)2 +m2(yi)2 = Ci ,i

Ci = 2m2 . (3.10)

We can rewrite the Ci in terms of phase space coordinates (yi, pi = 2f2(r)r2yi) as

Ci =1

4(pi)2 +m2(yi)2 . (3.11)

The integrals Ci are easily checked to be in involution. Only k of the k + 1 integrals Ci are

functionally independent and correspond to the angles on the k sphere. Altogether with the p+1

constants for the x coordinates and the Hamiltonian this gives in total k+p+2 constants of motion

which is the same as the number of degrees of freedom.

3.2 Non-integrability of string motion

Let us now study extended string motion in the geometry (3.4),(3.5). We shall choose a particular

pulsating string ansatz for the dependence of the string coordinates on the world-sheet directions

(, ): we shall assume that (i) only x0, r and two angles , of S2 Sk (with d22 = d2 +sin2 d2) are non-constant, and (ii) x0, r, depend only on while depends only on , i.e. 7

x0 = t(), r = r(), = (), = () . (3.12)

7Note that when only one of the angles of Sk is non-constant, 1-d truncations are not sufficient to establishnon-integrability, see below.

8

This ansatz is consistent with the conformal-gauge string equations of motion and the Virasoro

constraint (3.2). The remaining string equations of motion and the Virasoro constraint give

t = Ef2 , E = const , (3.13)

= = const , = , (3.14)

2 (f2r) = r(f

2)E2 + r(f2)r2 + r(r

2f2)(2 2 sin2 ) , (3.15) (f

2r2) = f2r22 sin cos , (3.16)E2 = r2 + r22 + 2r2 sin2 . (3.17)

Thus the string is wrapped (with winding number ) on a circle of S2 whose position in r and

changes with time. The equations for r and can be derived from the following effective Lagrangian

L = f2(r2 + r22 2r2 sin2 + E2

), (3.18)

with the corresponding Hamiltonian restricted to be zero by (3.17). We shall show that this 1d

Hamiltonian system is not integrable, implying non-integrability of string motion in the p-brane

background (3.4).

Let us choose as an invariant plane {(r, ; pr, p) : = pi2 , p = 0}, corresponding to the a stringwrapped on the equator of S2 and moving only in r. Then (3.17) gives r2 + 2r2 = E2 which is

readily solved by (assuming e.g. that r(0) = 0)

r = r() =E

sin() . (3.19)

According to the general method described in section 2, we have to show that small fluctuations

near this special solution are not integrable, or, more precisely, that the variational equation normal

to this surface of solutions parametrized by E and has no Liouvillian solutions along the curves

inside the surface.

For the invariant subspace { = pi2 , p = 0, r = r, pr = 2f2(r) r} the Hamiltonian vector field is

Xr = r, Xpr = pr = 4ff r2 + 2f2 r, X = 0, Xp = 0 (3.20)

and thus the normal direction to this plane is along and p. Expanding the Hamiltonian equation

for around = pi2 , = 0, r = r one obtains the NVE ( )

+ 2 cot()(1 +

E

sin()

f (E sin())

f(E sin())

) 2 = 0 . (3.21)

If f /f is a rational function (as is the case for (3.5)) this equation can be algebrized through the

9

change of variable x = sin() giving

+(2x x

1 x2 + 2E

f (E x)

f(E x)

) 1

1 x2 = 0. (3.22)

Using the explicit form of f(r) in (3.5) this NVE reduces to

+( 2x x

1 x2 2mnnQ

x(nQ+ Enxn

)) 11 x2 = 0. (3.23)

We shall assume that 6= 0 as otherwise the string is point-like and we go back to the case ofgeodesic motion.

