Irrelevant perturbations of 2D integrable models
Osaka, September 2019
Roberto Tateo

Mainly based on work in collaboration with:
M.Caselle, A. Cavaglià, R. Conti, D. Fioravanti, F. Gliozzi, L. Innella, S. Negro, I. Szécsényi
JHEP 1307 (2013) 071, JHEP 1610 (2016) 112, JHEP11(2018)007, JHEP 1902 (2019), andarXiv:1904.09141
1

S. Dubovsky, R. Flauger and V. Gorbenko, Solving the simplest theory of quantum gravity, JHEP
1209 (2012) 133;
F. A. Smirnov and A. B. Zamolodchikov, On space of integrable quantum field theories, Nucl.Phys.
B915 (2017);
L. McGough, M. Mezei, H. Verlinde, Moving the CFT into the bulk with TT, JHEP 1804 (2018)010;
O. Aharony, S. Datta, A. Giveon, Y. Jiang, D. Kutasov, Modular invariance and uniqueness of TT
deformed CFT, [arXiv:1808.02492];
A. Dei, A. Sfondrini, Integrable spin chain for stringy Wess-Zumino-Witten models, JHEP 1807(2018) 109;
G. Bonelli, N. Doroud, M. Zhu, TT-deformations in closed form, JHEP 1807 (2018) 109;
J. Cardy, The TT deformation of quantum field theory as random geometry, [arXiv:1801.06895]
L. Santilli, M. Tierz, Large N phase transition in TT-deformed 2d Yang-Mills theory on the sphere,
JHEP 1901 (2019) 054, [arXiv:1810.05404]
M, Guica, An integrable Lorentz-breaking deformation of two-dimensional CFTs, JHEP 1901
(2019) 054, [arXiv:1810.05404]
2

CFTUV

?
CFTIR

CFTIR
3

Exact S-matrix and the CDD ambiguity
Consider a relativistic integrable field theory with factorized scattering:

Castillejo-Dalitz-Dyson ambiguity:

The simplest possibility, consistent with the crossing and unitarity relations is:

4The sine-Gordon NLIE
[1991: Klümper-Batchelor-Pearce, 1992: Destri-DeVega, 1996: Fioravanti-Ravanini- et al. ]For the ground state
contours appear for excited states.

and, but more more complicatedand5replacing

we getwith

Therefore:

6

and

Then

which allows to compute the exact form of the τ-deformed energy level once its
R-dependence is known at τ = 0. The result is:therefore7

From (1.2), it follows that the solution to (1.1) can be written, in impli

2
We now have an implicit form of the solution of theEinviscid
equationwitha(R0addition(R, ⌧ ) Burgers
P 2 (R) =E 2 (R0 )with
P 2the),source term:with the additional constraint

E(R, ⌧ ) at
, fixed R0 ,) obta
As extensively diat fixed R0 , obtainable directly from (1.3) (cf. with the s = 1 case of (to the TT̄-deform
As extensively discussed in [18] (see also [19]), the solutions to the clas
dependent coordto the TT̄-deformed Lagrangians [2, 18, 20, 21] are obtained from the(Typical
τ=0 finite-volume
spectrum)dependent
coordinate transformation
⇣
⌘
µ
µ
µ
⌫
e
dx =
y =with(y 1 , y 2 ) = (x0 , t0
⌫ + ⌧ T ⌫ (y) dy ,( @⌧ R =

with

!

0 1
✏µ⌫ =
where T µ⌫
1 (y)
0 is
µ⌫
ically defined asµ⌫where T (y) is the Hilbert stress-energy tensor associated to the unde
ically defined as
q
@L
(y)
2
p
g
T µ⌫ (y) = p,
g =and det(gµ⌫is) , theLg (y)
g @gµ⌫
coupled to gravit
where, ceff= c – 24Δ andis the“effective
centralcharge”
of
the
UV
CFT
state.
Lg (y) is the undeformed Lagrangian in the set ofas cartesian coor
coupled to gravity through the metric gµ⌫
8 . As shown in [18], equationasTeµ⌫ (y) =g µ ✏ ⇢ ✏ ⌫ T ⇢ (y) ,9where E (±) (R, t) denote the canonical expressions for I (±) , evaluated on the solutions of
the deformed NLIE system:
The CFT case Z
"Z
#
⇣
⌘
⇣
⌘
M
d✓ ±✓
d✓ ±✓
(±)
f (±) (✓)
f (±) (✓)
E
(R,
t)
=
e
ln
1
+
e
e
ln
1
+
e
(2.10).
An extra CDD factor2couples
left (-) with right (+) movers andany NLIE or TBA(±) 2⇡i(±) 2⇡iC1C
equation leads to a pair ofcoupled algebraic equations: 2
Equations (2.9) reveal that the deformation can be interpreted as a redefinition of thelength-parameters appearing in the NLIEs, R ! R + 2tE (±) (R, t). Consistency with (2.5)
then yields the following conditions:
R ! R + 2⌧ E (±) (R, ⌧ )(2.11)
ceff= c – 24 Δ(primary), obtained by an energy-dependent shift:
✓
◆
✓
◆
n
c
/24
n̄
c
/24
0
0
e↵
e↵
( )
E (+) (R, ⌧ ) = 2⇡
,
E
(R,
⌧
)
=
2⇡
(2.12).
R + 2⌧ E ( ) (R, ⌧ )R + 2⌧ E (+) (R, ⌧ )These are precisely the relations found in [9] starting from (generic) TBA equations andTheimply
total thatenergy:the energy levels have the form [7, 9]:E(R, ⌧ ) = E (+) (R, ⌧ ) + E ( ) (R, ⌧ )
s
✓
◆2
⇣
⌘
2
R
R
2⇡
ce↵
2⇡(n0 n̄0 )
=
+
+
n
+
n̄
+
,
(2.13)
0
0
2⌧
4⌧ 2
⌧
12
R
2⇡(n0 n̄0 )P (R) = E (+) (R) E ( ) (R) =.
(2.14)
R
which matches the form of the (D=26, ceff=24) Nambu Goto spectrum, for a genericAs reviewed
in thewhereintroduction,ce↵ =D 2 this coincides with the spectrum of the
CFT,with τ=1/(2s),s is thefor
stringtension.
10
Nambu-Goto string in D-dimensional target space obtained through light-cone quantization(for more comments on this relation, see the Conclusions).

