Scale invariance of the η -deformed AdS 5 × S 5 superstring, T-duality and modified type II equations

We consider the ABF background underlying the η -deformed AdS 5 × S 5 sigma model. This background fails to satisfy the standard IIB supergravity equations which indicates that the corresponding sigma model is not Weyl invariant, i.e. does not define a critical string theory in the usual sense. We argue that the ABF background should still define a UV finite theory on a flat 2d world-sheet implying that the η -deformed model is scale invariant. This property follows from the formal relation via T-duality between the η -deformed model and the one defined by an exact type IIB supergravity solution that has 6 isometries albeit broken by a linear dilaton. We find that the ABF background satisfies candidate type IIB scale invariance conditions which for the R-R field strengths are of the second order in derivatives. Surprisingly, we also find that the ABF background obeys an interesting modification of the standard IIB supergravity equations that are first order in derivatives of R-R fields. These modified equations explicitly depend on Killing vectors of the ABF background and, although not universal, they imply the universal scale invariance conditions. Moreover, we show that it is precisely the non-isometric dilaton of the T-dual solution that leads, after T-duality, to modification of type II equations from their standard form. We conjecture that the modified equations should follow from κ -symmetry of the η -deformed model. All our observations apply also to η -deformations of AdS 3 × S 3 × T 4 and AdS 2 × S 2 × T 6 models.


