Elegant ideas that do not work - Dur - Elegant.pdfINTRODUCTION OBJETS D’ATFT NUPTIALS REVERSE...

Not just...

Elegant ideas that do not work

Craig Robertson

20 January 2014

http://www.youtube.com/watch?v=tlx2PgESXhs

INTRODUCTION OBJETS D’ATFT NUPTIALS REVERSE MITOSIS FAIL SUMMARY

OUTLINE

already using this trick?

1 INTRODUCTION

Everything is on the lineAffine Toda field theory

2 OBJETS D’ATFTSolitonsBoundariesDefects

3 TIL CRAIG FINISHING HIS PHD DO US PART

Quantum transmission4 DESPERATELY SEEKING SPONSORSHIP

Conjoined twin coverup5 ELEGANT IDEAS THAT DO NOT WORK

Offensive materialMy research

6 SUMMARY

ConclusionsOutlook


LIFE IN LINELAND

sAlice sBob

Shocked at the latest Home Office immigration scandalinvented by the tabloids, Alice and Bob both decide that todayis a day to move towards the right.


LIFE IN LINELAND

sAlice sBob

Alice is going faster than Bob and decides that she willovertake him.


DEATH IN LINELAND

That was a bad idea. Alice and Bob were smashed into Npieces. Overtaking ought to be forbidden in this Lineland.


INTEGRABLE FIELD THEORIES IN 1+1 DIMENSIONS

Integrable field theories have some special properties:• Infinite number of conserved charges• Factorised scattering and Yang–Baxter relations,

Sn→n → ΠS2→2

• Bootstrap relations• Often possess solitons

• Overtaking is possible without injury.Stupid ‘no line here to end’ nonsenseDue to the existence of exact scattering matrices, we can seeexactly what happens to Alice and to Bob when they collide. Ifthey both survive unscathed then it’s a sign that we have anintegrable field theory.


INTEGRABLE FIELD THEORIES IN 1+1 DIMENSIONS

Integrable field theories have some special properties:• Infinite number of conserved charges• Factorised scattering and Yang–Baxter relations,

Sn→n → ΠS2→2

• Bootstrap relations• Often possess solitons• Overtaking is possible without injury.

Stupid ‘no line here to end’ nonsenseDue to the existence of exact scattering matrices, we can seeexactly what happens to Alice and to Bob when they collide. Ifthey both survive unscathed then it’s a sign that we have anintegrable field theory.


AFFINE TODA FIELD THEORY (ATFT)We can associate a 1+1 dimensional relativistic field theory

L =12

u · u− 12

u′ · u′ −U(u)

to any affine Dynkin diagram. The ATFT potential is

U(u) =m2

β2

r∑j=0

nj

(eβαj·u − 1

)n0 = 1, α0 = −

r∑j=1

njαj

• m is a mass scale- we’ll just set m = 1.

• β is the coupling strength.

• r is the rank of the Lie algebra, with α0 the lowest root in the rootspace. The marks {ni} are characteristic of the underlyingalgebra.


AFFINE TODA FIELD THEORY (ATFT)

Here are some celebrity* affine Dynkin diagrams

* Anything on Wikipedia must be famous, right?


AFFINE TODA FIELD THEORY (ATFT)

We will now specialise to the a(1)r series of ATFTs associated to the

su(r + 1) root spaces.

e e e e e e ee e e e eQQQ �

�� Q

QQ

a(1)11

The marks are all ni = 1 so the Lagrangian simplifies to

L =12

u · u− 12

u′ · u′ − 1β2

r∑j=0

(eβαj·u − 1

)

anew line

The field u then lives in the su(r + 1) root space.


WHAT ARE SOLITONS?

Manton & Sutcliffe: ‘stable, particle-like objects, with finitemass and smooth structure’.

of the Top Sol course

Drazin & Johnson type: localised solutions of non-linearequations which retain their form even after interactions*.

that’s quite inelegantly put

The solitons in this talk fit both descriptions.

and

* Alice and Bob would be glad.


SOME SLIDES DEMAND TO BE RE-USED

A rather standard example of a soliton is the classicalsine-Gordon kink.

