Pohlmeyer Reduction, the Dressing Method and Classical ... · The project HAPPEN has started on...

IntroductionElliptic and Dressed Elliptic String Solutions

Features of the Dressed Elliptic StringsFuture Extensions

Pohlmeyer Reduction, the Dressing Method and ClassicalString Solutions on R× S2

Georgios PastrasNSCR Demokritos

based onarXiv:1907.04817 and arXiv:1907.08508

in collaboration with Dimitrios Katsinis and Ioannis Mitsoulas

INPP Annual Meeting,NSCR Demokritos, November 15th 2019

Georgios PastrasPohlmeyer Reduction, the Dressing Method and Classical String Solutions on R × S2



Project HAPPEN overview

The project HAPPEN has started on November the 2nd 2018 and will run untilNovember the 1st 2020.

It has a budget of 182ke(50ke+ overhead used so far)

Three post-docs (G. Pastras, G. Linardopoulos, I. Mitsoulas) are working on theproject.Technical progress:

- WP1 dressed minimal surfaces: Research has advanced and results are expected to bepublished within the next few months.

- WP1a dressed elliptic strings: Research has been completed and results are publishedin two papers, Eur.Phys.J. C79 (2019) no.10, 869, JHEP 1909 (2019) 106.

- WP2 entanglement in thermal field theory: Research has been completed and resultsare published in two papers, arXiv:1907.04817, arXiv:1907.08508, both to appear inJHEP.

- WP3 Holographic RG flow of entanglement: Research has been completed and resultsare published in one paper, arXiv:1910.06680.

- WP4 Equivalence between entanglement thermodynamics and Einstein equations: Thisis going to be the main goal of the second year of the program.

- WP5 Dissemination: 5 papers, 4 conference presentations and our websitehappen.inp.demokritos.gr/


http://happen.inp.demokritos.gr/



1 Introduction

2 Elliptic and Dressed Elliptic String SolutionsElliptic SolutionsThe Dressed Elliptic SolutionsThe Sine-Gordon Counterparts

3 Features of the Dressed Elliptic StringsThe Sine-Gordon CounterpartsClosed StringsStability of the SeedsSpike InteractionsEnergy and Angular Momentum

4 Future Extensions




Section 1

Introduction




Classical string solutions have shed light to several aspects of the holographicduality.

The dispersion relations of the classical strings are related to the anomalousdimensions of operators in the dual CFT. 1

They also serve to develop some intuition on the dynamics of the classical systemwhose quantum version is the only known mathematically consistent theory ofquantum gravity

1S. Frolov and A. A. Tseytlin, Nucl. Phys. B 668, 77 (2003) [hep-th/0304255]N. Beisert, J. A. Minahan, M. Staudacher and K. Zarembo, JHEP 0309, 010 (2003) [hep-th/0306139]S. Frolov and A. A. Tseytlin, Phys. Lett. B 570, 96 (2003) [hep-th/0306143]




In this work:

We focus on strings propagating on R× S2, which are Pohlmeyer reducible to thesine-Gordon equation.

We invert Pohlmeyer reduction and construct systematically the solutions withelliptic SG counterparts

Then we perform a Backlund transformation on the side of the SG equation andfind new “dressed” string solutions

The new solutions have several interesting features

They have interacting spikes.

There are interesting interrelations between properties of the strings and their SGcounterparts.

The dressed solutions reveal the stability properties of their seeds.

The energy and angular momenta of the dressed solutions have severalqualitative features that could be detectable on the side of the boundary CFT.




Elliptic SolutionsThe Dressed Elliptic SolutionsThe Sine-Gordon Counterparts

Section 2

Elliptic and Dressed Elliptic String Solutions





The Elliptic Strings

In a previous work2 we took advantage of Pohlmeyer reduction to systematicallyconstruct the Elliptic String Solutions on R× S2.The action for strings propagating on R× S2, written as a Polyakov action is

S = T∫

d𝜉+d𝜉−((𝜕+X) · (𝜕−X) + 𝜆

(X · X − R2

)).

The system is integrable. A signature of the system’s integrability is the fact that it canbe reduced to an symmetric space sine-Gordon system, in our case thesine-Gordon equation3. Defining the reduced field as the angle between the vectors𝜕+X and 𝜕−X (

𝜕+X)·(𝜕−X

):= f+′f−′ cos𝜙.

It is easy to show that the Pohlmeyer field obeys the sine-Gordon equation

𝜕+𝜕−𝜙 = 𝜇2 sin𝜙.