Introduction
The study of integrable deformations of the AdS 5 × S 5 superstring sigma model is an important direction in the search for new solvable examples of AdS/CFT duality.An interesting oneparameter integrable generalisation of the classical AdS 5 × S 5 Green-Schwarz action related to the quantum deformation of the underlying supergroup symmetry was found in [1].Just from the construction of this "η-model" (based on a particular current-current deformation of the supercoset action [2] generalising the bosonic model of [3]) there is no a priori reason why it should define a scale invariant (UV finite) 2d theory and, moreover, why it should preserve the conformal (Weyl) invariance and hence still correspond to a consistent superstring theory as the undeformed AdS 5 × S 5 model does. 1   The only indication in this direction is that the η-model action, like the original AdS 5 × S 5 action, is invariant under a version of fermionic κ-symmetry [1], which reduces the number of fermions by half.However, the usual claim that κ-symmetry implies the corresponding action can be interpreted as that of a GS superstring propagating in a background that is a consistent type II supergravity solution (and hence defines a consistent critical superstring theory) assumes that the κ-symmetry is of the standard GS "projector" form [5].This is most probably not the case for the η-model at higher orders in fermions.Indeed, it was found in [6,7] that the target space background corresponding to the η-model action [1], interpreted as a GS action, does not represent a type IIB supergravity solution.
Starting with the GS Lagrangian written in superspace form (Z M = (x m , θ α )) one can solve the standard type II superspace constraints and Bianchi identities for E(Z), B(Z) (which imply the supergravity equations) in order to express the GS action in terms of component fields.One then observes that the dilaton ϕ (which is part of the dilaton superfield Φ(Z) that is introduced in the process of solving the constraints) enters the world-sheet action (i) in the combination F = e ϕ F with the R-R field strengths starting at order θ 2 and (ii) via derivatives ∂ m ϕ starting at order θ 4 (see [8] and references therein).This action has classical Weyl invariance and κ-symmetry, which will be broken, in general, by quantum corrections.As for the bosonic string [9], to cancel the 2d stress tensor trace anomaly requires adding the familiar 1-loop dilaton counterterm ∼ d 2 z √ hR (2) Φ(Z) (see [10,11] and references therein). 2  The case relevant to our discussion below is a special isometric type II solution for which the metric G mn , B-field B mn and R-R fields F m 1 ...mn are invariant while ϕ is linear in the isometric directions.In this case the GS action will depend on the isometric coordinates only through their 2d derivatives and can thus be T-dualised.As we shall see, in this case the T-dual model will be scale invariant but may not be Weyl invariant (one may not be able to cancel the Weyl anomaly by a local counterterm), i.e. may not correspond to a type II supergravity solution.
The ABF background [6,7] includes the 10d metric G, the B-field and the R-R fields F n (n = 1, 3, 5) that are extracted from the quadratic fermionic part of the action of [1] put into the usual GS form, For the standard GS action in a type IIB supergravity background F n are interpreted as the products of the dilaton and the R-R field strengths F n = e ϕ F n but in the η-model case there is no independent information about the dilaton, and there exists no dilaton field that completes G, B, F n of the ABF background to a type IIB solution [7].
While not solving the standard type IIB equations directly this ABF background still turns out to be very special: it is related by T-duality to an exact type IIB supergravity solution [12,13].The latter HT background involves a non-diagonal metric Ĝ, an imaginary 5-form F5 and the dilaton φ, and the T-duality applied in all 6 isometric directions acts only on the fields Ĝ and F5 = e φ F5 entering the corresponding GS action (1.2) on a flat 2d background.The GS action for any type II solution (and thus for the HT background) should be Weyl invariant and, in particular, scale invariant.As the T-duality applied to the GS action [14] is a simple path integral transformation, the T-duality relation between the ABF and HT backgrounds implies [12] that the η-model action should define a scale invariant 2d theory at least to 1-loop order.
However, there may be a problem with Weyl invariance for the η-model action on a curved 2d background.The HT dilaton φ has a term linearly depending on the isometric directions of Ĝ and F5 and thus one cannot directly apply the standard T-duality transformation rules [15] to the full HT background to get a full T-dual supergravity solution, and thus the Weyl invariance of the T-dual sigma model requires further investigation. 3This is of course consistent with the observation [7] that the ABF background does not satisfy the full set of type IIB supergravity equations.
The aim of the present paper is to further clarify and extend these observations.We shall demonstrate that the relation by formal T-duality between the ABF and HT backgrounds implies that the former, while not a supergravity solution, should satisfy the following two generalisations or "modifications" of the type II supergravity equations: (i) the scale invariance conditions for the type II superstring sigma model (with equations on the R-R fields F being of 2nd order in derivatives) (ii) a set of equations that are structurally similar to those of type II supergravity (with 1storder equations for the R-R fields F) but involving, instead of derivatives of the dilaton, a certain co-vector Z m playing now the role of the dilaton one-form and a Killing vector I m responsible for the "modification" of the equations from their standard form. 4  While the scale invariance conditions are universal, the second set of equations (which we shall refer to as "I-modified" type II equations) only apply to particular backgrounds with isometric G, B and F-fields, which are related by formal T-duality to a type II solution ( Ĝ, B, F, φ) with the dilaton φ containing a term linear in the isometric coordinates.Such a dilaton background, breaking isometries by a linear term only, is special.As the type II supergravity equations written in terms of the F-fields only depend on the dilaton through its derivatives, they remain independent of the isometric directions.As a result, the standard type II supergravity equations for the T-dual solution ( Ĝ, B, F, φ) can be re-interpreted as certain modified type II equations 3 The 1-loop Weyl invariance conditions of the NSR or GS type II superstring sigma model are believed to be equivalent to the field equations of type II supergravity.While this is a well-established fact in the NS-NS sector [9,16] this was never demonstrated directly with the R-R couplings included (for some related work, mostly for the heterotic string, see [10,11,17]).Given that the linearised equations for all the supergravity fields follow from the condition of marginality of the corresponding NSR vertex operators and that the type II action is a leading term in the string effective action reconstructed from the superstring S-matrix on flat space, it is usually assumed that the superstring sigma model defining consistent critical string theories should correspond (to leading order in α ′ ) to backgrounds that solve the 10d supergravity equations. 4In what follows we shall not distinguish between co-vectors and vectors, referring to both X m and Xm as vectors.
for the original fields (G, B, F), also depending on the vectors Z and I.The Killing vector I m dependence is fixed by the term linear in isometric coordinates in the dilaton, while the vector Z m is determined by applying the standard T-duality rules to the part of the dilaton independent of the isometric coordinates. 5t is possible to express the modified equations for the NS-NS fields in terms of just one single vector X m = Z m + I m , which is the vector that appears in the scale invariance conditions.The superstring scale invariance conditions generalise the familiar one-loop scale invariance conditions for the bosonic sigma model with couplings G mn and B mn (cf.(1.2)) Here the terms involving X m and Y m do not contribute to on-shell UV divergences or, equivalently, reflect the freedom of renormalisation by reparametrisations and B-field gauge transformations.The X m terms drop out of the action if the sigma model field x m is subject to the classical equations, or, equivalently, they can be absorbed in a field renormalisation, The origin of the X k H kmn term can be understood either by starting with a counterterm proportional to (D m X n + D n X m )∂ a x n ∂ b x n + ..., integrating by parts and using the equations of motion for x m , or by observing that B mn transforms under a combination of reparametrisations and gauge transformations as m or Y m drop out of the sigma model action upon integration by parts.
The Weyl invariance conditions are equivalent to the vanishing of the trace of the 2d stress tensor operator on a curved 2d background.For the NSR type II superstring sigma model they can be satisfied provided one adds the dilaton term ∼ R (2) ϕ(x) [9,18,19,20,16]: they are a stronger form of the scale invariance conditions (1.3), (1.4) with X m and Y m no longer arbitrary, but given by The Weyl invariance equations (1.3),(1.4),(1.5)imply the "central charge" identity [9,21] i.e. that the effective dilaton "β-function" is a constant (which should be zero in critical string theory).The full set of Weyl invariance equations for G, B and ϕ follows from the effective action with the same form as the NS-NS sector of the type II supergravity action (F ≡ e ϕ F ) where we have indicated the presence of the R-R field strength terms for future reference.
The generalisation of the scale invariance conditions to the presence of R-R fields is given by (1.3), (1.4) with extra FF terms, together with a set of second-derivative equations for the R-R fields F that directly enter the GS action (1.2), 1  2 D 2 F + ... = X∂F + F∂X.Here the r.h.s.stands for reparametrisation (Lie derivative) terms with the same X-vector as in (1.3), (1.4) and dots indicate non-linear terms.In the special case when X m = ∂ m ϕ these equations are the consequence of the type IIB equations or Weyl invariance conditions, which are 1st order in F = e −ϕ F, i.e. d ⋆ F + ... = 0 and dF + ... = 0.6 These universal scale invariance conditions will be satisfied by the ABF background for a particular choice of the vectors X m and Y m .
To explain the origin of the second "I-modified" set of equations let us first ignore the R-R fields and assume that there exists the following metric-dilaton background that solves the Weyl invariance equations (i.e.R mn + 2D m D n ϕ = 0, βϕ =const) Here the metric has an isometry which is broken by the linear term in the dilaton (c = const).
Examples of such non-trivial solutions7 can be found by taking special limits of gauged WZW backgrounds [13].T-dualising this metric, we find a diagonal metric G and B-field, i.e.The dilaton equation βϕ = 0 for the original background (1.8) also can be rewritten as the following generalised equation (cf.(1.6)) that is satisfied for the T-dual background.
The T-dual background (G, B) defines a sigma model that is scale invariant on a flat 2d background (satisfying equations (1.3), (1.4) with Y m = X m ) but which is not Weyl invariant.The trace of stress tensor ) (up to terms proportional to the x m equations of motion).This cannot be cancelled by a local counterterm (the classical dilaton term) unless X m = ∂ m ϕ, Y m = 0 [18,19], which is not the case for the ABF background.The sigma models based on (1.9) (with explicit backgrounds given below) thus represent particular examples of 2d scale invariant theories that avoid the Zamolodchikov-Polchinski theorem [22] due to their non-compactness (and/or non-unitarity related to the presence of time-like directions).It thus remains unclear if such backgrounds related by formal T-duality to Weyl invariant models (1.8) can also be associated somehow with a consistent critical string theory.
As we shall see below, a similar generalisation of the full set of the bosonic type II supergravity equations also exists in the presence of R-R fields F n that have the same isometries as the metric (i.e. when (1.8) is extended to an analog of the HT solution [12]).Thus in general, given a type II solution with non-isometric linear dilaton there will be an associated ("T-dual" or ABF-like) background solving such a modified set of type II equations.
The rest of this paper is organised as follows.In section 2 we shall present the general scale invariance conditions for the couplings G, B of the sigma model (1.2) that generalise (1.3), (1.4) to the presence of the R-R fields F and show that there exist such vectors X = Z + I and Y that these equations are satisfied by the ABF background.In section 3 we shall derive a modification of the standard 1st-order IIB supergravity equations of the R-R fluxes that is "driven" by the special isometry vector I and which are satisfied by the ABF background.In section 4 we shall show that combining these 1st-order equations one can find 2nd-order equations for F that have the right structure (when generalised to arbitrary vector X) to be interpreted as scale invariance conditions on the R-R couplings.In section 5 we explain how the standard type II supergravity equations for a solution with the dilaton linear along the isometric directions is mapped to the modified equations for T-dual solution.
Our notation and some useful relations are summarised in Appendix A. In Appendix B we present the explicit form of the ABF background and the T-dual type IIB HT solution.Appendix C contains the derivation of the identity ∂ m βX = 0 from the modified type II equations which is closely related to the on-shell conservation of R-R stress tensor.In Appendix D, starting with the modified type II equations, we derive the 2nd-order equations for the R-R fields that are candidates for the corresponding scale invariance conditions.In Appendix E we remark on an alternative derivation of the relation (2.13) for the vector Z, which plays the role of the dilaton one-form in the modified equations.In Appendix F we summarise the analogs of the ABF and HT backgrounds in the AdS 2 × S 2 × T 6 and AdS 3 × S 3 × T 4 cases and give the corresponding expressions for the vectors X, Y and I that solve the scale invariance and modified type II equations.In Appendix G we explain how the 2nd-order equations for the R-R couplings F emerge as the one-loop conditions of scale invariance for the GS sigma model (1.2).

