Ultraviolet properties of noncommutative non-linear sigma-models in two dimensions

We discuss the ultra-violet properties of bosonic and supersymmetric noncommutative non-linear sigma-models in two dimensions, both with and without a Wess-Zumino-Witten term.

There has been a great deal of recent interest in noncommutative (NC) quantum field theories, stimulated by their connection with string theory and M -theory; for a review and comprehensive list of references see Ref. [1]. Most of this interest has focussed on four-dimensional theories. However, since two-dimensional theories have often been used as laboratories for investigating general properties of quantum field theories, it is natural to extend the discussion to this arena. Two-dimensional non-commutative non-linear σmodels have been discussed in Refs. [2] [3]. A particularly interesting case to consider, by virtue of its conformal invariance properties, is the Wess-Zumino-Witten (WZW) model. This has been studied in the NC case in Refs. [2] [4]. The NC WZW term is also discussed in Ref. [5] and the Kac-Moody algebra associated with the NC WZW model has been investigated in Ref. [6]. Moreover, its renormalisation has been carried out at one-loop order [7]. Our purpose in this paper is to continue the program of perturbative investigation of the NC WZW model, and also the NC version of the principal chiral model (i.e. the theory defined on a group manifold without the WZW term). We show how results for the NC U N WZW, and also principal chiral, model may be obtained from the leading-N term in the corresponding result for the commutative SU N theory.
Firstly we discuss the elements of the construction of NC field theories. The algebra of functions on a noncommutative space is isomorphic to the algebra of functions on a commutative space with coordinates x µ , with the product f * g(x) defined as follows where Θ is a real antisymmetric matrix. Quantum field theories analogous to the corresponding commuting theories are now straightforward to define, with * -products replacing ordinary products. In particular the noncommutative two-dimensional Wess-Zumino-Witten (WZW) model is defined by where as usual Σ is the boundary of a three-dimensional manifold B, and g is a map from Σ (or its extension B) into U N . (Note that SU N is not a group under the * -product, whereas U N is.) ǫ µνρ is the three-dimensional alternating symbol. A subscript * in Eq. (2) indicates that every product of fields within the corresponding brackets is a * -product.
We assume that the co-ordinates x 0 , x 1 on the worldsheet are non-commutative, but the extended co-ordinate x 2 on the manifold B commutes with the others. The group-valued field g is defined as where φ is in the Lie algebra of U N . φ can be expanded as where T a , a = 1, . . . N 2 − 1 are the generators of SU N , T 0 = 2 N 1 N , and with our conventions The U N structure constants f abc are totally antisymmetric, with f 0ab = 0 and f abc , a = 1, . . . N 2 − 1 being the structure constants of SU N . The commutative version of the theory is the sum of the commutative SU N theory together with a free scalar field. Later on we compare the β-function for λ in Eq. (2) with the corresponding β-function for the commutative SU N theory.
The ultra-violet properties of the NC WZW model may be investigated using the background field method. We expand the field g around a classical background g c as g = g c * g q , and express g q in terms of a quantum fluctuation π as The expansion of the action may then be effected straightforwardly [8]; we readily obtain where with ǫ µν the two-dimensional alternating symbol, and then derive an expansion in terms of π by using (together with a similar relation with λ → −tλ). The difference between noncommutative and commutative theories at the level of Feynman diagrams is that in the NC case, Feynman diagrams can acquire momentum dependent phase factors arising from the *product. If such a factor contains a loop momentum, the UV divergence for that loop is suppressed. Since the π are adjoint fields in U N , the detailed discussion is simplified by using the diagrammatic notation originally introduced by 't Hooft [9], where we represent a π A B propagator by a double line as in Fig. 1, the arrow pointing towards the upper index. We start by considering conformal invariance properties of the WZW model. At the critical point the NC WZW model becomes conformally invariant, as discussed in Ref. [6]. In the commutative case the result can be derived straightforwardly starting from the commutative version of Eq. (7)[8](for the generalisation to an arbitrary parallelised manifold see Ref. [11]). We sketch the proof here. Feynman diagrams are constructed with vertices derived from the expansion of Eq. (7) in terms of π; the propagator, derived from the term in Eq. (7) quadratic in π, is simply η µν k 2 . Note that there are two sorts of vertex; those with one derivative acting on a quantum field π and one factor of g −1 c ∂ ν g c , (Type A) and those with two derivatives acting on π and no factors of g −1 c ∂ ν g c (Type B). Each vertex contains a factor of P µν , although by symmetry only the η µν or the ǫ µν in P µν contributes to the Type B vertices with even or odd numbers of πs respectively. The contributions to the renormalisation of λ arise from logarithmically divergent diagrams, which contain two Type A vertices and an arbitrary number of Type B vertices. (The WZW term is not renormalised [12].) If one is using dimensional regularisation, it is necessary to have a prescription for products of ǫ tensors, valid in d = 2 dimensions. The simplest is to define [13] ǫ µρ ǫ ρν = δ µ ν , the contraction here being effected by the d-dimensional metric. (This definition leads to conformal invariance at the critical point (Eq. (10)) without the necessity of additional finite counter-terms.) Crucial is that we now have in d dimensions After performing all the Feynman integrals and implementing all the resulting tensor algebra, the final result is proportional to P µρ P ν ρ Tr[∂ µ g c g −1 c ∂ ν g c g −1 c ] * . At the critical point Eq. (10) we have P µρ P ν ρ = 0, and therefore there are no corrections to λ. Clearly, since this proof relies only on the tensor structure of the vertices and not on the details of the Feynman diagrams, it is unaffected by the additional phase factors present in the NC case.
