Yang Baxter and anisotropic sigma and lambda models, cyclic RG and exact Smatrices
HJE
Baxter and anisotropic sigma and lambda models, cyclic RG and exact Smatrices
Calan Appadu 0 1
Timothy J. Hollowood 0 1
Dafydd Price 0 1
Daniel C. Thompson 0 1
0 Swansea , SA2 8PP , U.K
1 Department of Physics, Swansea University
Integrable deformation of SU(2) sigma and lambda models are considered at the classical and quantum levels. These are the YangBaxter and XXZtype anisotropic deformations. The XXZ type deformations are UV safe in one regime, while in another regime, like the YangBaxter deformations, they exhibit cyclic RG behaviour. The associated a ne quantum group symmetry, realized classically at the Poisson bracket level, has q a complex phase in the UV safe regime and q real in the cyclic RG regime, where q is an RG invariant. Based on the symmetries and RG Smatrices to describe the scattering of states in the lambda models, from which the sigma models follow by taking a limit and nonabelian Tduality. In the cyclic RG regimes, the Smatrices are periodic functions of rapidity, at large rapidity, and in the YangBaxter case violate parity.
Integrable Field Theories; Quantum Groups; Sigma Models

Yang
1 Introduction
2
3
2.1
2.2
3.1
3.2
3.3
3.4
4.1
4.2
4.3
5.1
5.2
5.3
5.4
5.5
Classical SU(2) sigma models
Lax connection and Poisson brackets
Nonlocal charges and in nite symmetries
Classical SU(2) lambda models
Target spaces
Lax formalism Poisson structure and symmetries Nonlocal charges and in nite symmetries
4
Renormalization group ow
ow
The sigma model RG
Yang Baxter lambda model RG
ow
Anisotropic XXZ lambda model RG
ow
5
Quantum group Smatrix: q complex phase
Quantum group Smatrix: q real
The RSOS Smatrix
High energy limit
The Smatrix proposals
6
Discussion
A Lambda spacetimes
1
Introduction
Sigma models are fascinating because they are the building blocks of string worldsheet
theories but also they share many of the features of QFTs in higher dimensions in a
simpler context. And within the space of sigma models, the ones that are integrable have
the additional lure of tractability.
The key examples are the Principal Chiral Models (PCM), whose target spaces are
group manifolds G. There is a Gvalued eld f and the action can be written1
S =
2
(1.1)
1We take x = t x and so for vectors A = A0
A1 along with A = 12 (A0
A1).
{ 1 {
outgoing particles as well as the internal quantum numbers i; j; k; l.
The PCM can appear as a bosonic subsector of a consistent string theory CFT background,
e.g. the D1D5 near horizon geometry, providing a modern holographic motivation for
studying this theory. The more prosaic view, which we adopt here, is that PCMs are an
exceptionally informative 1 + 1dimensional QFTs exhibiting asymptotic freedom in the
running coupling ( ) and a dynamically generated mass gap.
The action given in eq. (1.1) manifests a GL
GR global symmetry, f ! U f V . A
feature that makes the PCM tractable is that it is classically integrable and the GL
GR
symmetry is part of a much larger classical Yangian Y (gL)
Y (gR) symmetry generated
by nonlocal charges.2
At the quantum level this integrability persists leading to the factorization of its
Smatrix [2, 3]. This means that it is completely determined by 2 ! 2 body processes which
preserve the individual momenta, as illustrated in gure 1. The states are labelled by their
rapidity
and by internal quantum numbers i; j; : : :. For example, in the SU(N ) PCM,
there are N
1 particle multiplets with mass ma = m sin( a=N ), a =; 1; 2; : : : ; N
1,
and each multiplet transforms in the [!a]
[!a] representation of the GL
GR symmetry,
where !a are the highest weight vectors of the ath fundamental representation.3 The 2body
Smatrix has the characteristic product form [4]:
S( ) = SGL ( )
SGR ( ) ;
(1.2)
where
= 1
2. The product form re ects the fact that the states transform in a product
of representations of GL and GR. The Smatrix building block SG( ) is Ginvariant, in fact
Yangian invariant, and is built from a rational solution of the YangBaxter Equation.4
Since the PCM is asymptotically free and its spectrum is massive and dynamically
generated, directly connecting the conjectured quantum Smatrix picture to the Lagrangian
description in eq. (1.1) is subtle. Nonetheless, consistency checks can be made by studying
the theory in a regime in which perturbation theory can be employed and compared against
the factorized Smatrix. The study of the exact solution of the model was initiated in the
classic works [4{7]. As a byproduct of the successful comparison of Thermodynamic Bethe
2A concise introduction to this symmetry can be found in [1]. There are also an in nite number of local
conserved charges which include and energy and momentum.
3For the groups SO(N ) the representations are actually reducible combinations.
4For the higher rank groups, the product form of the Smatrix must be multiplied by a scalar factor to
provide the bound state poles.
{ 2 {
Ansatz and perturbative calculations of the free energy in a background charge one obtains
an exact expression for the mass gap.5
A natural question to ask, is whether the PCM can be deformed in a way that preserves
integrability? For the case SU(2)  which we will concentrate on in this work  there are
several ways to do this, while for higher rank groups the possibilities appear to be more
limited. We will concentrate on the deformations that preserve one of the chiral symmetries,
SU(2)L, say. Deformation which preserve the SU(2)L symmetry can be written
where
is endomorphism of the Lie algebra,
T a =
abT b. A fascinating problem is
to determine systematically which choices of
lead to integrable models both classically
and quantum mechanically.
For the particular case of G = SU(2), there are anisoptropic type deformations that
involve in the most general case three di erent couplings
components of the SU(2)L current J
general deformation of this type as [12, 13]
= P
a J aT a,6 we can write the action for the most
T a =
a 1T a. Introducing the
S =
(1.3)
HJEP09(217)35
S =
where we have de ned the RLie bracket
[Ra; Rb]
R[a; b]R =
c2[a; b] ;
[a; b]R = [Ra; b] + [a; Rb] ;
{ 3 {
for all a; b in the Lie algebra and where c is a free parameter. The action of the deformed
theory is de ned by taking in (1.3)
=
1(1
R) 1
;
5The case of G = SU(2) viewed as the O(4) model was done in [8], with the extension to SU(N )
in [9, 10] and other Lie algebras in [11].
iT 2)=p2 and the alternative decomposition J = J3T 3 + J+T
+ J T +.
ab. So for SU(2), T a = i a p
= 2 where
6Throughout the paper we use a basis fT ag that are antihermitian and normalized so that Tr(T aT b) =
a are the Pauli matrices. In addition, we de ne T
= (T 1
where
is the real deformation parameter. YB deformations of this type can be de ned
for an arbitrary group and in general the deformed theories have a KalbRamond
eld
which correspond to the terms odd in R when the operator (1
R) 1 is expanded in
powers of R.
generality, can be written as
For SU(2) there is a single class of deformations of this type which, without loss of
(1.8)
(1.9)
(1.10)
1. In this case, one can show that the KalbRamond eld is
a total derivative and  at least with periodic boundary conditions  the YB and XXZ
sigma theories are equivalent with
in the regime with
two parameter Fateev model [18].7 We will not consider this more general deformation any
further and focus on deformations that preserve the SU(2)L symmetry because these cases
have an associated lambda model.
The lambda models are a completely di erent class of integrable deformations of the
PCM. In fact of each of the sigma models, whether PCM, XXZ, XYZ or YB, i.e. all having
an SU(2)L symmetry, have an associated lambda model that inherits the integrability of the
parent sigma model. Motivated by the process of nonabelian Tduality in string theory,
each sigma model whose target space is a G group manifold with GL global symmetry
has an associated lambda model.8 The de nition of the lambda model associated to the
SU(2) PCM go back to [19] but in a more general context are best constructed by Sfetsos's
gauging procedure [20]:
1. Write down a theory which is the sum of the actions of the sigma model eq. (1.3)
and a WZW model for a Gvalued eld F .
7The matching of parameters (de ned after eq. (76) of [18]) is given by
r
u
2 = (`u 1 + 1) ;
2 = (ru 1 + 1) ;
= u :
8There are also examples associated to symmetric space quotients G=H that we will not consider here.
`
u
{ 4 {
2. Gauge the joint G symmetry, which acts on the WZW
U F U 1 and the sigma model eld by left action f ! U f .
3. Gauge x the G symmetry by setting the sigma model eld f = 1.
eld by vector action F !