Scale invariance conditions and modified type II equations: NS-NS sector
The scale invariance conditions for the bosonic sigma model (1.3),(1.4)have a straightforward generalisation to the GS superstring case with non-zero R-R couplings F = e ϕ F (see Appendix G).The θFθ∂x∂x terms in the GS action (1.2) should lead to one-loop diagrams (with one bosonic and one fermionic line) contributing logarithmic UV divergences ∼ FF∂x∂x.These terms will produce extra FF terms in the β-functions in (1.3) and (1.4).In particular, the analog of the Einstein equation (1.3) should pick up the R-R stress tensor term and the B-field equation (1.4), the FF term as in the II supergravity equations. 10This is expected as for X m = ∂ m ϕ, Y m = 0 the resulting equations are the Weyl invariance equations that should be equivalent to the type II supergravity equations.
The scale invariance equations for the F-fields (to be discussed in section 4) will not, however, have the familiar supergravity form of 1st-order equations for F (these should follow from the Weyl invariance conditions).Instead they will be of 2nd order, D 2 F + ... = X-dependent terms, and for X m = ∂ m ϕ will be a consequence of the 1st-order supergravity equations.
Explicitly, the scale invariance conditions (1.3) and (1.4) generalise to Here F m , F mnk , F mnklp are R-R fields of type IIB supergravity (for notation see Appendix A).
For X m = ∂ m ϕ, Y m = 0 these equations follow from type IIB supergravity action (1.7).T mn is the familiar stress tensor that follows from the type IIB action (1.7) upon variation over G mn .11 As was noted in [12], the existence of the HT solution related to the ABF background by Tduality, suggests that the GS sigma model for the latter defined on a flat 2d background should be scale invariant (at least to leading, 1-loop, order).Our key observation is that indeed there exist vectors X m and Y m such that eqs.(2.1) and (2.2) are satisfied for the ABF background (B.1).The vector X m required to satisfy (2.1) turns out to be (see Appendix B for notation) X m can be split in the following way where I m = 5 i=0 c i (I (i) ) m .The index i labels the 6 isometric directions y i = (t, ψ 2 , ψ 1 , φ, ϕ 2 , ϕ 1 ) of the 10d ABF metric and c i are arbitrary constant coefficients.(I (i) ) m are the 6 independent commuting Killing vectors of the ABF background: the Lie derivatives of the G, B and F-fields in [6] along I m all vanish.If we split the coordinates as x m = (y i , x µ ) where µ = 1, 2, 3, 4 labels the non-isometric directions x µ = (ζ, ρ, ξ, r), then The vector Y m required to satisfy (2.2) on the ABF background is found to be12 dr . (2.8) We observe that if we fix c i in (2.5) to the following specific values then Y m and X m coincide Y m = X m . (2.10) The next surprising observation is that for these specially chosen values of c i in (2.9) the vector X m satisfies also a direct generalisation (1.11) of the dilaton equation As we shall show in Appendix C this βX satisfies the generalisation of the dilaton identity (1.6) (2.12) The reason for this particular choice of c i in (2.9) can be traced to the form of the linear terms in the dilaton φ of the T-dual HT solution (B.3).That is the presence of the I-term in X m in (2.6) reflects the presence of the non-isometric linear terms in φ.Therefore, these terms drive the modification of the equations satisfied by the ABF background from their standard type II form.In this sense the Z m part of X m may be interpreted as the analog of ∂ m ϕ in the modified equations.Indeed, one can check that for I m in (2.7) with c i chosen as in (2.9) the following relation is satisfied This may be interpreted as a modified "dilaton Bianchi identity": if I m is formally set to zero then Z m becomes a derivative of a scalar, ∂ m ϕ.In general, assuming that I m represents an isometry of the B-field, i.e. the Lie derivative (L I B) where ∂ m ϕ term represents the trivial "zero-mode" solution.In the particular case of the ABF background with Z m and I m given by (2.5),(2.6),(2.7)and c i fixed as in (2.9) we find (2.16) The scalar ϕ in (2.16) is precisely the one that is found [12] by applying the standard T-duality transformation rule to the isometric part of the dilaton φ of the HT solution in (B.3) (cf.(1.10)).
3 Modified type II equations: first-order equations for R-R couplings Let us now explore what modification of the type IIB equations for the R-R couplings is satisfied by the ABF background.
The standard equations of type IIB supergravity [27] in the R-R sector written in terms of the rescaled F = e ϕ F field strengths are pairs of dynamical equations and Bianchi identities (see Appendix A for notation)15 Here Z = Z m dx m = dϕ is the dilaton one-form.The five-form F 5 is also required to satisfy the self-duality equation ⋆F 5 = F 5 which implies the equivalence of the first and second equation in (3.3).
An a priori surprising observation is that there exist direct generalisations of the 1st-order equations (3.1)-(3.3)involving Z = Z m dx m and I = I m dx m in (2.5), (2.6), with fixed values of the coefficients c i as given in (2.9), which are solved by the ABF background (B.1).Explicitly, the equations for the one-form F 1 in (B.1) are We have added the condition I m F m = 0 as an independent equation on F 1 . 16  Similarly, the equations that generalise (3.2) and are satisfied for the three-form F 3 in (B.1) are found to be of H3, F1 and F5.The choice of sign of H3 or B can be changed by parity. 16Alternatively, one can derive this equation from the Bianchi equation (3.5), the invariance of F1 under the isometry, the orthogonality of I and Z, and the condition that Z is not an exact one-form.Indeed, multiplying (3.5) by I m one finds ∂n(I m Fm) − ZnI m Fm = 0 Thus, if I m Fm ̸ = 0 then Z = d ln(I m Fm).We find, however, it more convenient to add I m Fm = 0 as an independent equation, and infer from it the orthogonality of I and Z.
The equations satisfied by F 5 of the ABF background are found to be These two are equivalent in view of the self-duality of F 5 .
These modified equations (3.4)-(3.9)reduce back to (3.1),(3.2),(3.3)if we drop all terms with I m and assume that dZ = 0, i.e. if we set The structure of (3.4)-(3.9)supports the interpretation of Z as a generalised "dilaton one-form", while the isometry vector I effectively drives the deformation of the standard type IIB equations.
An interesting observation is that there exist certain combinations of the equations (3.4)-(3.9)that depend on Z and I only through the combination X = Z + I, which entered the NS-NS equations of the previous section.These are found by adding together equations of equal form degree, for example, the equation of motion for the R-R three-form and the Bianchi identity for the R-R one-form.The resulting X-dependent equations are given by Using the self-duality of F 5 the last equation can be also written as As will be discussed below, these three equations are already sufficient for deriving candidates for the scale invariance equations for the F-fields, which are 2nd order in derivatives.
It is useful to rewrite (3.1)-(3.3) in the notation of forms (see Appendix A for conventions).To do so we introduce the dual forms defined by Then the complete set of the type II supergravity equations for R-R strengths and Bianchi identities (3.1)-(3.3) is given by17 where Z = dϕ.
The "I-modified" equations (3.4)-(3.9)are given by18 Due to (3.15) the two equations in (3.16) are equivalent and the same is true for (3.17).
Let us note that the deformed R-R equations (3.17) together with the relation (2.13) or dZ + ι I H 3 = 0 imply the following relation Thus the condition that the F-fields are invariant under the isometry I is equivalent to the condition I•Z = 0, which is clearly satisfied for the ABF background as is evident from (2.5),(2.7).