5. Identification of the perturbing operator

Let us now establish a direct link between the CDD-factor introduced in (2Start from the equation:
operator. We start from equation (4.12):@t En (R, t) = En (R, t)@R En (R, t) + R1 Pn (R)2 ,and use the relations
where En (R, t) denotes the n-th excited state of the system on a ring of size
nient to consider the theory as defined on a cylinder or torus, with Euclide(x, y) ⇠ (x + R, y). The expectation values of the components of the streson the eigenstates of the Hamiltonian defined on constant-y slices, satisfy [since

En (R, t) =Rhn|Tyy |ni, @R En (R, t) =hn|Txx |ni, Pn =iRhn|Tso that (5.1) can be rewritten as@t En (R, t) = R (hn|Tyy |nihn|Txx |ni

then
with

=

R
⇡2

hn|T |nihn|T̄ |ni

hn|Txy |nihn|Txy |ni)hn|⇥|nihn|⇥|ni .

In (5.3), we have used the standard conversion between the energy-mocomponents in Euclidean and complex coordinates,
Txx =12⇡ (T̄ + T

1
i
2⇥) , Tyy = 2⇡
(T̄ + T + 2⇥) , Txy = 2⇡(T̄
11

One of the main results of [18] is the important identity

Zamolodchikov's

composite operator fulfills the following factorization property:

Putting all this information together:

Therefore, up to total derivatives:12:L( ~E(R,, 0) =⌧ )@=~ · @¯~ , ~+= (1
1 ⌧ F(2.17)
2˜⌧
0
B = |@ ~ ⇥ @ | =det [hµ⌫ ] .
4 proved in [13] that the deformed Lagrangi
itwasBoson field theories with withgeneric⌧˜ = ⌧potential(1 bosonic⌧ F0 ), thatis a reparametr
would like to extend the result (2.16)
togeneric
interactingof L
Infeld model or, equivalently,
theLagrangians
Nambu-Goto¯~ 2parameter ⌧ in the energy di↵erences E✓◆qFurthermore,
it was arguedin [7] that
(2.1
~1) ,
LV ( ~ , 0) = @ ~ L(
· @¯~~, +V((2.18)2~⌧) =1 + 1 + 4⌧ L( , 0) 4⌧ B =following
fundamental equation for the La2⌧( ~ ) is a generic derivative-independent potential. Instead of solving (2.14) using a[Bonelli-Doroud-Zhu, Conti-Negro-Iannella-RT@that= det[T⌧ L(⌧ )theµ⌫ (⌧ )
~ · @postulating~tive brute-force approach, as in [13],with
weevident
hproceed(2018)]µ⌫ = @µ by⌫ andwith µ, ⌫ 2 {1, 2} andBEuclidean= |@ ~ ⇥ coordina@¯~ |2 =initial condition
Here, we would like to extend L(the~ , result0) = @ ~(2.16)· @¯~ , to– the
4 – form:

it was proved in [13] that Vthe~ deformed~ La¯
L ( , 0) = @ · @Infeld model or, equivalently, the Nambuand the Hamiltonian density is a straightforward
generalization of the sin~
where V ( ) is a generic✓derivative-independentqAlso:reported in [20]1~, ⌧ ) =L( brute-force
1 + 1 + 4⌧0) 4⌧
perturbative
approach,asL(
in~ ,[13],
w
2⌧
✓
q

H ( ~ , ~⇡ , ⌧ ) =V ( ~)V

1

1
2 P 2( ~, ~~ , ~⇡ , 0) + 4¯+
1 + 1 + 4¯
⌧
H(
⌧
⇡
~
~
~
with
h
=
@
·
@
and
2¯
⌧
µ⌫
µ
⌫
⌧V( )

B = |@ ~ ⇥ @¯

– 4den–where H( ~ , ~⇡ , 0) = 14 | ~ 0 |2 |~⇡ |2 = T22 (0) is formally the Hamiltonian
extendthe result
undeformed theory, while P( ~ , ~⇡ ) = Here,~⇡ · ~ 0we= would
T1312like(⌧ ) tois theconservedmo(the form: (2.5).
of the deformed theory, following the convention