Applied to the deformed PCM de ned in eq. (1.3), the result of this procedure leads
to a deformation of a G WZW model written in the following way:
eld that can be integrated out. The rst term is the gauged WZW
model action [21{25] for a Gvalued eld F , where the whole vector G symmetry is gauged,
and k 2 Z is the level. What is crucial for us is that if the original sigma model is integrable
then so is the associated lambda model. There is also a sense that the original sigma model
is recovered in the limit k ! 1 along with a nonabelian Tduality [20]. It is noteworthy
that this relation is also seen quantum mechanically at the level of the Smatrix where
nonabelian Tduality manifests as an IRFtovertex transformation on the space of asymptotic
states [26].9
A fascinating question is to understand whether these integrable deformations persist
in the quantum theory and if so, what are their factorizable Smatrices. We have already
remarked that the PCM Smatrix takes the product form of two rational factors (1.2) that
manifest the Yangian Y (su(2)L) Y (su(2)R) symmetry. This form seems to generalize: the
XXZ models in the regime
<
lie in the class of \SS models" considered by Fateev [18],10
which have an Smatrix of the form [12]
In this expression, S( ; 0) is the Smatrix of the sineGordon theory with coupling11
S XXZ( ) = SSU(2)L ( )
S( ; 0) :
8
2
=
0
1 + 0
:
(1.13)
(1.14)
The tensor product form of the Smatrix in (1.13) will prove ubiquitous and deserves
some comment. Like the PCM Smatrix (1.2) it re ects the factor that the particle states
carry two sets of quantum numbers which under scattering are completely independent.
9It is worth remarking that at the classical level nonabelian T duality can be thought of as a canonical
transformation [27] while at the quantum level the IRFtovertex transformation can be thought of as a
change of basis in the Hilbert space [28{30]. It would be interesting to make the connection between the
two phenomena more explicit.
10In terms of Fateev's more general model with U(
1
)
U(
1
) symmetry and parameters (a; b; c; d), we have
a2 = u(u + `), b = 0 and c = d = `=2 and ? = (u + `) 1 and 3 = u 1. Then
? = ( ) 1 ;
3 = ( ) 1 ;
3 < ?
:
(1.12)
11Our 0 is 0=8 of Zamolodchikov and Zamolodchikov [55]. For us the breather spectrum is mn =
2M sin( n 0=2), n = 1; 2; : : : < 0 1.
{ 5 {
The XXZ deformation has broken the SU(2)R Yangian symmetry but rather than
disappearing it is deformed to an a ne quantum group Uq(s[u(2)) symmetry, where the
deformation parameter
in section 3. Note that for 0 < 1, the model has bound states that correspond to the
breathers of the sineGordon theory. In the present context, the nth breather transforms
as a singlet under Uq(su(2)) but as a reducible representation of SU(2)L corresponding to
the tensor product of n spin 12 representations.
The XXZ model of Fateev displays an important general feature of the integrable
deformations: Yangian symmetries generally get deformed into a ne quantum group
symmetries. The label \quantum" here might be thought a misnomer because the quantum
group symmetries are manifest in the classical theory at the Poisson bracket level [31{33].
This point deserves some comment. We shall show that the deformation parameter q does
indeed depend on ~ (or more precisely the coupling that plays the role of ~) as q = exp[ ~].
However, there is a consistent classical limit, where ~ ! 0 but the coupling constant
dependent quantity
! 1 such that q is xed. In addition, as part of the overall consistency
we will show that the q is an Renormalization Group (RG) invariant and so the quantum
group symmetries are well de ned in the quantum theory and the classical limit where it
becomes realized at the Poisson bracket level.
The lambda deformations also have a characteristic e ect on the Smatrix [26]. For the
PCM itself, the deformation changes the Smatrix block for the SU(2)L symmetry into an
a ne quantum group invariant block, but realized in the InteractionRoundaFace (IRF),
or RestrictedSolidOnSolid (RSOS), form:12
The original PCM Smatrix is recovered in the limit k ! 1, where the kink factor
becomes unrestricted, and then an IRFtovertex transformation which is the Smatrix
manifestation of nonabelian Tduality:
will describe all the integrable deformations of the PCM and this intuition will turn out
to be true. In this paper, we will concentrate on the XXZ and YB deformations of the
12This type of Smatrix block appears in the context of the \restricted sineGordon theory'" [28{30] and
also perturbed WZW models [34].
(1.16)
(1.17)
{ 6 {
SU(2) PCM and their associated lambda models at the quantum level and map out their
renormalization structure and their Smatrices and con rm the ubiquity of the product
form. Speci cally in this paper we:
1. Review the classical integrability of the deformed sigma models and establish some
new results for the Poisson brackets of the associated lambda models.
1. We show that the lambda models have quantum group symmetries in the classical
theory realized at the level of the Poisson brackets.
2. We then consider the RG ow of the sigma and lambda models at one loop order (so
in the lambda models to leading order in 1=k). We show that the XXZ models, both
sigma and lambda, have one regime which has UV safe ows, whereas in the other
regime there are cyclic RG type ows. The YB lambda model also has cyclic RG
ows.
tum groups are RG invariants.
3. We show that the quantum deformation parameters q of the classicallyrealized
quan4. Using the RG
ow and the structure of the classical symmetries, we propose
Smatrices to describe all the lambda models. For the examples with cyclic RG
ow,
the Smatrix has periodicity in the rapidity when the rapidity is large.
5. We then argue that Smatrices of the sigma models are obtained in the large k limit
after an IRFtovertex transformation.
In a follow up paper, we will address the question of whether the theories that we nd
with cyclic RG behaviour actually exists as QFTs in the continuum limit [35]. We will nd
that the continuum theories can be formulated as a Heisenberg XXZ spin chain. When the
RG
ow of the theory has a UV safe limit, the spin chain is critical and a continuum limit
can be de ned. On the contrary in the regime with cyclic RG
ows, the spin chain has a
gap and a continuum limit does not exist. Th conclusion would be that the theories with
cyclic RG behaviour only exist as e ective theories with an explicit cut o .
2
Classical SU(2) sigma models
In this section, we consider some of the aspects of the sigma models and in particular the
symmetries, that will inform our Smatrix hypotheses.
2.1
Lax connection and Poisson brackets
The most direct way to prove classical integrability is to write down the equations of motion
in Lax form, that is as the atness condition on an auxiliary connection that depends on
an additional free parameter, the spectral parameter,
+ L (z)] = 0 ;
(2.1)
for arbitrary z.
{ 7 {
If we de ne the SU(2)L invariant current J
motion along with the CartanMaurer identity of the YB deformed sigma models can be
written in Lax form with
a gauge transformation [32, 33, 36]. We note in passing that the Lax connection is valued
in the loop algebra s[u(2) = su(2)
C[z; z 1], the untwisted a nization of su(2) (with
vanishing centre). This can also be described as the a ne algebra with the homogeneous
gradation and we will denote it as s[u(2)h.
For the anisotropic models, the Lax connection take a characteristic form that
generalizes nicely as one goes through the hierarchy from PCM to XXZ to XYZ:
For the PCM, the functions wa(z) are rational
while for the XXZ case, with
a = ( ; ; ), the functions wa(z) are trigonometric (or
hyperbolic) [13],
where
For these theories, if we transform to a multiplicative spectral parameter z ! log z and
then expand in powers of z, it is noteworthy that the Lax connection takes values in the
twisted loop algebra, where the twist is an automorphism :
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(T 1;2) !
T 1;2 ;
(T 3) ! T 3 :
The Lie algebra splits into its eigenspaces under
and in the twisted loop algebra each
eigenspace receives a di erent scaling of the spectral parameter. The twisted loop algebra
thus has elements T 3z2n, T 1z2n+1 and T 2z2n+1, with n 2 Z. Since the automorphism
inner the twisted loop algebra is simply equal to original in another gradation, in this case
is
it called the principal gradation and we denote it s[u(2)p.
Finally, for the XYZ case, the functions wa(z) are elliptic functions
{ 8 {
where the Jacobi elliptic functions have an elliptic modulus
In addition,
Whilst these theories still have the SU(2)L symmetry, the SU(2)R symmetry is broken
to a
nite Z4 subgroup. The question as to whether the associated Yangian symmetry
becomes deformed is an interesting one that we do not tackle here. Note that since the
SU(2)L symmetry is preserved these theories are distinct from the general twoparameter
HJEP09(217)35
deformations considered in [18].
Note that the XXZ model in the regime
>
has the same equation of motion as the
YB model but the Lax connections are completely di erent. The relation between the two
formulations was considered in detail in [33].
As part of the standard formalism of integrability (e.g. see the book [38]), a key
structure is the Poisson bracket of the spatial component of the Lax connection L
L+
L .
This is sometimes called the Maillet algebra [39] and in general takes the form
fL1(x; z); L2(y; w)g = [r(z; w); L1(x; z) + L2(x; w)] (x
y)
[s(z; w); L1(x; z)
L2(y; w)] (x
y)
2s(z; w) 0(x
The notation means that the bracket acts on a product of su(2) modules V
the subscripts indicate which of the copies a quantity acts on: L1(z) = L (z)
L2(z) = 1
L (z). The tensor kernels r(z; w) and s(z; w) act on V
V .
In many cases, the kernels r and s can be written in the form
where (z) is known as the twist function and in many cases
=
Casimir tensor. For example, for the YB deformation with the de nition of the Lax
connection in [33], the kernels r and s take precisely this form with a twist function
Pa T a
T a is the
Note that here we include the factor 2
which plays the role of ~ in the quantum theory.