Second-order equations for R-R couplings as scale invariance conditions
Let us return to the discussion of the scale invariance conditions for the couplings of the GS sigma model (1.2) in section 2 and consider the equations for the R-R couplings F that should follow from the requirement of (1-loop) UV finiteness of the 2d model.One can argue that the conditions analogous to eqs.(2.1),(2.2) for the G and B-field couplings should have the form where we have omitted possible non-linear terms such as RF + DHF + ... on the l.h.s.The X-dependent Lie derivative term on the r.h.s.reflects, as in (2.1),(2.2), the reparametrisation (or off-shell x m -renormalisation) freedom.For example, starting with the linearized RG equation We shall discuss the computation of 1-loop logarithmic UV divergences for the GS action (1.2) in Appendix G clarifying the structure of β F .
Moreover, the same equations should follow also from the modified type II equations (3.4)-(3.9)or (3.17) (as, e.g., the ABF background that solves the modified equations should also be a solution of the scale invariance conditions).This should provide a non-trivial consistency check: after properly "squaring" (3.4)-(3.9) the dependence on the Z and I vectors in any candidate scale invariance equations should appear only through their sum X = Z + I as in (2.1),(2.2).
Starting from the modified type II equations (3.4)-(3.9)(which include the standard type IIB supergravity equations as a special case (3.10),I m = 0), let us outline the derivation of the 2nd order equations for the R-R couplings that should be equivalent to the scale invariance conditions 19 For example, using the NSR approach on a flat background we may consider the R-R vertex operators built out of spin operators and consider the linearised conditions for conformal invariance (marginality).Then dF = 0, d ⋆ F = 0 will follow (see, e.g., [28]) just like the usual transversality conditions on the graviton operator follow from the marginality conditions of the hmn(p)e ipx ∂x m ∂x n vertex.On a curved 2d background these are equivalent to the decoupling of derivatives ∂aρ of the conformal factor of the 2d metric (see, e.g., [29,30]).These conditions are stronger than just scale invariance which requires only "masslessness" for F n of the GS sigma model (1.2).To be a candidate for the scale invariance conditions these equations should have the following properties: (i) vanish on the supergravity equations (2.1),(2.2),(2.11),(3.1)-(3.3)with X = dϕ, Y = 0 (ii) depend on Z and I through X = Z + I (iii) depend on X through Lie derivatives. 20tarting with the modified equations (3.17) and acting with ⋆d⋆ on the first equation and d⋆ on the second and then using the modified equations (as described in Appendix D) we arrive at the following equation, which satisfies the above properties Here β B is the 2-form analog of (2.2), i.e.
This is then a candidate for the scale invariance equation for the R-R form F 2n+1 .
Using the identity where β G mn is defined in (2.1), we find that (4.2) becomes The dependence of these equations on X rather than separately on Z and I can be related to their close connection to the particular X-dependent combinations of the modified equations in (3.11),(3.12),(3.13),i.e. to (here n ∈ Z as in (3.17)) We also define as in (1.11),(2.2) Deconstructing the derivation in Appendix D, we find that the 2nd-order equation for the R-R fluxes (4.2) can also be written as Finally, let us present the explicit form of eq.(4.5) in components.For F 1 we find Using the identity D [m H npk] = 0 the term 1 6 D m H pnk F pnk in (4.9) can be replaced by 1 2 D p H mnk F pnk .The equation for F 3 may be written as while the equation for F 5 can be put into the form This expression is consistent with the self-duality of F 5 (in particular, the third and forth lines are manifestly dual to each other).
These 2nd-order equations for F 1 , F 3 and F 5 exhibit obvious structural similarities.In particular, they contain the expected Hodge-de Rham operator terms and the vector X only enters through the reparametrisation terms as in (4.1).The β G and β B terms in these equations are defined as in (2.1),(2.2) but can also be replaced by expressions on the r.h.s. of (2.1),(2.2).
As we shall discuss in Appendix G, similar equations come out of the computation of the one-loop beta-functions for the R-R couplings in the GS sigma model (1.2).

Origin of modified equations: T-duality relation to type II equations for backgrounds with non-isometric linear dilaton
Given a scale invariant sigma model in flat 2d space T-duality in an isometric direction should also produce a scale invariant sigma model.Similarly, given a Weyl invariant sigma model on curved 2d space with all couplings including the dilaton being isometric the T-dual background should also be Weyl invariant (provided the dilaton transforms in the usual way [31,32]).As discussed in the introduction, in general this is not so if the dilaton is not isometric: T-duality will still preserve scale invariance but not Weyl invariance.Thus given a solution of type II supergravity equations which has linear non-isometric term in the dilaton its T-duality image will no longer solve the standard type II equations but will satisfy instead a modified set of type II equations as discussed above.

