Renormalisability of non-homogeneous T-dualised sigma-models

The quantum equivalence between sigma-models and their non-abelian T-dualised partners is examined for a large class of four dimensional non-homogeneous and quasi-Einstein metrics with an isometry group SU(2) times U(1). We prove that the one-loop renormalisability of the initial torsionless sigma-models is equivalent to the one-loop renormalisability of the T-dualised torsionful model. For a subclass of Kahler original metrics, the dual partners are still Kahler (with torsion).


Introduction
The subject of target space duality, or T-duality, in String Theory and in Conformal Field Theory has generated much interest in recent years and extensive reviews covering abelian, non-abelian dualities and their applications to string theory and statistical physics are available in the literature [1,2,3]. The geometrical aspects of this duality can be found in [4]. T-duality provides a method for relating inequivalent string theories. First discovered for the case of σ-models with some abelian isometry, the concept of T-duality has been recently enlarged to theories with non-abelian isometries [5,6,7]. A very important and interesting property of T-duality applied on non-abelian isometry is that it can map a geometry with such isometries to another which has none. Therefore, non-abelian T-duality can not be inverted as in the abelian case.
By showing that T-duality is a canonical transformation [8,9,5], it was proved that theories in such way related where classically equivalent. Furthermore, this equivalence was still remaining at the one-loop level, in a strict renormalisability sense, in all the many example that have been tested up to now to this duality, with an emphasis put on SU(2) [10,11,1,12,13,7]. For example, this one-loop equivalence still remains for principal σ-models whatever strongly broken the right isometries may be [14]. The non-abelian dualisation of non-homogeneous metrics such as the Schwarzschild black hole or Taub-NUT was performed in [7], [12] and in [15]. We propose here the dualisation of the general SU(2) × U(1) metrics.
Problems arise when one addresses the question of the renormalisability of dualised theories beyond the one-loop order. It had been proved that even for the simplest (SU(2) × SU(2))/SU(2) principal σ-model, the dualised theory is not two-loop renormalisable, in the minimal dimensional scheme [16,17]. However, as shown in [18], a finite deformation at the order of the dualised metric is sufficient for recovering a two-loop renormalisability for this particular model. As it will be shown, the SU(2) × U(1) σ−models are not in general two-loop renormalisable, even though the one-loop renormalisability remains for their dual partners ! The content of this article is the following : in section 2, we recall the general expression of the SU(2) × U(1) metrics and set the notations. In section 3, we make a review of such metrics which give rise to one-loop renormalisable σ−models, as for example the celebrated Taub-NUT and Eguchi-Hanson metrics. In section 4, we show that only the particular metrics where homogeneity is recovered by some enhancement of the isometries are two-loop renormalisable. In section 5, we dualise the original theory and show in section 6 that the one-loop renormalisability survives during the dualisation process. When the original metric is Kähler, we investigate in section 7 if such a property is still present for the dual partner. Some concluding remarks are offered in section 8.
If ε = +1, the triplet of 1-forms σ is changed under infinitesimal transformations of su(2) L ⊕ su(2) R as Therefore σ is a SU(2) L singlet and a SU(2) R triplet. If β(t) = γ(t), the SU(2) R isometries will be broken down to a U(1) and the total isometry group of the metric will then be SU(2) L ×U(1). Indeed, in order to keep the metric invariant, one then must have ǫ R = {0, 0, µ}. If ε = −1, σ is changed under infinitesimal transformations of su(2) L ⊕ su(2) R as and therefore the isometry group of the metric will be SU(2) R × U(1). The choice of ε switches also the autodual components of the Weyl tensor (W + ↔ W − ). In all cases, when β(t) = γ(t), the metric has for isometry group SU(2) L × SU(2) R and is conformally flat. It is then possible to define the σ-model corresponding to these metrics with {φ 0 = t, φ 1 = θ, φ 2 = ϕ, φ 3 = ψ}, and address the question of its one-loop and two-loop renormalisability.
In order to derive the Ricci tensor, we define the vierbein {e a |a ∈ {0, 1, 2, 3}} as In the absence of torsion, the condition for giving one-loop renormalisability is the quasi-Einstein property of the metric : where the Einstein constant λ will renormalise the coupling while the vector v will renormalise the field.

