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

Arkady Tseytlin

Stepanchuk, AT arXiv:1211.3727

1Monday, 10 December 12

• importance of knowing how to solve string theory in curved space - both closed and open string sector (e.g., description of D-branes)

• gauge-string duality: strings in AdS5 x S5 integrable system

• AdS5 x S5: ‘‘throat’’ region of D3-brane geometry; derivation of gauge-string duality -decoupling of asymptotically flat region + low energy approximation

• would be important to know string spectrum in full D3-brane background

2Monday, 10 December 12

3Monday, 10 December 12

4Monday, 10 December 12

• Importance of classical integrability: solvability of classical and quantum first-quantized string theory (2d CFT)

• string sigma model is conformally invariant integrability likely to survive at quantum level

• conformal symmetry here is gauge symmetry, not infinite global symmetry; need separate hidden symmetry for integrability

5Monday, 10 December 12

• Given curved background when string sigma-model is integrable? Finding Lax is hard... no general classification of integrable sigma-models

• Necessary condition: all consistent one-dim reductions of 2d model should be integrable mechanical systems

• In particular, geodesics of metric G should be integrable (obvious for group spaces or cosets)

• There are large classes of black-hole backgrounds with integrable geodesics but corresponding sigma-models will not in general be integrable

I =Rd

2x Gmn(X)@aXm

@

aX

n

6Monday, 10 December 12

• S2 =SO(3)/SO(2) sigma model integrability first discovered via relation (Pohlmeyer reduction) to sine-Gordon model

• any G/H coset sigma-model: Lax connection in terms of current components ( L= d + a j + b *j , etc)

• principal chiral model + WZ term: integrable; (gauged)WZW model is solvable explicitly

Examples:

7Monday, 10 December 12

Integrability is also preserved by some anisotropic deformations of the group space [Cherednik; Klimcik], the 2-sphere [Fateev, Onofri, Zamolodchikov] and the 3-sphere [Fateev; Lukyanov] Attempts to find “non-diagonal” generalisations of the principal chiral model by studying conditions for the existence of the corresponding zero curvature representation [Sochen; Mohammedi]Some models with WZ-like terms (with B-coupling) [Beisert, Luecker; Lukyanov]

Integrability rare for 2d sigma-models: G/H cosets, gauged WZW, some anisotropic versions, not much more known ... plus models related to these by 2d duality (T-duality) transformations

8Monday, 10 December 12

• key recent example: AdS5 x S5 is integrable conformal sigma model

• this is limit of D3-brane background what about full D3-brane? Integrability would have important implications...

9Monday, 10 December 12

10Monday, 10 December 12

•  Coset σ model

3.2 GS action for

General references are Metsaev-Tseytlin, Mc Loughlin thesis, and Callan plus Mc Lough-

lin. Here we follow the summary in [5]. Very useful is also the recent review by Zarembo

[1].

3.2.1 Group structure

String theory on can be defined via a -model on a coset space. Technically, the

coset model provides a simple way to couple to Ramond-Ramond fields. This construc-

tion is due to Metsaev and Tseytlin (in 1998) , following the corresponding construction

of superstring action in 10d flat space by Henneaux and Mezincescu. The coset space is

(3.1)

The bosonic part of is 1

(3.2)

Thus the bosonic part of the coset space is

(3.3)

The superalgebra su is a non compact real form of sl . This is the algebra of

supermatrices over complex numbers

(3.4)

with zero supertrace

(3.5)1The rank of , , and is .

31

3.2 GS action for

General references are Metsaev-Tseytlin, Mc Loughlin thesis, and Callan plus Mc Lough-

lin. Here we follow the summary in [5]. Very useful is also the recent review by Zarembo

[1].