Although we know a lot about the sine-Gordon equation, the Pohlmeyer reduction is anon-local and many-to-one mapping, making its inversion a non-trivial task.However, it can be inverted in the case of elliptic solutions.

2D. Katsinis, I. Mitsoulas and G.P., Eur.Phys.J. C78 (2018) no.11, 977, [arXiv:1805.09301 [hep-th]]3K. Pohlmeyer, Commun. Math. Phys. 46, 207 (1976)





The Elliptic String Solutions - Periodicity

Without posting the details of the derivation, the elliptic string solutions assume theform

t0/1 = R√

x2 − ℘ (a)𝜉0 + R√

x3 − ℘ (a)𝜉1,

cos 𝜃0/1 =

√x1 − ℘

(𝜉0/1 + 𝜔2

)x1 − ℘ (a)

,

𝜙0/1 = −sgn(Ima)√

x1 − ℘ (a)𝜉1/0 − Φ(𝜉0/1; a

),

where the quasi-periodic function Φ is defined as

Φ(𝜉; a) = −i2ln

𝜎 (𝜉 + 𝜔2 + a)𝜎 (𝜔2 − a)𝜎 (𝜉 + 𝜔2 − a)𝜎 (𝜔2 + a)

+ i𝜁 (a) 𝜉.





The Elliptic String Solutions - Rigid Rotation - Spikes

Writing down the Virasoro constraints in terms of the Pohlmeyer field, yields𝜕0X

2= R2𝜇2cos2 𝜙

2,

𝜕1X

2= R2𝜇2sin2 𝜙

2.

Thus, whenever the Pohlmeyer field equals an integer multiple of 2𝜋, thederivative 𝜕1X gets inverted and spikes emerge.

The elliptic string retain their shape as they move.

All well known solutions on the sphere4 emerge as special limits of thesesolutions, including giant magnons, GKP strings, BMN particle, giant hoops.

The elliptic strings can be classified to four classes 5, depending onwhich worldsheet coordinate the SG counterpart dependswhether the SG counterpart is an oscillatory or librating pendulum solution

4D. E. Berenstein, J. M. Maldacena and H. S. Nastase, JHEP 0204 (2002) 013 [hep-th/0202021]S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Nucl. Phys. B 636 (2002) 99 [hep-th/0204051]D. M. Hofman and J. M. Maldacena, J. Phys. A 39 (2006) 13095 [hep-th/0604135]

5K. Okamura and R. Suzuki, Phys. Rev. D 75 (2007) 046001 [hep-th/0609026]





The Elliptic String Solutions - Classification

static oscillatingcounterpart

static rotatingcounterpart

translationally invariant oscillatingcounterpart

translationally invariant rotatingcounterpart





The Dressing Method

The theories emerging after the Pohlmeyer reduction of the non-linear sigmamodels, which describe the propagation of classical strings in symmetric spaces,possess autoBacklund transformations, which connect pairs of solutions, viafirst order differential equations. These transformations are a manifestation of themodel’s integrability.

The dressing method6 is the direct analogue of the Backlundtransformations in the NLSM.

The simple formulation of the elliptic string solutions that emerged naturally viathe inversion of the Pohlemeyer reduction facilitates the application of thedressing method, although the seeds are highly non-trivial.

6J. P. Harnad, Y. Saint Aubin and S. Shnider, Commun. Math. Phys. 92 (1984) 329T. J. Hollowood and J. L. Miramontes, JHEP 0904 (2009) 060 [arXiv:0902.2405 [hep-th]]





The Dressed Elliptic StringsThe application of the dressing method to the elliptic seeds is highly technical7. Withoutposting the details of the derivation, the dressed solution with the simplest dressingfactor, which corresponds to the application of a single Backlund transformation, is

X ′ = U

√1

2X T+X−

sin 𝜃1 (X+ + X−) + cos 𝜃1X0 := U (sin 𝜃1X1 + cos 𝜃1X0) ,

where X+ = Ψ (𝜆1) 𝜃p, X− = 𝜃Ψ (𝜆1) 𝜃p. The matrix 𝜃 is simply diag{1, 1,−1}, thevector p is any vector obeying pT p = 0 and p = 𝜃p. The vectors ei are simply X0,X0 × X and X0 × (X0 × X), where X0 equals (0 0 1). The parameter 𝜆 = ei𝜃1

determines the position of the poles of the dressing factor and it is directly connected tothe Backlund parameter and the matrix Ψ is composed by the three vectors

E1 := cos(√

Δ𝜉0 − Φ(𝜉1; a

))e1 + sin

(√Δ𝜉0 − Φ

(𝜉1; a

))e2,

E2 := − cos(√

Δ𝜉0 − Φ(𝜉1; a

))e2 + sin

(√Δ𝜉0 − Φ

(𝜉1; a

))e1,

E3 := e3.