L =12

(∂u)2 − (1− cos u) =⇒ ∂2u + sin u = 0

The potential, U = 1− cos u = 2 sin2 (u2

), has minima at

u = 2nπ, n ∈ Z.Single soliton solutions interpolatebetween adjacent minima.

kinky and antikinky

The topological charge is given by

Q0 =u(∞)− u(−∞)

2π=

∫ ∞−∞

u′

2πdx ∈ Z


SOME SLIDES DEMAND TO BE RE-USED

The kink is a solitonthat interpolatesbetween 0 and 2π(Q0 = 1).

und

The kink centred at x = a (when t = 0) with rapidity θ has thesolution

u = 4 arctan(

ex cosh θ−t sinh θ−a)

This can be found by a Bogomolny argument.

i

Similarly Q0 = −1 gives an antikink.


HOW SOLITONS ENTER THE FRAY

The ATFT potential hides a dark secret. If the coupling is imaginary,β = iβ, the potential becomes

U =1β2

r∑j=0

(1− eiβαj·u

)

give me a space

This is invariant under u→ u + 2πβλ, where λ is a vector in the weight

space (λ · αi ∈ Z for all i). When r = 1 we have sine-Gordon theorybut the presence of multiple degenerate vacua is suggestive oftopological solitons for all r.


THE FADDEEV–ZAMOLODCHIKOV ALGEBRA

(FZ ALGEBRA)

We can represent a quantum soliton of species p (p ∈ {1, . . . , r})possessing topological charge labelled by the index i as anoperator

Api (θ)

at what cost?

The possible topological charges for a species p soliton in thequantum a(1)

r ATFT are the weights of the p-th fundamentalrepresentation.


THE FADDEEV–ZAMOLODCHIKOV ALGEBRA

(FZ ALGEBRA)

In the far past a two soliton state can be represented by

Api (θ1)Aq

j (θ2) where θ1 > θ2

and the algebra

The S-matrix evolves the state into the far future, resulting inthe FZ algebra

Api (θ1)Aq

j (θ2) = pqSklij (θ1 − θ2)Aq

k(θ2)Apl (θ1)

simplify

For simplicity, we will mainly focus on the case wherep = q = 1 and drop those superscripts and let θ12 = θ1 − θ2.


THE YANG–BAXTER EQUATIONS

Apart from an important prefactor, we can figure out the rest ofSkl

ij by using the Yang–Baxter equations. Factorised scattering ofa three soliton solution tells us:

��

��

��

≡��

��

��

��

Algebraically this is

Slmjk (θ23)Sns

il (θ13)Stunm(θ12) = Smn

ij (θ12)Snuil (θ13)Sst

ml(θ23)


THE S-MATRIXThe form of the scattering matrix between two species 1solitons in a(1)

r is:

Sjjjj(θ12) = ρ(θ12)

(Q−1 x2

1x2

2−Q x2

2x2

1

)Sjk

kj(θ12) = ρ(θ12)(

x21

x22− x2

2x2

1

)j 6= k

Sjkjk(θ12) = ρ(θ12)

(Q−1 −Q

)x2

1x2

2if l = j− k < 0

x22

x21

if l = j− k > 0

• Q = −eiπγ , where γ = 4π2

β2 − 1.

• xi = eγ(θi−∆). The quantity ∆ will be specified later, butplays no role here is only ratios of x appear.


a(1)r ON THE HALF-LINE

It is possible to restrict ATFT to the half-line x < 0 in a particularway that maintains integrability.

L = θ(−x)Lu + δ(x) (−B(u))

and

For a(1)r (r ≥ 2) maintaining integrability severely restricts the

form of the boundary potential B

B =1β2

r∑j=0

Ajeβ2 αj·u with either A2

j = 4 ∀j or Aj = 0 ∀j


WHAT IS A DEFECT?

Defects or impurities may bethought of as ‘internalboundary conditions’.