Simple examples
Here we shall make the origin of the modified equations explicit by showing that they represent the original type II equations for a solution with non-isometric linear dilaton, rewritten in terms of the fields of the T-dual background.To explain how this happens in simple terms let us first start with a bosonic background (1.8) with A µ = 0, i.e.
Then the corresponding Weyl anomaly coefficients have the following components under the xm = (ŷ, x µ ) splitting of coordinates We see that if c = 0, i.e. the dilaton is isometric, then the Weyl invariance conditions βG µν = 0, βϕ = 0 are invariant under T-duality in y, i.e. under â = −a → a, φ → φ + a or φ → φ.The c = −∂ ŷ φ dependent terms in (5.4) thus represent obstructions to mapping one Weyl invariant model to another.The T-dual metric then solves weaker, modified, equations with X m being (cf.(1.10)) Similar conclusions are reached in the case we have a non-diagonal metric in (1.8) (see the general discussion below).The presence of non-zero Âµ is in fact necessary to have a solution of the Weyl invariance conditions when c ̸ = 0 (cf.(5.4)) and the target space should thus be at least 3-dimensional.An example of such a solution was found in [13].It represents a limit of the background associated with the SO(4)/SO(3) gWZW model, which has the following metric and dilaton [33] where p = sin ψ, q = cos θ cos ψ and t, ψ, θ are angles of the coset parametrisation of the SO(4) group element.This background (which solves the Weyl invariance condition βG = 0 with βϕ = const) has no isometries.One option to generate an isometry is to set t = iz and then shift z by an infinite constant.Doing so we get linear dilaton in z, but the z direction decouples in the metric.A non-trivial alternative is to set ψ = iŷ and to shift ŷ by an infinite constant (which corresponds to infinite rescaling of p, q generating a scaling isometry in the metric (5.7)).
The resulting background (we drop an infinite constant in the dilaton) is therefore of the same type as in (1.8) and defines a conformal sigma model 22 Similar higher dimensional backgrounds can be constructed starting from SO(n)/SO(n − 1) gWZW models with n > 4 [13].
T-dualising the metric (5.9) along ŷ we get a (G, B) background that will solve the modified (G, B) equations (2.1),(2.2) with non-trivial X m = I m + Z m , where I y = −∂ ŷ φ and Z m is given by (2.14), with ϕ = φ − 1 2 log G ŷ ŷ.These modified equations will be the original Weyl invariance conditions rewritten in terms of the dual G and B-fields.
Given the 2d CFT in (5.9) with 2d stress-tensor defined taking into account the dilaton in (5.10), one may formally compactify ŷ and ask if this CFT has T-duality as a symmetry of its spectrum.The answer appears to be no as the 2d stress-tensor will not be invariant under T-duality (i.e.mapping momentum into winding modes). 23This is compatible with our expectation that formally T-dualising the metric (5.9) will not lead to a consistent CFT.
The same conclusions are reached for type II solutions with a linear dilaton and non-zero R-R fluxes (with isometric G, B and F n = e ϕ F n ), such as the HT solution dual to ABF background in the AdS 5 × S 5 case and its counterparts in the AdS 2 × S 2 × T 6 and AdS 3 × S 3 × T 4 cases discussed in Appendix F. Explicitly, in the case of a solution ( Ĝ, B, Fn , φ) with several isometries broken only by the linear dilaton term we will get a generalisation of the type II supergravity equations, depending on the following two vectors Z and I (cf.(2.6),(2.7),(2.14)) log Ĝŷ i ŷi .
(5.12) 22 The central charge for this d = 3 conformal model is given by c Here the scale of the space was set to one, so that α ′ is then the inverse of the WZW level k.This is in agreement with the usual count of the central charge for the SO(4)/SO(3) gWZW model c = 6k/(k + 4) − 3k/(k + 2) = 3 − 18/k + ..., which should be unchanged in the coordinate limit leading from (5.7) to (5.9). 23Given a free compact scalar CFT L = r 2 (∂ϕ) 2 with ϕ ≡ ϕ + 2π the spectrum of dimensions of primary operators (like e inϕ+im φ, etc.) is T-duality symmetric.If one formally adds a linear dilaton term q d 2 z √ h R (2) ϕ, or equivalently modifies the 2d stress tensor by q∂ 2 ϕ terms (which are invariant under shifts of ϕ and thus defined for a compact boson) then the T-duality symmetry of the spectrum is broken by extra terms ∼ qn.The formal symmetry would be restored in the "doubled" formulation if the linear dilaton term were given by qϕ + q φ where φ is the dual field (with Here G and B are the background fields of the T-dual background.
Below we shall illustrate these relations in the general case with one isometry.

NS-NS sector
We consider the following two d-dimensional backgrounds (here we use K µ instead of Âµ in (1.8)) where (5.16) The two (G, B) backgrounds in (5.13) and (5.14) are T-dual to each other and thus for c = ĉ = 0 solve the equivalent Weyl invariance equations (see, e.g., [36] and references therein).We will now show how this relation also extends to the more general case with linear dilatons.
Let us first consider the generalised dilaton equation (5.17) The question we want to address is: if the background (5.13) satisfies (5.17) does that imply that (5.14) satisfies (5.17).As the two backgrounds (5.13) and (5.14) are related by the obvious symmetry it is sufficient to compute the left-hand side of (5.17) for (5.13) and check that it is invariant (or at least covariant) under (5.18).
For (5.13) we have It will also be useful to define the following objects where we observe that h is invariant under (5.18).Now using the dimensional reduction formulae in Appendix A we find which is indeed invariant under (5.18).Therefore, if (5.17) is satisfied for background (5.13) it is satisfied for background (5.14) and vice versa. 24  Let us now turn to the modified metric and B-field equations to show that the two combinations appearing in (1.3) and (1.4) ) are covariant under the symmetry (5.18). 25Then if they vanish for the background (5.13) this implies that they vanish for (5.14) and vice versa.As (5.22) is symmetric and (5.23) is antisymmetric, we may just consider their difference which we can decompose into a part independent of c and ĉ (C mn ) and a part linear in c and ĉ (C (1) mn ) as mn + 2C (1)  mn , (5.25) C (1)  mn = D m X (1)  n + 1 2 X (1)p H mnp . (5.27) Here we have used the fact that all the c and ĉ dependence is contained in X = X (0) + X (1) , where X (0) is the c-and ĉ-independent and X (1) is the c-and ĉ-dependent part.Using the specific form of X for the background (5.13), as given in (5.19), we have on the background (5.13) C (1)  yy = e 2a (cA (5.30) Then using the map (5.18) between the backgrounds (5.13) and (5.where the left-hand side is evaluated on (5.13) and the right-hand side on (5.14).From these equalities it follows that the vanishing of the tensors (5.22),(5.23) on the background (5.13) implies their vanishing also on the background (5.14), and in this sense are covariant under T-duality.
Finally, let us note that the same analysis can be carried out with I µ and Îµ in the backgrounds (5.13) and (5.14) non-vanishing (i.e. when there are extra isometries in the x µ directions).The T-duality relation between the equations for (5.13) and (5.14) still holds with I µ = Îµ .The same is also true for the modified equations of motion for the R-R fields discussed in the following section.Furthermore, as I µ = Îµ should also be a Killing vector we still have Therefore, we can now use these relations to perform the sequence of T-dualities required to transform from the HT supergravity solution of [12] to the ABF background (B.1) and its AdS 3 × S 3 and AdS 2 × S 2 counterparts (F.4),(F.13).