Moreover, the proof in the commutative, and hence also the NC case, is equally valid for the supersymmetric theory (which will be defined explicitly later), as was emphasised in Ref. [14].
Another result for the commutative bosonic SU N case, which can be proved using conformal field theory arguments, is [15] ∂β λ ∂λ 2 We can expand Eq. (13) as This result can be interpreted as a perturbative loop expansion. Each term is leading order in N for the corresponding loop order. Therefore the result for the NC U N theory will be identical, and we deduce that Eq. (13) is also valid for the U N NC WZW model. The β-function β λ for the commutative WZW model has been computed up to three loops in Ref. [16]. After specialising to SU N , the result is leading order in N at this order and hence the result is identical in the NC U N case. For completeness, we quote it here: where η = kλ 2 4π and ρ = λ 2 N 4π . (In fact three-loop results have also been given in Ref. [17], but apparently in a different renormalisation scheme.) It is easy to verify that Eq. (15) is compatible with Eqs. (13), (14); notice that in taking the derivative with respect of λ 2 of Eq. (15), only the terms arising from hitting the (1 − η 2 ) factor survive because the result is to be evaluated at η 2 = 1.
Let us now turn to the supersymmetric case. The NC supersymmetric WZW model has the superspace action [18] where t ≡ x 2 , and θ α are the Grassman co-ordinates and G is now a superfield and a group element of U N , defined in terms of a superfield Φ as G = exp * (iΦ). The supercovariant derivative D is defined by and γ 3 = γ 0 γ 1 . In the commutative supersymmetric SU N case β λ is given through three loops by i.e. the two [19] and three [20] in other words the result for ∂β λ ∂λ 2 λ 2 = 4π k is one-loop exact. This result will clearly be equally valid in the NC case. Eq. (19) is consistent with the perturbative results through three loops, and predicts that β λ at four loops and beyond should be proportional to (1 − η 2 ) 2 .
Results have been presented at the 4-loop level [24] for a general N = 1 supersymmetric 1 In the torsion-free case this was shown for a general manifold in Refs. [21], [22] σ-model with torsion, but the specialisation to the group manifold case appears to be incorrect and we have been unable to verify Eq. (19) at this level.
In view of this uncertainty at four loops in the case of the WZW model, we now turn to the case of the NC σ-model defined on a group manifold without a WZW term-i.e.
the NC version of the principal chiral model. Once again, we can obtain the NC results for the β-functions for the group U N simply by picking out the leading N behaviour of the corresponding commutative results for SU N . We start with the bosonic case. Results are available for the general bosonic σ model at two [25][21], three [26] and four [27] loops, expressed in terms of the Riemann tensor for the target space metric. The results for the SU N case may be obtained by substituting the appropriate Riemannn tensor; in terms of general co-ordinatesφ k on the (commutative) group manifold, we have where f abc are the structure constants for SU N and e k a are the vielbeins for the metric on the group manifold, defined by e k a e l a = g kl , g kl e k a e l b = δ ab .
We find We deduce that the result in the NC U N case is given by Finally we turn to the case of the supersymmetric principal chiral model. As we already know from our earlier discussion of the WZW model, in this case the first non-zero contribution to the β-function beyond one loop appears at four loops [28].  (23); and we have deduced that in the supersymmetric version of this theory, the β-function vanishes from two through four loops. This tempts us to speculate that β λ may be one loop exact in this case, at least when using standard dimensional reduction. It is not clear however how to determine whether there are any general reasons why this result should persist at higher orders. The generally covariant methods used in the calculation of β-functions for general σ models as in Ref. [28] are not very well adapted for extracting the leading N behaviour; on the other hand, we have repeated the 4-loop N = 1 computation using the non-covariant expansion as in Eq. (7), and extracted the contributions which are planar in terms of 't Hooft's double line notation, but this does not seem to afford any general insights.