The sine-Gordon model

withand EoM(Lax consistency equation)
Deformed Conserved charges(expansion in the spectral parameter λ)14

A local change of coordinates

Where:
15

The kink16

Deformed breather
Start with the breather solution with envelope speed v = 0

17

t follows that the solution to (1.1) can be written in implicit form as
E 2 (R, ⌧ )E 2 (R, ⌧ )P 2 (R) = E 2 (R0 )P 2 (R0 ) ,modelsP 2 (R) Generic= E 2 (R0 ) TTbar-deformed
P 2 (R0 ) ,(1.4)

with the additional constraint

itional constraint@⌧ R =

E(R, ⌧ ) ,@⌧ R =

E(R, ⌧ ) ,(1.5)

at fixed R , obtainable directly from (1.3) (cf. with the s = 1 case of (5.48)).

0 from (1.3) (cf. with the s = 1 case of (5.48)).
obtainable directly
As extensively
discussed
inthe[17]where(see Talso
the thesolutionsto the classicalassoctµ⌫ (x,⌧ ) isclassicalnow
Hilbertstress-energy
tensorEoMs
associatedly discussed
in [17] (seealso [18]),
solutions
to[18]),
theEoMs associatedto Lagrangians
the TT̄-deformed[2, 17, from19, 20]the ⌧ = 0 ones by aeformed[2, 17, Lagrangians
19, 20] are obtained
theare⌧ =obtained
0 ones byfrom
a field2 @Lg (x, ⌧ )
µ⌫
T
(x,
⌧
)
=
,
p
dependent
coordinate transformation
oordinatetransformation
g @gµ⌫⌘⇣⌘ ⇣2in the set of cartesian coo
2
⌧Te
)1µis
the
Lagrangiandxµ dy=⌫anddy ⌫ , y= (y 1 , y(1.6)),
dxµ = µ⌫ + ⌧ Teµ⌫ (y), µ⌫Lyg+(x,=⌧(y,⌫y(y)) , deformedwithTeµ⌫ (y) =convention of [18], we will switch from cartesian to complex coordi
(
(
!
!
1
2
z = x + ix
w = y1 + i y2
0 1
,
⇢ µ
0 2(1.7)1 ,µ⇢gµ ✏ ⇢ T T
✏
,
✏
=
,
1
1
2
e (y)
⌫=
µ⌫ ✏ ⇢ T
(y)
g
(y)
✏
,
✏
=
,
z̄ = xix
w̄ = y
iy
⌫
µ⌫
⌫
10
µ⌫

10

µ⌫

and we shall denote with z = (z, z̄) and w = (w, w̄) the two set of) is the Hilbertµ⌫stress-energy tensor associated to the undeformed theory, canonwhere T (y) is the Hilbert stress-energy
associatedthe undeformed
theory,An importanttensor
link with
topologicalto
gravity
was noticed andstudiedcai
d asthat JTq gravity coupled to matter leads to a scattering phase match
ically defined as 2 @L (y)pgq
T µ⌫ (y) = p,µ⌫ TgT̄=perturbation, 2, p(1.8)and (1.9) were also obt
4–6]. Equations (1.6)@L(gg (y)2detµ⌫ )[1,g @gµ⌫ T (y) = p,
g = det (g ) ,the deformed
EoMs following, therefore,µ⌫a completely independentg @gµ⌫
to [19].
However, thefinal results
out to be fully consistent wi
s the undeformed Lagrangian in the setof Euclidean
coordinatesy, turnminimally
Lg (y)
the undeformed
Lagrangian
ininvertedthesetthatminim
thispaper,
argue
the18approachcoordinatesof [18] admits y,naturalgravityandthroughtheismetricgµ⌫ . Equation(1.6)
canwebeshallasof Euclidean
coupled to ⇣gravity through ⌘theingmetricgµ⌫ . families
Equation(1.6) can bedeformations
inverted asof classical a
to infiniteof geometric-type
µ

µ

µ

⌫

1

2

irrelevant operator (3.19) remains valid. In order to achieve a deeper understandin
Deformations
by conserved
currents
withhigher Lorentz
spin dedeformations,induced
it is probably
necessary tomove from
Hamiltonianto Lagrangian
Unfortunately, except for special cases, we were not able to pursue this task for gen
n the following sections we shall generalise the change of variables (2.7) to deformations built
Finally, considering (3.21) for k = 1, and given the geometric origin of these systems
om conserved currents with generic positive and negative integer spins. Conventionally, we
denote these theories as geometric TT̄s -deformed models.
0hall denote with s = |s| 0 (k = |k| 0) the absolute value of the spin, and set s = s  0
k 0 = 3.2
k  0).Deformations
We will discuss related
separately
inducedby conserved
currents
withtoperturbations
charges withnegativespin s <0
pin s = s > 0, s0 = s < 0 and s ! 0, providing explicit examples.