R-R sector
Let us now consider the case of non-zero isometric R-R fields F n .The contribution of the R-R fields to the metric and B-field equations appears in the usual unmodified form and hence we can focus our attention on the modified equations of motion for the R-R fields (3.17).Written in terms of the forms f k ≡ e −a/2 F k these take the following form (dropping the distinction between y and ŷ) where we have introduced (K = K µ dx µ , A = A µ dx µ ) which are related to each other under T-duality as Recall that the invariance of the R-R forms under the isometry along y requires c ĉ = 0 . (5.36) This follows from the condition I • Z = 0, which is implied by the invariance of R-R fields as We want to show that if f k satisfies the equation E k = 0, then fk satisfies Êk = 0. Taking into account the T-duality relations in Appendix A one finds (5.37)Here we used that δZ ∧ (dy + K) ∧ (dy + A) ∧ ι y f k−1 = 0 and ((dy (5.38) If we set E k = 0 and c = ĉ = 0 we get as expected.It remains to consider the c and ĉ dependent terms only (5.40) Thus, if E k = 0 then Êk = 0, i.e. the backgrounds (5.13) and (5.14) supplemented by R-R fields have their corresponding modified equations mapped into each other by this generalised T-duality.

Concluding remarks
There are several open problems and puzzling questions.First, it remains unclear if the scale invariant but arguably not Weyl invariant η-model can still be used to define a critical superstring theory.This might be possible in view of the existence of the λ-model [23] which is classically related to the η-model by the Poisson-Lie duality combined with an analytic continuation of the deformation parameter, and for which there is a candidate supergravity solution [24] (i.e. it should represent a Weyl invariant sigma model).In fact, a special limit [13] of this solution should be essentially equivalent to the HT solution [12]. 26Thus if the classical Poisson-Lie duality relation [25] between the η-model and λ-model [26,12] extends to the full quantum level there may be a way to associate a string theory to the ABF background.This might also require increasing the number of 2d fields (such as in a doubled or phase space formulation).Indeed, already at the classical level, establishing the connection between the two models calls for the use of the phase space formalism.The quantum η-model defined in terms of an extended number of fields (including, e.g., analogs of 2d gauge fields of the gWZW part of λ-model) may then be Weyl invariant, and integrating out extra fields might produce the GS action corresponding to the ABF background plus extra non-local terms required for restoring its Weyl invariance.
As we have seen above, the fact that the HT background solves the type IIB equations implies that the T-dual ABF background should satisfy the I-modified type II equations.These explicitly depend on the isometry vector I, whose origin can be traced to the presence of the linear term in the dilaton of the HT solution.One can ask whether these I-modified equations are Lagrangian, i.e. if they can be derived from the action principle.Answering this question may require the introduction of R-R potentials and understanding whether one should treat the vector I as an external source or as an auxiliary field with no physical degrees of freedom.In view of our analysis of T-duality in section 5, it would be interesting to know if there exist more general I-modified equations that are compatible with T-duality and have c ĉ ̸ = 0 in (5.16).One would also like to understand how the usual action of the T-duality group O(d, d) is modified.
In the present work we discussed only the I-modified equations for bosonic fields.It is an interesting question how the equations for the fermionic fields of type II theory are modified.Furthermore, if the I-modification destroys the local supersymmetry of type II theory one may ask if there is still any (hidden) symmetry of the I-modified equations for bosonic and fermionic fields.
To better understand the nature of the ABF background it would be important to derive the quartic fermionic action for the η-model of [1] and to show that the I-modified equations indeed follow from the κ-symmetry [1] of this action.Starting with the standard GS action for the HT solution [12] (which, as was mentioned in the Introduction, is invariant under shifts of the 6 isometric coordinates) and performing the T-dualities one will get θ 4 and higher terms in the η-model GS action depending on the vectors I and Z.These will originate from the dilaton, θ 4 ∂ϕ, etc., terms in the HT GS action.The resulting η-model action should still be invariant under the κ-symmetry defined in [1], however it is then probable that the structure of these transformations will deviate from those of the usual GS action.
The knowledge of the quartic fermionic action should also enable one to perform the full computation of one-loop divergences of the η-model in the R-R sector (completing our discussion in Appendix G) and hence check the agreement between the 2nd-order equations for R-R fields derived from the modified type II equations with the scale invariance beta-functions for F n .
It would also be important to attempt a direct analysis of the Weyl invariance conditions, which should lead to 1st-order conditions for R-R strengths equivalent to type II supergravity equations.More generally, one may study the one-loop renormalisation of a generic κ-symmetric sigma model with 8 bosonic and 8 fermionic degrees of freedom, and classify interaction terms for which the corresponding model is either conformal or scale invariant only.It is possible that the class of conformally invariant models may be bigger than just the usual type II GS superstring sigma models.
Finally, it would be interesting to perform a similar analysis for the deformations of AdS n ×S n backgrounds constructed from other solutions of the modified classical YB equation [1,39], or solutions of the classical (non-modified) YB equation, see, e.g., [40,41].In the latter case many of the resulting metrics and B-fields can be completed to full type II supergravity solutions, however it remains to verify that these completions are indeed realised by the supercoset action.Indeed, the analysis of [7] has shown that the large κ-limit of the η-model does not coincide with the AdS 5 × S 5 mirror sigma model [42] even though the bosonic part of the model does.

Conventions for forms
We have for any m-form Y and n-form Z on a manifold of dimension d where the antisymmetrisation is understood as In d dimensions with Lorentzian signature we have In particular for m = 1 and even d one has

Dimensional reduction formulae
Let us take the metric and B-field as in (5.13) where y is an isometric direction.It is useful to define We can now write the various d-dimensional quantities appearing in the modified type II equations in terms of (d − 1)-dimensional ones as follows (A.9) (A.11) Also, for a vector X = X y dy + X µ dx µ we have (A.12) (A.13) Here , ∇ µ is the covariant derivative with respect to the (d−1)-dimensional metric g µν with connection γ λ µν , and R µν and R are the (d − 1)-dimensional Ricci tensor and scalar respectively.