Δ is always real. When the seed is oscillatory it is always negative, whereas when theseed is rotating there is a range of 𝜃1 that sets Δ positive.

7D. Katsinis, I. Mitsoulas and G.P. Eur.Phys.J. C78 (2018) no.8, 668 [e-Print: arXiv:1806.07730 [hep-th]]





The Dressed Elliptic Strings SG CounterpartsThe sine-Gordon equation possesses the Backlund transformations

𝜕+𝜙+ 𝜙

2= a𝜇 sin

𝜙− 𝜙

2, 𝜕−

𝜙− 𝜙

2=

1a𝜇 sin

𝜙+ 𝜙

2,

The simplest dressing factor corresponds to a single Backlund transformation withparameter

a =

√−

m+

m−tan

𝜃1

2.

It is not difficult to perform this single Backlund transformation to find the sine-Gordoncounterparts of the dressed elliptic solutions. They are

𝜙 = 𝜙+ 4 arctan

[A + B

Dtanh

D𝜉1 + iΦ(𝜉0; a

)2

],

where

𝜙 = 2 arctan

(a − a−1

a + a−1tan

𝜙

2

)+ (2k − 1)𝜋 + sgn

(a2 − 1

)2𝜋

⌊𝜙

2𝜋+

12

⌋,

A = sc𝜇

2

√a2 + a−2 + 2 cos𝜙, B = −𝜕0

𝜙

2, Δ = −D2.




The Sine-Gordon CounterpartsClosed StringsStability of the SeedsSpike InteractionsEnergy and Angular Momentum

Section 3

Features of the Dressed Elliptic Strings





The Dressed Elliptic Strings - Epicycle Picture

The vector X1 is a unit vector, which is perpendicular to X0. Thus, the arcconnecting the endpoints of the vectors X and X ′ is equal to 𝜃1. In other words,the dressed string solution can be visualized as being drawn by a point inthe circumference of an epicycle of arc radius 𝜃1, which moves so that itscenter lies on the seed string solution.

The form of the new solution provides a nice geometric visualization of theaction of the dressing on the shape of the string.

This is an outcome of the form of the simplest dressing factor and not a specificproperty of the dressed elliptic solutions, but a generic property that holdswhenever the simplest dressing factor is adopted.

A further implication of the above is the fact that at the limit 𝜃1 → 0 the dressedsolution tends to the seed, whereas as 𝜃1 → 𝜋 the dressed solution tends to thereflection of the seed with respect to the origin of the enhanced space.





The Dressed Elliptic Strings - Epicycle Picture

seed with staticoscillating counterpart

seed with staticrotating counterpart

seed with translationally invariantoscillating counterpart

seed with translationally invariantrotating counterpart





The Sine-Gordon Counterparts - ClassificationFrom the form of the SG counterparts, it is evident that there is a qualitative differencebetween the solution with D2 > 0 and D2 < 0.

The former describe a localised kink-like disturbance propagating on top of anelliptic background, whereas the latter are a non-localised disturbances of theelliptic background.The former are (quasi-)periodic under translations by a single vector, whereas thelatter under two vectors forming a lattice.

𝜙

𝜉1

𝜉0

4𝜔1

−4𝜔14𝜔1

−4𝜔1

0

2𝜋

𝜙

𝜉1

𝜉0

2𝜔1

−2𝜔12𝜔1

−2𝜔1

−4𝜋

4𝜋





The Sine-Gordon Counterparts - D2 > 0 - Kink-Background Interaction

In the case D2 > 0, far away from the kink

limD𝜉1+iΦ(𝜉0;a)→±∞

sd𝜙 = 𝜙(𝜉0 ± a

)+ sd ((2k − 1)± sc)𝜋.

Therefore, the passage of the kink effectively causes a delay to the translationallyinvariant motion of the system equal to

Δ𝜉0 = 2 |a| .

In the case of a static background, the kink causes a displacement of the background.The energy of the kink, the energy density of the background and the delay causedby the kink are connected via an equation of state, in principle experimentallyverifiable in systems that realize the sine-Gordon equation

E2kink

64= ℘

(Δ𝜉0

2;

E2

3+ 𝜇4,

E3

(E2

9− 𝜇4

))−

E3,

P2kink

64= ℘

(Δ𝜉1

2;

E2

3+ 𝜇4,

E3

(E2

9− 𝜇4

))−

E3.