One-loop renormalisation
We will only consider metrics satisfying condition (2) so that the corresponding σ-models are one-loop renormalisable. Of course, as we want to keep the SU(2) symmetry while renormalising, we will only consider here vectors v that depends only on the t coordinate : v = v(t). As the expression of the SU(2) × U(1) metric (3) we chose does not mix dt, σ 1 , σ 2 and σ 3 , both the metric g and the Ricci tensor Ric will be diagonal in the {dt, σ 1 , σ 2 , σ 3 } basis and this will hold in the vierbein. As a consequence, D (a v b) must be also diagonal ; this is true only for vectors of the form v = v 0 (t) e 0 + ρ γ(t) e 3 . The constant ρ is arbitrary as γ(t) e 3 is in fact the form dual to the Killing ∂ ψ . We will take ρ = 0. In order to simplify matters, from now on, we will choose the coordinate t so that β(t) = t. The metric now writes All this being settled, the quasi-Einstein character of the metric (2) can now be expressed as a set of three non-linear differential equations which are : This system is difficult to solve, even though it can still be done for some limited cases as the Einstein one (v 0 = 0) and the quasi-Einstein Kähler one. It is possible to eliminate α(t) and v 0 (t) in the system (4), leading to a single, deeply non-linear, differential equation of the fourth order in γ(t). The general SU(2) × U(1) quasi-Einstein metric should therefore depend on four parameters.
In order to convince the reader of the large class of models that will be dualised, we will now give a short review of the SU(2) × U(1) Einstein and quasi-Einstein Kähler metrics.

Einstein metrics :
The metric g will be Einstein if Ric = λ g. It is possible to integrate the differential system (4) imposing v 0 = 0 and one gets A and B being the integration constants. This family contains many metrics of interest which we recall briefly. If A = 0, we recover the Kähler-Einstein extension of Eguchi-Hanson [20]. If A = 0 then g identifies with the large class of Einstein metrics derived by Carter [21]. By making the change of coordinates, one can have for g a more simple expression : Notice that as A and B are real constants, M and n can be both reals or pure imaginaries. Defining 2 n dψ = dΨ and taking the limit n → 0 gives the Schwarzschild metric with cosmological constant : Other limits of (6) lead to the Page metric on P 2 (C)#P 2 (C) and to the Taub-NUT metric.

Quasi-Einstein Kähler metrics :
These are the only SU(2) × U(1) quasi-Einstein metrics known up to now [22]. We suppose here that there is a choice of holomorphic coordinates on which the isometries SU(2) × U(1) act linearly. It happens that this hypothesis implies the integrability of the complex structure. A necessary condition of the Kähler property is the closing of the Kähler form : It is clear that this relation will hold iff β ′ (t) 2 = α(t) γ(t), i.e. α(t) = 1 γ(t) . It is then possible to solve system (4) and one gets for the metric and for the vector v : with where C and D are the integration constants.
In the limit C → 0, we have v = 0 and thus we are back to the Kähler-Einstein metrics, i.e. the Kähler-Einstein extension of Eguchi-Hanson (the correspondence between the parameters is then D = B) 1 .

Two-loop renormalisation
The two-loop divergences, first computed by Friedan [19], are In order to re-absorb these divergences, the counter-terms may come from the renormalisation of the coupling T and the fields φ, but also from the renormalisation of the parameters that were let in the metric at one-loop. For example, if one starts with the Einstein metric (6), one should allow for counter-terms renormalising the parameters M, n. In general, if we define such parameters as ρ c , the theory will be renormalisable at two loops iff one can find some vector v =ṽ(t) and some constantsλ and χ c such that We will show that, except for the few particular cases where the metric is homogeneous 2 , the SU(2) × U(1) Einstein and Kähler metrics do not give in a direct way two-loop renormalisable σ-models.

