folding defect affine toda field theories

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Folding defect affine Toda field theories

Craig Robertson

April 2013

Abstract

A folding process is applied to fused a(1)r defects to construct defects for the non-simply laced

affine Toda field theories ofc(1)n , d

(2)n and a

(2)2n at the classical level. Support for the hypothesis

that these defects are integrable in the folded theories is provided by the observation that

transmitted solitons retain their form. Further support is given by the demonstration thatenergy and momentum are conserved.

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arXiv:1304.3129v1

[hep-th]10Apr20

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Contents

1 Introduction 2

1.1 Affine Toda field theory and classical defects . . . . . . . . . . . . . . . . . . . . 31.2 Layout of this paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Bivalent foldings of a(1)r 5

2.1 The bivalent foldings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.1 Case 1: h even, k even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Case 2: h even, k odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.3 Case 3: h odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Solitons and folding 10

3.1 a(1)r solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Folding a(1)r solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Tau functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3.1 One soliton solution of a

(1)

r. . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3.2 Two soliton solution ofa(1)r . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3.3 Single solitons in the folded theories . . . . . . . . . . . . . . . . . . . . . 133.3.4 Other solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Energy-momentum of solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.1 Coupling dependence of energy, momentum and mass . . . . . . . . . . . 193.4.2 Calculation of energy and momentum of solitons . . . . . . . . . . . . . . 213.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Fusing defects 25

4.1 Two defect system and time delays . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 A fused a(1)r defect and energy-momentum conservation . . . . . . . . . . . . . . . 27

4.2.1 Energy conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2.2 Momentum conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 The folded defect 32

5.1 Folding the Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.2 Content of the defect equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.3 Time delay of solitons through the defect . . . . . . . . . . . . . . . . . . . . . . 34

5.3.1 Algebraic constraints with such d and d . . . . . . . . . . . . . . . . . . . 355.4 Energy conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.5 Momentum conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.5.1 Proof of the relations (5.22) and (5.23) . . . . . . . . . . . . . . . . . . . . 38

5.5.2 Proof of (5.27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6 Discussion 41

A Tau function identifications 43

A.1 c(1)n solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

A.2 d(2)n solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

A.3 a(2)2n solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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B a1 embeddings in a(1)2n1 and folded theories 45

B.1 p = n soliton as an a1 soliton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45B.2 A less obvious embedding ofa1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

C Equivalence of a priori and a posteriori folding and ways to write D 47

C.1 Gradients of the defect potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 48C.1.1 A more useful way to write D . . . . . . . . . . . . . . . . . . . . . . . . 51

C.2 Comparison of the a priori and a posteriori defect conditions . . . . . . . . . . . 52

D The folded defect in a(2)2 53

D.1 Energy conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55D.2 Momentum conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55D.3 Comparison to the type II defect . . . . . . . . . . . . . . . . . . . . . . . . . . . 56D.4 Algebraic constraints and a posteriori folding . . . . . . . . . . . . . . . . . . . . 57

1 Introduction

The affine Toda field theories (ATFTs) have seen an upsurge of interest due to the discovery

of integrable defects. Thus far only for the a(1)r [1, 2] and a

(2)2 [3] models do integrable defects

exist in the literature- even at the classical level- despite the search for defects in ATFT havingbeen initiated a decade ago [1]. Compare this to the discovery of solitons in ATFT, where the

construction of a(1)r and d

(1)4 solitons [4] was rapidly followed by solitons in the other models [5].

The different ATFTs have similar properties (e.g., they all possess soliton solutions [5]) so itis expected that defects should exist for all of the ATFTs- as such, an overarching goal in thisfield is to find and investigate the properties of all of the possible defects.

Folding, or reduction, is a powerful tool which allows properties of non-simply laced theo-ries to be found using properties of simply laced theories1 which are often easier to work with.Indeed, folding has previously been put to use to find, in the non-simply laced theories, solitons[5, 6] and also scattering matrices at the classical [7] and quantum [8] levels. Note that foldinghas not previously been applied to the study of defect ATFTs.

In this paper, a folding process, originally described in [9], is used to obtain candidate inte-

grable defects for the c(1)n , d

(2)n and a

(2)2n series of non-simply laced ATFTs by making use of

a(1)r solitons and defects. Two methods, which turn out to be linked, are used to suggest that

these defects should be classically integrable. Firstly, what happens when a soliton is sentthrough the defect; secondly, in what circumstances does the defect conserve energy and mo-mentum. An interesting by-product of the folding process is that it allows for the construction

of multisolitons, breathers and fusing rules for these folded theories (c(1)n , d

(2)n and a

(2)2n )- some-

thing not explicitly considered previously in the literature.

What this paper does not do is provide defects for any of the ATFTs which do not fall un-

der the umbrella of a(1)r , though it does suggest that if the appropriate defects can be found

1The simply laced ATFTs are the a(1)r ,d

(1)s ,e

(1)6 ,e

(1)7 and e

(1)8 theories. They are distinguished in having all roots

of the same length, conventionally

2; although in the a1 case the choice of = 1 is more conventional. Thenon-simply laced theories all have more than one length of root present.

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for the other simply laced theories (d(1)s ,e

(1)6 ,e

(1)7 and e

(1)8 ) then folding may be applied to give

the defects of the other non-simply laced theories. Note that only one type of defect is given

for each ofc(1)n , d

(2)n and a

(2)2n- there is no claim that other defects should not exist in these theories.

In order to set notation, and inform the reader unfamiliar with ATFT, a short summary of

ATFT with defects is given below.

1.1 Affine Toda field theory and classical defects

To each affine Dynkin diagram there is a corresponding affine Toda model [10]. In 1+1 dimen-sions a working definition of ATFT is given by the Lagrangian

L(u) = 12

u u 12

u u U(u) (1.1)

where u is an r component vector living in the root space described by the affine Dynkin diagramand is a Lorentz scalar. The potential is given by [4, 5]

U(u) =m2

2

rj=0

nj

ej u 1

. (1.2)

In (1.2), {i} (i = 1, . . . , r), are the positive simple roots of the root space while 0 =rj=1 njj is the lowest root of the root space, corresponding to the extra node of the affineDynkin diagram. It is the case that

rj=0 njj = 0, so conventionally n0 = 1, while the other

marks {nj} are characteristic of the underlying Lie algebra.The constant m sets a mass scale, which has no importance for the classical discussions hereinso will be set to unity, m = 1. is the coupling constant- it will be convenient to also set = 1but this will need to be justified first (see section 3.4). The potential here is chosen such that

the zero solution has zero energy.

The Euler-Lagrange equations applied to (1.1) give

u u = Uu (1.3)where Uu denotes the gradient of the potential with respect to the vector u.

Guided by the Lagrangian description of a sine-Gordon defect [1]; Bowcock, Corrigan andZambon suggested an Ansatz for defects in the other ATFTs in [2] (subsequently referred to asa type I defect [3])

L = (x)Lu + (x)Lv + (x)

12

uAu + uBv + 12

vCv D(u, v) . (1.4)

Equation (1.4) describes a defect situated at x = 0 with A,B and C constant matrices. Lu andLv are bulk Lagrangians of the form (1.1) for the fields u and v respectively. In practice u andv always correspond to the same root data.

The Euler-Lagrange equations applied to (1.4) give at x = 0 the conditions

u = Au + Bv Duv =

Cv + BTu + Dv (1.5)

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as well as the bulk equations of motion.

In [2] a defect Lax pair, taking into account (1.5), was constructed and it was shown that

only the a(1)r models may incorporate such a defect and remain integrable. It was subsequently

shown in [11] that (1.4) conserves modified energy and momentum only for a(1)r and that the

conditions imposed are the same as for the Lax pair- i.e., in this case energy-momentum con-servation implies integrability. From either approach it is found that

A = C = 1 B

D(u, v) = dr

j=0

e12j (BTu+Bv)

+ d1r

j=0

e12j B(uv)

(1.6)

and so the defect can be classified by B along with the rapidity parameter d. There are twopossibilities for B, which are

B =

rj=1

(j j+1) Tj (1.7)

and its transpose

B =r

j=1

(j j1) Tj (1.8)

where {i} are the fundamental weights of a(1)r , defined by i j = ij for i, j = 1, . . . , r and0 = 0. In this paper B will be taken to mean (1.7), with B

T used if the defect is classified by(1.8).

In [2] the effect of the defect on a one soliton solution was examined. Using a one solitonAnsatz (of species p, say) for u and for v revealed that a B defect gives v a time delay of

p =ie + d

p2

ie + dp2

(1.9)

while a BT defect gives a time delay of

p =ie d

p2

ie

d

p2

(1.10)

where is the rapidity of the soliton and = e2ir+1

. It should be noted that the time delay isdependent on the species of soliton. In particular, the species p soliton and the species h psoliton receive different delays, meaning that neither B nor BT type I defects are compatiblewith the folding considered in this paper.

The framework for defects was extended in [3] to include integrable defects for a(2)2 by means of

what is referred to as a type II defect with an Ansatz of the form

L = (x)Lu + (x)Lv + (x) (uv 2(u v) D(u,v,)) (1.11)

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where u and v are now scalar fields in either a1 or a(2)2 (the form of D depends upon which

theory is being looked at) while is a scalar field which only exists at the defect, known as anauxiliary field.

It was also noticed that a system containing a type I B defect and a type I BT defect with the

same parameter d in a(1)2 would give the same overall time delay to the species 1 and species 2solitons2. As such, solitons possessing the a

(2)2 symmetry in the field u would also possess it in

the field v. This opens up the possibility that (1.11) for a(2)2 somehow arises from fusing two

type I a(1)2 defects. This possibility is formulated and generalised to all ATFTs obtainable by

folding a(1)r in this paper.