3.2.1 Group structure

String theory on can be defined via a -model on a coset space. Technically, the

coset model provides a simple way to couple to Ramond-Ramond fields. This construc-

tion is due to Metsaev and Tseytlin (in 1998) , following the corresponding construction

of superstring action in 10d flat space by Henneaux and Mezincescu. The coset space is

(3.1)

The bosonic part of is 1

(3.2)

Thus the bosonic part of the coset space is

(3.3)

The superalgebra su is a non compact real form of sl . This is the algebra of

supermatrices over complex numbers

(3.4)

with zero supertrace

(3.5)1The rank of , , and is .

31

3.2 GS action for

General references are Metsaev-Tseytlin, Mc Loughlin thesis, and Callan plus Mc Lough-

lin. Here we follow the summary in [5]. Very useful is also the recent review by Zarembo

[1].

3.2.1 Group structure

String theory on can be defined via a -model on a coset space. Technically, the

coset model provides a simple way to couple to Ramond-Ramond fields. This construc-

tion is due to Metsaev and Tseytlin (in 1998) , following the corresponding construction

of superstring action in 10d flat space by Henneaux and Mezincescu. The coset space is

(3.1)

The bosonic part of is 1

(3.2)

Thus the bosonic part of the coset space is

(3.3)

The superalgebra su is a non compact real form of sl . This is the algebra of

supermatrices over complex numbers

(3.4)

with zero supertrace

(3.5)1The rank of , , and is .

31

symmetry

•  Internal 4° order automorphism

The real form is defined by

(3.6)

The identity is a central element. Removing it we obtain psu which has no repre-

sentation in terms of supermatrices.

The algebra su admits an automorphism (and on bosons). It takes

the form

(3.7)

which can be used to filter

g su g g g (3.8)

Notice that the 0-th degree part is

g so so (3.9)

3.2.2 The action

Let us consider an element . The Lie algebra valued form

(3.10)

is flat in the sense that

(3.11)

32

The real form is defined by

(3.6)

The identity is a central element. Removing it we obtain psu which has no repre-

sentation in terms of supermatrices.

The algebra su admits an automorphism (and on bosons). It takes

the form

(3.7)

which can be used to filter

g su g g g (3.8)

Notice that the 0-th degree part is

g so so (3.9)

3.2.2 The action

Let us consider an element . The Lie algebra valued form

(3.10)

is flat in the sense that

(3.11)

32

The real form is defined by

(3.6)

The identity is a central element. Removing it we obtain psu which has no repre-

sentation in terms of supermatrices.

The algebra su admits an automorphism (and on bosons). It takes

the form

(3.7)

which can be used to filter

g su g g g (3.8)

Notice that the 0-th degree part is

g so so (3.9)

3.2.2 The action

Let us consider an element . The Lie algebra valued form

(3.10)

is flat in the sense that

(3.11)

32The Lagrangian density is

L (3.12)

where is the worldsheet metric and . Right multiplication of by an element

of gives a local symmetry. The identity gives also invariance, so

we get a Lagrangian which depends on cosets as desired.

The term involves fermions. Imposing symmetry under left multiplication of by odd

elements we obtain and elimination of 16 fermionic degrees of freedom leaving

16 physical fermionic degrees of freedom ( -symmetry).

33

Metsaev, AT 1998

= 1

11Monday, 10 December 12

•  The string equations of motion can be put in Lax form

3.3 Classical integrability and algebraic curve

In full generality one wants to write the string equations of motion in the form

(3.13)

(3.14)

where is a -vector and are matrices, the Lax connection. The variable is the

spectral parameter. Compatibility requires to be flat

(3.15)

Once this happens, we can define the monodromy matrix

(3.16)

whose trace is independent on , due to flatness, and generates the conserved integrals of

motion.

Figure 3.1: The trace of the monodromy matrix.

34

z : spectral parameter

•  As usual, compatibility requires

3.3 Classical integrability and algebraic curve

In full generality one wants to write the string equations of motion in the form

(3.13)

(3.14)

where is a -vector and are matrices, the Lax connection. The variable is the

spectral parameter. Compatibility requires to be flat

(3.15)

Once this happens, we can define the monodromy matrix

(3.16)

whose trace is independent on , due to flatness, and generates the conserved integrals of

motion.