For the XXZ model in the trigonometric formulation, the r=s kernels take a similar
form, except that
depends on z and w, and the twist function
(z) =
2
1
p 2
sinh2 z
cosh2 z
:
{ 9 {
(2.9)
(2.10)
y) :
(2.11)
V and
1 and
(2.12)
(2.13)
(2.14)
(2.15)
Z 1
1
T (z) = Pexp
dx L (x; z)
(2.16)
Integrable eld theories have an in nite sets of both local (integrals of expressions local in
the elds and their derivatives) and nonlocal conserved charges. All these charges can be
extracted from the Lax connection. The local conserved charges include the energy and
momentum but the nonlocal ones are our central focus here because they generate some
remarkable in nite symmetries in the form of Yangians and quantum groups.
The nonlocal charges are encoded in the monodromy matrix, the parallel transport of
the Lax connection, along the spatial direction
which is conserved in time (in the in nite volume limit with appropriate fall o assumed).
We can think of T (z) as a generating function for the charges. It is natural to lift the
Poisson bracket on L (x; z) to the monodromy matrix. However, this is where a problem
arises as a result of the non ultralocality of the Poisson bracket: when the kernel s is
nontrivial the Poisson bracket of the monodromy matrix is illde ned due to the 0(x
y)
term in (2.11). This non ultralocality can lead to ordering ambiguities when considering
nested integrals in the expansion of the monodromy matrix and a violation of the Jacobi
identity for the monodromy matrix. One way to deal with the ambiguities is to use Maillet's
prescription [39]. This corresponds to lifting the Poisson bracket to the monodromy matrix
in the form
fT1(z); T2(w)g = [r(z; w); T1(z)T2(w)]
+ T1(z)s(z; w)T2(w)
T2(w)s(z; w)T1(w) :
It is remarkable that the non ultralocality and its associated ambiguities generally turn
out not to a ect the discussion of the Yangian and quantum group symmetries when they
are manifested at the classical level [36]. As we will see, there can also be quantum group
symmetries that can only be seen consistently at the quantum level.
The in nite symmetries are associated to the expansion of the monodromy matrix T (z)
around special points z which de ne nonlocal charges that generate Yangian or quantum
group symmetries. The general idea is as follows: generically the kernel r has a pole as
z ! w; however, there are special points z in the neighbourhood of which,
z = z + ;
w = z + ~;
the Poisson bracket algebra has a nite limit as and ~ are scaled to 0. The special points
can also be at in nity in which case one takes z =
1 and w = ~ 1.
If the r=s kernels take the form (2.12), then poles of the twist function are special points
(see [37] for a general analysis for these cases). For example, for the YB deformation with
twist function (2.13), there are poles at z =
i , around which
i
i
1 + 2
1 + 2
+ ~
~
3
X T a
a=1
T a + O( ) :
T a + O( ) ;
(2.17)
(2.18)
(2.19)
z = +i
z = i
+2
+1
0
.
.
.
1
2
Q2+
Q~+
Q+
Q+1
Q+2
.
.
.
Q32
Q31
Q2
Q1
Q
~
Q
Q 2
.
.
.
i . The blue/red and positive/negative graded charges are associated
to
i , respectively. The red and blue charges generate the a ne quantum group in homogenous
gradation and all the other charges are obtained by repeated Poisson brackets of these charges.
It has been shown that the charges de ned by expansing the monodromy matrix around
these special points generate a classical version of an a ne quantum group symmetry
Uq(s[u(2)) with a deformation parameter [32, 33, 36]
comes from the overall normalization of the action and plays the role of
~, and so is usually set to 1 in a classical analysis [32, 33, 36]. For us, pursuing a quantum
analysis, having the correct overall normalization is crucial because the correctly de ned q
is then an RG invariant. For YangBaxter deformations a similar result was obtained for
arbitrary groups and also symmetric space coset models in a now seminal paper [40].13
In the expansion of T (z) around z =
i , the charges are naturally are classi ed by
the order in which they appear [33] (positive/negative grade for z =
i ): see gure 2.
The U(
1
)R charge Q3, local in elds, is supplemented with nonlocal conserved charges
Q
that obey a (classical) quantum group Uq(su(2)) symmetry under the Poisson bracket
fQ+; Q g =
i
qQ3
q
q Q3
q 1
;
fQ ; Q3g =
iQ
:
(2.21)
In addition to these, one obtains generators Q~
symmetry (the extension is centreless since Q~ 3 =
associated to the a ne extension14 of this
Q3 and so the a ne algebra is actually
13For group case the result of [40] is that q = exp[ (1
2)3=2] with
the above after taking into account that the overall tension has been set as
= = 1
p
2 which matches
1 = (1 + 2)2. Although
not present focus it would be remiss not to mention that that a YangBaxter deformation of the
MetsaevTseytlin action for strings in AdS5
S
5 was constructed in [41, 42].
14Recall that the a ne extension s[u(2) supplements the Chevalley generators fE1; F1; H0g of su(2) with an
additional root and corresponding generators fE0; F0; H0g obeying the standard relations [Hi; Ej] = aijEj,
[Hi; Fj] =
aijFj and [Ei; Fj] =
ijHj together with the Serre relations. Here the generalised Cartan
matrix aij has o
diagonal elements equal
2. K = H0 + H1 is central and in what follows we will
a loop algebra). There are an in nite series of higher charges, but these can be recovered
by taking repeated Poisson brackets of the charges shown. The grading that is imposed
on the algebra by the order of the expansion that the charges appear around the special
points is precisely the homogeneous gradation s[u(2)h. The other important point is that
the full set of charges that generate the a ne quantum group are associated to a pair of
special points.
w = ~ 1, the kernels have the expansion
In the YB sigma model there is also a special point at in nity. Setting z =
In this case, the nonvanishing contribution is at O( ). The charges that are de ned by the
expansion of monodromy matrix around in nity generate an in nite Yangian symmetry
Y (su(2)L) that includes the global SU(2)L symmetry.
Now we turn to the anisotropic XXZ deformed sigma model with twist function (2.15).
In this case, the in nite symmetries are associated to the pole of the twist function at z = 0
and to the behaviour at
1. Before proceeding it is more convenient to transform to
multiplicative spectral parameter z ! log z in which case the twist function takes the form
The pole is now at z = 1, and expanding around it, we have
(z) =
2
1
p 2
The leading behaviour here is O( ) and so it indicative of a Yangian symmetry. In fact,
expanding around this pole gives the Yangian symmetry Y (su(2)L) in the trigonometric
formulation.
The special points at
1 map to z = 0; 1, around which
consider modules where K = 0, i.e. centreless representations for which s[u(2) becomes the loop algebra.
Note that we will not distinguish the real form sl(2) from su(2) where appropriate. This being the case,
representations are the tensor of an su(2) representation and functions of a variable z. There is a choice,
known as gradation, to be made as to the relative action in su(2) space and zspace. In the homogenous
gradation is E1 = T +, F1 = T , E0 = z2T , F0 = z 2
T +, H1 =
H0 = T 3. In the principal gradation
E1 = zT +, F1 = z 1
T , E0 = zT , F0 = z 1T +, H1 =
H0 = T 3. These gradations lift to the quantum
group deformation Uq(s[u(2)).
(2.22)
(2.23)
(2.24)
(2.25)
+2
+1
0
Q3 2
Q3 =
Q~ 3
~
Q
Q
1 (or 0; 1 with a multiplicative spectral parameter). The blue/red
and positive/negative graded charges are associated to
1, respectively. The red and blue charges
generate the a
ne quantum group in principal gradation and all the other charges are obtained by
repeated Poisson brackets of these charges.
The expansions in this case are associated to a quantum group symmetry with the same
deformation parameter (2.20) as in the YB case, once we identify parameters as in (1.9).
The charges emerge as illustrated in
gure 3 [33]. Once again there are an in nite set of
charges but the ones shown generate the a ne algebra and the higher charges are then
obtained by repeated Poisson brackets of the lower charges. The a ne algebra is now
revealed to be associated to the principal gradation s[u(2)p.
So although the YB and XXZ sigma models have the same equations of motion and
what seems like identical symmetries, a Yangian and an a ne quantum group, there is a
subtle di erence. The a ne quantum group for the YB is in the homogeneous gradation
while in the XXZ case it is in the principle gradation. This interpretation is consistent with
the spectral parameter rescaling of SU(2) generators found in [33] required to go between
the two expansion. For the YB deformation of arbitrary rank groups, for which only a
rational Lax description exists, the same homogenous gradation shows itself [36].
The existence of these symmetries at the classical level is important because they will
inform our search for the quantum Smatrices that describe the quantum versions of these
theories. The symmetries are summarized in table 1.
3
Classical SU(2) lambda models
The lambda model associated to a sigma model have been de ned in (1.11). The second
term in (1.11) vitiates the gauge symmetry and A
becomes an auxiliary Gaussian
eld.
Correspondingly, the equations of motion of A
change from
rst class to second class
constraints [43]:
F
T A+ ;
A ;
Model
YB
PCM
XXZ
Left symm.
