On dual description of the deformed O(N) sigma model

We study dual strong coupling description of integrability-preserving deformation of the O(N) sigma model. Dual theory is described by a coupled theory of Dirac fermions with four-fermion interaction and bosonic fields with exponential interactions. We claim that both theories share the same integrable structure and coincide as quantum field theories. We construct a solution of Ricci flow equation which behaves in the UV as a free theory perturbed by graviton operators and show that it coincides with the metric of the η-deformed O(N) sigma-model after T -duality transformation.


Introduction
Studying quantum field theories in the strong coupling regime is one of the most important problems of modern theoretical physics. In general, there are no methods for doing this, but a new tool arises when the quantum system admits the so called weak/strong coupling duality. In this case, the theory admits two perturbative expansions with coupling constants being inversely related. Typically, these descriptions use microscopic degrees of freedom and actions of completely different nature. One of the most notable examples of the duality, known as bosonization in two spacetime dimensions, was pioneered by Coleman and Mandelstam [1,2]. One part of the duality is described by a massive Dirac fermion with four-fermion interaction, while the other by a scalar field with the cosine potential, the celebrated sine-Gordon model. Both theories were shown to coincide at the level of perturbation theory in the mass parameter. Other examples of dualities usually mentioned in the literature include electric-magnetic dualities in supersymmetric gauge theories, various string dualities, AdS/CFT duality etc.
The main purpose of this paper is to motivate a weak/strong coupling duality between two integrable quantum field theories in two space-time dimensions. The former theory is the so called η-deformed O(N ) non-linear sigma model [3,4]. The latter is effectively described by the coupled theory of Dirac fermions with four-fermion interaction (of Thirring type) and bosonic fields with exponential interaction (of Toda type). This theory has been recently introduced in [5]. Strong coupling regime of the sigma-model coincides with the JHEP11(2018)139 perturbative regime of the Toda-Thirring model and vice versa. Below we formulate the results in more details.
We will treat the O(N ) sigma model via the coset construction, as a gauging of the two-dimensional principal chiral field model (PCF). The PCF is the non-linear sigma-model whose target space is a group manifold G. The action of the PCF model is where g is a map from the 2D space-time to the simple Lie group G and ∂ ± are the light-cone derivatives. This theory has a global G L × G R symmetry which acts as Moreover, the theory (1.1) is known to be classically integrable [6]. The simplest way to show the integrability is to notice that the left Noether current J ± = g∂ ± g −1 is both conserved and flat ∂ + J − + ∂ − J + = 0, These two equations can be rewritten as a compatibility condition known as a zero curvature representation [6] [D + , D − ] = 0, where D ± = ∂ ± − J ± 1 ± λ , (1.4) and λ is an arbitrary complex number (spectral parameter). As a simple consequence of the zero curvature representation (1.4), the Wilson loop built out of this connection generates an infinite tower of conserved quantities, the Integrals of Motion. This infinite set of quantities can be used to solve the model by means of the inverse scattering method. It has been argued in [7] and subsequently passed many consistency checks that the theory defined by the action (1.1) is integrable at the quantum level. Quantum systems possessing this property are usually handled within the framework of the Quantum Inverse Scattering Method [8]. However, this method is known to fail when applied to integrable non-linear sigma models directly. Only recently a considerable progress has been achieved in this direction (see [9] and discussions therein). Now, let H be the Lie subgroup of G and h be the corresponding Lie algebra, such that the quotient manifold is a symmetric space. We can define the sigma-model on this symmetric space by gauging the left symmetry of the action (1.1) (1.5) Clearly, this procedure breaks the left symmetry, but preserves the right one. It can be shown that the model is still enjoys the zero curvature representation and hence possesses the integrability property. In this paper we consider the case of O(N ) sigma-model, i.e. we take G = SO(N ) and H = SO(N − 1). Integrability of this sigma-model at the quantum level has been first demonstrated by Polyakov [10]. As QFT O(N ) sigma-model corresponds to an asymptotically free theory with a dynamically generated mass scale. It describes scattering of N mesons in the vector representation of the global O(N ) group [11,12]. The