In order to define perturbations induced by higher conserved currents with negati
we flip s ! s0 related
in the definitionof theJacobian.1 Deformations
to charges
withgeneralised
positive spins > 0 (3.2) and (3.3), obtainin!
¯ s 1derivatives
he natural generalisation of (2.7)
which ensuresthepartialis1
1 +commutativity
2⌧ T̄s+1 (w) of 2⌧
⇥
(w)
(s0 )
J
= (s0 )
,
!
!
! (w)
!
2⌧ ⇥s 1 (w) 1 + 2⌧ Ts+1 (w)
⇣
⌘
⇣
⌘
T
1
dw
dz
@w f
@f
(s)
(s)
2!
= J
,
= J
,
(8f
:
R
! R) ,(3.1)¯ s 1 (w)¯ 2⌧ ⇥dw̄
dz̄(s0 ) 1 @w̄ f1 + 2⌧ Ts+1 (w) @f
J=,2⌧ ⇥s 1 (w) 1 + 2⌧ T̄s+1 (w)herewith s0 = s < 0 , s = |s|
! and

!

1
@w @ w̄
1 + 2⌧ ⇥s 1 (w)2⌧ Ts+1 (w)(s0 )2J (s) ==,⇥¯ s (3.2)(w)(w) 12⌧+T̄2⌧
T̄
(w)
4⌧
⇥
(w)
(s)s+1¯ =@¯w̄1 + 2⌧ T
¯
s+1
s
1
1 (w) .
@w
(w)
1
+
2⌧
⇥
(w)
(w)
s+1
s 1
!
!
⇣
⌘
¯ s 1 (w) the2⌧spin-flip
1 to (3.22)–(3.23)
We arrived
by 1implementing@w z @w z̄+ 2⌧ ⇥
Ts+1 (w)symmetry (see, for examp
(s)
J
=
=
,
(3.3)
@w̄ z @w̄ z̄
2⌧ T̄s+1 (w) 1 + 2⌧ ⇥s 1¯(w)⇥s0 1 = T̄s+1 ,Ts0 +1 = 19
⇥s 1 ,

ith s >
0, corresponds to the following reflection property at the level of the 1-forms
which

¯
dz̄
@w̄ f
@f
•higherread theT̄-deformed
higherconserved currents
asincomponentsof (2• read the TT̄-deformed
conserved
currents
as⇥components
ofk(2.10)
zT
coordinates:
Tk+1(w(z))+T2⌧
Tk+1(w(z))⇥
0 (w(z))2 (w(z))
1 (w(z))Tk+1 (z, ⌧ ) =,where
Therefore, the natural generalization which ensurestheequalityof
mixed
derivatives(w(z))
Tcurrents+ 2⌧ Tk+1 (w(z))⇥in⇥k 1 (w0 (w(z))k+1 (w(z))• read the TT̄-deformed
higherconserved
ascomponentsof(2.10)zcoordinates:
Tk+1 (w(z))+2⌧T(w(z))⇥(w(z))⇥(w(z))
T(w(z))
0
2
k+1
k
1
Tk+1
(z,
⌧
)
=
!,
¯
Tk+1 (z, ⌧ ) = ⇥k 1 (w(z))!
(w(z))+
2⌧⇥(w(z))⇥(w(z))
T(w(z))
T̄(w(z))
2
0 ⇥
k
1
k+1
1
(w(z))
@
w̄
1
+
2⌧
(w)
2⌧
T
(w)
s
1
s+1
⇥k 1 (z, ⌧J) (s)= = @w.(
Tk+1 (w(z))=+ 2⌧ Tk+1
(w(z))⇥0 (w(z))⇥k (w(z))
T(w(z))
2 (w(z)) T,¯
1 (w(z))⇥(w(z))
+
2⌧
⇥
⇥
(w
0
k
1
k
1
k+1
Tk+1 (z, ⌧⇥) = @w,¯ @¯w̄
¯T̄s2 (w(z))¯ (w(z))(w(z))
Tk+11(w(z))
T̄s+1 (w)+ 2⌧ ⇥⇥k+12⌧(z, ⌧⇥)(s)k 1 (w(z))
k =(w)
1 (w(z)) ⇥02⌧
1 (w)
(w(z))⇥k 1 (z, ⌧ ) =.
(w(z))!
!
(2(w(z))¯⇥k 1 (w(z)) + 2⌧ ⇥k 1 (w(z))⇥(w(z))
T(w(z))
T̄(w(z))
⇣
⌘ 1
2
0
k+1
¯ s 1 (w) 2⌧ Ts+1 (w)
1 + 2⌧ ⇥⇥k 1 (z, ⌧ ) = @w z @w z̄.(2.11)
(s)
J
=
=
,
(
(w(z))
@
z
@
z̄
2⌧
T̄
(w)
1
+
2⌧
⇥
(w)
w̄
s+1 in the followings 1section we will explic
order to make the abovew̄strategy
more concrete,dw̄

(2.11)In order to make the above strategy more concrete, in the following sect
In
order
to
make
the
above
strategy
more concrete,in the following
sectionwe will
explicitly
cuss the TT̄-deformeddiscusshighercurrents
forthemasslessfreeboson
theoryandcomment
the TT̄-deformed higher currents for the massless free boson thewith s >
0,
Inthe
order
to make thehigher
above strategy
concrete,in thefollowing
section andwe will
explicitly
discuss
Tcases.
T̄-deformed
currentsmore
for themasslessfreeboson theory
commentonre generalmore
generalcases.
discuss
the TT̄-deformed higher currents for the massless free boson
theory and comment on
more general(s)cases.
2
¯

2

2.2

(w) =
1 + 2⌧ ⇥s 1 (w)more generalcases.