Bringing (3.23) to the normal form by changing the variable to (x) = g(x)(x), where g(x) is a

suitably chosen function, one obtains

+[ 2 + x24(x2 1)2

(m 1)mn2x2(1 +Bxn)2

+mn(n 1 (n 2)x2)x2(x2 1)(1 +Bxn)

] = 0 , B E

n

Qn. (3.24)

There are two special limits of (3.23) and (3.24)

Q 0 : + 2 + x2

4(x2 1)2 = 0 , (3.25)

Q : + 2x2 + x4 4(mn)2(x2 1)2 + 4mn(1 3x2 + 2x4)

4x2(x2 1)2 = 0 , (3.26)

corresponding to the flat space-time and AdS2 S2 for nm = 1.We can now apply the Kovacic algorithm8 to determine whether these NVEs do not admit

Liouvillian solutions and thus the identity component of their Galois group is not solvable which

would imply non-integrability.

For the special cases Q 0 and for Q with nm = 1 Liouvillian solutions are found whichis consistent with the fact that string motion in flat space-time and in AdS2 S2 is integrable.However, for a finite value of Q Liouvillian solutions do not exist for generic values of E, for the

cases of a p-brane background with p = 0, .., 6 and the intersecting brane backgrounds of (ii) and

(iii) mentioned above. This implies non-integrability of string motion in those special cases of the

general background (3.4),(3.5).

Finally, let us comment on the special case of a 7-brane in 10 dimensions when the transverse

space is 2-dimensional, i.e. the string motion is described by

L = f2(r)axax + f2(r)(ar

ar + r2aa). (3.27)

8We use Maples function kovacicsols.

10

It is easy to see that if we truncate this model to a 1-parameter system by taking each of the

string coordinates to be a function of or only then the corresponding equations always admit

constants of motion for x and , and the Virasoro condition then provides the solution for r.

To test integrability in this case one is to consider some more non-trivial truncations. Assuming

the spatial coordinates xi are constant the corresponding effective metric is 3-dimensional: ds2 =

f2(r)dt2 + f2(r)(dr2 + r2d2). Integrability of string motion in such a background deservesfurther study.

4 String motion in NS5-F1 background

Let us now include the possibility of a non-zero BMN coupling and consider the case of string motion

in a background produced by fundamental strings delocalised inside NS5 branes [3, 62, 66, 67] (here

we use the notation x0 = t, x1 = z and i, j = 2, ..., 5)

ds2 = H11 (r)(dt2 + dz2) + dxidxi +H5(r)(dr2 + r2d23) , (4.1)d23 = d

2 + sin2 d2 + cos2 d2 ,

B = H11 (r) dt dz +Q5 sin2 d d , (4.2)

H5 = 1 +Q5r2

, H1 = 1 +Q1r2

, (4.3)

with the dilation given by e2 = H1(r)H5(r) . The corresponding string Lagrangian in (3.1) L = (GMN+BMN )+Y

MYN (where = 0 1) is

L = r2

r2 +Q1+uv + +x

ixi +r2 +Q5

r2+rr

+ (r2 +Q5)(+ + sin

2 ++ cos2 +

)+Q5 sin

2 (+ +) , (4.4)u t+ z , v t+ z .

It interpolates between the flat space model for Q1, Q5 = 0 and the SL(2) SU(2) WZW model(plus 4 free directions) for Q1, Q5 . Both of these limits are obviously integrable.

Another integrable special case is Q1 = 0, Q5 when we get the SU(2) WZW model plusflat directions.

The opposite limit of Q5 = 0, Q1 is described (after a rescaling of coordinates) by L =r2 +uv + +rr+ r

2(++ sin2 ++ cos

2 +)+ free directions. Solving the

equations for u, v as ( = )

r2+u = f(+) , r2v = f(

) , (4.5)

11

one finds the effective Lagrangian for r being

L = +rr f()f(+)

r2+ r2(+ + sin

2 ++ cos2 +) . (4.6)

As we shall find below, this model is not integrable.9

Let us note that in the limit Q5 the action (4.4) reduces to a combination of the SU(2)WZW model, flat directions and the following 3-dimensional sigma model

