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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 949 (2019) 114798

www.elsevier.com/locate/nuclphysb

Integrable 2d sigma models: Quantum corrections to

geometry from RG ﬂow

Ben Hoare

, Nat Levine

b,∗

, Arkady A. Tseytlin

b,1

ETH Institut für Theoretische Physik, ETH Zürich, Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland

Blackett Laboratory, Imperial College, London SW7 2AZ, UK

Received 22

July 2019; accepted 9 October 2019

Available

online 11 October 2019

Editor: Stephan

Stieberger

Abstract

Classically

integrable σ -models are known to be solutions of the 1-loop RG equations, or “Ricci ﬂow”,

with only a few couplings running. In some of the simplest examples of integrable deformations we ﬁnd

that in order to preserve this property at 2 (and higher) loops the classical σ -model should be corrected

by quantum counterterms. The pattern is similar to that of effective σ-models associated to gauged WZW

theories. We consider in detail the examples of the η-deformation of S

(“sausage model”) and H

, as

well as the closely related λ-deformation of the SO(1, 2)/SO(2) coset. We also point out that similar

counterterms are required in order for non-abelian duality to commute with RG ﬂow beyond the 1-loop

order.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

Corresponding author.

E-mail

addresses: bhoare@ethz.ch (B. Hoare), n.levine17@imperial.ac.uk (N. Levine), tseytlin@imperial.ac.uk

(A.A. Tseytlin).

Also at the Institute of Theoretical and Mathematical Physics, MSU and Lebedev Institute, Moscow.

https://doi.org/10.1016/j.nuclphysb.2019.114798

The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

2 B. Hoare et al. / Nuclear Physics B 949 (2019) 114798

1. Introduction

The bosonic 2d σ -model

S =

4π



(x) ∂

∂

(1.1)

is a theory with an inﬁnite number of couplings (as can be seen, e.g., by Taylor expanding

(x) = δ



mn;k

···k

···x

) that run with RG scale according to the “generalized

Ricci ﬂow” equation [1,2]

=β

+∇

, (1.2)

mp q r

pqr

+... . (1.3)

Here ξ

(t) corresponds RG scale dependent diffeomorphisms, or equivalently ﬁeld redeﬁnitions

of x

In certain special cases, the RG ﬂow may be consistently restricted to a ﬁnite subset of cou-

plings. These σ -models may be called renormalizable in the usual sense. In the simplest case of

homogeneous spaces, for example group or symmetric spaces, with non-abelian global symme-

try, there is just one running coupling related to the curv

ature radius. More non-trivial examples

include models related by T-duality, for which the global symmetry becomes partly hidden. Start-

ing, e.g., with the σ -model on S

(of radius

√

h) and T-dualising along the isometry direction φ

one ﬁnds a dual metric

=h(dθ

+sin

θdφ

) →



=h(dθ

sin

) (1.4)

that no longer has a global SO(3) symmetry. Instead, this symmetry is hidden as a consequence

of integrability [3] and the dual metric still solves the 1-loop RG equations with only one coupling

h(t).

More generally

, it is now widely believed that renormalizability, or invariance under the RG

ﬂow, is closely linked with the integrability of a σ -model [4–6] (for some recent examples

see, e.g., [7–10]). One motivation for this, implicit in [4,5], is that a σ -model corresponding

to a gi

ven a quantum integrable S-matrix with a ﬁnite number of parameters should also be

parametrized by only a ﬁnite number of couplings. Another is that the conservation of inﬁnitely

many hidden symmetry charges should be enough to reduce the inﬁnite-dimensional RG ﬂow to

a ﬁnite-dimensional subspace.

While there are no kno

wn counterexamples to the conjecture that classically integrable

σ -models should be invariant under RG ﬂow, in most examples this has only been checked at

the leading 1-loop order in (1.2), (1.3), and there is no general proof.