Right symm.
Y (su(2))
Y (su(2))
Y (su(2))
Uq(s[u(2)h)
Y (su(2))
Uq(s[u(2)p)
is given in (2.20) in terms of the underlying coupling constants. The only (subtle) di erence between
the symmetries is that in the YB case, the a ne quantum group is naturally in homogeneous grade,
while in the anisotropic XXZ case it is in principal grade.
where
After integrating out the auxiliary eld A , we can write the resulting theory as
k Z
2
Sk; [F ] = k SWZW[F ] +
d
2 Tr F
This form makes it clear that as an expansion in
currentcurrent deformation of the WZW model:
1 the theory can be interpreted as a
k Z
2
Sk; [F ] = k SWZW[F ] +
d
2 Tr F
1
1 +
The implication is that if the couplings ow into the UV in such a way that
lambda model can be interpreted as a perturbed WZW CFT.
The equations of motion of the theory have a simple form when written in terms of
the auxiliary eld A :15
:
1
! 0, the
The isotropic lambda model associated to the PCM for which
=
1I, gives
= I + k 1
:
=
1I ;
=
k
k
= B 0
0
1 0 C ;
0 1
1
A
This is the model constructed and studied in [19].
The XXZ version of the model has
15Note that the transpose is de ned with respect to the trace: Tr(a b) = Tr( T a b).
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
This should be compared with the YB version of the model for which
=
k
k + 1
;
= I +
(I
1
k
R) 1 =
=
1
k
k
where
where
and where the original sigma model couplings are
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
HJEP09(217)35
Now we can see that the XXZ lambda model, even in the regime
> , i.e. > , where
the associated sigma models are equivalent up to a boundary term, is distinct from the YB
lambda model. In particular, the YB lambda model breaks parity symmetry explicitly as
can be seen from the fact that , entering the de nition eq. (1.11), is not symmetric.
There is also a XYZ lambda model for which
1
= diag( i ) with all i distinct, rst
constructed in [43]. This will be considered in more detail elsewhere [35].
3.1
Target spaces
With the group element parametrized as
F =
C + iS C
S S e
i
S S e i
C
iS C
!
;
where we have de ned Sx
sin x and Cx
models with target spaces of the following form
cos x, the lambda theories can viewed as sigma
k
A0
A5
0
1
2
^ d ;
1
2
A1 d 2 + A2 d 2 + A3d 2 + A4 d d
;
=
where Ai = Ai( ; ). The nontrivial dilaton is due to a determinant arising from
performing the Gaussian integration on the nonpropagating exgauge elds A in the path
integral. The exact functional forms are not particularly enlightening but are recorded in
appendix A. Here we note the feature, seen in other lambda deformations, that all the
coordinate dependence cancels in the expression for the dilaton beta function,16
~
= R + 4r2
2
(H3)2 :
Explicitly we nd that for the XXZ lambda model
1
12
1)
~XXZ =
2
4
( + 1)
in comparison to the result obtained for the YB lambda model in [44]
and
as
It is noteworthy that these come out as constant despite that fact, as we will discuss
later, the couplings ; ;
run under RG. This is a feature of lambda models and was
observed in the generalised gauged WZW models of Tseytlin [48]. This strongly suggests
that, like isotropic lambda deformations, both of these can give rise to complete solutions
of type II supergravity (i.e. de ne conformally invariant world sheet theories) when the
theory is complemented by a similarly deformed noncompact SL(2) WZW together with
Evidently since we have two functions of three variables one can force ~
XXZ and ~Y B
2 =
Later we will see this relation arising form identifying the RG invariants of the two models.
However, a more discerning comparison of ~XXZ and ~Y B can be made by recasting them
in their common sigma model variables (
and
) making use of eq. (1.9). The result
is striking: they do not match! This indicates that the XXZ and YB lambda theories
are not completely equivalent.
This may be surprising since the XXZ and YB sigma
models di ered only by a gauge transformation of the NS twoform. Under a conventional
Buscher Tdualization, one would expect this di erence to give rise to theories related by a
combination of di eomorphism and gauge transformations after dualization. However the
Sfetsos procedure we employed is not a dualization but instead a deformation and so there
is no reason a priori to expect such a relationship to be the case. The exception is in the
limit k ! 1, in which case the Sfetsos procedure reduces to nonabelian Tdualization;
indeed, in this limit we nd that the two expressions coincide
1
2
~YB
~XXZ
4
+ O(k 1) :
(3.18)
One may recognise this as the being exactly the expected scalar curvature of the anistropic
XXZ sigma model on the squashed sphere.
16To be precise ~ =
stress tensor 2 hTaai = ~ R(2) + : : : and
phism generated at leading order by the derivative of the dilaton [45{47].
1 G 1 G appears as a coe cient of the expectation value of the trace of the
4
i are related to the betafunctions of couplings via a di
eomor3.2
Both the XXZ and YB lambda models inherit the integrability of their mother sigma
models. This can be shown by constructing Lax representations of their equations of motion.
For the YB lambda model, the Lax connection was established in [44].
Let us
rst de ne
and functions of the spectral parameter z:
(z) ! z
z
2
1
;
In terms of the auxiliary gauge eld A , the Lax connection equals
L (z) = ( (z)
R)(1
R) 1A :
The sigma model limit is obtained by restoring
= k =(k
other constants xed. In this limit we have
+ 1) and taking k ! 1 with
A
becomes identi ed with J
and the Lax connection reduces to that of the YB sigma
model (2.2). Having made this connection, in order to facilitate an easier comparison to
the standard form Maillet algebra, it suits us henceforward to rede ne z ! 1=z for the YB
lambda model.
For the anisotropic XXZ lambda model, the Lax operator takes the form
3
X wa(
a=1
and
Note in the sigma model limit k !
1, wa(z) and
equivalents (2.3) and A
becomes identi ed with J .
reduce to their XXZ sigma model
cosh2
=
(1
)( + )
2 (
1
)
:
3.3
The Poisson brackets of the lambda models are inherited from the underlying WZW model
where the KacMoody (KM) currents are
J+ =
J
=
k
2
k
2
F
1A+F
A
;
1
F A F
1 + A+ ;
and whose Poisson brackets take the form of two commuting classical KM algebras [49]
J a(x); J b (y) = f abcJ c (y) (x
y)
J+a(x); J b (y) = 0 :
2
k ab 0(x
y) ;
In the present context, the f abc are the structure constants of the su(2) Lie algebra.
In the YB lambda model, the spatial component of the Lax connection is written in
terms of the KacMoody currents as [44]
L (x; z) = (c+(z) + d(z)R) J+(x) + (c (z) + d(z)R) J (x) ;
(3.26)
(3.27)
(3.28)
(3.29)
(3.30)
(3.31)
(3.32)
HJEP09(217)35
where
where
This is the twist function quoted in [26] for the isotropic lambda model.
d(z) =
2
k(1
2
k(1
2) (
2) (
(z)
+(z) +
(z)) ;
(z)
1) :
The way to extract the Maillet form of the Poisson bracket of L (x; z) is to think of a
change of variables on phase space from the KM currents J
to the Lax operator L (z)
and L (w), for a pair of generic points z and w. This yields precisely the form (2.11)
with kernels
b ;
#
b ;
g(z; w) =
d(z)d(w) + c (z)c (w)
c (z)d(w)
c (w)d(z)
;
and either sign on the righthand side can be taken.
recover the simpler form (2.12) with a twist function
There are two relevant limits to consider. The rst is
! 0, for which the r=s kernels
(z) =
k(1
2)(1 + )
2
2
(1
1
)2
z
2
(1 + )2z2 :
The other interesting limit, is the sigma model limit for which k ! 1, = k =(k +1),
with
and
xed:
T a +
T a +
b ;
These kernels provide a di erent realization of the Poisson bracket algebra of the YB sigma
model compared with [33] whose twist function we quoted in (2.13).
For the anisotropic XXZ lambda model, the spatial component of the Lax connection is
where
where
3
a=1
L (x; z) = X
fa(z)J+b (x) ga(z)J a(x) T a ;
f1(z) = f2(z) =
f3(z) =
g3(z) =
k(1
k(1
2
2
g1(z) = g2(z) =
2
2
1
k(1
s 2
2)
2)
k(1
s 2
1
r 2
r 2
1
1
2)
2)
2
2
2
2
2
2
2
2
coth( + z) coth(
z) ;
coth(
z) coth( + z) ;
csch( + z) csch(
z) ;
csch(
z) csch( + z) ;
and one nds that the r=s kernels are
b ;
(3.33)
(3.34)
(3.35)
(3.37)
(3.38)
and where the twist function is
(z) =
k(1 + )p1
2
p 2
2
2
2 +
+ (1
) cosh(2z)
(1 + ) cosh(2z)
:
The isotropic limit, involves taking
and z
1 and one can verify that this
gives (3.32). The sigma model limit yields (2.15).
!
In this case, the RG
ow appears to follow a cycle. However, the cycle passes outside the
perturbative regime (small
and
) and so it is not clear that the oneloop result can be
trusted. Theories with RG limit cycles have been the subject of a lot of interest and there
are several physical applications (see the review [57] and references therein).