L = 11 +Q1e2

+uv + Q5+ , ln r , (4.7)

This model is T-dual to a pp-wave model with an exponential potential function [34,51] and is thus

related to the SL(2) WZW model by a combination of T-dualities and a coordinate transformation

(it can be interpreted, upon a rescaling of u, v, as an exactly marginal deformation of the SL(2, R)

WZW model [35]). It should thus be integrable. Indeed, solving the equations for z t as in (4.5)we get the following effective Lagrangian for (cf. (4.6))

L = Q5+ f()f(+)(1 +Q1e2) . (4.8)

Since f()f(+) can be made constant by conformal redefinitions of (reflecting residual gauge

freedom in the conformal gauge) this model is equivalent to the Liouville theory, i.e. it is integrable.

Indeed, the results of our analysis below are consistent with this conclusion.

A point-like string does not couple to B-field, so geodesic motion in the background (4.1) can

be shown to be integrable in the same way as in the previous section. To study extended string

motion let us consider the following ansatz describing the probe string being stretched along the

fundamental string direction z (that may be assumed to be compactified to a circle) and the S3

angle , i.e.

t = t(), z = z(), xi = 0, r = r(), = (), = (), = () . (4.9)

Then the string equations are solved by

t = 1H1(r()) + 2 , z = 2 , = , =Q5

r2()H5(r()). (4.10)

9Let us note that the constant term in the harmonic function of the fundamental string background (1 in H1) canbe changed by a combination of T-duality, coordinate shift and another T-duality [51, 61], so that the fundamentalstring background is not integrable for any value of Q1 if it is not integrable for large Q1. A similar shift in theharmonic function can also be achieved for D-brane backgrounds but this involves S-duality (mapping e.g. F1 toD1) which is not a symmetry of the world-sheet string action; thus there is no contradiction with integrability of theD3-brane background at large Q.

12

The resulting 1d subsystem of equations for r and is described by the following effective Lagrangian

L = H5(r)[r2 + r22 2r2 sin2 Q252r2H25 (r) cos2

]+ 21H1(r) , (4.11)

which should be supplemented by the consequence of the Virasoro constraint

H5(r)[r2 + r22 + 2r2 sin2 +Q25

2r2H25 (r) cos2 ] 21H1(r) = 212 , (4.12)

implying that the Hamiltonian corresponding to (4.11) is equal to 212. This system admits the

special solution

=

2, r = r() , H5(r)( r

2 + 2r2) = 212 + 21H1(r) , (4.13)

which may be chosen as an invariant plane in the phase space. The corresponding Hamiltonian

vector field is

Xr = r, Xpr = 2 ( rH5(r)), X = 0, Xp = 0 . (4.14)

Expanding along the normal direction to this solution plane we find for the NVE ( = pi2 + )

+(H 5(r)H5(r)

+2

r

)(21H1(r)

H5(r)+

212H5(r)

2r2)1/2

2(1 Q

25

r4H25 (r)

) = 0. (4.15)

Changing the variable to x = r() we obtain the NVE in the normal form ( ddx)

(x) + U(x)(x) = 0 , (4.16)

where

U(x) =1

4x2

[Q5(Q5 2x2)(Q5 + x2)2

+4Q5x

4 2x6 + 2Q1q2(2Q5 + x2)(Q5 + x2)(Q1q2 + x2[Q5 + x2 q2(1 + )])

+3(Q1q

2 + x4)2

(Q1q2 + x2[Q5 x2 + q2(1 + )])2]= 0 , q 1

, 22

1. (4.17)

One finds that already for = 0, i.e. for 2 = 0, this NVE has no Liouvillian solutions for general

values of Q1 and Q5, implying non-integrability of string motion in the NS5-F1 background with

generic values of the charges. Our results are summarised in the table below.

Let us discuss some special cases. Taking Q1 = 0 or Q5 = 0 one can absorb the remaining brane

charge by rescaling x, q and 1. The resulting equations do not admit Liouvillian solutions for

arbitrary values of q which implies non-integrability of string motion in the NS5-brane or in the

fundamental string backgrounds.