�@x = 0

u v

invisible defects

The two fields to the left and to the right of the defect inprinciple belong to different theories, but we are looking atATFT and the fields u and v always belong to the same rootdata.

more invisible defects

All known ATFT defects stem from the ar algebra and it isknown that sine-Gordon defects are intimately related tosine-Gordon boundaries.


a(1)r TYPE I DEFECTS

Bowcock, Corrigan and Zambon found, for each r ≥ 2, tworelated ‘type I’ defects.The first I will call, perhaps quite egregiously, a ‘species 1 defect’

L = θ(−x)Lu + θ(x)Lv

+ δ(x)

(12

u(1− B)u + uBv +12

v(1− B)v−DB(u, v)

)

with integrability conserved when the defect potential is

DB(u, v) =dβ2

r∑j=0

eβ2 αj(BTu+Bv) +

1dβ2

r∑j=0

eβ2 αj·B(u−v)

The parameter d = e−η where η is the characteristic rapidity ofthe defect.



L = θ(−x)Lu + θ(x)Lv

+ δ(x)

(12

u(1− B)u + uBv +12

v(1− B)v−DB(u, v)

)

new line cinema

The species 1 defect is thus fully specified by the r× r matrix B,where

B = 2r∑

j=0

(λj − λj+1

)λT

j

with {λi} being the fundamental highest weights for thealgebra.



The other type I defect I suggestively call the ‘species r’ defect.It’s the same as the species 1 defect but with B switched with itstranspose BT everywhere.

more info needed

A type I defect is a place where u can sell its services (in anintegrable manner) to v directly without the need for anintermediary.


SOME THINGS TO NOTE

• Curiously, all integrable defects we know of conservemomentum. Are all momentum conserving defectsintegrable?

• Integrable defects are purely transmitting.• We have solitons and defects so it is natural to ask how

they interact.• That question becomes all the more natural when you

realise that the Euler-Lagrange for either species of defectgive a Backlund transformation ‘frozen’ at x = 0.


TYPE II DEFECTS

From nowhere in particular, Corrigan and Zambon came upwith a type II defect. It turns out that this is the result ofsticking a species 1 and a species r defect together

L = θ(−x)Lu + θ(x)Lv + δ(x)(−DB(u, χ)−DBT

(χ, v)

+12

u(1− B)u + (u− v)Bχ− 12

v(1− B)v)

bogus

In this case the auxiliary field χ is the middleman who carriesout the business between u and v.

and

Type III and higher defects are possible, but I don’t find thisclassification useful because I have big ideas that do not work.


WHEN A SOLITON LOVES A DEFECT

Soliton and Defect aredelighted to announce theirforthcoming wedding.

they

They would be

they

delighted if you

they

could join them in

they

this joyous celebration.

the

The date of the divorce

they

is yet to be announced.

newlinedo

R.S.V.P.

C. Robertson,Physicist: ”Noone can evermeasure theamazing depthof love a solitonfeels for itsperfect defect.Sometimes it’senough to makeme feel sick.”


QUANTUM TRANSMISSION AND MODERN LIFE

In the quantum theory a soliton passing through a defect isdescribed by a transmission matrix. In terms of the FZ algebra,a defect (of species 1, say), holding topological charge α, maybe represented by an operator Dα. The FZ algebra for solitontransmission is then

Ai(θ)Dα(η) = Tnλiα (θ − η)Dλ(η)An(θ)

and note

The transmission matrix T is infinite-dimensional as the defectcould start off with any topological charge α. Overalltopological charge is conserved, so I will suppress the defectcharge as much as possible in what follows.



Once we know the S-matrix we can find the T-matrix (apartfrom it’s prefactor) using another form of the Yang–Baxterequation, sometimes called the triangle relations.

��

��

≡

��

��

��

Which algebraically is

Tmj (θ1)Tn

k (θ2)Sstmn(θ12) = Snm

jk (θ12)Tsn(θ2)Tt

m(θ1)


QUANTUM TRANSMISSION AND MODERN LIFEAfter ‘some work’ we get the transmission matrix for a species1 soliton though a species 1 defect in a(1)

3 to be

T(θ) = h(θ)

Qλ·l1 0 0 x

x Qλ·l2 0 00 x Qλ·l3 00 0 x Qλ·l4

• λ is the outgoing topological charge of the defect.

• As before, Q = −eiπγ , where γ = 4π2

β2 − 1.

• x = eγ(θ−η−iπ2 ), tells us how likely a soliton is to change its

topological charge on transmission.• {li} are the weights of the 4 representation of a3 = su(4).

They are the possible topological charges of the species 1soliton.