The relation between the reality of q and RG behaviour seems to be quite general:
In the YB lambda model, the one loop beta functions are [44]
The dilaton beta function eq. (3.16) is invariant under this mapping.
We can use the RG invariant to eliminate
to get a single equation for :
!
1
2
log
(1 + 2) 3
1 + 2 2
:
(k2(
1
)2 + 4 2)(k2(
1
)2 + 4 2 2
)
:
d
d
=
1
4k
2
Integrating gives
implicitly in terms of the RG scale
2
tan 1 k(
1
)
2
+ tan 1 k (
2( 2 + 1)( 2 2 + 1)
It is important to note that in the lambda model, the loop counting parameter is the inverse
WZW model level k 1 and the beta function is exact at this order in k 1 as a function of
the couplings
and . One can readily verify that there is an RG invariant combination
Once again we see that the quantum deformation parameter that we established at the
classical level (3.43) is an RG invariant.
In the YB lambda model we note an important duality symmetry on couplings
and elds
constant shift:
!
1
;
k !
k ;
F ! F
This leaves the worldsheet action eq. (3.3) classically invariant and extends the similar
duality symmetry seen in the isotropic case. Under this transformation we note that the
target space metric and two form are necessarily invariant however the dilaton receives a
4(1 + )2
1) + 4 2
2 k
k2(
+ k log k2(
1
)2 + 4 2
1)2 + 4 2 2 = 4 log
:
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
(4.13)
1
0
1
2
2
1
0
1
2
HJEP09(217)35
the blue dot in the middle. The red curved is an example of a cyclic trajectory which has a jump
from
= +1 to
1 at
= 0 and a jump from
=
1 to
= +1.
gure 4. Apart from discontinuities at in nity, the ow
follows a cycle. The jumps are seen to be continuous in terms of the dual couplings
in (4.10) and so we interpret the ows as following a physically continuous set of theories.
In addition, the beta function (4.19) has a pole at
= 1 but the
ow is perfectly well
de ned through it. For a cyclic RG
ow, a key quantity is change in the energy scale
as the ow goes around one cycle [51{53]. This follows easily from (4.13): around a cycle
each of the arctan functions jump by
and so around a complete cycle the energy scale
changes by a factor
!
exp[ ] :
(4.14)
Given the famous ctheorem of Zamolodchikov [58], the presence of RG cycles may
come as some surprise since navely these seem to forbid the existence of a monotonic
function along the ow. To assuage anxiety we note the couplings as functions of scale are
multisheeted can this can allow for a (unbounded) monotonic function that jumps sheets as
a cycle is traversed (see [59] for a toy model exhibiting this fact). One may further wonder
about the robustness of these cycles as the oneloop RG equations are employed in domains
where the couplings are not small; however one should keep in mind that the loop counting
parameter k 1 does remain small. Nonetheless, further study is required to de nitively
conclude the existence of such behaviour; it may be that the theory in this domain should
be viewed only as an e ective theory with a cuto that is necessarily encountered before
an RG cycle can be completed. We will return to this in the next section and comment
further about this possibility in the conclusion.
ow in the XXZ lambda model. The RG
ow of the two couplings
follows from the general formula in [60]:
(4.15)
)
2(1
(1
2)( + 1)
)
2
2)2 :
;
02 =
k
)
2
:
0
2
! 4 (
1
)
;
Note in the sigma model limit, k ! 1 we get precisely the sigma model RG
ow (4.1)
when we use (3.8).
ow in this case also has an invariant
We have used the same notation 0 for the RG invariant here because in the sigma model
limit, (3.8) with k ! 1, we have
precisely, as it must be, the RG invariant of the sigma model (4.2).
The XXZ lambda model also has a duality symmetry that takes
!
1
;
!
1
;
k !
k ;
F ! F
The RG invariant is also invariant under this symmetry. These kinds of duality symmetries
have previously been investigated in the context of currentcurrent deformations of WZW
models in [61, 62].
equation for ,
There are two distinct types of RG ow that depend on whether 0 is real or imaginary
which are \UV safe" and \cyclic", respectively. The RG
ows are shown in
gure 5. We
can use the RG invariant to solve for
and substituting into (4.15), we can write a single
d
d
=
1
4k
(k2(
1
)
2
4 02)(k2(
1
)
2
)
04(1 + )2
:
(4.19)
We will soon exploit the fact that this is identical with the RG
ow equation for the YB
lambda model (4.12) with 0 ! i .
integral Se
= R p
Ge 2 ~
Since ~ [63, 64] can be thought of as a generalised central charge function (and its
the central charge action) is natural to study its property
along the RG
ows. In principle one simply needs to substitute the solution of the RG
equations into the expression (3.15). In practice given the implicit form for the solutions to
2.0
1.5
by the blue blob. The blue line is a line of UV
xed points. The green curve is a UV safe trajectory
that has 0 2 R. The red curve is a cyclic RG trajectory with 0 = i ,
2 R. The trajectory has
a jump in the coupling
from
1 to 1, but is continuous in the dual coupling 1= .
eq. (4.15) it is expedient to proceed numerically and study the evolution along for instance
the green and red trajectories of gure 5. On the UV safe trajectory one nds ~ decreases
monotonically except at one point (the saddle point in
gure 5 where
= 1,
= 1 )
where ~ jumps from
1 to +1. Similarly on the UV cyclic red trajectory ~ decreases
monotonically except at two points ( where
=
1 and
= 1 ). Being a function of cyclic
functions of RG time in this case ~ returns to itself after a complete cycle. Thus with
the exception of isolated points in which ~ is discontinuous, it is elsewhere monotonic.
Although these points look rather innocuous in the RG
ow  they are saddles in the ;
plane  they are distinguished from the sigma model perspective as locations in which the
determinant of the target space metric changes sign.
The UV safe regime, corresponds to 0 2 R, so quantum group parameter q a complex
phase. In this region, as the ow runs backwards towards the UV,
goes to zero while
goes to a constant that we denote
which is determined by the RG invariant via
These ows have a safe UV limit and in the UV, we can expand the couplings in powers
of q = ( = ) , where
=
k
k + 2 0
:
=
4
k(1 +
)
=
2
0 + k
;
(4.20)
(4.21)
and
 the lambda parameter  is the dynamically generated mass scale. The series are
R =
r 1
2k
:
p
R2 + R2 :
R2
The scalar eld determines the one of the components of the currents via
The critical line emerges because the model remains critical as we change the radius. This
corresponds to adding the term J+3J 3 to the action which is clearly equivalent to the
coupling at
= 0 for small . Adding J+1J 1 + J+2J 2 on top of this, gives an integrable
massive deformation corresponding to turning on
(for
> 0).
So in the UV limit,
! 0 and
goes to a constant
and one has
Bernard and LeClair [66] identify the Smatrix of the perturbed theory, the socalled
\fractional sineGordon" theory, as
S XXZ( ) = SRSOS( ; k)
S( ; 0) ;
(4.22)
(4.24)
(4.25)
(4.26)
(4.27)
(4.28)
where the second block is the sineGordon soliton Smatrix and the rst factor describes
additional kink quantum numbers of the states. The sineGordon Smatrix with coupling is
kR2
0 =
R2
R2 =
k(1
2
)
:
This is exactly the RG invariant we de ned in (4.16) and explains our earlier notation.
Now we turn to the regime of imaginary 0 = i , for
2 R. Note that the RG
equation (4.19) with 0 ! i is precisely the same as the RG equation (4.12) in the YB
= k=(2 )@ FF 1.
=
1
+ X
n=1
nq2n ;
=
1
X
n=1
nq2n 1 :
The points
=
varying and
= 0 parametrize a line of UV
xed points shown
in blue in
gure 5. For small couplings the action takes the form of a currentcurrent
perturbation of the WZW model,17
k SWZW[F ]
d2x
J+1J 1 +
J+2J 2 +
J+3J 3 :
(4.23)
The xed line corresponds to just turning on the J+3J 3 perturbation.
It is known that the SU(2) WZW model does lie on a line of xed points. In order
to see this, one uses the fact that the SU(2) WZW model at level k can be realized as a
compact scalar on a circle of radius R coupled to Zk parafermions [65, 66]. The WZW
point has the critical radius
lambda model. This is signi cant and suggests that YB lambda model and XXZ in the
cyclic regime are closely related.
The solution for
in terms of is he same as (4.13) and there is an RG cycle. A typical
trajectory is shown in gure 5 in red. Just as in the YB lambda model, the trajectory follows
a closed cycle which involves a jump from +1 to
1 in
which is continuous in the dual
coupling 1= .