1 + 2⌧ ⇥s 1 (w)The massless free
2.2boson
The massless free boson

4⌧ Ts+1 (w) T̄s+1 (w) ,(The massless free boson
2.2
The massless
free
boson
Considerthe massless
Lagrangianof a single
masslessboson
in theset ofnsiderConsider
the Lagrangianofasingleboson
fieldφ in
in
the
set
offield
complexcoordintheLagrangianofasinglemassless
boson
field
the
set
of
complexcoordinatesw
Consider the Lagrangian of a single massless boson field in the set of complex coordinatesw
Consider the Lagrangianof a single massless boson field in the set of complex coordinatesw–@8 – @
L(w) = @w @w̄ .
w
L(w)
=
.
(2
w
w̄
L(w)
=
@
@
.
(2.12)ww̄
L(w) = @w @w̄ .
(2.12)The EoMs are
e
EoMs
are
The EoMs
are areThe EoMs@w @w̄ = 0 ,@ww̄@w̄==0=,00 ,,@@ww@@w̄
(2.13)(2(2.13)
therefore, the components of (2.5) are somehow trivial. However, withthe componentsof (2.5)
somehow
trivial.However,
However,
without
further
external
therefore,
thecomponents
componentsaresomehowfurther
external
refore,therefore,
theof (2.5)(2.5)
arearesomehow
trivial.
However,
without
further
constraints,
there
existsantrivial.
infinitenumberofwithoutoptions
for thechoiceexteof ttherefore, some of thecomponentsof thestress-energy
tensorare 20trivial.
constraints,
there
an infinite
numberofoptions
options forthechoicethebasisbasisofconstraints,
thereexists
existsexists
an infinite
infinitenumberofofoptions
thechoiceofofthebasis
of conservedconserved
nstraints,
there
annumberthechoiceof theof conser
currents.
For
example,both forforcurrents.
For example,
currents.
Forexample,
example,bothbothrents. Forboth@w @w̄ = 0 ,(2.13)
therefore, thecomponents ofof(2.5)
are somehow
trivial. However, without further external
the components(2.5) aresomehow trivial. However, without further external
3.3 therefore,
The classicalBurgers-type
equations
There existsthere
an infinite numberof options forofthe
choice ofthe
the basis ofofconserved
currents.
constraints,
constraints, thereexists
existsananinfinite
infinitenumber
number ofoptions
optionsfor
for thechoicechoice of thethe basis
basis ofof conservedconserved
For
example,both
In currents.
this
section,we
considerForbothcurrents.
Forexample,
example,bothdeformations of the massless free boson theory induced by thcoordinate transformations (3.23)–(3.24),
and we derive the higher conserved currents of th
11
(POW)
(POW)
k+1
(POW)
(POW)(w) = 0 ,
k+1 ,
T
(w)
=
(@
)
⇥
(k
,,
(2.14)wk
11 the
T
(w)
=
(@
)
,
⇥
(w)most= 0 , general(k 22 N)N)level-k(2.14)of th
deformed models. k+1
Asalready
discussedin
sectioncurrent
w
k+1
k 2,
22
hierarchy
can be expressed in the form (2.17). While the structure of the deformed curren
4
4
and
and
doesandnot emerge clearly by working
withthe general combination (2.17), we observed that th
⇣
⌘
1+k
2
⇣
⌘
1 1 21+k 2
(KG)
(KG)
(KG)
(KG)
subset (2.15) is T
analyticallymucheasier
to
treat
since
not
mix
with
2
(w)
=
@
,
⇥
(w) =it0does
, (k
22 2N
+
(2.15)
w
T
(w)
=
@
,
⇥
(k
2N
+ 1)
1) ,, the others.
(2.15) Th
k+1
k
w
k+1
k 1 1 (w) = 0 ,22property allows to obtain compact expressions for the deformed currents which are formal
arearepossiblesets
ofofhigher
currents
since
they
possiblesets
higherconservedconserved
currents
since
theyfulfil
fulfil (2.6)(2.6) on-shell.
on-shell. In
In general,
general, anyanyidentical
to theexact
quantum
resultsof section
4.2.
linearcombinationarepossible
ofofhigher
conserved
currents. transformations
For simplicity, weread
will consider
the negative
linearcombination
ofthe
theform
form
Using(2.15)
insets(3.23)–(3.24),
the coordinate
explicitly
spin power-type set of currentsXk(s’k =−s≤0, s=|s|≥0):
!
X
(GEN)
(k)
(GEN)
(GEN)
(k) 1 kk j j j j(GEN)== cjcj (@(@
(2.16)
Tk+1(w)⇥kk 11 (w)(w) == 00 ,,
Tk+1 (w)ww )))s+1 @@ww , , 0⇥
1 ⌧ (@
(s0 )
w
J
=
,
(3.3
j=0
j=0
1
0
1 ⌧ (@w̄ )s+1!
automaticallydefinesk.k. Moreover,automaticallydefinesa aconservedconservedcurrent
currentwith
withspin
spin
Moreover,since
since thethe changechange of variables
s+1
0
1
⌧
(@
)
0
1
w
(1.6)
non-linear,diﬀerent
diﬀerent
choicesthecurrent
current(1.16)should,
should,s+1
at least
least
in principle,
principle, give(3.3
(1.6)
is is
non-linear,inin(1.16)
at
J (s ) choices= ofofthe. in0
1 ⌧ (@w̄ )rise
totallydiﬀerent
diﬀerentdeformations
deformationsofofthe
theoriginal
original
theory.
rise
toto
totally
theory.
4Thecurrents(2.15)cancanbebeobtained
obtainedasasthe
themasslessmasslesslimit
limitofofthe
theKlein-Gordon
Klein-Gordon hierarchy:hierarchy:Thesetsetof ofcurrents(2.15)
Repeatingthesamecomputation
performed
in section
2.2,
we
first express
(@w
1+k
T(KG)
1 1+k
m2 2 kk2 211 22(KG)
1m
2 2 of (KG)
2
terms of (@ , @¯ T)(KG),
bysolvingthesetequations
2
Tk+1
(w)
(w) =
(k222N
2N+
+1)
1) ,,
(w)
= = 2@w@w, , ⇥⇥@@ww, , (kk+1
k k 1 1(w) =
2
22