Note that:• That thing on the previous slide is the transmission matrix

for a soliton through an unexcited defect. The prefactorh(θ) does possess a pole so it is possible for a soliton to beabsorbed and excite the defect.

• Such an exciting wedlock of soliton and defect is of coursetemporary in the quantum theory. Divorce is inevitable.

• This particular transmission matrix obeys the classicalselection rules, but this isn’t generally the case. E.g., aspecies 1 soliton going through a species 3 defect in a(1)

3 canmake classically forbidden transitions in topologicalcharge.


THE WEDDING IS OFF!When it comes to transmission through a type II defectCorrigan and Zambon made the remarkable discovery thatinside the infinite-dimensional T-matrix certain solitonS-matrices can be found. In particular, in a(1)

3 :• The fundamental S-matrices 11S, 33S, 13S, 21S, 23S can all be

found in the type II transmission matrix.• Conspicuous by its absence is the S-matrix 22S, which

possesses ugliness of Medusan proportions.• It appears that the type II defect can, in a sense, become

either a species 1 or a species 3 (species r in general)soliton.

• If a soliton were a diploid cell, then the type I defectswould be gametes. . . probably.

• So, solitons are genetically closely related to defects! Wehad better call the wedding off!


MORE DARK SECRETS

It turns out that everything solitonycould, in principle, be described byspecies 1 solitons alone. This is due tothe existence of soliton fusing rules.

really?

What you thought was a species 2soliton is in fact a conjoined twin ofspecies 1 solitons.


WHEN YOU’RE TOO LAZY TO DRAW THE DIAGRAM

YOURSELF

Soliton fusing 11→ 2 occurs when thetwo species 1 solitons are placed at thesame location and have an imaginaryrapidity difference equal to i times thefusing angle 2π

r+1 . For the simplest theory

with fusing rules, a(1)2 , this is illustrated

quite well by the Mercedes–Benz logo.

but

Of course, I keep going on about a(1)3 ,

which is not quite so symmetric


SOLITON FUSING RULESWe could use fusing rules to generate all of the other solitonS-matrices from 11S, but let’s not. Let’s assume, wherevernecessary, that we know all of the soliton S-matrices.

and

The operator for a species 2 soliton A2(jk) in the FZ algebra is

A2(jk)(θ) = cjkA1

j (θ − iπr+1)A1

k(θ + iπr+1) + ckjA1

k(θ − iπr+1)A1

j (θ + iπr+1)

with no sum implied. The three-point couplings {cjk} dependon the theory (i.e., on r) and are generally known (in principle).

you too?

Using the transmission FZ algebra from 7 slides ago and theabove result allows us to find transmission matrices for species2 solitons from those of species 1 solitons

2T(ab)(jk) (θ)cab = cjk 1Ta

j (θ− iπr+1)1Tb

k(θ+ iπr+1)+ckj 1Ta

k(θ− iπr+1)1Tb

j (θ+ iπr+1)


SOLITON FUSING RULES

Note that:• In the quantum theory topological charge is conserved in

the fusing process.• As a check not worth doing, the somewhat more indexed

triangle relations verify that this all works.

• In a(1)2 an additional check comes from the crossing

symmetry sending a species 1 soliton to a species 2 soliton.• A fourth bullet-point can often be used to fill space when

desired.• I offer a shiny penny to whoever can tell me now where

I’m heading with this (you must be physically present atthe talk to qualify).


ELEGANT IDEAS THAT DO NOT WORK

• Organised religion• Makes claims without

evidence.• Historically enforced by

violent means.• Used for perfidious political

ends.

• Organised atheism• Makes claims without

evidence.• Historically enforced by

violent means.• Used for perfidious political

ends.

spacingand a bit more



There must be some evidence for some sort of religion, right?



• Democracy• True democracy resorts in diluted

power structures.• Electoral mandate bestows a sense

of self-entitlement.• Populist career politicians are most

likely to be elected.

• But it’s still the best we have

• Totalitarianism• Endemic corruption in many such

regimes.• Human rights not respected.• Widespread high-level education

is against the regime’s interests.

some space at the topand some more

andand further


SOME MORE THINGS THAT DO NOT WORK• Retired people

• Children


I TOLD YOU 22 SLIDES AGO

e e e e e e ee e e e eQQQ �

�� Q

QQ

0 1 2 . . .

r . . .

space needed?