5
In this section, we make informed conjectures for the Smatrices of the generalized lambda
and sigma models. In order to pin down the Smatrix there are some important pieces of
information to take into account:
1. The Smatrix of the isotropic lambda model associated to the PCM takes the form
of a product of the rational, i.e. Y (su(2)) invariant Smatrix, and an a ne quantum
group Uq(s[u(2)) RSOS kink Smatrix [19, 26]:
In the limit, k ! 1 the RSOS factor becomes the rational limit of the unrestricted
SSOS( ) which is itself the vertextoIRF transform of the SU(2) invariant Smatrix
block. This manifests at the Smatrix level that the k ! 1 limit of the lambda
model is the nonabelian Tdual of the PCM:
(5.1)
(5.2)
model
2. The XXZ sigma model with 0 2 R lies in the class of SS models of Fateev [18]. The
Smatrix is then known to have the product form (1.13) where 0 is the RG invariant
related to the UV limit of the coupling
as in (4.16).
3. The YB lambda model breaks parity while the XXZ model preserves parity.
4. As described in sections 2 and 3, the classical sigma and lambda models have Poisson
bracket realizations of the a ne quantum group Uq(s[u(2)) where q is related to the
RG invariants as in (2.20), (3.43) and (3.62).
5. For the theories with cyclic RG ow with a periodicity
e , it is expected that
!
the Smatrices at high energy have a periodicity in rapidity to match [51]:
S( +
) = S( ) ;
1) :
(5.3)
The intuition here is that in the UV at energy scales E
m, the RG cycle behaviour
requires that the theory has a discrete scaling symmetry E ! E exp(
). But for
a particle state with E
m, i.e.
1, we have E
me =2 and so the scaling
symmetry corresponds to a rapidity shift
!
5.1
Quantum group Smatrix: q complex phase
Before making our Smatrix conjectures, there are some general features of Smatrix theory
in the integrable context to take into account. Smatrices for relativistic integrable QFTs
with degenerate particle multiplets are built out of solutions to the YangBaxter equation,
for which quantum groups provide an algebraic framework. For present purposes, we will
be interested in the quantum group deformation of the a ne (loop) Lie algebra Uq(s[u(2)).
We start with the case when q is a complex phase in which case the Smatrix describes the
scattering of solitons in the sineGordon theory [55].
The Smatrix in an must satisfy some important identities (described, for example, in
the lectures [56]):
1. Factorization. Due to integrability, there is no particle production and the complete
Smatrix is determined by the 2 ! 2 body Smatrix elements, as illustrated in gure 1.
2. Analyticity. The Smatrix is an analytic function of the complexi ed rapidity with
poles along the imaginary axis 0 < Im
associated to stable bound states.
Since there is no particle creation in an integrable eld theory there are no particle
thresholds, however, there can be anomalous thresholds in the form of additional,
usually higher order, poles
3. Hermitian analyticity
HJEP09(217)35
4. Unitarity
5. Crossing
where C is the charge conjugation matrix.
Unitarity is implied by Hermitian analyticity and the braiding relation
which is more natural in the context of quantum groups.
Sikjl( ) = Skijl(
) :
X Sikjl( )Smkln( ) = im jn ;
kl
2 R :
Sikjl( ) = Ckk0 Sklj00i(i
)Cj0j1 = Sklji(i
) ;
X Sikjl( )Skml n(
kl
) = im jn ;
(5.4)
(5.5)
(5.6)
(5.7)
ei
q)ei
ej + ej
ei i > j ;
i = j ;
i < j ;
group Uq(s[u(2))
where, on a basis for V , e1
is a generator of the Hecke algebra (the commutant of the quantum group acting on tensor
products) and obeys
(T + q) (T
q 1) = 0 :
In (5.9), f ( ) is a scalar factor which is needed to ensure that the Smatrix satis es
the Smatrix constraints of crossing and unitarity. Based on matrix form of R, there are
four basic processes; identical particle, transmission and two kinds of re ection:
In the present context, the basis states jmi transform in the spin 12 representation of
su(2), or the quantum group Uq(su(2)), with m =
. The 2body Smatrix is a map, or
1
2
intertwiner,
S( ) :
V (
1
)
V ( 2)
! V ( 2)
V (
1
) ;
where V ( ) is the vector space spanned by the states j
; i. Here, the rapidity of the
states is i, and
= 1
2 is the rapidity di erence. The Smatrix takes the form
1
2
where x( ) = ec , c to be determined, and R(x) is the Rmatrix of the a ne quantum
HJEP09(217)35
The braiding relation (5.7) is automatically satis ed because the Hecke algebra
relation (5.12) implies
(5.8)
(5.9)
(5.10)
(5.11)
(5.12)
SI ( ) =
ST ( ) =
R
S ( ) =
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
= f ( )(xq
q 1
x 1) ;
= f ( )(x
x 1) ;
(5.13)
= f ( )x 1(q
as long as the scalar factor obeys
f ( )f (
) =
1
(xq
x 1q 1)(x 1q
xq 1) :
Unitarity then follows if the Smatrix is Hermitian analytic
SI ( ) = SI (
) ;
ST ( ) = ST (
) ;
R
R
) ;
providing the scalar factor satis es
Crossing symmetry requires that either
f ( ) =
f (
) :
x = q
=(i )
or
x = ( q) =(i ) :
(5.15)
(5.16)
(5.17)
(5.18)
(5.19)
(5.20)
(5.21)
(5.22)
(5.23)
HJEP09(217)35
It turns out that the resulting Smatrices are physically equivalent and so we choose the
former. However, with this choice some extra factors of
1 appear in the crossing symmetry
relation and charge conjugation operator; however, these are unobservable.18
Crossing symmetry implies where the charge conjugation operator acts as and there is a further constraint on the scalar factor:
SI ( ) = ST (i
) ;
R
S+( ) = q 1
SR (i
) ;
Cj
1
2 ; i =
iq 1=2
1
2 ; i
f ( ) = f (i
) :
R
R
In addition, if the theory is parity symmetric then one has an additional constraint on
the re ection amplitudes19
in (5.18), we have
For the sineGordon theory, the Smatrix was originally constructed in the seminal
work of Zamolodchikov and Zamolodchikov [55]. In this case, with the former choice
q = exp[ i = 0] ;
x( ) = exp[ = 0] :
18See the discussion in appendix C of [29] for details.
19Since parity
ips the spatial coordinate, the ordering of particles is interchanged. Parity also
ips
momenta pi = m sinh i and so sends i !
i. However the rapidity
in the Smatrix is the rapidity
di erence of particles and so remains unchanged under the combined action of ipping the order and
momenta of individual particles.
clearly violating (5.19).
formation on the states of the form [28{30]
Hermitian analyticity can, however, be restored by a simple rapiditydependent
trans
R
R
) ;
1
! x( ) 1=2
j
1
; i :
However, there is a problem: the Smatrix as written is not Hermitian analytic: the re
ection amplitudes are noncompliant because they satisfy
(5.24)
(5.25)
:
(5.28)
12 i, in
This transformation removes the factors of x 1 from the re ection amplitudes and restores
Hermitian analyticity.20 It has an algebraic interpretation of moving from the homogeneous
to the principal gradation of the a ne algebra s[u(2).21 The transmission and identical
amplitudes are insensitive to this change whereas in the principal gradation the re ection
amplitudes become
R
R
S+( ) = S ( ) = f ( )(q
q 1) ;
(5.26)
such that the resulting Smatrix now also describes a parity symmetric theory. Due to
the change in gradation, the Smatrix is now invariant under the a ne quantum group in
principal gradation Uq(s[u(2)p).
The result Smatrix is precisely the Smatrix of the solitons of the sineGordon theory
once we specify the scalar function f ( ). This is not determined uniquely by the
conditions (5.15), (5.17) and (5.21). However, we can invoke the concept of minimality meaning
that the solution has the minimal number of poles on the physical strip: 0 < Im
< . The
signi cance of this is that poles on the physical strip along the imaginary axis are usually
interpreted in terms of bound states propagating in either the direct or crossed channels.22
The minimal expression can be written in various ways, for example as
f ( ; 0) =
1
2 i
1
Y
( 2n0 + i 0 ) (1 + 2n 0 2 + i 0 ) ( 2n 0 1
n=1 ( 2n +01 + i 0 ) (1 + 2n 0 1 + i 0 ) ( 2n0
i 0 ) (1 + 2n 0 3
i 0 ) (1 + 2n 0 2
i )
i 0 )
0 : (5.27)
It is simple to show that this solves the conditions by computing its divisor. The other
important condition that this expression satis es is the Hermitian analyticity condition (5.17).
Another way to write the results that will be useful later is as the integral expression (valid
for 0 > 2)
f ( ; 0) =
q
1
q 1 exp 2
0
w
Z 1 dw cosh[ w( 0
2)=2] sin[w(i
)=2] sin[w =2]
20To ensure crossing symmetry charge conjugation needs to be modi ed so that Cj
j
agreement with the original construction of [55] but with the additional factors of
needed for the choice
made for q explained in [29].
a(z) ! U a(z2)U 1 where U = z
the transformation (5.25) on states.
21In more detail the change of grade can be achieved by a rescaling and conjugation on the loop algebra
ip2T 3 . Then with the identi cation z
2 = x, the conjugation is precisely
22There are also double poles which are explicable as anomalous thresholds.