1.2 Layout of this paper

Section 2 describes the foldings of a(1)r which will be used in this paper, which may be referred

to as bivalent foldings since the roots of the folded theories are obtained by identifying roots

of a

(1)

r pairwise (contrast with e.g., d

(1)

4 a(2)

2 by which one of the folded roots is obtainedby identifying four of the roots of d

(1)4 ). Such foldings were also considered in [9]. Section

3 describes how solitons of the folded theories may be obtained from a(1)r solitons via these

folding processes. The relevant Hirota tau functions [12] are obtained. Where the folding beingdone is non-canonical (in the sense of [9], i.e., foldings not found in [13]) the tau functions are

compared to those found in [6], which were obtained by canonical folding of d(1)s models. The

energy and momentum of solitons are then considered. An argument is given which suggeststhat at this classical level there is no need to consider just imaginary coupling. Section 4

describes a fused a(1)r defect and its effects on solitons and energy and momentum conservation.

Section 5 considers folded defects and how they affect the folded solitons. Conservation ofmodified energy and momentum is then proven for the folded defects. This section is perhaps

the most important, and opens up a number of possible future research directions. Section 6gives the conclusions and outlook for this discourse. The appendices contain some additional

calculations, including some explicit calculations in a(2)2 .

2 Bivalent foldings ofa(1)r

In this section the folding process used in later sections is formulated. It was noted in [9] that

numerous foldings of this type are possible for a given a(1)r and whilst other ways to fold a

(1)r are

considered in [9] and [14], no extra folded theories arise in this way. Here the bivalent foldingis done explicitly by means of a parameter k Z.

This paper places its foundations in the firm ground of a(1)r , so it is useful to first state some of

the properties of a(1)r . The roots of a

(1)r obey

i j = 2ij i(j+1) i(j1) (2.1)

for i, j = 0, . . . , r. It is useful to extend this range for i and j by identifying the indices modulo

the Coxeter number h. For a(1)r , h = r +1, so e.g., 2 r1. For a(1)r the roots conventionally

2P. Bowcock, private communication. Also mentioned in [3]

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all have length

2 so the Cartan matrix is given by

Cij =2

j j i j = i j

i.e., (2.1). Note that there is no sum over j. The eigenvalues of the Cartan matrix play an

important role in the construction of solitons.

Out of the r + 1 roots {0, . . . , r} any choice of r of them form a basis for the root space.Conventionally, the fundamental weights {i} are given by

i j = ij , i, j = 1, . . . , r

with 0 = 0 so they form a dual basis to {1, . . . , r}. However, it is equally valid to define aset of fundamental weights by e.g.,

i j = ij , i, j = 0, 1, . . . , q 1, q+ 1, . . . , r

with q = 0. Note then that in (1.7) and its transpose the sum may be extended to includej = 0 and then whichever set of fundamental weights is used B and BT remain the same asbefore. Note also that the relation

i = 2i i1 i+1is also valid for any choice of{i}.

The affine Toda field, u, lives on the a(1)r root space so a natural basis is given by the simple

roots {1, . . . , r}: u = u11 + . . . urr. However, one can consider an extended representation

u = 0u0 + 1u1 + . . . + rur

where only r of the r + 1 components are independent. Noting then that ui = i u, it isreasonable to consider uq = 0 for some q = 0 whilst retaining u0. This innocuous observationplays a role when considering the folding here in its full generality.

2.1 The bivalent foldings

The foldings considered here are bivalent in the sense that the roots of the folded theory will be

obtained by identifying all roots ofa(1)r pairwise. Specifically, the roots i and h+ki in a

(1)r will

be paired up for all i. Which folded theory is obtained depends upon whether h is even (r odd)

or odd (r even) and also on whether k is even or odd. That there are a number of distinct waysto identify the roots whilst obtaining a folded theory was already noted in [9]. Roots in thefolded theories will be labelled by .

2.1.1 Case 1: h even, k even

This case includes the only canonical folding in this paper (i.e., the only one also found in [13])

when k = 0, that is a(1)2n1 c(1)n (h = 2n). The following Dynkin diagrams illustrate this

folding.

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e

e e e e e

e

e e e e e

ZZ

ZZ

a(1)2n1

-6

?e u u u u u e

c(1)n

Figure 1: a(1)2n1 c(1)n

White nodes correspond to roots of length

2 and black nodes length 1. The nodes in the above

diagram have purposely not been labelled. For the a(1)2n1 picture 0 could really be placed any-

where, though conventionally it would be placed on the left. To go from a to c diagrammaticallythe nodes that are directly above/below each other are identified.Here two nodes are self-identified, so there exist two solutions to i = h+ki; the self-identifiednodes are k

2and h+k

2= k

2+n. Upon folding the labelling of nodes is no longer democratic

and to get the right Cartan matrix for the folded theory, the extra root 0 must be identifiedwith one of these self-identified nodes.

The choice made here is

0 = k2

implying that

i =k

2+i + k

2+2ni

2.

For the a(1)

2n1

field u fold by identifying u k2

= 0; uk2+i

= u k2+2ni

= i

2

for i = 1, . . . , n

1 and

uk2+n

= n. So

u =2n1j=0

ujj =n

j=0

jk

2+j+ k

2+2nj2

.

In above, note that when j = h2 = n, both a2n1 roots there are the same, so that term is

merely nn+k2

. The notation is that u is an unfolded field, and is a folded c(1)n field, so

=n

j=0

jj .

The claim is that, firstly, {i} are the roots of c(1)n and, secondly, that the Lagrangian (1.1) fora(1)2n1 folds to that of c

(1)n . For the first part of the claim, by (2.1)

i j =k

2+i+ k

2+2ni2

k

2+j+ k

2+2nj2

=1

2

2ij i(j+1) i(j1) + 2i(2nj) i(2n+1j) i(2n1j)

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so the inner products are

(i, j) i j(0, 0) 2

(s, s) s

= 0, n 1

(n, n) 2

(s, s + 1) s = 0, n 1 12(0, 1) 1(n 1, n) 1

with all other cases giving zero. These are the inner products of c(1)n , confirming the veracity of

the folding. With this observation it is clear that the kinetic terms of the Lagrangian ( 1.1) are

as required when folding u . Less obvious is the potential: for a(1)2n1, (1.2) gives

U(u) =

2n1j=0

(ej u 1)

since all of the marks of a(1)r obey nj = 1. This can be rewritten

U = eu k

2 +n1j=1

euk

2+j + euk

2+2nj

+ e

uk2+n 2n .

Note that k2+i j = k

2+2ni j = i j , so folding gives the potential

= ek2 +n

1

j=1

ek2+j + ek2j

+ ek2 +n 2n

= e

0 + 2n1j=1

e

j + e

n 2n

=n

j=0

nj

e

j 1

(2.2)

with {ni} the marks of the c(1)n algebra. Thus, whichever even k is used this folding prescriptionsends the a

(1)2n1 ATFT to the c

(1)n ATFT.

2.1.2 Case 2: h even, k odd

This case is quite different, in as much as all nodes are identified with one other. This case

gives a(1)2n1 d(2)n (despite the notation d(2)n has just n nodes instead of n + 1). There are

equivalent choices of 0 since the Dynkin diagram looks the same from both ends. The choice

0 = k+1

2+ k1

2 +2n

2 is made, which includes 0 when k = 1. The diagram that illustrate thisare

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e e e e e e

e e e e e ea(1)2n1

-6

?er u u u u re

d(2)n

Figure 2: a(1)2n1 d(2)n

The black-in-white node has length 12

.

The identification of roots in the folded theory is chosen to be

i =k+1

2 +i+ k1

2 +2ni2

and so

u =2n1j=0

ujj = n1j=0

jk+12 +j + k12 j

2=

n1j=0

jj

The normalisation of the d(2)n roots here is non-canonical so this could be viewed with suspicion,

however the inner products

(i, j) i j(0, 0) 12(s, s) s = 0, n 1(n, n) 12(s, s + 1) 12

are precisely half of the standard inner products of d(2)n . The remedy, if conventional normali-

sation is required, is simply to rescale the roots i

2i.

The potential obtained by this folding is

=n1j=0

ek+1

2 +j + ek1

2 +2nj

2n

= 2

n1j=0

nj

ej

1 (2.3)since all the marks are equal to one in d

(2)n . This potential obtained is twice the standard d

(2)n

potential. The factor of two may be removed in the action by an isotropic space-time rescaling:

x x2

, t t2

. Such a rescaling does not affect the kinetic terms, so the standard d(2)n ATFT

Lagrangian is obtained.

2.1.3 Case 3: h odd

This case covers a(1)2n a(2)2n, so h = 2n + 1. Since the indices on the roots of a(1)2n are identified

modulo this odd Coxeter number, there is no longer a notion of odd and even indices (e.g.,

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1 2n+2). Due to this, k odd and k even give the same folding, and so the choice will be madethat k is even (to ensure that all foldings are considered, a suitable range is k = 0, 2, 4, . . . , 4n).This case contains one self-identified node, k

2which will be chosen to correspond to 0.

e

e e e e e e

e e e e e e

ZZ

a(1)2n

-6

?e u u u u u re

a(2)2n

Figure 3: a(1)2n a(2)2n

There are n + 1 roots in the folded theory and the identification made is

i =k

2+i+ k

2+2n+1i2

.

The inner products are

(i, j) i j(0, 0) 2

(p, p) p = 0, n 1(n, n) 12(p, p + 1) p = 0 12(0, 1) 1

which are the correct inner products for a(2)2n.

The potential becomes

=n

j=0

nj

e

j 1

(2.4)

with n0 = 1 and ni = 2 for i = 0. Exactly as required for the folding to give a(2)2n ATFT.