As soon as I chose to name the defects species 1 and species r Iimplied strongly that there ought to be r fundamental defects ina(1)

r (but only two fit the type I description).

and

Well, there are r species of soliton and defects are related tosolitons, so why not r species of defect?

and

But the species of soliton are related by fusing rules. So, defectsought to have fusing rules too.


DEFECT FUSING RULES IN THE FZ ALGEBRA

Consider a species 1 soliton passing through a species 2 defect

A1i (θ)D2α(η) = 1

2Tnλiα (θ − η)D2λ(η)A1

n(θ)

We would like to find an expression for 12T.

still

Treating the defect operator like we did the soliton operator weshould have

D2α(η) = dβ,γD1β(η − iπr+1)D1 γ(η + iπ

r+1)

where α = β + γ, and dβ,γ is not known at all.


DEFECT FUSING RULES IN THE FZ ALGEBRAThe FZ algebra tells us that

12Tnλ

iα (θ − η)dδ,ε = 11Tjδ

iβ(θ − η + iπr+1)1

1Tnεjγ (θ − η − iπ

r+1)dβ,γ

• We don’t know what dδ,ε is to begin with, so it seems likewe can’t really say too much about 1

2T.• As long as we have some insight into 1

2T we can makeplausible guesses and check their validity using thetriangle relations (17 slides ago) and soliton fusing rules (amere 11 slides ago).

• In a(1)2 we already know 1

2T so we can find various relationsfor dδ,ε. The obvious answer is not consistent.

• In a(1)3 an expression for 1

2T would be a new thing. Defectfusing gives us a general ansatz for its form. It’s on thenext slide.


4 × 4 = 6, SO IT MUST WORK

The transmission matrix for a species 1 soliton through aspecies 2 defect in a(1)

3 , with some labels suppressed, takes theform

T(θ) = h2(θ)

Qλ·l1 0 x2g13(λ) xf14(λ)

xf21(λ) Qλ·l2 0 x2g24(λ)

x2g31(λ) xf32(λ) Qλ·l3 00 x2g42(λ) xf43(λ) Qλ·l4

with h2(θ) = h(θ − iπ

4 )h(θ + iπ4 ).

triangle

The triangle relations for this transmission matrix give 256constraints which should specify all the f s and gs. Most aretrivial, 28 are definitely not.



T(θ) = h2(θ)

Qλ·l1 0 x2g13(λ) xf14(λ)

xf21(λ) Qλ·l2 0 x2g24(λ)

x2g31(λ) xf32(λ) Qλ·l3 00 x2g42(λ) xf43(λ) Qλ·l4

Here are some guesses at the solution



T(θ) = h2(θ)

Qλ·l1 0 x2 xQλ·(l1+l4)

xQλ·(l1+l2) Qλ·l2 0 x2

x2 xQλ·(l2+l3) Qλ·l3 00 x2 xQλ·(l3+l4) Qλ·l4




T(θ) = h2(θ)

Qλ·l1 0 x2Q−λ·l2 xQλ·(l1+l4)

xQλ·(l1+l2) Qλ·l2 0 x2Q−λ·l3x2Q−λ·l4 xQλ·(l2+l3) Qλ·l3 0

0 x2Q−λ·l1 xQλ·(l3+l4) Qλ·l4




T(θ) = h2(θ)

Q−λ·l1 0 x2Q−λ·l2 xQλ·(l1+l4)

xQλ·(l1+l2) Q−λ·l2 0 x2Q−λ·l3x2Q−λ·l4 xQλ·(l2+l3) Q−λ·l3 0

0 x2Q−λ·l1 xQλ·(l3+l4) Q−λ·l4




T(θ) = h2(θ)

Qλ·l1 0 x2Qλ·Y13 0

0 Qλ·l2 0 x2Qλ·Y24

x2Qλ·Y31 0 Qλ·l3 00 x2Qλ·Y42 0 Qλ·l4




T(θ) = h2(θ)

Qλ·l1 0 x2Qλ·Y13 xQλ·Z14

xQλ·Z21 Qλ·l2 0 x2Qλ·Y24

x2Qλ·Y31 xQλ·Z32 Qλ·l3 00 x2Qλ·Y42 xQλ·Z43 Qλ·l4




None of these work. That leaves two possibilities:

1 There is a solution but it’s more complicated than any Ihave tried.

2 The whole idea is wrong. The local ducks were right!


CONCLUSIONS

What you could have learned from this talk:• Affine Toda field theory is a large and elegant subject,

amply demonstrated by the lack of publishable research inthis talk.