The Asymptotics of the Dressed Strings with D2 > 0

The asymptotics of the dressed strings are closely related to the asymptotics of theirSG counterparts. Far awat from the kink they look like a rotated version of the ellipticseed,

limΦ→±∞

𝜃0/1

(𝜎0, 𝜎1

)= 𝜃seed

(𝜎0, 𝜎1 ∓

a2𝜔1

𝛿𝜎0

),

limΦ→±∞

𝜙0/1

(𝜎0, 𝜎1

)= 𝜙seed

(𝜎0, 𝜎1 ∓

a2𝜔1

𝛿𝜎0

)±Δ𝜙0/1,

Δ𝜙0/1 = arg (ℓ+ iD) + arg 𝜎 (a + a) + i(𝜁(

a + 𝜔x3/2

)− 𝜁

(𝜔x3/2

))a.

It follows that these solutions approximate finite close strings not when the seedsare closed but when they obey

(n1𝛿𝜙+ 2sΦΔ𝜙) n2 = 2𝜋, n1, n2 ∈ Z.





Approximate Finite Closed Strings with D2 > 0

seed solutionrotated seed solutiondressed solution





Exact Infinite Closed Strings with D2 > 0

Had we not restricted to finite length strings, we could form infinite strings that obeyappropriate and exact periodicity conditions in the same sense as the single spikesolution.

For this purpose, both the seed must be closed and the rotation induced by thedressing Δ𝜙 should be a rational fraction of 2𝜋.





Finite Closed Strings with D2 < 0

In the case D2 < 0, one can show that there are finite closed dressed strings,whenever the 𝜎1 axis (the axis perpendicular to the physical time) coincides to adirection of the periodicity lattice of the sine-Gordon counterpart.

𝜉0

𝜉1

𝜎0

𝜎1

2𝜋iD 𝜉1

2𝜔1(𝜉0 + vtb𝜉

1)

𝜎1

segm

ent c

over

ing

the

clos

edst

ring





Finite Closed Strings with D2 > 0

In a similar manner there is a special case one can construct finite closed dressedstrings with D2 > 0. In this case, the 𝜎1 axis should coincide with the periodicityvector of the SG counterpart. This may happen only when the kink issuperluminal.

𝜉0

𝜉1

𝜎0

𝜎1

2𝜔1(𝜉0 + v0𝜉

1)

kink

posi

tion

asym

ptot

icre

gion

𝜎1

segm

ent c

over

ing

the

clos

edst

ring





Stability of the Seeds

The latter solutions can be used for an unconventional approach to discoverinstabilities of the seed elliptic solutions. These solutions tend asymptoticallyin time to an elliptic string solution, but they are not a small perturbation aroundthe latter. Such solutions are the analog, for example, in the case of the simplependulum, to the trajectories connecting asymptotically two consecutive unstablevacua. The existence of such a solution reveals that the elliptic solution, which isthe asymptotic limit of the latter, is unstable.

The special solutions of this kind emerge only when the kink propagating on topof an elliptic background in the sine-Gordon counterpart of the solution issuperluminal.The dressing method analysis gives identical results to those of a smallperturbation analysis, encouraging the use of the dressing method as a generaltool for the study of string solution stability.





Stability of the Seeds

𝜎0/( 𝛾𝛽D )

0131050< 0

𝜎0/( 𝛾D )

012310< 0

𝜎0/( 𝛾𝛽D )

0131050< 0

𝜎0/( 𝛾𝛽D )

011050200< 0





Spike Interactions

An interesting feature of the elliptic string solutions is the presence of singularpoints, i.e. the spikes8. However these string solutions do not change shape, andthus, the spikes never interact.Interacting spikes emerge in higher genus solutions. The simplest possible suchsolutions are those presented here.

We may observe two kinds of spike interactions.

Two spikes annihilate and regenerate at a different position.

A loop dissolves to two spikes and vice versa.8K. Okamura and R. Suzuki, Phys. Rev. D 75 (2007) 046001 [hep-th/0609026]





Spike Interactions - SG side

The above processes are quite simple to understand in the language of thesine-Gordon equation. Spike appear whenever 𝜙 = 2n𝜋. The shape of the kink altersperiodically as it advances in the elliptic background. As the shape changes, it ispossible that the solution ceases to cross a 𝜙 = 2n𝜋 horizontal line, or on the oppositemay start crossing such a line.Continuity ensures that whenever this happens, two points where the solution crossesa 𝜙 = 2n𝜋 line appear or disappear. It follows that spikes interact in pairs.