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scattering matrix for these mesons is strongly constrained by integrability, which implies, in particular, the absence of particle production and factorization of the multi-particles amplitudes into the product of the two-particle scattering. These requirements plus the conditions of crossing invariance and unitarity are so strong that allow one to compute the S-matrix exactly. The two-particle S-matrix for the O(N ) sigma-model has been found by Alexander and Alexey Zamolodchikov in their seminal paper [13]. It has an explicit form , and θ = θ 1 − θ 2 is the rapidity difference of the incoming states.
The S-matrix given above is known as the rational solution to the Yang-Baxter equation. It means that all matrix elements are rational functions of the rapidity (except for the prefactor). These solutions are known to be associated with the certain class of infinite dimensional algebras called the Yangians, in our case the Yangian of O(N ). It is known that the Yangians always admit one parametric deformation called the quantum affine group. The corresponding solution to the Yang-Baxter equation is the trigonometric solution, i.e. the matrix elements are expressed in terms of trigonometric functions. For example, the trigonometric deformation of the O(3) symmetric S-matrix has been found in [14]. It describes the scattering of two charged and one neutral particles (A ± , A 0 ). Explicitly, this S-matrix has the form (here λ is the deformation parameter) S ++ ++ (θ), S 00 00 (θ) = S +0 +0 (θ)+S +− −+ (θ), with all other matrix elements related by CPT-symmetry and crossing invariances. In the limit λ → 0 one recovers the rational S-matrix (1.6) for N = 3 in the basis (A + +iA − ) , A 0 . The generic O(N ) trigonomentric solution to the Yang Baxter equation has been constructed in [15,16]. It has the form very similar to (1.7). It is important that this S-matrix similarly to (1.7) depends on a continuous parameter λ such that in the limit λ → 0 one recovers Zamolodchikov's S-matrix of the O(N ) model (1.6).
It is tempting to find an appropriate deformation of the O(N ) sigma-model such that the scattering in the deformed theory is described by the trigonometric O(N ) S-matrix. This deformation, if exists, must correspond to the renormalizable quantum field theory. The one-loop renormalizability of the non-linear sigma model 1

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requires the target space metric G µν = G µν (X) to satisfy the Ricci flow equations [17] R µν = −Ġ µν , (1.9) where the derivative is taken with respect to the RG time t the logarithm of the scale. The solutions of this non-linear evolution equation corresponding to the local quantum field theory must be UV stable, i.e. has to have a smooth limit as t → −∞. In general, if one starts with an arbitrarily chosen metric at some intermediate scale t 0 , the solution will blow up before reaching the UV region. The existence of UV stable solutions is a rather nontrivial fact and there are only a few explicitly known examples satisfying this property. Most of them correspond to Einstein manifolds such as round sphere S N −1 in the case of O(N ) sigma-model. One of the exceptions is the one-parametric deformation of the round 2-sphere known as the "sausage" metric [18] Here φ is the U(1) isometry coordinate φ ∈ [0, 2π] and ζ ∈ [−1, 1] is the longitudinal coordinate along the sausage. The parameter κ is the running coupling constant κ = − tanh νt, (1.11) and ν is the deformation parameter. As ν goes to zero the metric (1.10) becomes that of the round 2-sphere. For generic ν the metric (1.10) can be visualized as an embedding of the sausage of length L = − √ νt and circumference r = κ ν into three-dimensional spacetime. In the UV limit t → −∞ the sausage looks like an infinite cylinder corresponding to the asymptotically free theory. Contrary, in the limit t → 0 the sausage behaves like a shrinking sphere, where the theory becomes strongly interacting and the perturbative expansion is no longer valid. It was conjectured and checked in [18] that the non-linear sigma model defined by the one-loop metric (1.10) provides an integrable deformation of the O(3) sigma model with the trigonometric S-matrix (1.7). The deformation parameter λ of the S-matrix is related to the parameter ν as (1.12) It is believed that the full theory including all higher loop corrections could be consistently extended starting from by the one-loop metric (1.10) in such a way that the resulting quantum field theory is integrable and described by the trigonometric S-matrix (1.7). Classical integrability of the sausage sigma model and its three-dimensional cousin has been demonstrated by Lukyanov [19]. As it became clear recently, the sausage sigma model belongs to a more general class of the so called η-deformed sigma models. It's all started with the seminal Klimčík's paper [3] where he suggested one-parametric deformation of the PCF action (1.1) preserving the integrability property. The deformation is done with the help of the linear skew-symmetric operator R on the complexified Lie algebra g of the group G which satisfies the modified Yang-Baxter relation