Thus it is shown that these foldings are valid, as expected from [9]. Given this, it is rea-

sonable to consider the construction of solitons for c(1)n , d(2)n and a(2)2n using these foldings (as thisis the only folding process considered in this paper, these ATFTs will be collectively referred toas the folded theories).

3 Solitons and folding

This section details the construction of solitons in the folded theories, all of which are a(1)r solitons

possessing a certain symmetry (namely i = k+hi). In this context, a(1)r solitons must be

introduced first.

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3.1 a(1)r solitons

Hirota methods [12] are used to find soliton solutions in the ATFTs. For this paper the starting

point will be a(1)r , as it is the theories obtained from folding this that are of interest. For a

general ATFT the soliton solutions may be written in the form [5]

u = 1

rj=0

jj ln j (3.1)

where i =2

ii with no sum implied and the ATFT corresponds to an affine Dynkin diagramwith r + 1 nodes.

The tau functions {j} depend only upon the root data (i.e., which ATFT is being consid-ered) and do not depend upon the coupling. In order to find the tau functions (3.1) is used inthe equation of motion (1.3) along with a decoupling [4]. In general the equation to be solvedis

i

ii 2i i i + 2i ni

rj=0

jj ij 1

2i = 0 . (3.2)

Note that the kinetic terms are always in Hirota bilinear form ii2i i i+2i = 12

D2t D2x

ii, but the potential is only in true bilinear form for a

(1)r . For a

(1)r , i = ni = 1 for all i, so (3.1)

becomes

u = 1

rj=0

j ln j (3.3)

while the equation the tau functions must obey simplifies greatly toii 2i i i + 2i = i1i+1 2i . (3.4)

3.2 Folding a(1)r solitons

In all cases folding is achieved by identifying i with k+hi, so to fold the a(1)r ATFT set

ui = uk+hi. In the soliton Ansatz (3.3) this is tantamount to having i = k+hi. Anya(1)r soliton with this property is also a soliton of the folded theory. A summary of the tau

function identifications required for the folded solitons to fit the Ansatz (3.1) is given here:

c(1)n d

(2)n a

(2)2n

0 k2

(k+12

) 12 k2

i i = 0, n 1, n k2+i

k+12 +i

k2+i

n1 k2+n1

(k+12 +n1

)12 k

2+n1n k

2+n (k

2+n)12

The details may be found in the appendix A.

The possibility of obtaining solitons in the folded theories is dependent on the relation i =k+hi having solutions for all i, so the functions will need to be examined. It is alreadyknown that for c

(1)n in the case k = 0 that such a relation is possible [5].

11


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3.3 Tau functions

3.3.1 One soliton solution of a(1)r

The tau functions for the one soliton solution (of species p) of a(1)r are of the form

j = 1 + pj

Ep

with = e2ir+1 , and p = 1, . . . , r (note that p = 0 just gives the trivial solution), so that

p encompasses the (r + 1)-th roots of unity (where r + 1 = h, the Coxeter number). Theexpression Ep is given by Ep = e

apxbpt+cp , cp is constant: its imaginary component determinesthe topological charge of the soliton under consideration- there are up to r+1 sectors of differingtopological charge [15, 6]. In order to avoid singularities in the soliton solution (avoiding j = 0)this imaginary part ofcp should be chosen carefully. The real part of cp gives the position of thecentre of mass of the soliton at time t = 0 and may be chosen arbitrarily. ap = p cosh andbp = p sinh with R() > 0 for a right-moving soliton ( is the rapidity- or at least R() is).p is taken to be the positive square root of the pth eigenvalue of the matrix (N C)ij = nii j .For a(1)r these are merely the ordered eigenvalues of the Cartan matrix of the (affine) Lie algebra,given by

2p = 4 sin2

p

r + 1

(3.5)

with most eigenvalues appearing twice. p = 0, 20 = 0 (in all cases), and p = n, 2n = 4, (in the

case r = 2n 1) are the only eigenvalues which arent degenerate. It should be noted that thesquare masses of the a

(1)r solitons are proportional to these eigenvalues, so it is not erroneous to

refer to {p} as the masses of the solitons. One thing to note is that the {p} in the foldedtheories obtained from a

(1)r in this paper are exactly the same, only that the degeneracy has

been removed (the masses of the basic soliton solutions in the folded theories are typically{2p}). This ought to be contrasted to the cases of d(2)n and a(2)2n when folded from d(1)s theories-there the eigenvalues ofN C for d

(1)s need to be rescaled to obtain those of the folded theories [6].

It will be convenient to rewrite the tau function in a slightly different way as

j = 1 + pj+pkEp (3.6)

which is equivalent to redefining the imaginary part of cp. For the one soliton solution this hasno purpose, but for the two soliton solution this reparametrisation will prove useful.

Note at this point that there is but one non-trivial soliton of a(1)r which is compatible with

folding (i.e., has j = h+kj), which is the n soliton of a(1)2n1 with k even. In that case

j = 1 + (1)jE = 2n+kj , since (1)2n+k = 1. This soliton is the species n single soliton ofc(1)n , as well as being a bona fide a1 soliton- more on a1 embeddings is included in appendix B.

3.3.2 Two soliton solution of a(1)r

The tau functions of a two soliton solution (one of type p; the other of type q) in a(1)r have the

form

j = 1 + pjEp +

qjEq + A(pq)(p+q)jEpEq (3.7)

12


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where there is now an interaction parameter

A(pq) = (ap aq)2 (bp bq)2 2pq

(ap + aq)2 (bp + bq)2 2p+q(3.8)

and ap = p cosh 1, aq = q cosh 2, etc. (i.e., there are two rapidities to be considered). If

p and q are the same, and both rapidities are the same, then (3.8) vanishes, and in fact (3.7)describes a one soliton solution (from this point of view the n soliton ofc

(1)n may also be thought

of as a two soliton solution of a(1)2n1).

For a(1)r solitons to also be solitons of the folded theories it must be the case that j = k+hj .

After folding (3.7) is to be interpreted as a one soliton solution, so the constituent solitons mustbe given the same centre of mass, R(cp) = R(cq), as well as the same rapidity, 1 = 2.Also, there is little chance of satisfying j = k+hj if the quadratic term in E depends on j:the conclusion is that p+q = 1 and so q = h p. Consequently, the solitons being paired uppossess the same mass, meaning that ap = aq, bp = bq. Thus, Ep and Eq may only differ in

I(c).

From (3.8), with 1 = 2,

A(p(hp)) = 22p

4a2p 4b2p=

4sin22pr+1

16sin2

pr+1

= cos2

p

r + 1

A . (3.9)

Thus, the tau functions compatible with folding (to a one soliton folded solution) all possessthe form

j = 1 +

pj + pkpj

Ep + ApkE2p (3.10)

which in a(1)r navely appears to have the interpretation of a species p a(1)r soliton joined to aspecies h p soliton which has its topological charge shifted into a different sector. It turns outhowever, that the topological charges of the folded solitons only depend on whether k is odd oreven (and even then only when h is even) and so all possibilities are actually encompassed by

k = 0 and k = 1. Nonetheless, this paper will continue to work with arbitrary k.

3.3.3 Single solitons in the folded theories

Here the folded solitons (3.10) with tau function identifications given in section 3.2 are comparedto the tau functions listed by McGhee in [6], thereby confirming that (3.10) are indeed the bonafide single soliton solutions of the folded theories

c(1)n tau functions

In this case k is even, and as identified in section 3.2, j = k2+j , so

j = 1 + pk2

pj + pj

Ep + ApkE2p .

This does not match [6] unless k = 0; however the freedom to shift I(cp) allows Ep pk2 Ep

to give

j = 1 + 2 cos

pj

n

Ep + cos

2

p

2nE2p

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which does match [6]. That changing k only shifts Ep clarifies the comment at the end of section3.3.2, i.e., any even k gives the same folded soliton possessing the same topological charges.

d(2)n tau functions

This case is significantly more interesting to verify as the folding process previously used to findd(2)n solitons made use of d

(1)n+1 solitons [5, 6], rather than those of a

(1)2n1 (note that d

(1)n+1 and

a(1)2n1 do possess the same Coxeter number h = 2n)- the identification of roots required for the

canonical folding d(1)n+1 d(2)n is given diagrammatically in the discussion, section 6.

In this case k is odd with 0 =

k+12

, n1 =

k+12 +n1

and j = k+12 +j

for j = 1, . . . , n 2.In general

k+12 +j

= 1 + pk2

p(j+

12 ) + p(j+

12 )

Ep + ApkE2p .

The k dependence can then be scaled away by Ep Eppk2

. Further scaling Ep by Ep EpA

= Epcos(p2n )

gives

k+12 +j

= 1 + 2cos

p(2j+1)2n

cosp2n

Ep + E2pwhich already matches j in [6] for j = 0, n 1 3. For the tau functions corresponding to theend nodes,

k+12

= 1 + 2Ep + E2p = (1 + Ep)

2

=

0 = 1 + Ep

and

k+12 +n1

= 1 + 2cos

p(2n1)2n

cosp2n

Ep + E2p= 1 + 2(1)pEp + E2p = (1 + (1)pEp)2= n1 = 1 + (1)pEp

in agreement with [6]. Note that the eigenvalues p of d(2)n are exactly those inherited from

a(1)2n1 so do not have to be rescaled, unlike with the canonical folding [6].

a(2)2n tau functions

This case is equally interesting, with the canonical folding being d(1)2n+2 a(2)2n, instead of

a(1)2n a(2)2n- the canonical folding is illustrated in the discussion.

In this case k may be taken to be even, with n =

k2+n

, whilst j = k2+j

for j = 0, . . . , n 1.From (3.10)

k2+j

= 1 + pk2

pj + pj

Ep + ApkE2p

3Note that [6] uses the wrong Coxeter number.