0

𝜋

2𝜋

𝜎1

𝜙

0

2𝜋

4𝜋

𝜋

3𝜋

𝜎1

𝜙





Spike Interactions - Topological ChargeThe closed dressed string solutions are characterized by a topological number N,

2𝜋N =

∫string

d𝜎𝜕𝜎𝜙, N ∈ Z.

In the case of the elliptic strings this is equal to the number of spikes. Spikeinteractions do not allow such an interpretation in general.The form of the spike interactions suggests that N is mapped to a conserved quantity,which receives ±1 contributions from each spike and ±2 from each loop.The turning number of the closed string cannot be defined due to the spikes. However,the string contains only this kind of non-smooth points. Therefore, the unorientedtangent to the string is continuous and an unoriented turning number t can bedefined. This is a member of the fundamental group of the mappings from S1 to RP1,i.e. 𝜋1

(RP1) = Z and receives the appropriate contributions.

It follows that t and N differ by an ever integer.Georgios Pastras

Pohlmeyer Reduction, the Dressing Method and Classical String Solutions on R × S2




Energy and Angular Momentum

The variation of the energy and angular momentum imposed by the dressing is

ΔE0/1 = ±2sΦn2TR𝜇2a√

x3/2 − ℘ (a),

ΔJ0/1 = 2sΦn21ℓ

(𝜁 (a) + x2/3a − D cos 𝜃1

),

These vanish for both the dressed solutions with D2 < 0 and the instabilities.The exact infinite dressed strings with D2 > 0 have obviously infinite energy andangular momentum. However, since they are the n1 → ∞ limit of the approximatesolutions, the difference of their energy and momentum to those of their elliptic seedsis well-defined. In other words the finite approximate closed dressed strings may serveas a regularization scheme for the exact infinite closed dressed strings.





Energy and Angular Momentum

tr. inv.osc. seed

tr. inv.rot. seed

staticosc. seed

staticrot. seed

𝛿 (E − J) 𝛿 (E − J)

𝛿 (E − J) 𝛿 (E − J)

(E−J)hop2

(E−J)hop2

𝜃1𝜃1

𝜃1

𝜃1

𝜃

𝜃

𝜃−𝜃+

𝜃− 𝜃+

infinite closed strings with D2 > 0finite closed strings with D2 > 0finite closed strings with D2 < 0





Energy and Angular Momentum - Qualitative Characteristics

There is an interesting bifurcation in the dispersion relation of the dressed stringsolution occurring at D2 = 0. When considering dressed strings whose seedshave rotating counterparts, the dispersion relation is a rather peculiar function ofthe angle 𝜃1; there is a range for 𝜃1 where the dispersion relation does not dependon the latter.

There is yet another interesting bifurcation of the form of the dispersion relationthat has to do with the presence of the instabilities.

Although the dispersion relations of the dressed strings are too complicatedexpressions to be directly verifiable in a holographically dual theory, the abovediscontinuities in the dispersion relations could be in principle detectable.




Section 4

Future Extensions




More complicated dressing factors can be applied, without further solving ofdifferential equations, to find solutions whose Pohlmeyer counterparts are severalkinks scattering on top of an elliptic background or even breather propagating onthe latter.

Similar techniques can be applied for strings propagating on other symmetricspaces, such as the dS, AdS or AdS×S or minimal surfaces in hyperbolicspaces9. The latter are interesting in the framework of the RT conjecture.

The nice geometric interpretation of the dressed string as being drawn by anepicycle of given radius whose center moves on the seed solution deservesfurther investigation in the case of strings propagating on other symmetric spaces.

9G. P., Eur.Phys.J. C77 (2017) no.11, 797 [arXiv:1612.03631 [hepth]]




The discovery of instabilities of the seed solutions through the dressing method isan interesting feature. Comparison with the results from linear perturbation impliesthat indeed the string without dressed instabilities are stable. Thus, the dressingmethod can be a more general tool for the study of string solution stability.

The discovery of the qualitative behaviour of the dispersion relation of the dressedstrings in the anomalous dimensions of operators of the boundary CFT isinteresting.

It can be shown that whenever the moduli a and a are a rational fraction of thecorresponding half-period, there are closed algebraic relations between thecharges. These deserve investigation on the side of the CFT.




This research is supported by the General Secretariat for Research and Technology(GSRT) and the Hellenic Foundation for Research and Innovation (HFRI) in theframework of the “First Post-doctoral researchers support”, which funds the program“Holographic APPlications of quantum ENtanglement (HAPPEN)”, based in NSCR“Demokritos”, under grant agreement No 2595.

Thank you for your attention!


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