Figure 3.1: The trace of the monodromy matrix.

34

•  The trace of the monodromy matrix is τ

3.3 Classical integrability and algebraic curve

In full generality one wants to write the string equations of motion in the form

(3.13)

(3.14)

where is a -vector and are matrices, the Lax connection. The variable is the

spectral parameter. Compatibility requires to be flat

(3.15)

Once this happens, we can define the monodromy matrix

(3.16)

whose trace is independent on , due to flatness, and generates the conserved integrals of

motion.

Figure 3.1: The trace of the monodromy matrix.

34

3.3 Classical integrability and algebraic curve

In full generality one wants to write the string equations of motion in the form

(3.13)

(3.14)

where is a -vector and are matrices, the Lax connection. The variable is the

spectral parameter. Compatibility requires to be flat

(3.15)

Once this happens, we can define the monodromy matrix

(3.16)

whose trace is independent on , due to flatness, and generates the conserved integrals of

motion.

Figure 3.1: The trace of the monodromy matrix.

34

infinite set of conserved charges

12Monday, 10 December 12

•  Such a Lax connection indeed exists

Actually, we shall be interested in the eigenvalues of which are invariant under

changes of the initial point of the string. As shown by [314], there is indeed a Lax connec-

tion for the string. This takes the form (we emphasize the dependence on the

spectral parameter)

(3.17)

where (in addition to the -symmetry condition we must impose

(3.18)

The eigenvalues of can be written

(3.19)

The first four are bosonic, the other four fermionic. The eigenvalues and have

singularities at and at those values where two eigenvalues coincide. A detailed

analysis of these singularities shows that they can be poles or branch points at collapsing

points plus essential singularities at . These are removed by considering

(3.20)

The eigenvalues

(3.21)

have only branch points or poles and lie thus on an algebraic curve 2.

It is interesting to see all this in more details in a special reduction leaving to most general

discussion of the full algebraic curve to [275].

2A very useful introduction is [299].

35

•  We have 4+4 eigenvalues (bosonic+fermionic)

Actually, we shall be interested in the eigenvalues of which are invariant under

changes of the initial point of the string. As shown by [314], there is indeed a Lax connec-

tion for the string. This takes the form (we emphasize the dependence on the

spectral parameter)

(3.17)

where (in addition to the -symmetry condition we must impose

(3.18)

The eigenvalues of can be written

(3.19)

The first four are bosonic, the other four fermionic. The eigenvalues and have

singularities at and at those values where two eigenvalues coincide. A detailed

analysis of these singularities shows that they can be poles or branch points at collapsing

points plus essential singularities at . These are removed by considering

(3.20)

The eigenvalues

(3.21)

have only branch points or poles and lie thus on an algebraic curve 2.

It is interesting to see all this in more details in a special reduction leaving to most general

discussion of the full algebraic curve to [275].

2A very useful introduction is [299].

35

•  The eigenvalues of

Actually, we shall be interested in the eigenvalues of which are invariant under

changes of the initial point of the string. As shown by [314], there is indeed a Lax connec-

tion for the string. This takes the form (we emphasize the dependence on the

spectral parameter)

(3.17)

where (in addition to the -symmetry condition we must impose

(3.18)

The eigenvalues of can be written

(3.19)

The first four are bosonic, the other four fermionic. The eigenvalues and have

singularities at and at those values where two eigenvalues coincide. A detailed

analysis of these singularities shows that they can be poles or branch points at collapsing

points plus essential singularities at . These are removed by considering

(3.20)

The eigenvalues

(3.21)

have only branch points or poles and lie thus on an algebraic curve 2.

It is interesting to see all this in more details in a special reduction leaving to most general

discussion of the full algebraic curve to [275].

2A very useful introduction is [299].

