New Scaling Limit for Fuzzy Spheres

Using a new scaling and a cut-off procedure, we show that φ theory on a deformed noncommutative R can be obtained from the corresponding theory on the fuzzy S × S. The star-product on this deformed R 4 is non-associative, and the theory naturally has an ultra-violet cut-off, Λ = 2/θ. We show that UV-IR mixing is absent to one loop, and that the β-function is the same as that of an ordinary φ theory. Noncommutative field theories provide a rich variety of interesting conceptual phenomena. The usual focus of research has been to try to understand the quantum behavior of theories defined on certain non-compact manifolds like R θ or compact ones like the fuzzy sphere S 2 F ,CP n etc. Theories on R θ have received considerable attention not only because they arise in string theory, but also because of their peculiar formal properties like UV-IR mixing [1]. Theories on compact noncommutative manifolds possess the attractive feature that they are simply finite-dimensional matrix models, and thus hold out the hope that they correspond to regularized versions of quantum field theories on ordinary manifolds. However, problems like UV-IR mixing are still present, although in a form different from their noncompact counterparts [2,3]. The fuzzy sphere S F can be “flattened” (by scaling the radius R and the cut-off l to infinity) to give the noncommutative plane, and UV-IR mixing re-emerges in the usual form [4]. Interestingly, as we will show in this article, if we start with a low energy sector (which we will define more precisely below) of the theory on S F ×S2 F , and simultaneously use a new scaling to flatten the spheres, we obtain a theory on a deformed Rθ (i.e. the star-product is non-associative) that has an ultra-violet cut-off Λ = 2/θ. Moreover, this theory shows no UV-IR mixing, as the noncommutative parameter θ and the UV cut-off Λ are intimately related. The fuzzy sphere is described by three matrices xi = θLi where Li’s are the generators of SU(2) for the spin l representation and θ has dimensions of length. The radius R of the sphere, θ and l are related: R = θl(l+1). The usual action for a matrix model on S F is S = R 2 2l+ 1 Tr ( [Li,Φ] [Li,Φ] R2 + μlΦ 2 + V [Φ] ) , (1) and has the right continuum limit as l → ∞. Because of the noncommutative nature of S F , there is a natural UV cut-off: the maximum energy Λmax is = 2l(2l + 1)/R . To get the theory on a noncommutative plane, the usual strategy [4] is to restrict to the neighbourhood of (say) the north pole (where L3 ≈ l), define the noncommutative coordinates as x 1,2 ≡x1,2 and then take both l and R to infinity with θ ′ = R √ l fixed, giving us the commutation relations [x 1 , x NC 2 ] = −iθ ′2. (2) In this limit, Λmax clearly diverges. [email protected] [email protected] 1 Here, we point out another scaling limit in which R, l → ∞ with noncommutative coordinates now given by X a = x F a / √ l, and θ = R/l fixed. This gives us [X 1 , X NC 2 ] = −iθ. (3) An immediate consequence of this scaling, as is obvious from the relation R = θl(l + 1) is that Λmax is finite and of order 1/θ, and there are no modes in the theory above this value. In other words, this scaling gives a theory on a noncommutative R where θ now has the physical interpretation of a UV cut-off, and thus making it appropriate for studying noncommutative field theories. It also ties in nicely with the properties of the star-product which acts like a cut-off, smoothly eliminating modes of characteristic length much smaller than θ [5, 6]. While the scaling (3) for obtaining Rθ is simply stated, obtaining the corresponding theory is somewhat subtle. We will need to modify the Laplacian on the fuzzy sphere to project out modes of momentum greater than 2 √ l. In other words, the theory on the noncommutative plane with UV cut-off θ is obtained by flattening not the full theory on the fuzzy sphere, but only a “low energy” sector. To get Rθ (which is our primary focus in this article), we work on S 2 F × S F and then take the scaling limit with θ = R/l fixed. By analogy with (1), the scalar theory with quartic self-interaction on S F × S F is S = R 2 a 2la + 1 R b 2lb + 1 TraTrb ( [L (a) i ,Φ] †[L (a) i ,Φ] R2 a + [L (b) i ,Φ] †[L (b) i ,Φ] R2 b + μlΦ 2 + λ4 4! Φ ) , Φ† = Φ. (4) where a and b label the first and the second sphere respectively, L (a,b) i ’s are the generators of rotation in spin la,b-dimensional representation of SU(2), and Φ is a (2la+1)×(2la+1)⊗(2lb+1)×(2lb+1) hermitian matrix. As la, lb go to infinity, we recover the scalar theory on an ordinary S 2 × S. Actually it is enough to set la = lb = l and Ra = Rb = R, and this corresponds in the limit to a noncommutative R 4 with a Euclidean R2×R2 metric. The general case simply corresponds to different deformation parameters in the two R factors [8] and the extension of all our results to this case is obvious. Following [7, 2] the field Φ can be expanded in terms of polarization operators [9] as follows Φ = (2l + 1) 2l