T-duality rules
Let us consider two isometric backgrounds related by T-duality, with the fields of the dual background denoted with hats.The metric and B-field will be taken in the form of (5.13), and we will also consider the isometric dilaton ϕ and the R-R field strengths F k ≡ e ϕ F k of type II theory The T-duality rules for the NS-NS fields are (see (A.6)) In terms of the forms A = A µ dx µ , K = K µ dx µ , H 2 = dK, H 3 = dB, and the corresponding hatted ones, one has27 To write the T-duality rules for the R-R fields it is convenient to introduce where ι y f k ≡ ι Iy f k , I m y = δ m y .Also using the assumption of invariance of the R-R forms under the isometry, L Iy f (k) = 0, one has (A.20)

B ABF background and T-dual HT solution
The ABF background [6,7] represents the couplings in the η-deformed AdS 5 × S 5 action [1] expanded to quadratic order in fermions and formally identified with a GS action.This background for the type IIB fields (G, B, F 1 , F 3 , F 5 ) (but not the dilaton which cannot be extracted from the DMV action, and, in fact, does not exist) is given by Here κ = 2η 1−η 2 is a continuous deformation parameter of the η-model: κ = 0 corresponds to the standard AdS 5 × S 5 solution [27].
ds 2 ≡ G mn (x)dx m dx n defines the 10d metric G and the sign of B-field is chosen as in [6], i.e. it corresponds to the sign in (1.2).F k ≡ e ϕ F k are effective R-R k-form strengths of type IIB theory that appear in the GS action.The self-duality equation satisfied by the R-R 5-form is where we order the coordinates as As found in [12], there exists an exact solution of the standard type IIB supergravity equations that is T-dual to the ABF background (provided we ignore the dilaton transformation).This HT background has the following explicit form28 When written in terms of the following "boosted"/"rotated" vielbein basis where M, N = 0, ..., 9 are flat tangent-space indices, and η M N is the Minkowski metric.

C Conservation of R-R stress tensor and dilaton beta function identity
Given a Weyl invariant sigma model the dilaton beta function βϕ in (1.6) represents a natural definition of the central charge: it appears as the coefficient of the R (2) -term in the expectation value of the trace of the stress tensor on a curved 2d background [19,20], and for this reason must be a constant [9]. 29n the case of the ABF background we found an analog of the dilaton beta-function and the equation βX = 0 was used in section 2 to determine the isometric part I of the diffeomorphism vector X.In this Appendix we reverse the logic and show that the modified type II equations for the NS-NS and R-R fields with the same vector X implies the constancy of βX .In other words, on the equations of motion (2.1)-(2.4),(3.4)-(3.9)governed by the vector (2.9) we have the dilaton beta-function identity ∂ m βX = 0.
To proceed, we first need to derive the conservation law for the R-R stress tensor T mn in (2.3) that should hold on the R-R equations of motion.First, consider Now using (3.4) and (3.5), we find Next, we have such that using (3.7) and (3.6), we obtain Finally, we need where we have used (3.8) and (3.9) and that F 2 5 = 0. Taking into account the self-duality of F 5 , which also implies that F m pqrs ε pqrsabcde = −24g m[a F bcdef ] , we find Combining (C.3),(C.5),(C.7)we find the following conservation law for the stress tensor T mn where K mn is defined in (2.4).We would like to rewrite this formula in terms of X = Z + I.We have Further, we use (2.1) and (2.2) (with Y = X) to write Notice that due to the properties of which implies the following identity Now using (2.13), we obtain we find Thus, the conservation law (C.8)acquires the following form depending only on the vector X Here, using (2.2), the tensor K kn can be eliminated such that the r.h.s. of (C.16) is written solely in terms of H 3 and X.
Now we ready to show the constancy of βX .We have from (C.1) Since D [m H nkl] = 0 this can be rewritten as Furthermore, using we may combine the terms in (C.18) as Finally, using eq.( 2.1) we have where the r.h.s.vanishes due to the conservation law (C.16).This proves that βX is a constant (actually zero) on the modified equations of motion.The same is then true also in the spacial case of the standard type IIB supergravity equations (i.e. in the limit (3.10)) with the R-R strengths non-zero. 30 Derivation of second-order equations for R-R strengths from modified type II equations Here we present the derivation of the 2nd-order equations for the R-R field strengths, which, as discussed in section 4, are candidates for the scale invariance conditions of the GS sigma model, starting with the modified type II equations (3.4)-(3.9)or (3.17), i.e. (n ∈ Z) Our aim is to derive (4.2).Acting on the first equation by ⋆d⋆ and on the second equation by d⋆ we get Taking the sum of these equations and using ⋆(I ∧ ⋆Z n ) = ι I Z n , we find where we have used (3.18): The terms on the first line are the same as in (4.2), so we consider the last line in (D.3) to get 6) The two terms on the first line are the same as in (4.2).To derive the remaining terms of (4.2), we use the relations (D.7) which transform the last two lines of (D.6) into Now using the identities (D.10)This leads precisely to (4.2).