35

lie on an algebraic curve with only poles or branch points in z

13Monday, 10 December 12

14Monday, 10 December 12

• Aim: study question of integrability of natural generalizations of AdS models

• p-brane backgrounds interpolate between flat space and - integrable deformations ?

• most symmetric D3-brane geometry has flat space and AdS5 x S5 as integrable asymptotics - what about intermediate region?

• Method: study integrability of 1-d reductions necessary condition

AdSn ⇥ Sn

15Monday, 10 December 12

can not be solved in quadratures

we will find that there exists a simple 1-d truncation of equations of motion which is not integrable

Non-integrability of 1-d Hamiltonian systems is shown using approach based on Kovacic algorithm

similar approach for less symmetric cases was used by Basu and Pando-Zayas (2011)

16Monday, 10 December 12

Integrability of classical Hamiltonian systems:related to behaviour of variations around phase space curves Ziglin (1982): necessary conditions for existence of additional functionally independent integrals of motion in terms of monodromy group properties of equations for small variations around phase space curves

Morales-Ruiz et al (1994): integrability implies that identity component G0 of the differential Galois group of variational equations normal to an integrable plane of solutions must be abelian: necessary integrability condition

17Monday, 10 December 12

Need analyse the normal variational equation (NVE)special class of solutions of NVE - Liouvillian solutions: functions of exponentials, logarithms, algebraic expressions and their integrals existence of such solutions <--> G0 is solvable if NVE has no Liouvillian solutions then G0 is non-solvable (and thus non-abelian) -> non-integrabilityfor equations with rational coefficients Liouvillian solutions can be determined by the Kovacic (1996) algorithm no Liouvillian solutions of NVE by the Kovacic algorithm -> non-integrability of Hamiltonian system

18Monday, 10 December 12

To prove non-integrability of a Hamiltonian system: demonstrate that given an integrable subsystem defined by some invariant plane in the phase space, the corresponding NVEs are not integrable in quadratures and thus do not admit sufficiently many conserved quantities.[consistent with the usual definition of integrability in the sense of Liouville: equations of motion should be solvable in quadratures, i.e. in terms of Liouvillian functions]

Steps to prove non-integrability:• Choose an invariant plane of solutions• Find NVEs - variational equations normal to invariant plane• Show that algebrized NVEs have no Liouvillian solutions using the Kovacic algorithm (implemented numerically)

19Monday, 10 December 12

require that the coefficients in this equation are rational functions of x

first need to algebrize the NVE - rewrite it as a differential equation with rational coefficients

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Example: non-integrable system

Invariant plane:

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Classical string motion in curved background

p-brane background:

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p-brane solutions of 10d supergravity equations: generalisation of extremal black holes like RN one

solution of

supported by extended source at the origin:by p-brane action -- analog of point-like particle source

get D-brane background in the case of F = RR flux

Rmn � 12gmnR = Fm. . . .Fn. . . + . . .

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Case with nm = 1 describes interpolation between flat space (Q = 0) and the AdS(p+2) × S(k) space (Q → ∞)Special cases:(i) n = 4, m =1/4, k = 5, p = 3: D3-brane interpolating between flat space and AdS5 × S5

(ii) n = 2, m = 1/2, k = 3, p = 1: D5-D1 intersection (or NS5-F1) with Q5 = Q1 = Q interpolating between flat 10d space and AdS3 × S3 × T4

(iii) n = 1, m = 1, k = 2, p = 0: 4 equal-charge D3-brane intersection interpolating between

flat 10d space and AdS2 × S2 × T6 (generalised Bertotti-Robinson)

1. point-like string (i.e. geodesic) motion is integrable. 2. extended string motion is not integrable for generic values of Q, despite integrability in the limits Q = 0 and Q = ∞.