13

In the limit Q1 , Q5 , in which the non-trivial part of the string action becomes that ofthe SL(2, R) SU(2) WZW model, one finds Liouvillian solutions in agreement with the expectedintegrability. The same applies to the case of Q1 0, Q5 described by the SU(2) WZWmodel plus free fields and, in fact, to the model with Q5 and arbitrary Q1 described by(4.7)(4.8).

In the opposite case of Q5 0, Q1 described by (4.6) integrability appears to be absent aswe did not find Liouvillian solutions (see table).10

Since 2d duality transformations of string coordinates preserve (non)integrability, similar conclu-

sions can be reached for string backgrounds related to this NS5-F1 background or to its limits via

T-dualities (and coordinate transformations). In particular, this applies to the pp-wave background

related to the fundamental string by T-duality [51] which is studied in in Appendix B.

5 Concluding remarks

We have shown that for various p-brane backgrounds, for which the string sigma model interpo-

lates between integrable flat and coset or WZW models and which has integrable geodesics, the

corresponding extended classical string motion is not integrable in general.

To demonstrate non-integrability we considered particular extended string motion for which the

dynamics reduces to an effective one-dimensional system. The latter has special integrable subclasses

of solutions, but perturbing near them leads to linear second-order differential equations that cannot

be solved in quadratures.

It would be interesting to understand better why switching on a non-zero D3-brane charge or

moving away from the throat (AdS5 S5) region leads to a breakdown of string integrability.Together with similar previous results [6,7,49] this supports the expectation that integrability of

classical string motion is a rare phenomenon. String integrability is thus a much more restrictive

constraint than integrability of particle motion. Still, in the absence of a general classification of

integrable 2d sigma models one cannot rule out the possibility that there are still interesting exam-

ples of integrable backgrounds (that cannot be obtained from known solvable cases by coordinate

transformations combined with T-dualities, cf. [29, 30]), that remain to be discovered.

10Let us note that the results about the existence of Liouvillian solutions turn out to be independent of whether thetruncated solutions contains any contribution from the B-flux of the fundamental string solution ( 6= 0 or = 0),reflecting a special choice of our ansatz (4.9).

14

limit U(x) in NVE (x) + U(x)(x) = 0, Q5Q1 rescalingQ5 0 Q1 0 2q2(1+)+x24(q2(1+)x2)2 -Q5 Q1 (q2)(x2+q2(2x2)4(x2q2(1+x2))2 Q1Q5 Q1 0 14x2 -Q5 0 Q1 Q21q4+10Q1q2x4+x82Q1q4x2+2q2x64(Q1q2xx5+q2x3)2 -

Q5 Q1 6= 0 (q2(1+))(x2q2(2+x2(1+)))4(x2q2(1+x2(1+)))2x Q1x,

1,2 Q11,2

Q5 6= 0 Q1

[2q4(x2 2) + 21q2(1 + x2)2(1 + 5x2) 21q4(1 + 5x2 + x4)

+ x2((1 + x2)2 q2)((1 + x2)2 + q2(2x2 1))]

[4(1 + x2)2(x2 + x4 q2(1 + x2))2

]1

x Q5x,1,2

Q51,2

Q5 0 Q1 6= 0 x8+q4(12x2(1+))+2q2x4(5+x2(1+))4x2(x4q2(1+x2(1+)))2x Q1x,

1,2 Q11,2

Q5 6= 0 Q1 0 (1+x2)4+2q2(x21)(1+x2)2(1+)q4(2x21)(1+)24x2(1+x2)2(1+x2q2(1+))2x Q5x,

1,2 Q51,2

limit spacetime2-form contributes to the

truncated systemNVE has Liouvillian

solutions = 0 6= 0 = 0 6= 0

Q5 0 Q1 0 R1,9 Q5 Q1 AdS3S3R4 Q5 Q1 0 R1,6 S3 Q5 0 Q1 - Q5 Q1 6= 0 - Q5 6= 0 Q1 - Q5 0 Q1 6= 0 F1 Q5 6= 0 Q1 0 NS5

Acknowledgments

AAT is grateful to G. Gibbons for a useful comment. He also thanks K. Sfetsos for pointing out

possible connection to ref. [68]. This work was supported by the STFC grant ST/J000353/1. A.A.T.

was also supported by the ERC Advanced grant No.290456.