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so, shifting Ep Eppk2

cos( p2n+1)results in

k2+j

= 1 + 2cos

2pj2n+1

cos p2n+1

Ep + E

2p

which matches j in [6] when j = n (though ordering is reversed- note that a(2)2n is self-dual) 4.When j = n the result is

k2+n

= 1 + 2cos2np2n+1

cos

p2n+1

Ep + E2p= 1 + 2(1)pEp + E2p = (1 + (1)pEp)2= n = 1 + (1)pEp .

This matches [6] but with the ordering reversed. Again the eigenvalues obtained need not berescaled.

Thus the previously known single solitons of each of the folded theories are recovered. The

relative simplicity of a(1)r allows easier construction of the multisoliton solutions than via d

(1)s ,

which would otherwise be necessary.

3.3.4 Other solutions

Once the basic soliton solutions of the folded models have been found, multisolitons in these

models can be constructed; however, this requires knowledge of the generally complicated inter-action parameters of the folded model. This problem may be obviated by instead constructing

these multisolitons in the a(1)r model. This was noted in [16], but was only applied to c

(1)n .

A formula for multisoliton solutions in a(1)r is [4]

j =1

1=0

. . .

1N=0

exp

N

i=1

i ln

pijEpi

+

1i


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Then, by (3.11), the four soliton a(1)r solution has tau functions

j = 1 +

pj + p(kj)

Ep +

qj + q(kj)

Eq + A(12)pkE2p + A

(34)qkE2q

+ A(13)pj+qj + A(14)pj+q(kj) + A(23)p(kj)+qj + A(24)p(kj)+q(kj)EpEq+ A(12)pk

A(13)A(23)qj + A(14)A(24)q(kj)

E2pEq

+ A(34)qk

A(13)A(14)pj + A(23)A(24)p(kj)

EpE2q

+ A(12)A(13)A(14)A(23)A(24)A(34)pk+qkE2pE2q .

This is a solution of the folded theory provided that j = k+hj . For this to be possible it mustbe the case that

A(13) = A(24)

A(14) = A(23)

and so the two soliton folded solution has

j = 1 +

pj + p(kj)

Ep +

qj + q(kj)

Eq + A(12)pkE2p + A

(34)qkE2q

+ A(13)

pj+qj + p(kj)+q(kj)

EpEq + A(14)

pj+q(kj) + p(kj)+qj

EpEq

+ A(12)A(13)A(14)pk

qj + q(kj)

E2pEq

+ A(34)A(13)A(14)qk

pj + p(kj)

EpE2q

+ A(12)A(34)

A(13)2

A(14)2

pk+qkE2pE2q . (3.12)

Note that there is no need to convert these tau functions to those that are conventional from(3.1) as these tau functions with the Ansatz (3.3) and appropriate identification of roots arethe solitons of the folded theories. Note that (3.12) contains four interaction parameters- a factthat is not obvious, should one wish to construct folded solitons using the folded theory as astarting point.

Using that ap = p cosh 1, aq = q cosh 2, bp = p sinh 1, bq = q sinh 2 and denoting thetwo rapidities by 1 = + and 2 = gives the interaction parameters as

A(13) =2pq + (p + q)

2 sinh2 (p q)2 cosh2 (

p+

q)2 cosh2

(

p q

)2 sinh2

2p+q

(3.13)

A(14) =2p+q + (p + q)

2 sinh2 (p q)2 cosh2 (p + q)2 cosh

2 (p q)2 sinh2 2pq(3.14)

A(12) = cos2

p

r + 1

(3.15)

A(34) = cos2

q

r + 1

. (3.16)

Among these two soliton solutions, an interesting case is when the relative rapidity 12 = 2between the solitons is imaginary. This can result in a breather (if p = q), and can also resultin an interaction parameter being singular (namely (3.13)) resulting in fusion.

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Fusing rules in the folded theories

It is known, certainly in the simply laced theories, that the rapidities of a two soliton solutionmay be chosen such that fusion occurs, resulting in a one soliton solution [4, 17, 18]. The twosoliton solution (3.12) is used here to show that fusion occurs in the same vein as in the simplylaced theories.

In the folded theory a one soliton solution of species s has the form

j = 1 +

sj + sksj

E+ cos2

s

r + 1

skE2 (3.17)

and so the aim of this section is to show that (3.12) may be reduced to this form. For thisto be the case it will be necessary that the highest order term in ( 3.12), (

A) pk+qkE2pE

2q , is

equal to the highest order term in (3.17), cos2

sr+1

skE2. This necessitates the identifications

s = p + q (identified modulo h) and E = EpEq. This still leaves unwanted terms in (3.12), but

shifting Ep A(13)12

Ep and Eq A(13)12

Eq gives

j = 1 +

A(13) 1

2

pj + p(kj)

Ep +

A(13) 1

2

qj + q(kj)

Eq

+

A(13) 12

A(12)pkE2p +

A(13) 12

A(34)qkE2q

+

pj+qj + p(kj)+q(kj)

EpEq + A(14)

A(13)

12

pj+q(kj) + p(kj)+qj

EpEq

+ A(12)A(14)

A(13) 1

2pk

qj + q(kj)

E2pEq

+ A(34)A(14) A(13) 12

qk pj + p(kj)EpE2q

+ A(12)A(34)

A(14)2

pk+qkE2pE2q

so all the unwanted terms may be removed provided that the relative rapidity between the pand q solitons is chosen such that there is a pole in A(13), which leaves the tau functions

j = 1 +

sj + s(kj)

E + A(12)A(34)

A(14)2

skE2

and so it needs to be shown that A(12)A(34)

A(14)2

= cos2

sr+1

and that E is valid for a

species s soliton, i.e., E = es cosh xs sinh t+c.

For there to be a pole in A(13) the denominator of (3.13) must vanish, resulting in

= i (p + q)2(r + 1)

i . (3.18)

The choice of the plus sign is made in what follows (the sign makes no real difference). Therapidity difference between the solitons 2 is i times the fusing angle. Having fixed the effect

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on A(14) is

A(14) =cos

(p+q)r+1

+ cos(2)

cos

(pq)r+1

+ cos(2)

=cos

sr+1

cos

pr+1

cos

qr+1

and so indeed

A(12)A(34)

A(14)2

= cos2

s

r + 1

is satisfied as required.

The final step in checking that fusion has occurred is to show that E has the desired form,

E = escosh

ssinh +c

,

E = EpEq = exp [(p cosh( + ) + q cosh( )) x (p sinh( + ) + q sinh( )) t + cp + cq]

so it must be the case that c = cp + cq while the prefactor of x gives

p cosh( + ) + q cosh( ) = 4 sin

(p + q)

2(r + 1)

cos

(p q)2(r + 1)

cos

(p + q)

2(r + 1)

cosh

+ 4i cos

(p + q)

2(r + 1)

sin

(p q)2(r + 1)

sin

(p + q)

2(r + 1)

sinh

= s cosh

+ i (p q)2(r + 1)

and similarly the prefactor of t gives s sinh

+ i(pq)2(r+1)

. Thus,

= + i(p q)2(r + 1)

is the expression for the rapidity of the one soliton solution which arises from such a fusion.Note that an alternative choice of rapidities

p = + iq

r + 1q =

i

p

r + 1

will result in the species p + q (modulo h) soliton obtained by fusion to have rapidity .

Breathers in the folded theories

The existence of breather solutions in ATFT has been known for some time [17] and have been

considered in Hirota form for a(1)r [19] and d

(1)4 [20]. In this section the breathers of the folded

theories c(1)n , d

(2)n and a

(2)2n are constructed- they are a

(1)r breathers which respect j = k+hj .

Equation (3.12) can describe folded breather solutions. This will happen when the constituentsolitons are of the same species p = q with the same centre of mass

R(c

1) =

R(c

2) (where

18


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Ep E1 and Eq E2) and with imaginary rapidity difference, = i (where 1 = + ,2 = ). Note that 2 must be less than the fusing angle.

Taking p = q and such gives simple forms for the interaction parameters

A(12) = A(34) = cos2 p

r + 1

A(13) =sin2

cos2

pr+1

cos2

A(14) =cos2

pr+1

sin2

cos2

so there are three interaction parameters involved and the tau functions take the form

j = 1 +

pj + p(kj)

(E1 + E2) + A(12)pk(E21 + E

22)

+ A(13)

2pj + 2p(kj)

E1E2 + 2A(14)2pkE1E1

+ A(12)A(13)A(14)pk

pj + p(kj)

E1E2(E1 + E2)

+

A(12)A(13)A(14)2

2pk(E1E2)2 .

These tau functions have spacetime dependence through E1 + E2, E21 + E

22 and E1E2. Denoting

C = c1+c22 and i =c1c2

2 and taking the rest frame = 0 gives

E1 + E2 ep cosx+Ccos(p sin t + )

E

2

1 + E

2

2 e2p cosx+2C

cos(2p sin t + 2)E1E2 e2p cosx+2C

so it is readily seen that all time dependence in the rest frame is oscillatory, as expected.

Note also, that in the cases of d(2)n and a

(2)2n these breather tau functions are also the tau

functions of particular breathers in d(1)s .

The validity of the results here is further evidenced by consideration of the soliton energiesand momenta.

3.4 Energy-momentum of solitons

3.4.1 Coupling dependence of energy, momentum and mass

Consider the Lagrangian for bulk ATFT (1.1) at general coupling with m = 1

L = 12

u u 12

u u + 12

rj=0

nj

ej u 1

.

Then the energy and momentum densities are obtained via the stress tensor

T =L

(u)u L (3.19)

19


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where is the flat metric of 1+1 dimensional space. In standard coordinates used here x0 = t,x1 = x so 00 = 1, 11 = 1, 10 = 01 = 0; though lightcone coordinates can be more useful inexamining conserved charges. In particular,

T00 =1

2u2 +

1

2u2 + U(u) = E

T11 =1

2u2 +

1

2u2 U(u)

T01 = T10 = uu = Pwhere Eis the energy density and Pthe momentum density.