Einstein metrics :
In the vierbein basis, one can compute the two-loop divergences for the metric given in (6) and find : Quite surprisingly, the two-loop divergences are conformal to the original metric. Relation (9) in the vierbein basis becomes were E ai is defined by e a = E ai dφ i . As for the one-loop renormalisation conditions (4), this last relation gives us three equations. These can easily be reduced to two by eliminatingṽ.
The remaining equations will only depend on the variable t and on the constantsλ, χ n and χ M . As these must vanish irrespectively of the values taken by t, one can show that they will be verified in only two particular cases where M and n are fixed such that In both cases, (9) will be satisfied withλ = λ 2 3 and χ M = χ n =ṽ = 0, but it is not surprising as these choice for M and n are the one which enlarge the SU(2) × U(1) isometries to SO (5), making the metric homogeneous (de Sitter metric).

Kähler metrics :
Proceeding as for the Einstein metrics, one can compute the two-loop divergence using the metric (8). Once again, the parameters C and D must have special values for the action to be two-loop renormalisable. Indeed, one must have (C = 2 λ, D = 0) or (C → 0, D = 0). In the first case, we recover flat space. In the second case, we get the Fubiny-Study metric on P 2 (C) and its non compact partner which are also two-loop renormalisable withλ = 2 3 λ 2 andṽ = 0.
The Einstein and Kähler metrics with no more isometries than SU(2) × U(1) are therefore not renormalisable in the minimal scheme at two loops. This could of course be cured by adding some infinite deformation of the metric itself as in D. Friedan's approach to σ models quantisation, but it is the author belief that a finite deformation keeping the isometries, as explained in [18], would be sufficient 3 .

The dual metric
We dualise the initial metric (3) over the SU(2) isometries, keeping aside the U(1). Practically, it consists in dualising the three dimensional metric [15] leaving the term α(t) dt 2 unchanged. If we define the new fields of the dual metric λ i , i ∈ {1, 2, 3}, the dual theory of g 3 will writes, in light-cone coordinates : After the following change in coordinates : one has for the total dual metricĝ = α(t) dt 2 +Ĝ 3(ij) dλ i dλ j : where ∆ = y 2 t + r 2 + t 2 γ(t) .
The torsion is defined by T = 1 2 dH where H = 1 2Ĝ 3[ij] dλ i ∧ dλ j is the torsion potential 2-form : We defineĝ ij as the tensor associated to the metric (10) andĥ ij as the torsion potential. Let G ij =ĝ ij +ĥ ij andRic be the new Ricci tensor which is not symmetric anymore because of the presence of torsion in the dualised model. Eventually, the dualised action of our SU(2) × U(1) theory is, in light-cone coordinates : where the coordinates are {φ 0 = t,φ 1 = r,φ 2 = y,φ 3 = z}. It could be useful to notice that detĝ = t 2 y 2 ∆ 2 α(t) γ(t) .
It was proved in [12] that the dualised Eguchi-Hanson model is conformally flat. We have checked that, in the class studied here, this is the only case where the Weyl tensor vanishes.

The SO(3) dual of Schwarzschild :
In the special case λ = 0, we recover the SO(3) dual of Schwarzschild which was one of the first examples for non-abelian duality [7]. While making n → 0, the torsion potential 2-form H (11) writes as d Ψ dr 2n + O(n), and therefore, as H is only defined up to a total derivative, the torsion vanishes, which is consistent with the result found in [7].
We will now address the question of the one loop renormalisability of the dual theoryŜ.