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With the operator R at hand the deformed action PCF is defined as where η is the deformation parameter, at η = 0 we return to the original action (1.1). For η = 0 the global symmetry group is broken down to G L × N , where N is the subgroup of G R which commutes with the operator R. In the virtue of the relation (1.13) the left Noether current is flat and hence the deformed theory admits the zero curvature representation leading to the classical integrability. We note, that since the left global symmetry G L is unaffected one can still apply the coset construction. As it was shown in [4] the resulting theory is integrable provided the quotient space G/H is a symmetric space. The action of the η-deformed coset sigma model can be written in the form [4] where R g = Ad g • R • Ad g −1 and P c is the projection on the coset space. So far we have not specified an explicit form of the operator R satisfying the bilinear relation (1.13). The common choice which has been widely elaborated in the literature since [3] requires the standard decomposition of the Lie algebra g g = c ⊕ α>0 g α ⊕ α>0 g −α , (1.17) with the operator R defined by Below, it will be convenient to work with the action (1.16) in an explicit coordinate system. We will use the standard basis for SO(N ) algebra and set T 12 , T 34 , T 56 , . . . to be the Cartan subalgebra. The SO(N − 1) subalgebra is spanned by the generators T ab with a, b = 1. First non-trivial case corresponds to the SO(3)/SO(2) coset. We use the following parametrization for the coset representative g 2 . Exactly at the value λ = 1 2 the model reduces to three non-interacting particles (A 0 , A + , A − ), which can be effectively described by free scalar Φ and Dirac fermion ψ fields of the same mass. For the interacting theory, as can be shown order by order, the perturbative expansion of the S-matrix (1.7) coincides with the one obtained from the Langrangian 2 (1.21) provided that the parameter λ from the S-matrix (1.7) and the parameter b from the Lagrangian (1.21) are related according to λ = 1 2(1+b 2 ) . The existence of the dual description of the sausage sigma model is a rather miraculous phenomenon. In this paper, we partially shed light on the origin of this relation. The short explanation: this is the integrability who stands behind the duality. More precisely, in some limit both theories can be shown to share the same integrable structure and the integrability property is so strong that the theories stay equivalent even beyond the limit. Here we should stress that the Lagrangian (1.21) is exact in the parameter b, meaning that in the perturbative region b → 0 or λ → 1 2 it describes the full theory. In the strong coupling regime b → ∞ the Lagrangian (1.21) is certainly useless, but instead, we have a sigma model description with the metric (1.10), which, however, is accurate only up to one-loop order.
The outline of the paper is the following. In section 2 we introduce following [5] the generalization of the Lagrangian (1.21) depending on a continuous parameter b for arbitrary N . We treat this theory as a perturbed conformal field theory and find a part of an infinite conformal symmetry, a system of local Integrals of Motion, which survives the perturbation and has the O(N − 1) symmetry in the limit b → ∞. We study corresponding conformal field theory in section 3. In section 4, we study very peculiar property of the system under consideration which is the generalization of the b → 1 b symmetry of Toda theories. Namely, we use the fact that the same system of Integrals of Motion can be obtained from the perturbed CFT of sigma model type. This theory, however, is not UV finite and requires renomalization. Treating it in the vicinity of the trivial fixed point, we study the Ricci flow equations (1.9) with the prescribed UV asymptotic and find an exact solution for the metric. We show that this metric coincides with the metric for the η-deformed coset sigmamodel (1.16) after a coordinate change and T -duality transformations. In appendices we collect important facts and present details of some computations.

Dual Lagrangians
Let us start with known results and consider the theory defined by the Lagrangian density (1.21). The last term − m 2 8πb 2 sinh 2 (bΦ), (2.1)

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is of order m 2 and plays the role of the counter-term. It can be shown that with this choice the theory does not have any divergencies and becomes UV finite. One may use Coleman-Mandelstam boson-fermion duality [1,2] and rewrite the theory in the form The theory (2.3) is a special integrable perturbation of Sine-Liouville conformal field theory [21]. Namely, it means that it possesses infinitely many local Integrals of Motion of odd spins One can check this statement perturbatively in the mass parameter m where G 2k (∂ϕ, ∂Φ) is a holomorphic differential polynomial of degree 2k. For example, Higher-spin densities can be defined from the requirement that they commute with the perturbation Clearly, this equation is satisfied by I free 2 , but its validity for k > 1 is highly non-trivial. The integrability of the model is closely tied to the special combination of the fields in the exponents.
Motivated by the explicit form of the Lagrangian (2.3) we will study more general setting. Let ϕ = (ϕ 1 , . . . , ϕ N −1 ) be the N − 1 component bosonic field and consider the theory defined by the Lagrangian density where (α 1 , . . . , α N ) is a given set of vectors, 3 which is required to have maximal rank. We are only interested in the quantum field theories of this form with infinitely many Integrals of Motion. In the leading order in Λ this constrains the Integrals of Motion I free k to obey I free k , e (αr,ϕ) dz = 0, for all r = 1, . . . , N. (2.9)