Now, consider soliton solutions at general coupling (3.1) for an r component affine Toda field

u = 1

ri=0

jj ln j =1

u

with u denoting the soliton solution obtained if is chosen to be unity (note that the tau func-tions do not depend upon the coupling). Then, for such a solution the energy and momentumdensities are

E(u) = 12

1

2u u + 1

2u u +

j

nj

ej u 1 = 1

2E(u) (3.20)

P(u) = 12

u u = 12

P(u) . (3.21)

Since the coupling dependence just boils down to a prefactor 12

, it is clear that the total energy

and momentum in the bulk, which are just the integrals of (3.20) and (3.21) over space, similarlyonly depends on the coupling via the same prefactor

E(u) =1

2E(u) , P(u) =

1

2P(u) .

In [4] the coupling was taken to be imaginary, = i (so E(u) = 12

E(u), P(u) = 12

P(u))

and the energy and momentum of a species p soliton in a(1)r were found to be

E(u) =2h

2ap =

4(r + 1)

2sin

p

r + 1

cosh()

P(u) = 2h2

bp = 4(r + 1)2

sin

pr + 1

sinh()

which are both real and positive, provided that the rapidity is real, and give the soliton mass

M =

E2 P2 = 2h2

p =4(r + 1)

2sin

p

r + 1

.

Since E(u) and P(u) are both positive in the imaginary coupling case the conclusion is thatE(u) and P(u), the energy and momentum for = 1, are both real but negative (the same istrue for all real couplings).

20


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From a purely classical standpoint, there is no a priori reason not to allow unit coupling, = 1. If this causes the wrong sign of energy and momentum to appear one might supposethat the definitions of energy and momentum used are wrong. Classically, there is no prob-lem with rescaling the action- rescaling does not affect the Euler-Lagrange equations obtained(provided there is no boundary contribution to the action). The particular choice

L =

2

||2

(1.1)

=

2

||21

2u u 1

2u u 1

2

rj=0

nj

ej u 1

(3.22)will result in real, positive energy and momentum for solitons with any complex coupling. For

the species p soliton of a(1)r the expectation is

E =2h

||2 ap =2h

||2 p cosh() (3.23)

P = 2h

||2 bp = 2h

||2 p sinh() (3.24)

M =2h

||2 p =4(r + 1)

||2 sin

p

r + 1

. (3.25)

3.4.2 Calculation of energy and momentum of solitons

A construction first found in [21] and applied in a similar way to here in [19] involves theobservation that for solitons (3.19) may be written in the form

T =

2 C (3.26)such that

T00 = C (3.27)T11 = C (3.28)T01 = T10 = +C . (3.29)

In particular this means that

2C = T00 T11 = 2U = 2||2r

i=0

ni

eiu 1

where the rescaled Lagrangian (3.22) has been used. Up to a constant, the solution to this isgiven by

C = 2||2r

j=0

j ln j (3.30)

since

2C = 2||2r

j=0

jjj 2j j j + 2j

2j

= 2

|

|2

ri=0

ni

eiu 1

21


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where (3.2) has been used.

Having C is useful as it allows the energy and momentum to be calculated much more eas-ily. That is,

E = T

00dx = Cx= (3.31)P =

T01dx =

Cx=

(3.32)

so it is clear that the actual values of E and P depend upon the particular form of the taufunctions of the soliton/multisoliton being considered.

3.4.3 Results

In what follows the coupling is scaled to = 1 and the context will be a(1)r so j = 1 j.

Noteworthy is the observation that C does not change when the theory being considered is one

of those bivalently folded from a(1)r : i.e., if j = k+hj in a(1)r then

C = 2r

j=0

ln j = 2j

j ln j

where the primed identifications are those of the folded theory and the range of j in the foldedtheory depends upon which folding is being considered- the number of terms is the number ofnodes on the folded Dynkin diagram (see appendix A for tau function identifications).With this observation it is clear that folding in this manner does not change the energy-

momentum of the solitons- the a(1)r soliton with the symmetry j = k+hj has the correct

energy-momentum for the folded theory.

Single solitons of a(1)r

The tau functions are given by (3.6), so for a species p soliton

j = 1 + pj+pkEp =

jj

=ap

pj+pkEp

1 + pj+pkEpand

j

j=

bppj+pkEp1 + pj+pkEp

.

For a soliton with real rapidity ap is a positive quantity and for a fixed time Ep = Keapx for

some constant K. Thus,

limx

jj

= 0

limx

j

j= 0

limx

jj

= ap

limx

j

j= bp

22


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implying that

C| = 2r

j=0

ap = 2(r + 1)ap = 2hap

C| = 2r

j=0

bp = 2hbp

and so

E = 2hap = 2hp cosh()

P = 2hbp = 2hp sinh()

M = 2hp

in agreement with (3.23), (3.24) and (3.25).

Double soliton solution in a(1)r

The tau functions are given by (3.7) for the case of a two soliton solution arising from a speciesp and species q single solitons. As long as the interaction parameter is non-zero (it is only zero

for p = q = n in a(1)2n1 which is a single soliton solution) and ap and aq are both positive, the

limit ofj and its derivatives will be dominated by the EpEq term as x as it will grow thefastest. Thus it is found that

limx

jj

= 0

limx

j

j= 0

limxjj = a

p + aq

limx

j

j= bp bq

giving

C| = 2r

j=0

(ap + aq) = 2h(ap + aq)

C|

= 2r

j=0

(bp + bq) = 2h(bp + bq)

and so

E = 2h(ap + aq)

P = 2h(bp + bq)

M2 = (2h)2

2p + 2q + 2pq cosh(p q)

.

Thus, it is seen that the square mass of the solution is relativistically invariant (depending as itdoes only on the difference of rapidities) and, for real rapidities, is at a minimum when p = q-in that case the result is M = M

p+ M

q.

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Folded solitons

With the exception of p = q = n in c(1)n the folded single solitons are all given by two soliton

solutions of a(1)r with q = h p and p = q = . The result is then simply

E = 4hap

P = 4hbp

M = 4hp .

I.e., the folded single solitons have twice the masses of the a(1)r single solitons. This is to be ex-

pected because, from the a(1)r point of view, two a

(1)r solitons are placed together with the same

mass and rapidity.

For two soliton solutions in the folded theories the relevant tau functions are given by ( 3.12)which for large x will again be dominated by the highest order term in E, namely E2pE

2q , so

limxjj = 0

limx

j

j= 0

limx

jj

= 2ap + 2aq

limx

j

j= 2bp 2bq

resulting in

E = 2h(2ap + 2aq) = 4h(p cosh p + q cosh q) (3.33)

P = 4h(p sinh p + q sinh q) (3.34)

M2 = 4 (2h)2 (2p + 2q + 2pq cosh(p q)) . (3.35)

Of particular interest are the breather solutions and the fusing rules, both of which occurat imaginary rapidity difference. For breathers the requirement is p = q so p = q, whilep = + i and q = i. So, if is real the dominant term in the tau function as x is once again the E2pE

2q term, giving

E = 8hp cosh cos

P = 8hp sinh cos

while the mass of the breather is

M = 4cos (2hp) = 4 cos Mp

and thus it is seen that this is indeed a bound state of four a(1)r solitons, all of the same mass

Mp, since the mass M < 4Mp when || > 0.

Fusion of two folded solitons will occur when p q = i(p+q)h so the choice will be made that

p = + iq

h, q = i p

h

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so (3.33) and (3.34) give

E = 4h(p cosh p + q cosh q) = 2hp+q cosh

P = 4h(p sinh p + q cosh q) = 2hp+q sinh .

For calculating the mass it must be noted that 2p + 2q + 2pq cosh(i(p+q)

h ) = 2p+q and so itfollows that

M = 4hp+q

which is precisely the mass of the species p + q folded soliton- this is entirely expected and addsweight to the assertion that fusion has occured.

In summary, the masses are given by

Type of soliton highest order of E m = mass2ha(1)r type p 1 p

most folded single solitons, type p 2 2pbreather of two type p folded solitons 4 4p cos

These solitons are directly proportional to the masses of the fundamental Toda particles (dif-fering by a factor of 2h) and possess the same fusing rules.

In subsequent sections the rescaled Lagrangian (3.22) will not be used but rather there is areversion to use of (1.1), though with = 1. This shall not cause any issues as only energy andmomentum differences will be looked at.

4 Fusing defects

This section deals with fusing defects, i.e., combining two defects (both of type I in the languageof [3]) to form a single (type II) defect. There are two kinds of (type I) integrable defect known

in a(1)r [2] which are referred to in the introduction as B and BT defects. One kind of defect may

be transformed into the other kind by considering the roots of the algebra {i} in a reversedorder- this is reminiscent of the folding procedure which identifies ui with uk+hi.

4.1 Two defect system and time delays

The starting point here will be consideration of a(1)r with two unfused defects: a B defect at

x = a < 0 and a BT defect at x = 0.

u v

x = a x = 0

Figure 4: Two unfused defects in a(1)r

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The Lagrange density describing this system may be written as

L = (a x)Lu + (a x)

1

2uAu + uB +

1

2A D(1)(u, )

+ (x

a)(

x)

L + (x)

1

2

A + BTv

1

2

vAv

D(2)(, v)+ (x)Lv

(4.1)

where A = 1 B and B + BT = 2 with Lu being a Lagrange density in the form of (1.1); Lvand L similarly. In section 2 it was argued that B may be written

B = 2r

j=0

(j j+1) Tj (4.2)

where q = 0 for some q (depending on how the fundamental weights are defined) and BT is

simply the transpose of B.

