ЖЭТФ, 2019, том 156, вып. 4 (10), стр. 667-670

CLASSICAL AND QUANTUM INTEGRABLE SIGMA MODELS.

RICCI FLOW, “NICE DUALITY” AND PERTURBED RATIONAL

CONFORMAL FIELD THEORIES

V. Fateev^*

Université Montpellier 2, Laboratoire Charles Coulomb, UMR 5221

F-34095, Montpellier, France

Landau Institute for Theoretical Physics

142432, Chernogolovka, Moscow Region, Russia

Received March 10, 2019,

revised version March 10, 2019

Accepted for publication March 12, 2019

Contribution for the JETP special issue in honor of I. M. Khalatnikov’s 100th anniversary

DOI: 10.1134/S0044451019100080

terious and needs to be further analyzed. Such analysis

crucially simplifies for two-dimensional integrable rela-

tivistic systems. These theories besides the Lagrangian

Duality plays an important role in the analysis of

formulation possess also an unambiguous definition in

statistical, quantum field, and string theory systems.

terms of factorized scattering theory, which contains

Usually, it maps a weak coupling region of one theory

all information about off- shell data of quantum the-

to the strong coupling region of the other and makes it

ory. These data allow the use of non-perturbative

possible to use perturbative, semiclassical, and renor-

methods for the calculation of observables in integrable

malization group methods in different regions of the

field theories. The comparison of the observables cal-

coupling constant. For example, the well known du-

culated from the scattering data and from the pertur-

ality between Sine-Gordon and massive Thirring mo-

bative, semiclassical or renormalization group analysis

dels [1, 2] together with integrability plays an impor-

based on the Lagrangian formulation makes it possible

tant role for the justification of exact scattering matrix

in some cases to justify the existence of two different

[3] in these theories. Another well known example of

(dual) Lagrangian representations of a quantum theory.

the duality in two dimensional integrable systems is the

weak-strong coupling flow from affine Toda theories to

The two particle factorized scattering matrix is a

the same theories with dual affine Lie algebra [4-6].

rather rigid object. It is constrained by the global sym-

The phenomenon of electric-magnetic duality in four

metries, factorization equation and unitarity and cross-

dimensional N = 4 supersymmetric gauge theories con-

ing symmetry relations. After solving of these equa-

jectured in [7,8] and developed for N = 2 theories in [9]

tions the scattering matrix S can contain one (or more)

(and in many subsequent papers) opens the possibility

free parameter. At some value of this parameter λ = λ₀

for the non-perturbative analysis of the spectrum and

the scattering matrix S (λ₀) becomes the identity ma-

phase structure in supersymmetric gauge field theories.

trix and has a regular expansion around this point. In

The remarkable field/string duality [10,11] leads to the

many cases this expansion can be associated with the

unification of the ideas and methods for the analysis of

perturbative expansion of some Lagrangian theory with

these seemingly different quantum systems.

parameter b near some free point. Sometimes there is

a second point λ = λ₁ where S (λ) reduces to identity

While known for many years the phenomenon of

matrix and admits a regular expansion in (λ - λ₁). If

duality in quantum field theory still looks rather mys-

this expansion can be associated with the perturbative

expansion of another local Lagrangian at small coupling

^* E-mail: vladfateev@gmail.com

γ = γ(b), then the two different Lagrangians describe

667

V. Fateev

ЖЭТФ, том 156, вып. 4 (10), 2019

the same theory, which has two different (dual) pertur-

axion, linearly coupled with the density of topological

bative regimes.

charge), this SM becomes integrable and its deformed

A more interesting situation occurs when S (λ) has

version has the “nice” duality property. We study this

a regular expansion in (λ - λ₀) which agrees with per-

theory in the main part of this paper. Here, we say a

turbative expansion in b of some field theory with local

few words about non-integrable CP (n - 1) SMs.

action A(b), but at the point λ₁ the S-matrix tends

These models were intensively studied during 70-80s

to some “rational” scattering matrix corresponding to

due to their’s similarity with four-dimensional SU(n)

the S-matrix of a non-linear sigma model on a sym-

gauge QFTs. Namely, the CP (n - 1) SMs and SU(n)

metric space. Near the point λ₁ it can be considered as

gauge theories are asymptotically free, possess instan-

a deformation of a symmetric scattering. In this case

tons, and manifest the phenomenon of confinement¹⁾.

it is natural to search for the dual theory as sigma

It is natural that CP (n - 1) SMs served as baby-

model with target space looking as a deformed sym-

laboratory for the analysis of SU(n) gauge theory, in

metric space. The metric and other characteristics of

particular, for analysis of instanton contributions [19]

sigma model on the manifold is subject to very rigid

and lattice simulations [20].