A Integrability and differential Galois theory

Integrability in the sense of Liouville implies that the equations of motion are solvable in quadra-

tures, i.e. the solutions are combinations of integrals of rational functions, exponentials, logarithms

15

and algebraic expressions. Solutions of this type are referred to as Liouvillian functions. Thus the

question of integrability is related to the question of when differential equations admit only Liou-

villian solutions. This is answered by differential Galois theory and the main integrability theorem,

which our analysis relies on, can be stated as follows [69]:

Theorem Let H be a Hamiltonian defined on a phase space manifold M with an associated

Hamiltonian vector field X. Let P be a submanifold of M which is invariant under the flow of

X with the invariant curves and let X|P be completely integrable. Also let G be the differentialGalois group of the variational equation of the flow of X|P along normal to P . If X is completelyintegrable on M then the largest connected algebraic subgroup of G which contains the identity is

abelian.

Since differential Galois theory is not frequently used in physics we shall give a brief introduction

and illustrate some basic concepts on a simple example below. We shall follow ref. [70]. First, let

us give some basic definitions:

1. A differential field is a field F with a derivation DF , which is an additive map DF : F Fthus satisfying DF (ab) = DF (a)b+ aDF (b) , a, b F .

2. A differential homomorphism/isomorphism between two differential fields F1 and F2 is a ho-

momorphism/isomorphism f : F1 F2 which satisfies DF2(f(a)) = f(DF1(a)) , a F1.

3. A linear differential operator L is defined by

L(y) = y(n) + an1y(n1) + ...+ a0y (A.1)

where y(n) = DnF (y) and ai F .

4. A differential field E with derivation DE is called a differential field extension of a differential

field F with derivation DF iff E F and the restriction of DE to F coincides with DF .

5. Elements a F whose derivative vanishes are called constants of F . The subfield of constantsof F is denoted by CF .

In order to investigate the types of solutions a differential equation admits one has to encode the

relations and symmetries of the independent solutions. Relations can be encoded in the Picard-

Vessiot extension of a differential field which includes the solutions of L(y) = 0:

Definition An extension E F is called a Picard-Vessiot extension of F for L(y) = 0 iff:E is generated over F as a differential field by the set of solutions of L(y) = 0 in E;

E and F have the same constants;

L(y) = 0 has n solutions in E linearly independent over the constants.

16

One can show that the Picard-Vessiot extension always exists for differential fields with an alge-

braically closed field of constants and this extension is unique up to isomorphisms.

Symmetries of the solution space are linear transformations which, applied to a solution, give a

new solution. The simplest formulation is in terms of automorphisms of the Picard-Vessiot extension

and leads to the notion of the differential Galois group:

Definition For a differential field F with the Picard-Vessiot extension E F the differential Galoisgroup G(E/F ) is the group of differential automorphisms : E E whose restriction to F is theidentity map, i.e.

G(E/F ) = { : E E | (a) = a a F } .

A solution y(x) of a linear differential equation L(y) = 0 is called

algebraic if it is a root of a polynomial over F ;

primitive if y(x) F , i.e. y(x) = x f(z) dz for f(x) F ;exponential if y

(x)y(x) F , i.e. y(x) = exp

( xf(z) dz

)with f(z) F .