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These equalities (see appendix A) can be treated as both, the equation for unknown Integrals of Motion I free k and for the set of vectors (α 1 , . . . , α N ). In fact, they are so strong, that the solution exists only if the Gram matrix Γ r,s = (α r , α s ) takes a very special form. We will be interested in the case where all roots α r are fermionic, meaning that (α r , α r ) = −1, for all r. (2.10) Moreover, we assume that only odd-spin Integrals of Motion are present: I free 2k = 0, because this is what we expect for the theory which has O(N ) symmetry in some limit. With these conditions specified one can find a series of solutions for every N ≥ 3 which depends on a continuous parameter b (see appendix A and [22] for more details). Since the cases N = 2n + 1 and N = 2n + 2 are different we consider them separately. Below, we restrict ourself to N > 4. Two cases N = 3 and N = 4 are special and we comment on them in the end of the section.
The Gram matrix (degenerate) of the vectors (α 1 , . . . , α N ) has to be of a very special form. It will be convenient to introduce Dynkin-like picture in order to represent it. We will use the following conventions while the crossed circles not connected by an edge correspond to orthogonal vectors.
The case N = 2n + 1. The root system corresponds to the diagram It is convenient to introduce Cartesian coordinates in R 2n : (E i , e i ), i = 1, . . . , n such that 12) and the vectors α r have the form where β = √ 1 + b 2 . It follows from (2.13) that (2.14)

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In accordance with this representation it is convenient to define Φ k = (ϕ, E k ) and φ k = (ϕ, e k ). We will study the theory in the region where b → 0. One has to add counterterms in order to regularize the UV behavior. 4 Using the representation (2.8) and the boson-fermion correspondence [1,2] the Lagrangian (2.8) can be rewritten as [5] where m = Λ/4π. The last term in (2.15) plays the role of the counter-term. We note that this part of the theory coincides with the affine B ∨ n Toda field theory (the Dynkin diagram has the orientation of the arrows opposite to B n ). One immediately sees that there are n pairs of charged particles ψ k ,ψ k and one scalar particle Φ n of the same mass M = m + O(b 2 ). In order to describe the rest of the spectrum one has to diagonalize the It is easy to check that which implies that the masses of the remaining (n − 1) particles have the form The case N = 2n + 2. The root system corresponds to the diagram

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In this case we introduce the Cartesian coordinates in R 2n+1 : E i , i = 1, . . . , n and e j , j = 1, . . . , n + 1 such that (2.20) and the vectors α r have the form (2.21) Using the boson-fermion correspondence [1,2] the Lagrangian (2.8) for the O(2n+2) model with n ≥ 1 can be rewritten as [5] We note that the purely bosonic part of the theory coincides with the affine C n Toda field theory. The spectrum consists of 2n+2 charged particlesψ k , ψ k of the mass M = m+O(b 2 ). The spectrum of Toda part of the theory consists of n particles with We conclude this section with a brief remark concerning O(3) and O(4) deformed sigma models and their dual descriptions. For O(3) case the dual Lagrangian is given by (1.21) which has, after bosonization, four exponents in the perturbation, not 3 as expected. This is a peculiar property of N = 3 which does not happen for N > 3. The case N = 4 is also special. Because of an exceptional isomorphism S 3 ≃ SU(2), the O(4) model can be treated equally as a coset sigma model or as a PCF model as well. According to Klimčík [23] PCF admits two-parametric integrable deformation so that in particular O(4) sigma model can be doubly deformed while preserving integrability. This deformation and its dual description has been found in [24]. 5 In these notes we consider O(4) model as a coset sigma model, which in fact coincides with equal parameter deformation α = β of the PCF.