4

!
2 22 2 .!
withLagrangian
L(KG)
(w)
with
Lagrangian
L(KG)
(w)
==
@w@w @w̄@w̄ ++mm
.
0
@
@
w
(s )
=J

!

8
<@ = 21@w s+1
1 ⌧ (@ )w.

, @w̄ )T

(3.3

(s)
2⌧
@f , @w̄ f = Fs
2⌧
@f , (3.3
s
s
s
2
s
2
T
(w(z))+
2⌧T(w(z))⇥(w(z))⇥(w(z))Ts 1
s+1 (w(z))
k+1
k+1
k 1
(s)
Tk+12(z, ⌧ ) =,(s)
It follows
thatitthe
deformed
EoMs are(w(z))
8f : R! R. immediately
From (3.35),
follows
immediatelythat the deformed EoMs are
Furthermore,
let us consider the level-k Hamiltonia¯⇥(w(z))+
2⌧⇥(w(z))⇥(w(z))
Tk+1 (w(z)) T̄s+1 (w(z))
s 1
k 1
k 1
(s)
⇥k 1 (z, ⌧ ) =. (3.3@ @¯ (s)= (w(z))0 , (s0 )
(s0 )
(s0 )
Hk (z, ⌧ ) = Ik (z, ⌧ ) + Īk (z, ⌧ )
1holomorphic and (3.9)
whichreflects
the
fact
that
the
s
<
0
perturbations
of
CFT’s
do
not
mix
the
which reflects the fact that the s < 0 perturbations of CFT’s domix the H
holomorphic= notk (w(z)) + 2⌧ a(s) (w(z))
anti-holomorphic sectors. The level-k Hamiltonian and momentum deformed
densities are:anti-holomorphic
fromand(3.32).
urthermore,
let us sectors,
consider as
thealreadylevel-k emerged
Hamiltonian
momentum density5
Using the technique described in section 2, we can now derive the deformed curren
(s0 )
(s0 )
(s0 )
(s0 )
(s0 )
(s0 )
Pk (z, ⌧ ) = Ik (z, ⌧ ) Īk (z, ⌧ )Hkthe(z,⌧ ) = Ik (z,⌧ )(3.35)+ Īk into(z, ⌧ )(3.27), we obtain
Plugging
diﬀerential
map


1

✓
◆ 1
=
+ 2⌧
s+1
s+1
k (w(z))= (s)sHk (w(z)) + 2⌧(s T++,k+ c.c.
,(3.10)
0P
1)(w(z))(@ T̄) ,s (w(z))(s) (w(z))
(s0 )
(s
)
Where
where
defined (w(z)) 1 + Fs
Ts+1(z, ⌧we)=2⌧, ⇥s 1 (z, ⌧ ) = 0 ,s
2⌧ (s + 1)
s
2
5
0
0
0
0
In (s
the) following, “c.c.”(s0denotes
the replacement{T(z, ⌧ ),
 (s ) ✓
(s )
(s )
)◆
(s0 )
s+1
s+1
¯
¯
0
0
(z,
⌧
)
,
Ī
(z,
⌧
)
=
T̄
(z,
⌧
)
+
⇥
(z,
⌧
)
,
(3.12)
I
(z,
⌧
)
=
T
(z,
⌧
)
+
⇥
s
(s
+
1)
(
@
)
(s ) k 0
(s
k+1
k0 1
k
k+1
k )1
0
¯
(s ) ⌧ ) =(s )(s ) FsT̄s+1, ⇥s 1 (z, ⌧ ) = 0 ,Pk(z, (z,⌧ ) = 2⌧Ik (s(z,⌧ ) Ī1k +(z,⌧ ) 2⌧+ 1)
ss (s0 ) 2(s0 )¯ combinations
(w)theT±,n(w)show= Tn+1
±⇥
(w)
,
T̄
(w)
=
T̄
(w)
±
⇥ 1 (w), generic(3.13)
Moreover, one
can
that
I
(z,
⌧
)
and
Ī
⌧ ) of
le
n 1
±,n k
n+1
k n(z,
1
(3.3
with
= (s)
Pk (w(z))+
2⌧ T ,k (w(z))
T̄ equations
c.c. ,(3.11),s (w(z))
deformed
currentsaresolutionstothefollowing
algebraic–9–
and(w(z))@ w f = Fs