The defect potentials are given by

D(1) = dr

j=0

e12j(BTu+B)

+1

d

rj=0

e12jB(u)

(4.3)

D(2) = dr

j=0

e12j(BTv+B)

+1

d

rj=0

e12jB(v)

(4.4)

where d and d are free parameters, referred to here as the defect parameters- they may beinterpreted as relating to the rapidities of solitons when p and p have poles/zeroes.

In addition to the bulk affine Toda equations of motion, the Euler-Lagrange equations of (4.1)give the defect conditions

u = Au + B Du |x=a (4.5) = A + BTu + D(1) |x=a (4.6) = A + BTv D(2) |x=0 (4.7)v = Av + B + Dv |x=0 . (4.8)

In (4.5), (4.8) and subsequently, where D is not given a superscript it will be taken to mean

D = D

(1)

+ D

(2)

.

Consider evolving a right-moving species p soliton through this system. An Ansatz is madeas in [2] that the soliton should retain its form, only picking up a time delay and phase shiftfrom the defect, so consideration of the defect on the left means that

u = r

j=0

j ln uj

= r

j=0

j ln j

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where the tau functions have the form of (3.6)

uj (a) = 1 + pj+pkEp(a)

j (a) = 1 +

pj+pkpEp(a) .

By considering (4.5) and (4.6) (take the inner product of each with i) it is found, as in [2]that p is given by (1.9). The time delay to the soliton from this defect is then given byt = 1

bpln |p| while the phase of p may be absorbed into I(cp) (possibly affecting the topo-

logical charge). Note the possibility of a pole in (1.9) hints at the possibility of solitons beingabsorbed and emitted- this change in soliton number is not surprising as the x = a defect

conditions give a Backlund transformation for a(1)r (the x = 0 defect conditions give another

Backlund transformation).

The other defect at x = 0 produces a similar delay

j (0) = 1 + pj+pkpEp(0)

vj (0) = 1 + pj+pkppEp(0)

where p is given by (1.10). The field on the right, v then differs from what it would in the nodefect case by the factor pp.

The general two soliton solution ofa(1)r has tau functions given by (3.7). The consituent solitons

(species p and species q) are delayed by the defects independently so the effect of the defects is

uj = 1 + pjEp +

qjEq + A(pq)(p+q)jEpEq

j = 1 + pjpEp + qjqEq + A(pq)(p+q)jpqEpEq

vj = 1 + pjppEp +

qjqqEq + A(pq)(p+q)jpqpqEpEq . (4.9)

This has been verified the lowest non-trivial order (i.e., consideration of (4.5)-(4.8) at orderEpEq). Its veracity may also be argued in terms of the commutability of B acklund transforma-tions: evolving the species p and species q solitons through a defect and then combining into atwo soliton solution ought to give the same as combining into a two soliton solution and thenevolving through the defect.

It turns out that (4.9) retains its relevance when the defects are fused and folded, this is

argued in the following sections.

4.2 A fused a(1)r defect and energy-momentum conservation

Fusing is done at the Lagrangian level by taking a 0 in (4.1), leaving

L = (x)Lu + (x)Lv+ (x)

1

2uAu + uB vB 1

2vAv D(1) D(2)

. (4.10)

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x = 0

u v

Figure 5: The fused a(1)r defect

There no longer exists any bulk for the field ; it is effectively trapped in the defect and hencemay be referred to as an auxiliary field. Note also that (4.10) bears some similarity to (1.11)though there is no direct coupling of the bulk fields u and v- this is to be expected here as thefused defect arises from two separated defects and locality dictates that u could only couple tov via the field . In fact, [22] and [23] use a modified type II Ansatz which eliminates the directcoupling of the bulk fields found in [3].

The Euler Lagrange equations of (4.10) give now the bulk equations for u and v as well asthe defect conditions at x = 0

u = Au + B Du (4.11)v = Av + B + Dv (4.12)

BTu + D(1) = BTv D(2) . (4.13)

There are now three vector equations instead of the four of (4.5)-(4.8) and it is seen that (4.13)also arises from identification (4.6) and (4.7) when a = 0. Thus it follows that any solution to(4.5)-(4.8) with a = 0 must also solve (4.11), (4.12) and (4.13) and so the time delays previouslyfound and (4.9) remain valid for the fused defect.

Conservation of a modified energy and momentum for the fused defect is now shown- thisstrongly indicates classical integrability of the fused defect, but is mainly done in order to setthe scene for energy-momentum conservation of the folded defects.

4.2.1 Energy conservation

Consider the bulk contribution to the energy, i.e., the integral of T00 of the stress tensor whichderives from (4.10).

E =

0

1

2u u + 1

2u u + U dx +

0

1

2v v + 1

2v v + V dx

where U and V are the bulk potentials for the fields u and v respectively (the field , beforefusing, would have a potential denoted X). Taking the time derivative of this and using thebulk equations of motion; u u = Uu, v v = Vv; gives

E = u u v v |x=0

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provided that the fields are constant at spatial infinity. This is now amenable to the use of thedefect conditions (4.11), (4.12) and (4.13)

E = u u v v|x=0= u (Au + B Du) v (Av + B + Dv)=

BTu BTv u Du v Dv

= u Du v Dv D = D

So, the quantity E + D, the bulk plus the defect energy, is conserved. In some sense this istrivial, as applying a Legendre transformation to the Lagrange density ( 4.10) and performingspatial integration gives the quantity E+ D as the Hamiltonian of the system.

It would appear that energy conservation makes no constraint on D, but there is one sub-tle implication in interpreting D as an energy contribution. The energy can be viewed as thesum of a positive and a negative helicity term, i.e., the sum of the lightcone momenta P+ andP (with a possible factor depending on notation)- hence, the defect potential D (indeed D(1)

and D(2) separately- from energy conservation of type I defects) may also be written as thesum of a positive and a negative helicity term. The form of D(1) and D(2) are already knownand given by (4.3) and (4.4) and each is already written in two parts- the interpretation here isthat the terms with prefactor d and d have positive helicity while the d1 and d1 terms havenegative helicity- the opposite labelling is equally possible. This interpretation of the helicitiesis a useful guide for the momentum conservation argument. Momentum is given by the differ-ence of the lightcone momenta P+ P, so the defect contribution to momentum is expectedto consist of the difference of a positive and a negative helicity term.

4.2.2 Momentum conservationHere it is shown that the fused defect conserves momentum. Navely one might not expectany sort of momentum conservation from a defect ATFT given that placing the defect explicitly

breaks spatial translation invariance. Nonetheless, [11] shows that not only do a(1)r defects admit

a modified conserved momentum, the conservation of energy and momentum alone is enoughto specify the Lagrangian with the result being that the defect is integrable, given the Lax pairanalysis of[2]. It is not guaranteed that momentum conservation in other types of defect impliesintegrability, but given the similarities of the different ATFTs, there is reason to believe thatthis is true.

The bulk contribution to the momentum is given by the integral of T01, so

P =

0

u u dx +0

v v dx .

Taking the time derivative and using the bulk equations of motion gives

P =1

2u u + 1

2u u U 1

2v v 1

2v v + V |x=0 (4.14)

so long as the fields and potentials are constant (in vacuum) at spatial infinity. The aim againis to show that there is a conserved quantity by showing that P is a total time derivative of a

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quantity that exists at x = 0, so the first step is to remove the spatial derivatives via (4.11) and(4.12) to obtain

u u = uAT + BT Du (Au + B Du)= uATAu + BTB + D2u + 2 B

TAu

2uATDu

2 BTDu (4.15)

and similarly

v v = vAT + BT + Dv (Av + B + Dv)= vATAv + BTB + D2v + 2 B

TAv + 2vATDv + 2 BTDv . (4.16)

Noting that ATA = BTB 1 = BBT 1, BTA = ABT, (4.15) and (4.16) give

u u v v = u BBT 1 u v BBT 1 v + 2 A BTu BTv+ D2u D2v 2uATDu 2vATDv 2 B

TDu + BTDv

= u

BBT 1 u v BBT 1 v + D2u D2v 2uATDu 2vATDv 2

BTDu + AD

(1) + B

TDv + AD(2)

(4.17)

where (4.13) has been used to substitute for BTu BTv. It is clear that (4.17) contains theterms u u + v v, cancelling terms in the momentum expression, but there are still termswhich are quadratic in time derivatives to deal with, i.e., uBBTu vBBTv; these terms can bedealt with by squaring and equating both sides of (4.13). As a general principle, it is sensibleto pair up D(1) with the field u and D(2) with the field v which is why (4.13) is presented in itsgiven form. Squaring both sides of (4.13) gives

uBBT

u vBBT

v = 2uBD(1)

2vBD(2)

D(1)

D(1)

+ D(2)

D(2)

(4.18)

and consequently, (4.14) reduces to

P =1

2

u u v v + u u v v U + V

= U + V+ 12

D2u D(1)2 + D(2)2 D2v

u

ATDu + BD(1)

v

ATDv + BD

(2)

BTDu + AD(1) + B

TDv + AD(2)

. (4.19)

For (4.19) to be a total time derivative strong constraints must be placed on the form of D.Fortunately, the form of D here is already known from [2] and the fusing process so it becomesa matter of testing that D has the right properties.