24Monday, 10 December 12

Complete integrability of geodesic motionThe symmetries of the metric 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 spherical symmetry does not a priori provide sufficiently many conserved quantities

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Non-integrability of string motionchoose a particular “pulsating string” ansatz for the dependence of the string coordinates on the world-sheet directions (τ,σ): we shall assume that (i) only x0, r and two angles φ, θ of S2 ⊂ Sk (with dΩ2 = dθ2 + sin2 θ dφ2) are non-constant, and (ii) x0, r, θ depend only on τ while φ depends only on σ, i.e.

27Monday, 10 December 12

String is wrapped (with winding number ν) on a circle of S2 whose position in r and θ changes with time. The equations for r and θ can be derived from the following effective Lagrangian

We shall show that this 1d Hamiltonian system is not integrable, implying non-integrability of string motion in the p-brane background

Let us choose as an invariant plane {(r,θ;pr,pθ) : θ = π, pθ = 0}, corresponding to the a string wrapped on the equator of S2 and moving only in r

28Monday, 10 December 12

Using the explicit form of f(r)

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Bringing to the normal form by changing the variable to ξ(x) = g(x)η(x), where g(x) is a suitably chosen function, one obtains

now apply Kovacic algorithm to determine if there are Liouvillian solutions -> identity component of Galois group is not solvable -> non-integrabilityspecial cases Q → 0 and for Q → ∞ with nm = 1: Liouvillian solutions found-> consistent with integrability in flat space and in AdS2 × S2 But for finite value of Q Liouvillian solutions do not exist for generic E,ν for p-brane background with p = 0, .., 6 and the intersecting brane cases:

non-integrability of string motion

30Monday, 10 December 12

String motion in NS5-F1 backgroundinclude non-zero B- coupling: string motion in background produced by fundamental strings delocalised inside NS5 branes

31Monday, 10 December 12

String Lagrangian

interpolates between flat space model for Q1,Q5 = 0 and

SL(2) × SU(2) WZW model (plus 4 free directions) for Q1, Q5 → ∞:

both limits are obviously integrable.

Another integrable special case is Q1 = 0, Q5 → ∞ when get the SU(2) WZW model + flat

The opposite limit of Q5 = 0, Q1 → ∞ is described by

32Monday, 10 December 12

resulting effective Lagrangian

turns out to be non-integrable

in the limit Q5 → ∞ reduces to a combination of SU(2) WZW model, flat directions and 3-dimensional sigma model:

solving the equations for u,v=z ± t: get Liouville model -- integrable:

33Monday, 10 December 12

related to SL(2) WZW by T-dualities and coordinate transformation: an exactly marginal deformation of the SL(2,R) WZW

Geodesic motion is again integrable (no coupling to B-field)Extended string motion: consider ansatz

The resulting 1d subsystem described by effective Lagrangian

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Virasoro constraint:

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NVE has no Liouvillian solutions for general Q1 and Q5:non-integrability of string motion in the NS5-F1 background

36Monday, 10 December 12

1. Q1 = 0 or Q5 = 0: no Liouvillian solutions: non-integrability of string motion in the NS5 or NS1 backgrounds2. Q1 → ∞, Q5 → ∞: string action becomes SL(2,R) × SU(2) WZW modelLiouvillian solutions in agreement with expected integrability3. Q1=any, Q5 → ∞: integrable deformation of SL2 WZW + SU2 WZW4. Q5=0, Q1 → ∞: no Liouvillian solutions, not integrable

Special cases:

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Conclusions1. p-brane backgrounds: string sigma model interpolates between integrable flat and coset or WZW models -- integrable geodesics but string motion is not integrable in general2. as 2d duality transformations preserve (non)integrability, similar conclusions for backgrounds related to NS5-F1 background via T-dualities, e.g., pp-wave backgrounds 3. method to demonstrate non-integrability: particular string motion when dynamics reduces to effective one-dim system for which linear second-order differential perturbation equations cannot be solved in quadratures 3. supports expectation that integrability of classical string motion is rare: string integrability is much more restrictive constraint than particle integrability; no general classification... new examples? 4. How to perturb near integrable cases in a useful way?

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