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3 Conformal field theory, reflection operator The theory (2.8) can be viewed as a perturbation of the conformal field theory obtained from (2.8) by throwing away the last term e (α N ,ϕ) from the Lagrangian. It is convenient to couple the conformal field theory to the background metric. The action in the curved space-time has the form where g ab is the metric on a surface and R is its scalar curvature. The vectors α r in (3.1) are exactly the same as in (2.13) and (2.21). Clearly, they form a basis. The Gram matrix of these vectors depends on a continuous parameter b and according to our conventions corresponds to the graph The vector ρ in (3.1) has been chosen in such a way that the theory enjoys the conformal invariance. Explicitly it has the form whereα r is the dual basis: (α r ,α s ) = δ rs . This conformal field theory has the central charge (3.4) The corresponding conformal algebra is generated by the stress-energy tensor and the spin 4 field. Its explicit form is too complicated to be presented here. This algebra coincides with the chiral local algebra of the coset CFT [25] Consider the problem of computation of correlation functions in the theory (3.1). For simplicity, we assume the geometry of the two-sphere. Using the well-known trick [26] one can show that the n-point correlation function of the exponential fields V a 1 (ξ 1 ,ξ 1 ) . . . V an (ξ n ,ξ n ) , where V a = e (a,ϕ) , (3.6)

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being considered as a function of the total charge a = a 1 + · · · + a n has multiple poles at the values where m j 's are some non-negative integer numbers. The multiple residues at these poles are proportional to the free-field correlation functions Res V a 1 (ξ 1 ,ξ 1 ) . . . V an (ξ n ,ξ n ) where S j = e (α j ,ϕ(ξ,ξ)) d 2 ξ. The multiple integral in the r.h.s. in (3.8) converges in the domain Outside this domain this integral should be understood as an analytical continuation. A useful tool for analytical continuation is the well known identity for Coulomb integrals [27] D n (x) This relation holds for any n and m and for all −1 < p j < 0. The last condition is equivalent to the absence of singularity at infinity, which is always the case for correlation functions (3.8). It can be used for computation of many interesting Coulomb integrals appearing in CFT (see for example [28]). With the help of the relation (3.10) we can massage the free-field correlation function in the r.h.s. of (3.8). It is convenient to define operators R r which act on a space of free-field correlation functions as follows (3.13) We note that the operator R r is just an application of the integral identity (3.10) to the contribution of the fermionic root α r . It changes the roots α s →α s as well as shifts the charges of the exponential fields according to a → a + α r . (3.14) As explained in [22], the operator R r can be lifted to the operator, called fermionic reflection operator, acting on the total "off-shell" correlation functions. Therefore it serves as an isomorphism between different conformal field theories, corresponding to different root systems. In particular, it establishes an isomorphism between W -algebras in different realizations. Applying different R r 's one can reduce the number of fermionic roots. Actually, if α s (s = r) is a fermionic root then the transformed rootα s is a bosonic one (i.e. (α s ,α s ) = −1) and vice versa. It will be convenient to draw bosonic roots as circles an use conventions similar to the Dynkin ones. Namely, One can show, that starting from the root system corresponding to the diagram (2.19) and applying successively transformation R r one can reduce it to the diagram with one fermionic root and N − 2 bosonic ones. For generic value of N there are only two possibilities: • First, corresponds to the operator This transformation sends (2.19) to the coupled root system of D and A type α 1 +α 3 α 2 +α 3 α 4 +α 5 α 6 +α 7 α 8 +α 9 α 0 −b 2 1+b 2 α 3 +α 4 α 5 +α 6 (3.16)

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where α 0 = −(α 3 + · · · + α N −1 ), for N = 2n, −(α 3 + · · · + α N −2 ), for N = 2n + 1. (3.17) • Another corresponds to the operator and sends the system to the C and A coupled system The operator R r establishes the relation between different correlation functions and provides an isomorphism between W -algebras in different realizations. From the considerations above, it is natural to conjecture that the corresponding W -algebra coincides with W (D(n + 1|n)) for N = 2n + 2, W (D(n|n)) for N = 2n + 1, (3.21) where D(m|n) ∼ OSP (2m|2n) is the corresponding superalgebra. We note that the case N = 4 corresponds to the superalgebra D(2|1) which is exceptional since it admits one parametric deformation D(2|1) → D(2|1, α). On the sigma model side it corresponds to the sigma model with S 3 = SU(2) target space which is a group manifold and hence it admits a two-parametric integrable deformation due to Klimcik [23]. See ref. [29] for more details on this model, its dual description and relation to the superalgebra D(2|1, α).