With the identification of helicities of terms in D made in section 4.2.1, as well as D being

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the sum of D(1) and D(2), the defect potential, (4.3) plus (4.4), splits into four parts given by

D(1)+ = dr

j=0

e12j(BTu+B)

rj=0

aj

D(2)+

= d

rj=0

e

12j(B+BTv)

r

j=0aj

D(1) =1

d

rj=0

e12jB(u)

rj=0

bj

D(2) =1

d

rj=0

e12jB

T(v)

=1

d

rj=0

e12jB(v)

rj=0

bj . (4.20)

Note now that D(2) has precisely the same form as D(1) only with v instead of u; consequentlyany property of D(2) may be drawn on by analogy to the corresponding property of D(1). Notenow that

D(1)+u =r

j=0

1

2Bjaj , D

(1)+ =

rj=0

1

2BTjaj

D(1)u =r

j=0

1

2BTjbj , D

(1) =

rj=0

1

2BTjbj

with similar expressions involving D(2). This gives the relations

BTD+u = BD(1)+

Du = D(1)BTD+v = BD(2)+

Dv = D(2) (4.21)and so, using (4.21), (4.19) reduces to

P = U + V+ 12

D2u D(1)2 + D(2)2 D2v

u D+u Du v D+v Dv D+ D .

so, in order for momentum conservation to work and give a result that is rather expected, D

must be quadratically related to U and V such that 1

2D2u D

(1)2 + D

(2)2

D2v = U V.

Noting that

iBTj =

2 if i = j,

2 if i = j 10 otherwise

iBj =

2 if i = j,

2 if i = j + 10 otherwise

and using the relations (4.21) and the form of D in (4.20) it is seen that

1

2

D2u D(1)2

= D+u D

u D(1)+ D(1) =

1

2

ri,j=0

iBTjaibj =

i

aibi i

aibi+1 = U X

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where X is the would-be bulk potential for . A similar relation holds for D(2), implying that

D(2)2 D2v = X V and so it is true that 12

D2u D(1)2 + D(2)2 D2v

= U V which finally

gives

P = D+ D

meaning that there is a conserved momentum given by P + D+ D. It is conjectured thatfor the folded case that the same quantity, with the bulk fields u and v replaced in D by theirfolded counterparts, is conserved, i.e., the aim is to take P and show that it in fact is equal to

D+ D

. This turns out to be true, but the calculation is a lot more involved than for

the unfolded case.

One thing to note is that an analogous analysis holds if in the first instance in ( 4.1) the Bdefect is taken to be to the right of the BT defect. This perhaps should come as no sur-prise as one may appeal to commutability given that the defect conditions describe B acklundtransformations.

5 The folded defect

This section discusses of the folding of the fused defect system and by consideration of solitontime delays then energy and momentum conservation it strongly suggests the existence of clas-

sically integrable c(1)n , d

(2)n and a

(2)2n defects. The process and the outcome are perhaps the two

most salient aspects of this paper.

5.1 Folding the Lagrangian

There is from the outset an ambiguity over what is meant by the folding of the fused defect

system of section 4.2; this ambiguity is investigated in appendix C. Here the folding being con-sidered is a priori folding: folding the Lagrangian (4.10).

The aim is to have the folded system describe a defect in the folded theory. In the bulkthat means that the fields u and v must be folded, but at the defect it is not clear what shouldbe done with the auxiliary field . Consideration of the time delays of soliton solutions, section5.3, suggests that the field should not be folded. One might then expect the folded Lagrangianto be the same as (4.10) but with u and v . This is true but there is at least one thingthat differs from nave expectations which occurs with the self-couplings at the defect: uAu and

vAv.

Note that all of the folded roots, i, resulting from bivalent foldings have the form of l+k+hl2 ,with i = k2 + l for c

(1)n and a

(2)2n and i =

k+12 + l for d

(2)n . Now observe that

(i + h+ki)B(j + h+kj) = iBj + h+kiBh+kj + iBh+kj + h+kiBj

=1

2(iBj + h+kiBh+kj + iBh+kj + h+kiBj)

+1

2

h+kiBTh+kj + iBTj + h+kiBTj + iBTh+kj

= (i + h+ki) (j + h+kj)

where use has been made of the fact that iBj = h+kiBTh+kj (= 2i,j 2i,j+1) and alsothat B + BT = 2.

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Since A = 1 B it must then be the case that

iAj =

i j iBj = 0

and thus

A = A = 0 . (5.1)

Thus, the folded defects contain no self-coupling kinetic terms at x = 0. This means that theLagrangian (4.10) folds to

L = (x)L + (x)L+ (x)

B B D(1)(, ) D(2)(, )

(5.2)

which fits the modified type II framework of [22].

In vector form the Euler-Lagrange equations are thus

= pB D (5.3) = pB + D (5.4)

BT + D(1) = BT D(2) (5.5)

The subscript p found in (5.3) and (5.4) in front of B denotes projected. What is meant bythis is explained in appendix C. Also tackled in appendix C is the relation between the folded

and unfolded gradients, i.e., comparison ofDu to D, the results of this analysis are summarisedin the next section.

5.2 Content of the defect equations

In appendix C the equations (5.3), (5.4) and (5.5) are examined in greater depth. Here a sum-mary is given of the most salient features, as the full detail here would obscure the clarity ofthe following sections.

Universal expressions are found for D and D which are

D+ =

h1j=0

1

4

Bj + BTk+hj

aj (5.6)

D =h1j=0

1

4

BTj + Bk+hj

bj (5.7)

D+ =h1j=0

1

4

Bj + B

Tk+hj

aj (5.8)

D =h1j=0

1

4

BTj + Bk+hj

bj (5.9)

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while the gradient of D with respect to is given by

D+ =j

1

2BTj (aj + aj) (5.10)

D =j

1

2BTj bj + bj . (5.11)

Examination of the components of (5.5) gives the algebraic constraints

Di + Dk+h1i = 0

which may be put in the form

ai ak+hi + ai ak+hi bi + bk+hi bi + bk+hi = 0 . (5.12)

5.3 Time delay of solitons through the defect

The picture here is similar to that of section 4.2 as regards how fusing defects affects the timedelays of the solitons. There it is argued that since any solution to (4.5)-(4.8) with a = 0 alsosolves the fused equations (4.11), (4.12) and (4.13), the tau functions pick up factors givenby (4.9). From the argument of a posteriori folding given in appendix C it is also true thatany solution to (4.11), (4.12) and (4.13) which has the symmetry of a folded field, namelyui = uk+hi, vi = vk+hi, must then satisfy the a posteriori folded equations (C.1), (C.2) and(C.3). Consequently, (4.9) should be re-examined in the case of a folded soliton on the left, i.e.,q = h p, Eq = pkEp. This gives

j =

uj = 1 +

pj + pkpj

Ep + A

kjE2p (5.13)

j = 1 +

pj

p + pkpj

hp

Ep + Akj

phpE2p (5.14)

j

?= vj = 1 +

pjpp +

qjhphp

Ep + AkjphpphpE2p (5.15)

where A = cos2

pr+1

and the delay factors are given by

p =ie + d

p2

ie + dp2

, hp =ie d

p2

ie dp2

p =ie d

p2

ie

d

p2

, hp =ie + d

p2

ie + dp2

. (5.16)

It is clear that (5.13) represents a folded soliton, this is the choice that is made. Equally clear

is that j = k+hj , since p = hp except for p = n in c(1)n - consequently it is not possibleto fold the auxiliary field , despite the algebraic constraints (5.12) implying that may bethought of as having the same number of degrees of freedom as a folded field has.

What is desired is that the field on the right, v represents a folded soliton, i.e., vj = vk+hj , so

that v may be replaced by and the a posteriori defect equations satisfied. For this to be trueit must be the case that

pp = hph

p

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in which case every may be absorbed into the definition of Ep. Note that this condition is the

same condition as having the a(1)r single soliton species p and species h p solutions receiving

the same delay through the fused defect. Thus, the condition for the soliton on the right of thedefect to be in the folded theory is

0 = pp hphp = 1denom.

e2

p p d2 d2where denom. is the common denominator obtained by multiplying all of the denominators of(5.16) together. So the defect represented by the Lagrangian (5.2) is only likely to be integrableif

d = d (5.17)as this is what is required for the soliton solution (5.15) to be compatible with folding. Therefore,there are two possibilities then that give folded solitons for

When d =

d it is the case that pp = h

ph

p = 1. i.e., all of the solitons receive

a trivial time delay- the defect does nothing. Indeed in this case if = is imposedthen the defect part of the Lagrangian (5.2) vanishes- so there is no defect there. Theinterpretation of this is that the second defect is the anti-defect of the first- fusing themcauses annihilation.

When d = d there is a non-trivial time delay (different for each p) so this should representa bona fide defect which does not destroy the form of the solitons (a strong constraint,suggesting that the defect is integrable). Note that if and d are real then the time-delayis real too- there is no change of topological charge.

5.3.1 Algebraic constraints with such d and d

The algebraic constraints from (5.12) have the undesirable property of mixing helicities forgeneral d and d. As such, only if the positive and negative helicity terms in (5.12) separatelyvanish will there be no mixing of helicities.Before folding the terms in D are

ai = deuiui+1+ii1

ai = devivi+1+ii1

bi = d1euiui1i+i1

bi = d1evivi1i+i1 .

If d = d thenai ak+hi + ai ak+hi = d

euiui+1+ii1 euk+hiuk+h+1i+k+hik+h1ievivi+1+ii1 evk+hivk+h+1i+k+hik+h1i

bi bk+hi + bi bk+hi = d1

euiui1i+i1 euk+hiuk+h1ik+hi+k+h1ievivi1i+i1 evk+hivk+h1ik+hi+k+h1i

where the and are strictly correlated- the top set of relations refer to d = d, while thebottom to d =

d.