The aim of the present paper is to address what happens be

yond the leading 1-loop order in

some of the simplest non-trivial examples of integrable σ -models with 2d target spaces without

non-abelian global symmetry. We shall see that the classical σ -model metric should generally

be deformed in order to remain a solution of the higher

-loop RG equation (1.1) with the same

number of running couplings. This order-by-order deformation of the metric may be interpreted

as a result of adding local “counterterms” required to preserve the integrability at the quantum

Here we ignore the B-ﬁeld coupling for simplicity and absorb the loop-counting parameter

h = α



into the target

space metric G

B. Hoare et al. / Nuclear Physics B 949 (2019) 114798 3

level (i.e. to satisfy “Ward identities” for hidden symmetries). Indeed, in cases with a UV ﬁxed

point and a known underlying quantum S-matrix, the deformed σ -model may be reconstructable

from a dual massive model (cf. [5,11]).

One e

xample where such a deformation is expected is the model deﬁned by



in (1.4), which

is T-dual to the σ -model on S

. Given the metric ds

= h[dx

+ M(x)dy

] the standard

T-duality rule

M = M

−1

is known to be modiﬁed at the 2-loop level [12](see also [13–15])

M = M

−1



1 +

−1

∂

log M ∂

log M



, (1.5)

from which one can straightforwardly determine the corresponding quantum correction to the

dual metric in (1.4).

Another known example of such quantum corrections is provided by the special integrable

σ -models corresponding to gauged WZW models. These models are scale invariant, i.e. ﬁxed-

point solutions of the RG equation (1.2). The conformal invariance of the underlying quan-

tum gWZW theory allows one to ﬁnd an exact form of the σ -model metric (and B-ﬁeld)

[16–20].

The simplest example is provided by the exact counterpart [16]of the familiar classical

SL(2, R)/U(1) gWZW metric [24,25]

=k(dr

+tanh

rdy

) → ds

=(k − 2)





coth

r −



−1



. (1.6)

As was checked directly in [26], the leading k

−1

correction in (1.6)is precisely the one required

to solve the 2-loop scale invariance equation implied by (1.2), (1.3)(see also [27]for a 4-loop

test of (1.6)).

The quantum deformation of the effective σ -model metric associated with gWZW models is

also essential for quantum integrability of generalized sine-Gordon models [28]. In particular,

as was shown in [29]the ﬁnite counterterm required to be added to the action of the complex

sine-Gordon model L = k



(∂x)

+ tan

x(∂y)

− m

sin



to ensure factorization of the cor-

responding S-matrix at the 1-loop level [30,31]is precisely the same as the k

−1

term in the

SU(2)/U (1) analog of the exact metric in (1.6).

One wa

y to understand the origin of the quantum correction in (1.6)is to note that integrating

out the 2d gauge ﬁeld A

in the gWZW model leads to a non-trivial determinant that can be

computed exactly [12,32](see also [33])



[dA]e



√

gMA

=exp



4π



√



−

∂

log M∂

log M −

(2)

log M





. (1.7)

In (1.4)we have x = θ and M = sin

θ so that (1.5) implies csc

θ →csc

θ(1 + 2h

−1

cot

θ +O(h

−2

)).

This is true in a particular scheme in which the “tachyon” equation is not modiﬁed (for a discussion and examples

see [21–23]). Here the level k plays the role of the inverse coupling (and loop-counting parameter) h used above.

In this Weyl invariant case the vector ξ

can be written as 2∂

 where the exact dilaton ﬁeld  is given by e

−2

sinh 2r



coth

r −



1/2

Here the integral is understood to be over 2d scalars u, v deﬁned by A

= ∂

u, A

−

= ∂

−

v and having measures

M = M(x(z)). For generality we assumed a curved 2d background with curvature R

(2)

(ignoring trivial M -independent

factor in (1.7)). Note that in the discussion of the abelian T-duality transformation at the path integral level in [34]the

∂

log M∂

log M term was missed but, in fact, it cancels against similar contribution coming from another determi-

nant

involved [32]so that the end result at the leading order is just the dilaton shift found in [34]. The analog of the

∂

log M∂

log M term survives, however, in the case of non-abelian duality (see below).

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