Definition The solution y(x) is called Liouvillian if there exist differential extensions Ei, i = 0, ..,m

F = E0 E1 ... Em = E

such that y(x) = ym(x) E, Ei = Ei1(yi(x)) and yi(x) is algebraic, primitive or exponential overEi1.

We will be interested in second order homogeneous linear ordinary differential equations

y(x) + a(x)y(x) + b(x)y(x) = 0 , (A.2)

where a(x), b(x) F and F = C(x) is the differential field of rational functions over complexnumbers. The equation (A.2) can be cast into the normal form

y(x) + U(x) y(x) = 0 , (A.3)

U(x) = 14a2(x) 1

2a(x) + b(x) , U(x) F (A.4)

by the transformation y(x) y(x) exp (12 x

a(z)dz).

Let us identify the corresponding Galois group. Let E be a Picard-Vessiot extension of F = C(x)

and z1(x), z2(x) E be two independent solutions of (A.3). The Galois group G(E/F ) consistsof differential automorphisms which act on the space of solutions of (A.3), i.e. map solutions to

17

solutions. Thus

(zi(x)) = Cijzj(x) , i, j = 1, 2 . (A.5)

Defining the Wronskian w(z1(x), z2(x)) = det( z1(x) z2(x)z1(x) z

2(x)

)we obtain

(w(z1(x), z2(x))) = detC w(z1(x), z2(x)) (A.6)

where detC is the determinant of the Cij matrix in (A.5). The Wronskian is a constant since

w(x) = z1(x)z2 (x) z2(x)z1 (x) = 0 by virtue of (A.3), i.e. w(x) F . Therefore, (w(x)) = w(x),

giving detC = 1. Thus the Galois group for (A.3) is a subgroup of SL(2,C). For more general

equations the Galois group is a subgroup of GL(n,C).

Let us now address the main question: whether a differential equation can be solved in terms

of Liouvillian functions. Since the minimal differential extension generated by all solutions is the

Picard-Vessiot extension we therefore require that the Picard-Vessiot extension is generated by a

tower of Liouvillian extensions. By the fundamental theorem of differential Galois theory this tower

of extensions is related to subgroups of the Galois group through a bijective correspondence. One

can show that for the Picard-Vessiot extension to be Liouvillian, the identity-connected component

of the Galois group G0 has to be solvable (see theorem 25 in [56]).11

A simple example is provided by the equation y(x) + 2xy(x) = 0 which has the normal form

y(x) (x2 + 1)y(x) = 0. (A.7)

For this equation F = C(x) is the field of rational functions over complex numbers. The solution

space may be represented as C (e1

2x2) C( e 12x2 ex2). The Picard-Vessiot extension is E =

C(x, e1

2x2 , ex

2

,ex

2

) and the tower of extensions of E is

C(x) C(x, e

1

2x2) C

(x, e

1

2x2 , ex

2) E = C

(x, e

1

2x2 , ex

2

,

ex

2).

Then the Galois group G(E/F ) and its identity component G0 are

G = G0 ={( a 0

c 1/a

), a 6= 0, c C

}(A.8)

This is a solvable group since it is a subgroup of SL(2,C) whose algebraic subgroups are all solvable

11A group is solvable if it has a subnormal series whose factor groups are all abelian, that is, if there are subgroups{1} G1 ... Gn = G such that Gk1 is normal in Gk (i.e. invariant under conjugation) and Gk/Gk1 is anabelian group for all k = 1, ..., n.

18

except for SL(2,C) itself.

B String motion in pp-wave background

Here we shall study string motion in a pp-wave metric (i = 1, .., d)

ds2 = dudv +H(u, x) du2 + dxidxi . (B.1)

In conformal gauge the equation for v reads +u = 0 and is solved by u = f(+)+ f(). We can

fix the residual conformal symmetry by choosing u = p . Then v is determined from the Virasoro

constraints and we get the following effective Lagrangian for xi

L = xixi xixi + p2H(u, x), (B.2)

which describes light-cone gauge string motion in a potential.