The sineGordon Smatrix also has an RSOS cousin, the restricted sineGordon
Smatrix [28{30] which is associated to case when q is a rootofunity. Details of this will
emerge in section 5.3.
Quantum group Smatrix: q real
Given that the YB and XXZ in the regime
have a quantum group parameter
= ] that is real, implies that we also need an Smatrix that will be a close
cousin of the sineGordon Smatrix but with this real value of q. On top of this, since
the resulting theories have a cyclic RG behaviour, heuristic arguments suggest that the
Smatrix should have a periodicity in real rapidity [51]:23
This periodicity requires more than a simple analytic continuation of couplings. Such
an Smatrix was constructed in [51] built on the same quantum group Rmatrix as the
sineGordon Smatrix but now with real q. Crossing symmetry now requires that
S( +
) = S( ) :
x = exp[ i = ] :
such that
equal:
The Rmatrix now has a periodicity under shifts
(more precisely up to
!
some minus signs). This periodicity can be inherited by the Smatrix if the scalar factor is
f ( +
; ) = f ( ; ) :
The situation with Hermitian analyticity is di erent from the real 0 regime: both the
Smatrix in the principal and homogeneous gradations are Hermitian analytic as long as
the scalar factor satis es (5.17). In principle grade, the two refection amplitudes S
While in homogeneous grade, the two refection amplitudes di er:
R
S ( )
SR( ) = f ( ; )(q
R
S ( ) = f ( ; )x 1(q
To complete the construction we must specify the scalar factor. Note that simply taking
the analytic continuation of the sineGordon scalar factor (5.27) from 0 ! i
would not
have the requisite periodicity (5.31) or satisfy the Hermitian analyticity constraint (5.17).
On the contrary, the minimal solution to the constraints can be written as the convergent
product [51]
f ( ; ) = q Y1 (1
n=1 (1
q4nx 2)(1
q4nx2)(1
q4n+2x2)
q4n 2x 2) :
Note that this immediately satis es (5.17) and is manifestly periodic under
!
.
23We follow the convention of [51] and allow the Smatrix to actually change up to some minus signs over
a period. Note that Smatrices with a real periodicity in rapidity cannot have bound states but can have
an in nite set of resonance poles [51, 54].
(5.29)
(5.30)
(5.31)
R are
(5.32)
(5.33)
(5.34)
So there are two consistent Smatrices Sh( ; ) and Sp( ; ), associated to the
homogeneous and principal gradations, respectively. It is important that the Smatrix that uses
the homogeneous gradation of the a ne quantum group, breaks parity SR+( ) 6= SR ( ),
whereas the principal gradation case preserves parity.
The other important point to emphasize here is that when q is real, the Smatrix
associated to the a ne quantum group Uq(s[u(2)) automatically has the periodicity in real
rapidity that matches the heuristic proposal of [51] that theories with cyclic RG behaviour
should have just such a periodicity at high centreofmass energy. But note that the
Smatrix goes beyond this because it has the periodicity for any centreofmass energy.
In order construct our Smatrices we will also need a piece to handle the kink quantum
numbers of the states. This is precisely the RSOS kink Smatrix of the restricted
sineGordon theory [28{30]. It is built out of a solution of the YangBaxter Equation, or
more precisely the startriangle relation, that plays the role of Boltzmann weights in an
Interaction Round a Face (IFR) statistical model, e.g. see [67].
In the IRF Smatrix, the states are kinks Kab( ) and states are labelled by the vacua
a; b on either side. The vacua (the local heights of the statistical model) are associated to
representations of Uq(su(2)) so to spins a; b; : : : 2 f0; 21 ; 1; 32 ; : : :g. When q, the quantum
group a parameter is a root of unit,
q = exp
i =(k + 2) ;
(5.35)
HJEP09(217)35
there is a restricted model, where the spins are restricted to lie in the set of integrable
representations of level
k, so a; b; : : : 2 f0; 12 ; 1; : : : ; k2 g. A basis of states in the Hilbert space
with N kinks is labelled by a sequence faN+1; aN ; : : : ; a1g, which has the interpretation
of a fusion path, so the spin aj+1 representation must appear in the tensor product of the
aj representation with the spin 12 representation (truncated by the level restriction). This
means that there is an adajency condition aj+1 = aj
The analogue of the Rmatrix, is an intertwiner W between 2kink states [67]:
X W
a
b
jKab(
1
)Kbc( 2)i
c u jKad( 2)Kdc(
1
)i ;
(5.36)
where u = =(i ) and
= 1
2. These intertwiners satisfy the star triangle relation [67].
The solution of the star triangle relation W (u) is the raw fodder from which one
fashions the RSOS Smatrix for kinks states. There are 3 basic types of nonvanishing
elements that take the form
W Ba 1
a uC =
W Ba
a uC =
0
a
a
a
1
2
1
2
1
2
1
2
1
A
1
A
[1]
u]
[1
[u]
[1]
p[2a + 2][2a]
[2a + 1]
;
W Ba
0
a
a uC
A
=
[ (2a + 1) + u]
[ (2a + 1)]
;
(5.37)
where we have de ned
The W intertwiner satis es some identities that are important for the Smatrix that
we going to build [67]: (i) the initial condition
u=(k + 2) :
!
W
c 0
= bd ;
e
(ii) rotational symmetry
and (iii) inversion relation
W
c 1
u
s [2b + 1][2d + 1]
c
b u
X
d
W
c u
W
c
u
[1
u][1 + u]
[1]2
be :
The alert reader will recognize that the rotational symmetry and inversion relation as
protoidentities for crossing symmetry and braiding unitarity, respectively.
When k is generic (i.e. not an integer), the local heights a; b; : : : are valued in 12 Z and the
Boltzmann weights W (u) de ne the SOS statistical model. However, when k is an integer
there is consistent restriction of the local heights to the
nite set f0; 12 ; 1; : : : ; k2 g. The
restriction is consistent because [0] = [k + 2] = 0 so consequently W (u) cannot propagate a
kink state with admissible local heights jKab(
1
)Kbc( 2)i with a; b; c 2 f0; 12 ; 1; : : : ; k2 g into
one with an inadmissible local height jKad( 2)Kdc(
1
)i with d 62 f0; 12 ; 1; : : : ; k2 g, in practice
d = 0 or k2 + 1, due to the adjacency condition. This is guaranteed if [0] = [k + 2] = 0.
In order to make a consistent Smatrix,
b
SRSOS( ; k) = v( )W
c u( ) ;
(5.42)
one has to construct a suitable scalar factor v( ) in order that the Smatrix is unitary and
crossing symmetric. The scalar factor must satisfy
v( ) = v(i
) ;
v( )v(
) =
sin2( =(k + 2))
sin(( + i )=(k + 2)) sin((
i )=(k + 2))
: (5.43)
One can readily verify that the solution to these conditions can be expressed in terms
of the usual sineGordon scalar factor in (5.28) with 0 = k + 2, up to a constant factor:
v( ; k) = (q
q 1)f ( ; k + 2) :
(5.44)
where f ( ; 0) is de ned in (5.27).
(5.38)
(5.39)
(5.40)
(5.41)
HJEP09(217)35
The RSOS kink Smatrix has a good limit k ! 1, the SOS limit, as long as the local
heights are suitably shifted, a ! k4 + a, etc, before the limit is taken. So the idea is that
one takes the local heights well away from the end points a = 0 and a = k2 as k ! 1. In
that limit, one can easily verify that the Smatrix becomes identical to the rational SU(2)
Smatrix with a simple mapping between the kinks of the SOS picture and states of the
spin 12 representation:
Ka+ 12 :a( )
! j "; i ;
K
a 12 ;a( )
! j #; i :
(5.45)
This is an IRFtovertex transformation which relies on the fact that the N kink Hilbert
space of unrestricted paths of length N faN+1; aN ; : : : ; a1g, is isomorphic to the N spin 12
particle Hilbert space for a xed a1; e.g.
3
2
3
2
1
2
a + 1; a + ; a + 1; a + ; a + 1; a + ; a
! j #"#"""i ;
(5.46)
etc.
Finally, we can compare our Smatrices by writing down an integral representations of
the identical particle amplitude, which for the RSOS case means
jKa 1;a 12 (
1
)Ka 12 ;a( 2)i ! jKa 1;a 12 ( 2)Ka 12 ;a(
1
)i :
Note that this particular amplitude does not depend on the right vacuum a.
For the q a complex phase  the sineGordon case  we have
For the case q real, the Smatrix of [51], we have
SI ( ; 0) = exp i
Z 1 dw sin[w ] sinh[ w( 0
1)=2]
0
w
SI ( ; ) = exp i = + i X1 2
n=1
sin[2n = ]
n 1 + exp[2 n= ]
:
:
SI;RSOS( ; ; k) = exp i
Z 1 dw
0
sin[w ] sinh[ w(k + 1)=2]
w cosh[ w=2] sinh[ w(k + 2)=2]
;
(5.50)
Finally for the RSOS case just constructed
which is simply (5.48) with 0 ! k + 2.