• Some objects which can be found in ATFT without ruiningintegrability are solitons, boundaries and defects.

• Some soliton scattering matrices are hidden in type IIdefect transmission matrices.

• Solitons possess fusing rules, linking the different speciesof soliton together.

• Defects could similarly possess fusing rules.


OUTLOOK

Where could one go from here:

• If the defect fusing rules work then one could look for 22Shidden inside a T-matrix somewhere.

• Some investigation into the semiclassical limit of thetransmission matrices is probably in order.

• If I can’t get the defect fusing to work then I could alwaysdo something with sine-Gordon - that always works.

• Defect-defect scattering is a natural thing to investigate.

• I could try to find a d(1)4 defect.


FURTHER READINGT.J. Hollowood, Solitons in affine Toda field theories, Nucl. Phys. B384 (1992)523

P. Bowcock, E. Corrigan, C. Zambon, Affine Toda field theories with defects,JHEP 01(2004)056; arXiv:hep-th/0401020

E. Corrigan, C. Zambon, On purely transmitting defects in affine Toda fieldtheory, JHEP 07(2007)001; arXiv:0705.1066

E. Corrigan, C. Zambon, Comments on defects in the a(r) Toda field theories, JPhys. A42 (2009) 204008 ; arXiv:0902.1307

E. Corrigan, C. Zambon, Integrable defects in affine Toda field theory and infinitedimensional representations of quantum groups, Nucl. Phys. B848 (2011) 545 ;arXiv:1012.4186

R.A. Hall, Affine Toda solitons and fusing rules, PhD thesis, Durham (1994);etheses.dur.ac.uk/5596/

W.M. McGhee, On the topological charges of the affine Toda solitons, PhD thesis,Durham (1994); etheses.dur.ac.uk/5482/

http://arxiv.org/abs/hep-th/0401020

http://arxiv.org/abs/arXiv:0705.1066



http://etheses.dur.ac.uk/5596/

http://etheses.dur.ac.uk/5482/


Should you have any cogent questions about this talk please e-mail me.

mailto:[email protected]

APPENDIX

CLASSICAL SOLITONS - HIROTA ANSATZ

Hollowood was the first to construct a(1)r affine Toda solitons

using Hirota methods. With imaginary coupling, β = iβ,solitons have the form

u = − 1iγ

r∑j=0

αj ln τj

new line

By using the a(1)r equations of motion and a decoupling the tau

functions must then obey

τjτj − τj2 − τ ′′j τj + τ ′2j = τj−1τj+1 − τ 2

j

APPENDIX

SINGLE SOLITONSIn a(1)

r ATFT there are r different species of single soliton. The speciesp single soliton has its tau function given by

τj = 1 + ωpjEp

man

withEp = eapx−bpt+cp

so

where

• ω = e2πir+1

• ap = mp cosh θ, bp = mp sinh θ

• R(cp) gives the centre of mass of the soliton and I(cp) relates tothe topological charge.

• The mass of the soliton is Mp = 2hmp with mp = 2 sin(

πpr+1

).

APPENDIX

MULTISOLITONS

The N soliton solution for a(1)r is given by the tau functions

τj =

1∑µ1=0

. . .

1∑µN=0

exp

N∑l=1

µl ln(ωpljEpl

)+

∑1≤l<m≤N

µlµm ln(

A(plpm))

which depend upon a single type of interaction parameter

A(pq) = −(ap − aq)

2 − (bp − bq)2 −m2

p−q

(ap + aq)2 − (bp + bq)2 −m2p+q

APPENDIX

DEFECT CONDITIONS

Applying the Euler–Lagrange equations to the defectLagrangian gives the usual bulk ATFT equations of motion aswell as as defect equations at x = 0. For a species 1 defect:

u′ = (1− B)u + Bv−DBu

v′ = −(1− B)v + BTu + DBv

And for a species r defect:

u′ = −(1− BT)u + BTv−DBT

u

v′ = (1− BT)v + Bu + DBT

v

andalso

In either case the conditions give a Backlund transformation.