with5

Ik (z)Īk (z)
(s0 )
(s0 )
¯
Ik “c.c.”(z, ⌧denotes) =Hk⇣(w)
,
Ī
(z,
⌧
)
=
In the following,the=replacement
{T(z,
⌧
),
⇥(z,
⌧
)}
!
{
T̄(z,
⌧
),
⇥(z,
⌘k , Pkk (w) = Ik (w)⇣ ◆ Īk (w)⌧ )}.
⌘k , (3.14)(
Ik (w)+ Īk (w)
,
✓
(s0 )
(s0 )
1 1 + 12⌧ Is (z,n⌧ ) 1 1 2
n 1 1 + 2⌧ Īs (z, ⌧ )Fn (x)with= n Fn 1,,...,; , ,...,; x , (n 2 N {0}) . (3.3

n+1 n+1n+1 n nnwhich generalise (3.41).
¯(s0 ) (w) .
0)
22 + ⇥The level-k
deformed
currentsare
Ik (w)=
Tk+1(w) + ⇥
, Īk (w) =T̄k+1 (w)
(3.15)(sk 1 (w)k19) +– '(z̄,From (3.36), it follows that (z, ⌧ ) = '(z,–⌧✓¯✓ ⌧ ) and therefore T◆◆(z, ⌧ ) and T̄s+1 (z,s+1k0(s + 1)s+1

±R

convenzione
conj +-, =dirloanchei a parolee i±2⇡n, (i: = 1, 2, . . . , N ) ,2 discussed in2 [13, 21], taught us that, at least formally, (+)
JT̄ examples
the
j=1
I (R)
I ( ) (R)ns for the quantized spectra already emerge at classical level afterI(z) replacing!
, Ī(z) !
.
R
R
5.1
Thethescattering
phase
sitiestheir averagevalueAnsatz
overvolumeequations
R RT:
introdottoqui lathewithasymptotic
Bethe
(BA)
for our
Implementing (3.44) in (3.42)
gives models. The two body scatt+-, dirlo anche a parole :plitudes involving rightandcaseleft movers
are:(±)
In theof) TT̄s -deformed
CFT, the results
0 ,±)Rk Ik of(R)section 3.3,
(+)
(
(s
I (R)
I (R)
Ik
(R, ⌧ ) = ⇣⌘k , mod
0
I(z)
!
,
Ī(z)
!
. to (5.8)(3.44)
0 ,±))
additional
phase
factors
describing
thequantisationofthewith s =(ss0 ) < 0. The
only
consistentsolutions
are
(s
0
m̂
(s )
R0
R + 02⌧ Is(R, ⌧ )
0R±(✓ s0 ✓ 0 )
0
(✓,
✓
)
=
0
,
(✓,
✓
)
=
±⌧
ˆ
e
,
(s
=
s
<
0) .
(
s the holomorphic and anti-Holomor(±,⌥)gives(±,±) of (3.45), where
tum Nversion(±)44) in (3.42)
2
X whichcoincides(±)(±) with the CFT quantum result (5.80) of section 5.3.
m̂ ±✓(±)m̂
(±)±(✓i thes0 interaction.
✓j )togetherbylevel-kHamiltoniani±R 0 e± ⌧ ˆs0Rk I (±)
e (R)= ±2⇡ni Then
, (i the,= 1, 2,. . . , N (±)),.
(5.9)an(s2,±)
milarly, Istartingfromfind: the, algebraic equations (3.45)2(3.49)•kwe
Although(3.45) were derived for s < 0 case, it
(R,
⌧
)
=
⇣
⌘k
j=1
k
factoriseas0(s ,±))
R + 2⌧ Is jecture(R, ⌧that) they can be extended also to s 0. From the reflection pThesame
relationscanbeobtained
NLIE
introducing
the
phase
factor:The twom̂(s) are the(s) byequations
0 asymptotic±(✓froms ✓ 0the)0Equations (5.9)
Bethe
Ansatz(BA)forourmodels.
0
0
find:
(s )
(s
,+)
(s0 , ) (
(✓,
✓
)
=
±⌧
ˆ
e
,
(✓,
✓
)
=
0
,
(s
=
s
>
0)
.
s
(±)
(±,⌥)
(±,±)
k
with
CFT quantum
result (5.80)
Hk movers(z,(s,±)⌧ )are:= Ik(z,R⌧ )Ik⌦ (R)I + I ⌦ Ik(
2 of section
bodythescattering
amplitudes
involvingright 5.3.
and leftI(R, ⌧ ) =0 ⇣⌘k , (s0 , )(s0 ) k(s ,+)(s,⌥)
equations(3.45) were
derivedfor s <strongly
0 case,it issupport
naturaltoconR(z,+that2⌧I¯s⌦the⌧I)⌦ Ik th
P
(z,
⌧
)
=
I
⌧
)
I(R,classical0)0 )section,
eebraic
results
in
this
the
idea
m̂
0
0
(spresented
(s
k
k
0
0
±(✓ s ✓ )0✓ ) =to0s, 0.(±,±)(✓, the✓)=±⌧ ˆproperty, we(s = s < 0) .
(5.10)
s0 e
can be extended
From
reflection(3.26),(±,⌥) (✓, also
2
which
again
match
the
s
>
0
quantum
resultquotedin(5.86).
roduced in this paper through a field dependent change of coordinates, can be consist