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such that with some local V (1,2) k (z). This fact can be checked by explicit computation. We note that V 1,2 (w) is defined modulo total derivative, i.e. up to a shift α 1 → α 1 + ζβ 12 . For our theory the fields V i,j can be combined into two groups Using the coordinate representation (2.13) and (2.21) we find that the corresponding exponents have the form  At this point, it is useful to recall the sausage model. As it was discussed in the introduction, in the deep UV the sausage looks like a very long cylinder with ends on both sides. Therefore, it is natural to expect the metric to be only slightly perturbed from being flat if we are not too close to the edge of the sausage. A careful inspection of the operator (4.2) (together with its anti-holomorphic counterpart) suggests that exactly these operators are natural candidates for this perturbation from the flat metric. 6 The same reasoning makes it obvious that these operators only correspond to the first linear perturbation and one has to include additional corrections in order to describe the region near the edge of the sausage. Also, one could worry that higher-order curvature correction in the renormalization of the metrics will become important near the cups of the sausage. However, as we will see, these corrections are suppressed in the limit b → ∞.
Keeping in mind the considerations above, we consider the theory which might be the dual sigma model description of the original theory (2.8), (2.15), (2.22). By . . . we mean possible counterterms. We note that the theory (4.7) has P symmetry, but lacks C symmetry for N > 4. As we will see below, by performing the T -duality one

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can make the metric real, but introduce pure imaginary B-field, exactly as follows from the general action (1.16). Now we come to the important point. We note that the theory (4.7) is non-renormalizable in a strong sense. One has to add infinite number of counter terms to ensure renormalizability. Let us study this question in the one-loop approximation. The parameter b plays the role of the coupling constant in our theory. In order to probe the semiclassical region, we rescale ϕ = bX (4.8) and consider the limit b → ∞. Introducing standard notation α ′ = 2 b 2 , we rewrite (4.7) as a series in with the vectors a i and b ij being defined through the asymptotic The subleading terms in the expansion (4.10) has to be chosen to ensure the one-loop renormalizability of the theory. This condition leads to the RG group flow equation 7 where dot corresponds to the derivative with respect to the RG time t which is proportional to the log(Λ UV ). The function Ψ in this equation is more or less arbitrary and can be chosen at will. It describes the effect of possible RG-time dependent gradient diffeomorphisms. 8 It means that the solution to (4.12) is always a pair: the flowing metric and the coordinate frame. In this sense, Ψ is a gauge. There are at least two gauges used in the literature. First, is the so-called Hamilton gauge, in which Ψ = 0. Another is a Friedan's gauge, such that Ψ satisfied additional constraint where c 0 is a constant, which can always been set to zero by a linear shift of the function Ψ → Ψ − c 0 t. In this frame Ψ can be treated as a dilaton field. We find it convenient not to use any of these gauges, but rather properly adjust Ψ in order to reduce the non-linearity of the equation (4.12). Now we proceed to the problem of our interest. Namely, we are looking for the solution to (4.12) with the UV asymptotic (4.10) prescribed by the bare action (4.7). As we will see, one can use the gauge freedom to set the determinant of the metric to a constant. Below we will study in details two cases: N = 5 and N = 6. We will show that after a proper change of variables and T -dualities the corresponding solutions coincide with the ones following from the general action (1.16).

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In (4.20) the functions F k depend on x 1 and t only: F k = F k (x 1 , t). Moreover, equations (4.12) are compatible provided that The function F 7 is arbitrary and corresponds to the choice of gauge. We can choose it to ensure the additional relation With this choice the problem (4.12) has a unique solution satisfying the asymptotic condition (4.16). It is given by The other functions F k are obtained from the constraints (4.21) and (4.22). There is a choice of coordinates, such that the metric G µν has a particularly simple form. Namely, we perform a shift This shift diagonalizes the (1, 3) part of the metric. Moreover, it is convenient to introduce new coordinates ζ, θ, φ 1 , φ 2 and the function κ(t) by the following equations , tanh (4.25) In these coordinates the metric has the form (after rescaling ds 2 → 4νds 2 , t → 4νt + log 2) (4.26) The metric (4.26) can be found in [30]. It satisfies Ricci flow equation (4.12) with the function Ψ given by and κ = − tanh(3νt). Moreover (4.27) satisfies