E Derivation of "Bianchi identity" for Z
Here we observe that the modified "Bianchi identity for the dilaton" (2.13) that holds for the ABF background may be derived more generally from the Bianchi equations for F k , the invariance of the R-R fields under the isometry L I F k = 0, the conditions F 1 ∧ F 3 ̸ = 0, F 1 ∧ F 5 ̸ = 0 or F 3 ∧ F 5 ̸ = 0 and the condition ι I F 1 = 0. Starting from (see (3.4)-(3.9)) we take the differential of (E.1) and use (E.2) to give31 A similar analysis of the Bianchi equations for F 3 and F 5 gives F Deformed AdS 3 × S 3 and AdS 2 × S 2 cases In the deformed AdS 3 × S 3 case the (complete) T-dual HT background [12] consists of just the metric, dilaton and a single R-R 3-form flux, and therefore has a simple embedding into Type IIB supergravity -one just needs to add 4 extra toroidal dimensions.Explicitly, this background which is T-dual to the η-deformed AdS 3 × S 3 background (cf.[34,35]) is given by When written in terms of the "boosted"/"rotated" vielbein basis [12] e As in the AdS 5 × S 5 case, the dilaton and the R-R flux F 3 depend on the isometric directions of the metric, but this dependence is such that e ϕ F = F is invariant under the isometries.Therefore, we can formally T-dualise the metric and F to find the following analog of the ABF background (cf.(B.1)) (F.4) As in the AdS 5 × S 5 case, it turns out that there exist vectors X and Y such that the scale invariance conditions for the metric and B-field (2.1),(2.2) are satisfied (cf.(2.5),(2.8)) The parameters c i and k a are eight arbitrary constants parametrising the Killing vector part of X m , while Y is defined up to a total derivative.As in the AdS 5 × S 5 case, we may split the vector X into two parts: I, containing the 8 commuting Killing vectors, and Z, which contains the rest.If we fix the constants c i and k a as so that Y m = X m then the equations (2.10),(2.11),(2.13),(3.4)-(3.9)are all satisfied, and hence the background (F.4) solves the same system of equations as the ABF background (B.1).Finally, we find ϕ in (2.14) is given by The parameters c i and k a are eight arbitrary constants parametrising the Killing vector part of X m , while Y is defined up to a total derivative.Here X m is given just by I m (i.e. the sum of commuting Killing vectors) and thus Z m = 0.If we fix the constants c i and k a as so that Y m = X m then the equations (2.10),(2.11),(2.13),(3.4)-(3.9)are all satisfied, i.e. the background (F.13) solves the same system of equations as the ABF background (B.1) in the AdS 5 × S 5 case.Finally, here we find that ϕ in (2.14) is given by ϕ = 0 .(F.17) As in the AdS 5 ×S 5 case, the coefficients in (F.7),(F.16)are equal to (minus) the corresponding coefficients of the isometric coordinates in the linear terms of the dual dilatons φ of the T-dual HT backgrounds [12].Furthermore, the "dilatons" ϕ in (F.8),(F.17)are again found by applying the standard T-duality rules to the remaining parts (depending only on non-isometric coordinates) of the dilatons φ of the T-dual solutions.Therefore, these examples also fit into the general picture described in section the main text.
G Second-order equations for R-R fields from scale invariance conditions for type II GS sigma model In this Appendix we shall expand on the discussion in section 4 and explain how the 2nd-order equations for the R-R couplings F such as (4.9)-(4.11)can emerge as the one-loop conditions of scale invariance (UV finiteness) of the GS sigma model (1.2).While we will not compute the beta-functions for R-R couplings in full, our aim will be to illustrate how the relevant structures come out of logarithmically divergent parts of the corresponding one-loop Feynman graphs. 32  We shall consider the type IIB GS sigma model [5] with couplings representing a generic type IIB superspace background subject to constraints required for κ-symmetry: we will assume κsymmetry to be able to gauge fix it but otherwise will keep the R-R fields unconstrained.The GS sigma model action expanded in powers of fermions may be written as (see, e.g., [14,8] 32 Previous studies of the UV finiteness conditions of the GS string [10,11] did not include R-R couplings, but special cases of AdS5 × S 5 [37] and pp-wave backgrounds [38] were explicitly discussed.The vanishing of the beta-functions for the R-R couplings was not checked as the fermionic coordinate was assumed to have trivial background. 33In this Appendix we use α, β, γ, ... In (G.5) the indices X and Y stand for multi-indices of the same type as the one carried by the fermions.The 2 × 2 matrices appearing in L 2f are s ≡ s where F n = e −ϕ F n are not required a priori to be field strengths. 34  We shall first fix the κ-symmetry gauge θ 1 = θ 2 and also consider flat 2d space (or, equivalently, fix the conformal gauge for 2d diffeomorphisms) and then expand (x, θ) near some background values (x, Θ).The aim will then be to compute the one-loop UV divergences that renormalise the R-R couplings in the quadratic fermionic term (G.3), i.e. ( F = F(x)) Here the classical term LF 2f and the expected divergent term δ LF 2f are decomposed into parityeven and parity-odd parts containing the linearly-independent combinations of antisymmetrised products of Dirac matrices.The combinations E and O should then represent the R-R betafunctions that should be set to zero modulo use of equations of motion on (x, Θ) or modulo target space (super)reparametrisations. A further contribution to the two-fermion divergence is the first term in δL 2f ; it should be proportional to the NS-NS fields (vielbein and H) beta-functions and thus should contain terms independent of the R-R fields.
The normal coordinate expansion of the R-R tensor fields ( F ≡ F(x)) takes a simpler form using tangent space indices: Note that the beta-functions for the couplings F a 1 ...an and F µ 1 ...µn are related by extra terms involving beta-functions of vielbein e a µ or the metric G µν ; this is related to the presence of β G terms in (4.5) or (4.9)-(4.11).
The expanded Lagrangian (G.1) has the following structure: where Y 0F and Y 2f contain zero and two background fermions Θ, respectively.We have also defined ρ α ≡ ζ a α Γ a .It will be sufficient to further assume that the induced metric is trivial, i.e.The tensors H are given iteratively by:  The same applies to integrals that lead to derivatives acting on two of the three vertices.
Using the coefficients (G.22) extracted from the expanded Lagrangian (G.21), one finds that the divergent term δL 1 in (G.24) is given by where Σ o , Σ e were defined in (G.9) and I 0 ∼ 1 ϵ is the UV pole factor.Comparing (G.32) to the corresponding terms in the classical action (G.8) one can read off the contributions to the beta-functions for the R-R couplings.Projecting onto the independent set of Dirac matrices Γ a 1 ...an we indeed observe the presence of the Hodge-de Rham operator terms as in (4.9)-(4.11).There are also similar terms depending on the H 3 field strength and its derivatives.The UV singular terms in δL There are apparently also other UV singular terms that do not appear in (4.9)-(4.11):terms containing two Ψ αβ a factors are independent of the R-R fields and contain only the H 3 strength and factors of the curvature tensor.We expect such terms to combine into the beta-function of the NS-NS fields entering the couplings (G.10) and thus yield the same scale invariance conditions as in eqs (2.1),(2.2).Moreover, all terms in δL 2 , δL 3 and δL 4 which do not contain Ψ αβ a are bilinear in R-R fields and all terms in δL 3 contain at least one additional factor of either H 3 flux or the curvature tensor.We expect such terms to cancel or to vanish upon use of the NS-NS scale invariance conditions (2.1),(2.2).

. 9 )
For c = 0 (i.e. when φ is isometric) these fields together with the T-duality transformed dilaton ϕ = φ − â would solve the standard Weyl invariance equations (1.3),(1.4)with X m = ∂ m ϕ, Y m = 0.For non-zero c the equation Rmn + 2 Dm Dn φ = 0 (for the original background (1.8)) expressed in terms of the dual fields G, B will contain additional c-dependent terms obstructing (for non-constant a(x)) the introduction of a new dilaton scalar.Still, they can be put in a more general form R mn + D m X n + D n X m = 0 with a special vector X given by 8 X m dx m ≡ I m dx m + Z m dx m = c e −2a dy + ∂ µ ( φ − â) + c Âµ dx µ .(1.10)

. 28 ) 24
Using the formulae inAppendix A we find the following relations for C This generalises the usual (c = ĉ = 0) discussion of the T-duality invariance of the string effective action (1.7) with √ G e −2ϕ = √ g e −2φ .

14 ) 34
Let us make a comment on different normalizations of RR field strength F5 used in the literature.In this paper we use the now standard normalization in which the type IIB Lagrangian is L = e −ϕ R − 1 4•5!F 2 5 + ....With the selfduality condition on F5 imposed at the level of equations of motion the analog of the Einstein equation becomes as in (2.1),(2.3):Rmn = 1 4•4! e 2ϕ FmpqrsF pqrs n + ....In particular, in the case of unit-radius AdS5 × S 5 solution with Rmn AdS = −4gmn AdS one has e ϕ F5 = 4(ϵ5 + ϵ * 5