e

i

±⌧

ˆs0

and
their
actionon genericmulti-particle
state(±) find:
kS-matrix
Similarly,
startingfrom(3.46)
weantised within
the
exact
approachthroughtheintroduction
specific•Itisstraightforward
to
check
that,
with
the
replacementof(3.44),(3.42)Lo
lea
R
I
(R)
(s,±)
kto massive field theories is:Thenaturalgeneralisation
Ik(R, ⌧ ) = ⇣,(3.46)⌘kthegeneralised
Burgers-type(5.45)–(5.48),
holdalso fo(s,⌥)
aking phase(s)factors.0 R +Thenaturalof0equations
andfor a(+)
massive(+) (4.11)(+) which
¯m̂
(+)
( ±(✓
)⌧ ) generalisation(+)(s)( (4.10)
)
0)
2⌧
I
(R,
s
✓
s
|N±⌧ , In
N|N
iAppendix,|N✓ ) in|✓,s✓>
✓N(5.11)i⌧ ⌦ |✓⌧ ⌦(✓,(✓, ✓ ) =
ˆsfact,e i⌧as=, in(±,±)=ithe0⌧ ,=CFT
(s1=
shown
case
equations
2 0), .. . . , (5.45)–(5.48)(±,⌥)
ory is:2ch the s > 0 quantum result quoted(s)in (5.86).
thesimpler
0 form0

9 ((✓, ✓strongly) = ⌧ msupport
sinh(✓s ✓(s,⌥)) that, limit,sin
The results presented in
section,the
idea
thedeterminedclassical
theories
is, thisin theasymptotically
the
R
!
1
by
(s,±)
(s,±)
@⌧leads
Ik(R,) = 2Is of (R, ⌧ ) @R Ik(R, ⌧ ) , (s > 0)
ard to check that, with the replacement (3.44), (3.42)
to a⌧solution
ntroduced in this paper through a field dependent(s0changeof coordinates,
canbe consistently
01,±)(s0 ,±)(s0 ,±)
hurgers-type
asymptotic
BA equations
equations(5.45)–(5.48), which hold also
formassivemodels.
@ ⌧ Ik 0(R, ⌧ ) = 2Is0
(R, ⌧ ) @R Ik0 N (±)(R, ⌧ ) , (s0 < 0quantised within the exact S-matrix approach with0 introduction of specific Lorentz
breakingX(±)ˆ
in Appendix, in the CFT case equations(s)(5.45)–(5.48)
canberecast
into
k
(±)
±k✓
(s,+) (s ,±)
(s)
(s,
)
@ theory
A |N
i
N
Iand
imassive
e is:with Ik =ofIk(5.10)
I¯k (R,= I⌧k)|N. a23
phase factors. The natural X
generalisation(5.11)
for
field
⌧ =
k and

R
m sinh(✓i ) +
⌧m s
(s,±)
(s,⌥)

sinh(✓
(s,±)
(s)
0i

s ✓j ) = 2⇡ni ,

2
(n
2
Z
,
i
=
1, 2,i=1. . . , N) .
0 i(

For simplicity, we will consider the sets (2.14)–(2.15) separately. Following the strategy described in the previous section, we shall first derive the diﬀerential map, which means to¯ )T , 8f : R2 ! R. Setting f = in the second exexpress (@w f, @w̄ f )T in terms of (@f, @f
pression of (2.7), we first write (@w , @w̄ )T as a function of (@ , @¯ )T by solving the set ofalgebraic equations
8
!
!
@w
<@ =
@
@w
1 ⌧ (@w )2=J!
,(2.17)
@w̄
:@¯ =
@¯
@w̄
2
1 ⌧ (@w̄ )where we used

T2 (w) =1(@w )2 ,2(2.18)⇥0 (w) = 0 .

The solution to (2.17) is@w = @
where we defined

1
4⌧

✓

1+S
@¯

◆2

@¯ ,

@w̄ =

1
4⌧

q
S = 1 + 4⌧ @ @¯ .

✓

1+S
@

◆2

@ + @¯ ,Now, plugging (2.19) into the second expression of (2.7), we find the24diﬀerential map
✓
◆2
✓
◆2
1
1+S
1
1+S

(2.19)(2.20)

25

THANK YOU !

26