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and hence Ψ can be regarded as a dilaton field. Now we perform the T-duality in the φ 2 isometry direction making the metric diagonal but generating a non-zero pure imaginary B-field In this form the metric and the B-field are exactly the same as follows from the general action (1.16) with the choice of coordinates in the coset as in appendix B. We note that the metric (4.29) and the B-field (4.30) satisfy where V µ and ω = ω µ dx µ are given by In the limit ν → 0 the metric (4.29) approaches the metric of the round four-sphere, while the B-field, the vector V µ and the one-form ω vanish.

Metric for the deformed O(6) model
The model in this case is five-dimensional and defined by the graph The vectors α r can be parameterized as follows (β = √ 1 + b 2 ) where (E 1 , E 2 , e 1 , e 2 , e 3 ) is an orthonormal basis in R 5 with coordinates (x 1 , x 2 , x 3 , x 4 , x 5 ).
In this case we are looking for the solution to the problem (4.12) in the form Then one can check that the solution stays within the following anzatz where the functions F k = F k (x 1 , t) do not depend on x 2 . The function F 7 in (4.37) is arbitrary. We use this freedom to set the determinant of the matrix G to a t-dependent constant. Now the solution is unique and explicitly given by (here X = e and lengthy expression for F 6 which can be obtained from the constraint det G = 1−XY 1+XY . Now we change the coordinates (κ = − tanh t)

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In these coordinates the metric has the form (ds 2 → 4ds 2 , t → 4νt) This metric is T -dual to the metric from [30] in the φ 3 isometry direction. The metric (4.41) satisfies Ricci flow equations (4.12) with Ψ and κ given by We can now perform the T -duality in the φ 2 isometry making the metric diagonal In this form the metric and the B-field coincide with the ones obtained from the deformed action (1.16) in [31]. The metric (4.43) and the B-field (4.44) satisfy (4.31) with At the end of this section we note, that the theory with N = 2n + 1 is self-dual with respect to the transformation b 2 → −1 − b 2 , while the theory with N = 2n + 2 is not. It is interesting to study the theory with N = 2n + 2 in this dual domain of parameters b 2 → −1 − b 2 . In particular, it will be interesting to construct the corresponding solution of Ricci flow equation and relate it to some deformed coset sigma model. For interested reader, we present some preliminary results for N = 6 in appendix C.

Conclusions
Our results for O(N ) sigma models has to be extended for other integrable sigma models including supersymmetric ones. An immediate generalization is related to the superalgebra sl(n|n). In this case, the arguments of our paper can be repeated with mild modification. We plan to return to this question in a future publication. As another example of the duality, it would be interesting to study the integrable deformations of strings on various supergravity backgrounds and find their dual description.

Acknowledgments
The results of section 3 and 4 as well as of the appendix C are obtained by A.L. The results of sections 2 and of appendices A and B are obtained by L.S. The introduction was written in close collaboration between the participants of this publication. A

A Integrals of Motion
Here we will study the set of commutativity equations (2.9) in more details. Namely, let ϕ(z) be the (N −1) component bosonic chiral field ϕ(z) = (ϕ 1 (z), . . . , ϕ N −1 (z)) normalized according to the operator product expansion We note that (2.9) is trivially satisfied by the first Integral of Motion Nontrivial equations appear when we consider Integral of Motion of higher spin. We assume that I free 2 = 0 and The density G 4 (z) is defined up to a total derivative. It can be represented as with unknown tensors A ijkl , B ijk and C ij with prescribed symmetry. The OPE of the field (A.4) with the exponential field V α (w) = e (α,ϕ(w)) , α = (α 1 , . . . , α N −1 ) has the form

B Parametrization of the group elements
The action of the deformed coset sigma model has the form S = 1 2 Tr g∂ + g −1 (c) We choose the basis in the Lie algebra SO(N ) to be (T ab ) ij def = δ ai δ bj − δ bi δ aj , (B.2) and SO(N − 1) subalgebra is chosen to be T ab with a, b = 1. Now we have to choose some parametrization of the group element. 9
(C. 6) In our case the special symmetry holds and K(z), cn(z|m), sn(z|m) are the standard elliptic functions.

JHEP11(2018)139
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