conditions, namely non-linear renormalization group

The spectrum of the CP (n - 1) SMs in 1/n ap-

(RG) equations [12]. If one has found the solution of

proach was studied in [21]. It was shown that besides

RG equations which gives the observables in the sigma

the basic particles, which are the only particles in the

model theory, coinciding with those derived from the

integrable version of this model, one has also the parti-

factorized S-matrix theory one can conclude that field

cles which are their bound states, confined by Coulomb

theory with the action A (b) is dual to a sigma model

forces. The addition of fermion (axion) produces an

on the deformed symmetric space. The short distance

essential restructure of the spectrum. Of course, the

behaviour of such theory can be studied by RG and

influence of the axion on the spectrum of gauge the-

conformal field theory (CFT) methods. The agreement

ory is a more interesting and much more complicated

of the CFT data, derived from the action A(b) (consi-

problem.

dered as a perturbed CFT) with the data derived from

The spectrum of the deformed non-integrable

RG for sigma model gives an additional important test

CP (n - 1) SMs seems to be qualitatively the same as

for the duality (”nice” duality).

that of the undeformed models and can not be studied

The analysis of integrable quantum SMs on the de-

by perturbative methods. Due to the “nice” duality,

formed symmetric spaces and their dualities started in

in the integrable CP (n - 1) SMs with axion there

the papers [13-15]. Later in the papers [16, 17] the ge-

is a weak coupling region. In this region the basic

neral classical SMs on the deformed groups and cosets

particles also form the bound state, which disappears

manifolds have been constructed. Unfortunately, not

from the spectrum outside the perturbative region. It

all these SMs, integrable classically, are integrable in

is possible however that in a non-integrable QFT they

quantum case. In particular it happens for the cosets

survive in the strong coupling (SM) regime. We hope

having U (1) group in denominator (see for example

to return to this problem in a future publication.

[18]). In many cases this situation can be improved by

This main paper (see JETP 129, № 10) is organized

introduction of additional quantum degrees of freedom,

as follows. In section 2, we describe the basic CFTs,

which are invisible in the classical limit.

which can be formulated in terms of 2n - 1 bosonic

We say that an integrable SM has the “nice” dua-

fields, and their primary fields are the exponents of

lity if the dual integrable QFT has the weak coupling

these fields. We calculate the reflection amplitudes in

region. This implies that we can study this theory by

these CFTs which are important for the calculation of

different methods (perturbation theory, RG and CFT

UV asymptotics in perturbed CFTs. These amplitudes

analysis) in different regimes. Note that the SMs with

serve also for identification of CFTs in different repre-

the property of “nice” duality form a very small sub-

sentations. In particular, for justification of dual SM

space in the space of all integrable quantum SMs on

representations.

the deformed symmetric spaces and such SMs with ad-

ditional quantum degrees freedom.

In section 3, we explain the general properties of

deformed CP (n - 1) SMs with fermion and write the

A simple (but rather non-trivial) example is pro-

action of perturbed CFTs, constructed in section 2. We

vided by the CP (n - 1) SM. This model is integrable

classically but non-integrable (for n > 2) at the quan-

tum level. After adding the massless fermion inter-

¹⁾ For n > 2, both theories have non-topological classical so-

acting with U (1) gauge field on CP (n - 1) (massless

lutions, the role of which is not clear at the moment.

668

ЖЭТФ, том 156, вып. 4 (10), 2019

Classical and quantum integrable sigma models. ..

conjecture that these QFTs provide a dual description

with topological parameter and integrable deformed

of deformed CP (n - 1) SMs with fermion.

CP (n - 1) with fermion (axion). We consider more

In section 4, we represent the action of dual QFT

general CFTs represented by the cosets G_m × G_l/Gm+l

in the form suitable for the perturbation theory in pa-

and study their deformations by different fields in

rameter b. We provide non-local integrals of motion

different regions of integers m, l, and h (Coxeter num-

which form the Borel subalgebra of SU(n)_q and gener-

ber of G). We study their RG propertied and show

ate SU(n)_q symmetry of the scattering theory. We de-

that these QFTs provide an independent description

scribe the spectrum and scattering theory of this QFT.

of a large variety of SMs on the deformed symmetric

In section 5, we use the Bethe Ansatz approach to

spaces. Finally, we give a simple inequality which

derive the exact relations between the parameters of

provides a necessary condition for a sigma model to

action and scattering theory in this QFT. We calculate

have “nice” duality.

the observables, which can be compared with the ob-

servables calculated using the dual SM description of

The full text of this paper is published in the English

our QFT.

version of JETP.

In section 6, we consider classical and quantum inte-

grable SMs on the deformed symmetric spaces. We dis-

cuss Ricci flows in these SMs and the relation between

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