One familiar example is the Ricci flat space with H(x) = ijxixj, ii = 0. Another is the pp-wave

limit of the AdS5 S5 background for which H(x) = xixi [44]. Here the string motion takes placein a quadratic potential and is obviously integrable. Penrose limits of the brane backgrounds we

consider in this paper also take the pp-wave metric form with H(u, x) = hij(u)xixj (see e.g. [46]). In

the light-cone gauge string motion takes place in a quadratic potential with a -dependent coefficient

and it is solvable [47] but formally the resulting model is not integrable.12

There are also other integrable examples with H(x) corresponding, for example, to the Liouville

or Toda potential.

The case that we shall study below is the pp-wave solution T-dual to the fundamental string

[51, 66] for which H is a harmonic function

H(x) = 1 +Q

rd2, r2 = xixi . (B.3)

Not surprisingly, the conclusions will be the same as for the fundamental string in section 4: the

geodesic motion is integrable but an extended string motion is not.

In the point-like string limit (B.2) takes the form (we set xi = ryi with ~y2 = 1)

L = r2 + r2~y2 + p2H(r) + (~y2 1) . (B.4)

The corresponding geodesic motion is completely integrable since we have d1 constants of motionfrom the coordinates yi (see section 3.1) plus the Hamiltonian.

12For the relevant case of h 12

the Kovacic algorithm gives no Liouvillian solutions for string modes that dependon .

19

Let us now consider an extended string moving only in a 3-space dxidxi = dr2+r2(d2+sin2 d2)

and choose the following ansatz (similar to the one in (3.12))

r = r(), = () = , = () . (B.5)

Then the effective Lagrangian for r and takes the form

L = r2 + r22 2r2 sin2 + p2H(r) . (B.6)

If we also assume that v = 0 (i.e. the string also moves in the light-cone direction) then the Virasoro

condition for (B.1) is equivalent to the vanishing of the Hamiltonian corresponding to (B.6)

r2 + r22 + 2r2 sin2 p2H(r) = 0 . (B.7)

Restricting motion to the invariant plane {(r, pr , , p) : = pi2 , p = 0} gives a one-dimensionalintegrable system parametrised by and p with the Hamiltonian as the constant of motion. Imposing

the zero Hamiltonian constraint (B.7) gives the solution for r

r2 + 2r2 = p2H(r) , r = r() . (B.8)

The Hamiltonian vector field on the plane of these solutions is

Xr = r, Xpr = 2r = 22r + p2H (r), X = 0, Xp = 0 (B.9)

and the direction normal to this plane is along and p. Expanding the equation of motion for

gives the NVE ( = = pi2 )

+ 2r

r 2 = 0 . (B.10)

Changing the independent variable r = r() one obtains ( ddr )

+(2r+

12p

2H (r) 2rp2H(r) 2r2

)

2

p2H(r) 2r2 = 0 (B.11)

Bringing this equation to the normal form via the change of variable (r) = g(r)(r) we get

+ U(r) = 0 , (B.12)

U(r) =(d 2)q4Q[(d 6)Qr2 + 4(d 3)rd]+ 4q2rd[(d2 2d+ 2)Qr2 + 2rd]+ 4r2+2d

16[q2(Qr2 + rd) r2+d]2 = 0 ,

20

where q p . For 2 < d < 13 the Kovacic algorithm does not yield Liouvillian solutions for generalfinite non-zero values of Q and q which implies non-integrability.

Note that in the limit Q 0, when the metric becomes flat, we get

+2q2 + r2

4(q2 r2)2 = 0, (B.13)

which has Liouvillian solutions in agreement with with flat space integrability.

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1 Introduction2 On the non-integrability of classical Hamiltonian systems3 String motion in p-brane backgrounds3.1 Complete integrability of geodesic motion3.2 Non-integrability of string motion

4 String motion in NS5-F1 background5 Concluding remarksA Integrability and differential Galois theoryB String motion in pp-wave background