One-loop renormalisation of the dual metric
We want to prove that the one-loop renormalisation property does survive to the dualisation process. In other words, if the torsionless action (1) is quasi-Einstein, then so is the action (12). In the presence of torsion, this now means that one can find some constantλ and some vectorsv andŵ such thatR This equality gives a system of equations much more complicated than (4), but what is important is that now α(t) and γ(t) are not considered as unknown functions. Furthermore, as we suppose the original metric to be quasi-Einstein, the system (4) is assumed to be verified and one can easily derive from it, in an algebraic way, the three functions A, B and C such that : The procedure is the following : we choose some ansatz forλ,v andŵ and express relation (14). Then, in this last expression, we replace each occurrence of α ′ (t), γ ′ (t) and γ ′′ (t) by its expression in (15) and check if (14) holds. We have checked that (14) is verified taking where X is defined by X = r ∂ r + y ∂ y .
Conversely, let us now suppose thatλ,v andŵ are defined by (16) where λ and v are supposed to be arbitrary. It is possible to show that if (14) holds, then the original metric is quasi-Einstein with Ric ij = λ g ij + D (i v j) . In order to demonstate this, we first define the three functions f A (t), f B (t) and f C (t) such that : Assuming that (14) holds, and after having replaced each occurence of α ′ (t), γ ′ (t) and γ ′′ (t) by its value in (17), we get some equation system where the unknowns are the functions f X (t). As this last system must hold irrespectively of the values taken by r and y which are free variables, one can then prove that f A (t) = f B (t) = f C (t) = 0. This shows that (15) holds and therefore the quasi-Einstein property of the original metric.

Remarks :
• The cosmological constant does not change through the dualisation process as it was already proved for T-dualised homogeneous metrics [14]. That means that the coupling will renormalise in exactly the same way that in the initial theory : the one-loop Callan-Symanzik β function is the same for the initial and dualised SU(2) × U(1) theories.
• As one could expect, the coordinate t which was a spectator coordinate during the dualisation process plays a special role :ŵ t = 0 and, up to the D t log ∆ term,v t and v t are equal.
• The SU(2) symmetries where lost during the dualisation process, so at the end, there is just a U(1) symmetry left and therefore the Killing ∂ z is unique. Indeed,v andŵ are defined up to this Killing vector, which dual 1-form is K = y 2 t γ(t) ∆ dz. One then has • One can adress the question of the unicity ofλ,v andŵ which satisfy (14). There will be multiple solutions if one can find some Λ, V and W such that On the one hand,ŵ alone is obviously defined up to a gradient whilev andŵ together are defined up to the Killing vector K ; on the other hand, equivalence (18) shows that if multiple solutions exist forλ andv in the dualised metric, then such ambiguity will appear for the original metric. We have checked that, in our case of SU(2) × U(1) metrics, only flat metric leads to such possibilities 4 . Therefore, except for this trivial original metric and up to the already noticed freedom inv andŵ, (16) is the unique solution of (14).
• The SO(3) dual of the Schwarzschild metric (13) gives us a nice example of a torsionless quasi-Einstein metric with a U(1) as minimal isometry.

Conservation of the Kähler property
Bakas and Sfetsos decribed, for SUSY applications, how the complex structures were changed when hyper-Kähler metrics were T-dualised [24]. We propose here to show that when one starts with the original metric (8), the dual partner is still Kähler. If we defineσ i = −Ĝ si dφ s , it is possible to write the dual metric of (8) under the specific shapê g = 1 γ(t) dt 2 + t (σ 1 2 +σ 2 2 ) + γ(t)σ 3 2 .
One can then check that the 2-form is a Kähler form with torsion for the dual metric. Indeed, for the almost complex structurê J , we have where D is the covariant derivative with torsion. One should notice here that, in the presence of torsion, the closing condition on the Kähler form is replaced by dρ = (⋆ dH) ∧ρ .
The torsion potential 2-form H is given by the equation (11).

Concluding remarks
We have considered all of the four dimensional non homogeneous metrics with an isometry group SU(2) × U(1). We have shown that the dual partners are quasi-Einstein (with torsion) iff the original metrics are quasi-Einstein (without torsion). Let us emphasize that this was possible despite the fact that the explicit form of these metrics are not all known yet.
In [17], it was proven that, in the minimal dimensional scheme, the dualised SU(2) principal σ−model is not two-loop renormalisable although this property holds for its original model. Here, the one-loop renormalisability remains although the starting models are not in general two-loop renormalisable. This is another suggestion that the renormalisability beyond one loop for the original and dualised models are not linked. Indeed, it is our ansatz that for the dualised models investigated here, one could still define a proper theory up to two loops. This could be achieved by adding some finite deformation to the dualised metric, as it was done in [18] for the SU(2) principal σ−model, irrespectively of the two-loop renormalisability of the original theory.