5.4
High energy limit
The nal information we will need when we establish our Smatrix conjectures is the high
centreofmass energy limit of the trigonometric Smatrices. This is just the large rapidity
limit, i.e. the limit of large x de ned in (5.18) . In order to take the limit, we focus on the
identical particle amplitude SI which can be written (by rearranging the arguments of the
gamma functions in (5.27)) as
SI ( ; 0) = Y
1
i + 2j 0) (
2
2
i + 1 + j 1 0)
2
( 2i + 12 + 2j 0) ( 2i + 12 + j 2 1 0)
(
i + 12 + 2j 0) (
2
2i + 12 + j 2 1 0)
:
(5.47)
(5.48)
(5.49)
(5.51)
0
)
10
5
0
5
10
value of 0. The key feature is that for large
the amplitude saturates.
Note that this amplitude is also valid in the RSOS version of the Smatrix with 0 ! k + 2.
The amplitude is a phase which we plot in
gure 6. The important point is that for
large enough
the amplitude saturates. In order calculate the asymptotic value we simply
apply Stirling's formula to the expression above:
i log SI ( ; 0)
! 4
=
2
( 0
1
1) X
1
j=1 (2 0j)2 + ( =(2 ))2 +
0) +
This means that the while Smatrix has a very simple limit proportional to the Hecke
algebra generator
S( ; 0)
1
! ei (1+ 0)=2T 1 :
(5.52)
(5.53)
Note that the RSOS kink Smatrix also has such a universal high energy limit, where now
T 1 is realized in the kink Hilbert space.
5.5
The Smatrix proposals
In this section, based on all the information and constraints, we make our proposals for
the Smatrices of the lambda and sigma models.
We begin with the XXZ lambda model in the regime with 0 2 R, i.e. the
quantum group parameter a complex phase.
Our proposal is that the Smatrix in this
regime, is precisely the fractional sineGordon Smatrix (4.27) proposed by Bernard and
LeClair [66]. The theory in this regime has a pair of a ne quantum group symmetries with
q = exp[ i = 0], for the sineGordon factor, and q = exp[ i =(k + 2)] for the RSOS factor.
The Smatrices for the sigma model follows in the limit k ! 1 and a nonabelian
Tduality which has the e ect of replacing the RSOS Smatrix piece with the rational SU(2)
Smatrix as shown in (5.2) and one recovered the Smatrix of the anisotropic XXZ sigma
model in (1.13).
Now we turn to the YB lambda model and the XXZ model in the regime 0 = i ,
i.e. where the quantum group parameter q = exp[
= ] is real. In these case the RG ows
are cyclic. This suggest that the Smatrices are based on the pieces Sh( ; ) and Sp( ; )
constructed in section 5.2. There is also a natural explanation for the existence of the
two distinct Smatrices based on the gradation because the YB lambda model is not parity
symmetric and this matches the Smatrix for the homogeneous gradation. Correspondingly
the principal gradation Smatrix is parity preserving as is the XXZ model.
To make a complete Smatrix we need to consider an appropriate RSOS kink Smatrix
factor. The only choice consistent with the sigma model and the classical symmetries is the
RSOS Smatrix piece SRSOS( ; k). However, this Smatrix does not have the periodicity
. The resolution is here is that the periodicity is only expected to appear in the
limit of large centreofmass energy and we have shown in section 5.4 that the trigonometric
Smatrix become constant at high energy. So the heuristic requirement that the Smatrices
of theories with cyclic RG behaviour should have a periodicity in rapidity at high energy
is actually satis ed.
Hence, we make our conjectures; for the YB lambda model
while for the XXZ lambda model
S YB( ) = SRSOS( ; k)
Sh( ; ) ;
S XXZ( ) = SRSOS( ; k)
Sp( ; ) :
The sigma model limit, involves taking k ! 1 along with an IRFtovertex
transformation,
S YB( ) = SSU(2)L ( )
Sh( ; ) ;
S XXZ( ) = SSU(2)L ( ; )
Sp( ; ) ;
respectively. These Smatrices exhibit the Yangian Y (su(2)) symmetry and also have the
periodicity in rapidity
at high energy.
6
Discussion
In this work we have considered the deformations of the SU(2) PCM that preserve
integrability. The class of deformations focused on, preserved an SU(2) symmetry and so
there are associated lambda models. We showed that the lambda models also have a ne
quantum group symmetries realized at the classical Poison bracket level. The are many
questions remaining. In particular, for the YB deformations and anisotropic ones with
for the associated lambda model), the RG
ow follows a cycle in coupling
constant space. So these theories have a mass gap but no xed point in the UV to de ne a
continuum limit. So the main question is: is the UV of these theories well de ned? There
are two pieces of evidence to suggest that these theories actually are only de ned with an
explicit UV cut o of the order of the mass scale of the particle states.
(5.54)
(5.55)
(5.56)
The rst, described in [51] for the case k = 1, comes from de ning the QFT as the
continuum limit of a spin chain. The anisotropic XXZlambda models with
< , so with q
in (3.62) a complex phase, can be regularized by the XXZ Heisenberg spin chain [35] with
spins of angular momentum j = k2 and with spin chain anisotropy
= cos
k + 0
(6.1)
where 0 is the RG invariant (4.16). The spin chain in this regime is critical and
consequently it is possible to take a continuum limit. The physical excitations and their
Smatrix agree precisely with our conjectured Smatrix, the fractional sineGordon Smatrix
in (4.27). Now if we try a similar spin chain description of the
case, then the XXZ
spin chain now lies in the
1 regime. In this regime the spin chain has a mass gap
and so there is no way to take a continuum limit. Even so, we shall show in [35], that the
excitations have an Smatrix that is a close relative of the Smatrix (5.55) It is possible to
create a hierarchy between the inverse lattice spacing and excitation mass only in the limit
of large . So this suggests that the RG cycle is never actually traversed in the UV before
the UV cut o is reached.
The second piece of evidence, again for the case k = 1 for the anisotropic XXZ lambda
model in the cyclic RG regime, is presented in [52]. The idea is to use nite size e ects to
compute the e ective central charge. It is shown that for the case when the theory has a
mass gap, the relevant case here, the nitesize e ects do indeed have a periodic behaviour
consistent with the beta function analysis but in the deep UV the nitesize central charge
either has a singularity or is ill de ned in the very deep UV. Again this suggests that in
the cyclic RG regime, the theories only make sense with an explicit UV cut o .
The other issue which is interesting to consider is how these issues play out in larger
groups. We have already pointed out that the anisotropic models are special to SU(2) and
they do not appear to admit generalizations to an arbitrary Lie group. However, the
YangBaxter deformation do lift to an arbitrary group and one can speculate that the sigma and
lambda models once again have a cyclic RG behaviour. We show this is the case in [35].
We go on to show that there is a natural conjecture for the Smatrix which is rather novel.
For the case SU(N ), it is related to the Smatrix constructed in [68] but like the Smatrix
S( ; ) considered here is periodic in rapidity. What is novel about the resulting Smatrix is
that it exhibits an in nite set of unstable resonance poles thus providing an example of the
\Russian Doll" phenomena described in [52]. Unlike the SU(2) example described there,
the Smatrix we construct satis es all the Smatrix axioms including hermitian analyticity.
Finally there is a generalization of the anisotropic models that we have mentioned
in the introduction, namely the XYZ model. The lambda model of this should have an
Smatrix that is related to the elliptic Smatrix of Zamolodchikov [69].
Acknowledgments
CA and DP are supported by STFC studentships. TJH is supported in part by the STFC
grant ST/L000369/1. DCT is supported by a Royal Society University Research Fellowship
Generalised Dualities in String Theory and Holography UF 150185. We would like to thank
Saskia Demulder, Kostas Sfetsos, Graham Shore, Kostas Siampos and Benoit Vicedo for
useful discussions and Arkady Tseytlin for interesting correspondence.
A
Lambda spacetimes
The lambda theories can viewed as sigma models with target spaces of the following form
A0 = c(
A2 = c(
A3 = c(
A4 =
A5 = 2c2(
+2
1)2( + 1)S2
1)2( + 1)S2 S2
2 2(
1
)
A0 =
c ( + 1) (C2 (
A1 = c 2C2 (
1
)C2 (
A2 = c(
A3 = c(
A4 = c(
A5 =
1)S2 (C2 (
1
)
1)(
1 S2 S2
)S2 S2
4c2(
S
1)S2
C2( (5
( + 1)(
3 ) + (3
) +
) +
2C2
2 4 2
3 2
2 + 1
8
2((
2) + 4 + 2
2 2 2 + 2 + 1 S3 S
For the XXZ lambda model we nd (with c =
1 ):
1) + 2C2 ( + 1)C2 (
( + 1)( + ) + ( + 1)( + 1)2
5 ) ) + ( + 1)
1) 2
1 + 3 +
1)
)S3 S
2
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
k
A0
A5
0
^ d ;
ds2 =
H3 = k A2 d ^ d
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C S ) (
C S + C )
2 + 1
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