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Folding the defect field theory, in all cases considered involves identifying ui = uk+hi andvi = vk+hi. Rather than identify the components of and , due to the subtlety in folding toc(1)n (normally ui =

i2 , but in a

(1)2n1 c(1)n the identification is u k

2+n = k

2+n), the folded fields

will be left in terms of

{ui

}and

{vi

}, thus

ai ak+hi + ai ak+hi = d

euiui+1+ii1 euiui1+k+hik+h1ievivi+1+ii1 evivi1+k+hik+h1i

whilst

bi bk+hi + bi bk+hi = d1

euiui1i+i1 euiui+1k+hi+k+h1ievivi1i+i1 evivi+1k+hi+k+h1i

= d1ei+i1k+hi+k+h1i

euiui1+k+hik+h1i euiui+1+ii1

evi

vi

1+

k+h

ik+h

1

i evi

vi+1

+i

i

1

= d2ei+i1k+hi+k+h1i (ai ak+hi + ai ak+hi) .

Thus, for either identification, d = d

bi bk+hi + bi bk+hi = d2ei+i1k+hi+k+h1i (ai ak+hi + ai ak+hi)

and so the positive and negative helicity algebraic constraints are equivalent

ai ak+hi + ai ak+hi = 0 bi bk+hi + bi bk+hi = 0 .

In light of (5.12) then, the algebraic constraints become

D+i + D+k+h1i

= 0

Di + Dk+h1i

= 0

resulting in

ai ak+hi + ai ak+hi = 0 (5.18)bi bk+hi + bi bk+hi = 0 . (5.19)

That the condition for solitons to be able to pass through the defect in the folded model withoutlosing their form, i.e., (5.17), is precisely what is needed to prevent the algebraic constraints frommixing helicities is one of the most interesting outcomes of this analysis. This is particularlysalient as the momentum conservation argument in section 5.5 relies upon (5.18) and (5.19) beingseparately true- this is the link between the two methods that is alluded to in the introduction.From the consideration of the solitons, it is thus assumed for the purposes on showing energyand momentum conservation that (5.17) is true, d = d.

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5.4 Energy conservation

For both energy and momentum conservation, the a priori folded defect equations; (5.3), (5.4)and (5.5); will be used. The process here is analogous to that of the energy conservation of thefused defect in section 4.2.1. The bulk energy is then modified by the defect such that

E = |x=0= pB D pB D=

BT BT

D D

= D D D = Dso once again the quantity E+ D is conserved. Note that it is safe to drop the p subscript aboveas and both lie in the folded root space.

5.5 Momentum conservation

The momentum conservation by itself is the most involved of the calculations presented in thispaper, but much simplified by employment of the relations (5.6)-(5.11) of section 5.2 along withthe helicity conserving algebraic constraints (5.18) and (5.19). The bulk momentum is modifiedsuch that

P =1

2

+

+ (5.20)

where and are the folded bulk potentials.

The first step in showing this momentum is conserved is to use the squares of (5.3) and (5.4),giving

= pBT D (pB D)= pB

TpB 2 BTD + D2

and

= pBT + D (pB + D)= pB

TpB + 2 B

TD + D2 .

In both cases it is not known what the first term, pBTpB, is in general; nor indeed is it even

clear what it means. However, this issue drops out of the momentum conservation argument

since = 2 BTD + BTD+ D2 D2 . (5.21)

At this stage progress can be made by anticipating the final answer to be P =

D+ D

which requires a term D+ D . It is evident from the defect conditions; (5.3), (5.4)and (5.5); that the only place where terms dependent on can appear in (5.20) stems from . The conclusion is then that

BTD+ + BTD+ = D

+ (5.22)

BTD + BTD =

D . (5.23)

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If (5.22) and (5.23) are true then (5.5) may be rewritten, noting that BT is invertible, as

+ D+ D = D+ + D (5.24)

Note that the left-hand side of (5.24) is not equal

BT1 times the left-hand side of (5.5)-

some rearrangement is necessary. Squaring both sides of (5.24) gives

= 2

D+ D

2

D+ D

D+ D2

+

D+ D2

(5.25)

and hence (5.21) and (5.25) may be combined to give

+ = 2

D+ D

+ 4D+ D 4D+ D

to wit (5.20) reduces to

P = D+ + D + 2D+ D 2D+ D + . (5.26)

Therefore, a modified momentum of P + D+ D is conserved, provided2D+ D

2D+ D = (5.27)

This argument would prove momentum conservation but there are three equations which mustbe proven which rely upon the particular form of D, i.e., (5.22), (5.23) and (5.27).

5.5.1 Proof of the relations (5.22) and (5.23)

In order to prove (5.22) and (5.23), all of the relations (5.6)-(5.11) will be required, along with

the algebraic constraints (5.18) and (5.19). The first step is to note (5.10), given by

D+ =i

1

2BTj (aj + aj)

and to put the left-hand side of (5.22) into this form. Thus,

BTD+ + BTD+ =

j

1

4BT

Baj + Baj + B

Tak+hj + BTak+hj

j

=i

1

4BT

(B + BT)aj + (B + B

T)aj + BT(ak+hj aj) + BT(ak+hj aj)

j

=j

12

BTi (aj + aj) +j

14

BTBT (ak+hj aj + ak+hj aj) i

=j

1

2BTj (aj + aj) = D

+

where B + BT = 2 has been used along with (5.18).

The situation for (5.23) is analogous:

D = j

1

2BTj

bj + bj

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while

BTD + BTD =

j

1

4BT

BTbj + B

Tbj + Bbk+hj + Bbk+hj

j

= j

1

4BT (B + BT)bj + (B + BT)bj + B(bk+hj

bj) + B(bk+hj

bj)j

=j

1

2BTj

bj + bj

+j

1

4BTB

bk+hj bj + bk+hj bj

j

=j

1

2BTj

bj + bj

= D .

5.5.2 Proof of (5.27)

Here the final relation for momentum conservation is proven

2D+

D 2D+

D = .

Before folding, the bulk potentials are given by

U(u) =r

j=0

ajbj

X() =r

j=0

ajbj+1 =r

j=0

aj bj+1

V =r

j=0

aj

bj

and so

U X = 12

ri,j=0

iBTjaibj

V X = 12

ri,j=0

iBTj aibj

resulting in

U V = 12

ri,j=0

iBTj

aibj aibj

. (5.28)

The equation (5.28) can now be folded to give the right-hand side of (5.27) as

= 12

ri,j=0

iBTj

aibj aibj

(5.29)

where ai, ai, bi and bi are now written in terms of the folded fields.

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The left-hand side of (5.27) can be rewritten using (5.6)-(5.9) to give

2D+ D 2D+ D =

i,j

1

8

aibj aibj

iB

TBTj + k+hiBBk+hj

+iBTBk+h

j + k+h

iBB

Tj=i,j

1

8

aibj aibj

iB

T(BT + B)j + k+hiB(B + BT)k+hj iBTBj

k+hiBBTk+hj + iBTBk+hj + k+hiBBTj

=i,j

1

4

aibj aibj

iB

Tj + k+hiBk+hj

+i,j

1

8

aibj aibj

(i k+hi) BBT (k+hj j)

=1

2 i,j

aibj aibj iB

Tj+

i,j

Mij aibj aibj

= +i,j

Mij

aibj aibj

where use has been made of the facts that k+hiBk+hj = iBTj and that B + BT = 2. Inthe last step (5.29) has been used. The quantity Mij is defined as

Mij =1

8(i k+hi) BBT (k+hj j)

so what now remains to be shown is that

i,j

Mij

aibj aibj = 0 . (5.30)

First note that

M(k+hi)j = Mi(k+hj) = Mij .

Using (5.18) then implies that

i

Mijai =i

Mijak+hi i

Mij ai +i

Mij ak+hi

= i

M(k+hi)jak+hi i

Mij ai i

M(k+hi)j ak+hi

= i

Mijai 2i

Mij ai

=i

Mijai = i

Mij ai

while (5.19) analogously gives

j

Mijbj = j

Mij bj .

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Thus, (5.30) is true since

i,j

Mijaibj = i,j

Mij aibj = +i,j

Mij aibj

and hence

2D+ D 2D+ D = .

In conclusion, (5.26) reduces to P =

D+ D

and so P+D+D is a conserved quantity,certainly when d = d. Hence, energy and momentum are conserved by the folded defects. Mo-mentum conservation in particular is a strong constraint, perhaps even implying integrability, so

there is more than a hope that (5.2) with d = d represents integrable defects for c(1)n , d

(2)n and a

(2)2n.

It is perhaps the soliton time delay argument that gives the stronger evidence of integrabil-ity: preservation of the forms of the solitons by the defect hints at the conservation of aninfinite number of charges.

6 Discussion

Conclusions

The main conclusion of this paper is that certain a(1)r defects may be folded to give defects in

the folded models of c(1)n , d

(2)n and a

(2)2n with Lagrangian description in the form (5.2). There are

two strong reasons to believe that such defects are integrable.

Firstly, energy and momentum conservation, when applied to a type I a(1)r defect, are inthemselves enough to force the defect to be integrable [11]. It is certainly plausible thatthe same holds for these folded defects.

Secondly, there is the rather simpler argument of classical scattering of solitons off thedefect. This argument alone is suggestive of integrability as the solitons retain their formand so ought to conserve an infinite number of charges.

Though not the main focus, this paper also furthers the work of [9] by using the relatively simple

arena of a(1)r to construct solitons and breathers in c

(1)n , d

(2)n and a

(2)2n. The breather solutions in

particular might not have been published before.

There do remain a couple of loose ends at the classical level. One regards the interpreta-tion of the auxiliary field, , in the folded defect. Since the algebraic constraints may be usedto reduce the number of degrees of freedom of to that of a folded field, it seems highly likelythat one may construct similar defects by starting in the folded model, i.e., without any refer-

ence to a(1)r . Another is that it is as yet unclear whether or not the folded defect conditions

((5.3),(5.4),(5.5)) represent a Backlund transformation in the folded theory- such as was found

for a(1)r type I defects in [2].

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Future directions

One clear extension of this work is to consider the transmission matrices of the folded defectsin the quantum p

folding defect affine toda field theories

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