APPENDIX

CLASSICAL TRANSMISSION AND OLD-FASHIONED

MARRIAGE

Using the single species p soliton solution from 14 slides ago,and applying the defect conditions from 4 slides ago you caneasily, in your head, figure out the effect a defect has on thesoliton.

However

Just in case, heaven forbid, you forgot, the soliton tau functionfor u is

τuj = 1 + ωpjEp

and after passing through the defect (in a rightwards direction)the tau function for v is

τ vj = 1 + ωpjzpEp

APPENDIX


MARRIAGE

The delay factor zp that a species p soliton receives from aspecies 1 defect is

zp =ieη−θ + ω

p2

ieη−θ + ω−p2

with something similar but omitted for a species r defect.

explain delay factor

The delay factor zp shows us that the species of soliton does notchange, and that the soliton is delayed/advanced and receivesa phase shift (possibly changing the topological charge) by thedefect.

APPENDIX


MARRIAGEThere are a couple of things to note about this:• Different solitons receive different delay factors, and in

multisoliton solutions the constituent single solitons aredelayed independently.

• Note that the delay factor has a pole (and a zero) in it. Fora species p = r+1

2 soliton (when r is odd) this pole lies onthe real line. When the pole is hit the soliton is infinitelydelayed - i.e., the soliton becomes married to the defectforever.

• Evidently a soliton may avoid marriage by dancingaround the pole.

• Since defect conditions are Backlund transformations, itshould not be surprising that defects can change thesoliton number of the system.

APPENDIX

CLASSICAL SOLITON FUSING RULES

In terms of tau functions, we can show that a two-solitonsolution (both of species 1) becomes a one-soliton solution (ofspecies 2) when the fusing conditions are met. The two-solitonsolution has tau function

τj = 1 + ωjE (A)−12 + ωjE (A)−

12 + ω2jEE

where the spacetime dependence and topological charge of thefirst soliton is described by E and of the second soliton by E.

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The interaction parameter is

A = − (a− a)2 − (b− b)2

(a + a)2 − (b + b)2 −m22

APPENDIX

CLASSICAL SOLITON FUSING RULES

τj = 1 + ωjE (A)−12 + ωjE (A)−

12 + ω2jEE

and

The interaction parameter has a pole precisely at the fusingangle θ − θ = 2π

r+1 , at this rapidity difference the tau functionbecomes

τj = 1 + ω2jE2

which is precisely the species 2 single soliton solution.

• Note that the for r ≥ 3 the fusing process may break theconservation of topological charge!

• Thus, a(1)3 is the simplest of these theories to have ‘missing

charges’.• Classical soliton fusing also shows up in the delay factors:

z2(θ) = z1(θ − iπr+1)z1(θ + iπ

r+1).

APPENDIX

CLASSICAL DEFECT FUSING RULESConsider again the delay factor for a species 1 soliton through aspecies 1 defect in a(1)

r

z1 =ieη−θ + e

iπr+1

ieη−θ + e−iπ

r+1

• It was noted that soliton fusing can be seen in the delayfactors:

z2(θ) = z1(θ − iπr+1)z1(θ + iπ

r+1)

• But z1 clearly doesn’t just depend on θ, but rather on θ − η.• Thus we also have

z2(η) = z1(η + iπr+1)z1(η − iπ

r+1)

This now describes a species 1 soliton passing through aspecies 2 defect.

APPENDIX

CLASSICAL DEFECT FUSING RULES

The other way to describe a classical defect is by its Lagrangian.

so what?

In a(1)2 we have Lagrangians for the two different fundamental

defects. Do the fusing rules also work at the Lagrangian level?

Yes!

APPENDIX

CLASSICAL DEFECT FUSING RULES

The other way to describe a classical defect is by its Lagrangian.

so what?

In a(1)2 we have Lagrangians for the two different fundamental

defects. Do the fusing rules also work at the Lagrangian level?Yes!

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