Field theory on a fuzzy noncommutative sphere can be considered as a
particular matrix approximation of field theory on the standard commutative
sphere. We investigate from this point of view the scalar $\phi^4$ theory. We
demonstrate that the UV/IR mixing problems of this theory are localized to the
tadpole diagrams and can be removed by an appropiate (fuzzy) normal ordering of
the $\phi^4$ vertex. The perturbative expansion of this theory reduces in the
commutative limit to that on the commutative sphere.Comment: 6 pages, LaTeX2e, Talk given at the NATO Advanced Research Workshop
  on Confiment, Topology, and other Non-Perturbative Aspects of QCD, Stara
  Lesna, Slovakia, Jan. 21-27, 200

AP Balachandran

BP Dolan

C Chong-Sun

H Grosse

J Madore

P Presnajder

S Minwalla

S Vaidya

English

arXiv

Field theory on a fuzzy noncommutative sphere can be considered as a particular

matrix approximation of field theory on the standard commutative sphere.We investigate

from this point of view the scalar Ø4 theory. We demonstrate that the UV/IR mixing

problems of this theory are localized to the tadpole diagrams and can be removed by an

appropiate (fuzzy) normal ordering of the Ø4 vertex. The perturbative expansion of this

theory reduces in the commutative limit to that on the commutative sphere

Dolan, Brian P.

O'Connor, Denjoe

Presnajder, Peter

MURAL - Maynooth University Research Archive Library

arXiv:hep-th/0204219v1  25 Apr 2002MATRIX MODELS ON THE FUZZY SPHEREBRIAN P. DOLANDepartment of Mathematical Physics,National University of Ireland, Maynooth, IrelandDENJOE O’CONNORSchool of Theoretical Physics,Dublin Intistitute of Advanced Study, Dublin, IrelandAND PETER PRESˇNAJDERDepartment of Theoretical Physics,Comenius University, Bratislava, SlovakiaAbstract.Field theory on a fuzzy noncommutative sphere can be considered as a particularmatrix approximation of field theory on the standard commutative sphere. We investigatefrom this point of view the scalar φ4 theory. We demonstrate that the UV/IR mixingproblems of this theory are localized to the tadpole diagrams and can be removed by anappropiate (fuzzy) normal ordering of the φ4 vertex. The perturbative expansion of thistheory reduces in the commutative limit to that on the commutative sphere.1. IntroductionThe field-theoretical models possessing a finite number of field modes can bepresented as matrix models. To this class of models belong, e.g., the modelson a finite cubic lattice subjected to some periodic boundary conditions.Another class of such models represent models on a fuzzy sphere, [1]-[3].The basic idea behind fuzzy sphere is quite simple:(i) The standard sphere S2 with radius R is a co-adjoint orbit possessinga Poisson bracket, and it can be considered as a particular ”phase-space”with finite volume 4πR2;(ii) After ”quantization” (deformation), this ”phase-space” is effectivelydivided to cells of the size ∼ ρ2 (ρ is a deformation parameter), each repre-senting one field mode. Thus, the total number of field modes is N ∼ R/ρ.The resulting ”quantum” space is known as the fuzzy sphere S2F , for detailssee [4].2In the framework of lattice models it is of great interest to investigatetheir properties in the continuum limit when, in a proper sense, the latticespacing a approaches zero: a→ 0. For models on a fuzzy sphere the commu-tative limit ρ→ 0 (or equivalently, N →∞) plays a similar role. A simplescalar field model on a fuzzy sphere with φ4 interaction was proposed in[1].A potential problem for this program has emerged due to the phe-nomenon of UV/IR mixing, [5]. The problem was discussed on a noncom-mutative Moyal space, where fields possess an infinite number of modes,and consequently a regularization procedure is needed. The phenomenonappears to be generic for noncommutative spaces.It was pointed out in [6] that the UV/IR mixing is present in the φ4model with naive action introduced in [1]. This work was followed by [7],where the one-loop contribution to the two-point vertex function was cal-culated explicitely, and it was shown that in the commutative limit there isa finite correction to the expected commutative contribution. Moreover, inthe planar limit it incorporates the UV/IR mixing singularity of the Moyalspace.The implications of this result are very serious for the program of us-ing matrix model approximation to continuum field theories. Recently, in[8] we found that there is a quite natural solution and that the problemdisappears when the interaction term is properly (fuzzy) normal ordered.The resulting action is therefore the correct starting point for fuzzy latticeapproximations.In the next section we briefly describe the Euclidean scalar field withφ4 interaction on a commutative sphere. Then we present its fuzzy versionand formulate all essential results found in [8]. The last section contains abrief summary.2. Model on a standard commutative sphere.We consider a real Φ(~n) defined on a usual sphere S2 with radius R:~n = (n1, n2, n3), ~n2 = R2. The Euclidean field action for Φ(~n) with Φ4(~n)interaction is given asS[Φ] =∫dn [12(LiΦ)2 +r2Φ2 +λ4!Φ4] , (1)where dn = sin θdθdϕ is the standard measure on S2 normalized to 4π, andLi = iεijknj∂nk , i, j, k = 1, 2, 3, are the usual generators of rotations.The Euclidean QFT can be formulated, e.g., in terms of functionalintegrals. If F [Φ] is some functional depending on field Φ = Φ(~n), then3its quantum average is defined by〈F [Φ]〉 =∫DΦ e−S[Φ]F [Φ] . (2)Here the functional measure DΦ is normalized by 〈1〉 = 1. It can be shownthat the model specified by (1) and (2) can be formulated rigorously, andmoreover, that it possesses the perturbative expansion in λ which is Borelsummable.Therefore, we can restrict ourselves to the standard perturbation ex-pansion of the model based on on (1). It is well known that it contains onlyfinite Feynman diagrams except the tadpole diagram which contributiongL(r) =14πL∑l=02l + 1l(l + 1) + r(3)is logharithmically divergent in the cut-off parameter L. In interactionpicture the interaction term in the action is normal orderedSint[Φ] =∫dnλ4!: Φ4 : , (4)where the normal ordering is specified by: Φ4 : = Φ4 − 12g(L, t)Φ2 . (5)The parameters r and t are related by r = t − (λ/2)gL(t). This proce-dure eliminates all tadpole diagrams, and the resulting perturbative seriescontains only finite Feynman diagrams.3. Model on the fuzzy sphereThe standard sphere S2 with radius R is fully characterized by its Cartesiancoordinates ni, i = 1, 2, 3, satisfying relations[ni, nj] = ninj − njni = 0 ,∑in2i = R2 . (6)In the noncommutative case one replaces these commuting coordinates byoperators nˆi, i = 1, 2, 3, satisfying relations[nˆi, nˆj ] = nˆinˆj − nˆjnˆi = iρεijknˆk ,∑inˆ2i = R2 . (7)If R2/ρ2 = L(L + 1) with L positive integer, then nˆi, i = 1, 2, 3, can berealized as (L + 1) × (L + 1) matrices (in fact, conditions (7) determine4the spin s = L/2 representation of SU(2) group). The scalar field is now apolynomial in nˆi, i.e. a general (L+ 1)× (L+ 1) matrix Φˆ = Φ(nˆ).The fuzzy analogs Lˆi, i = 1, 2, 3, of the differential operators Li, act onan arbitrary matrix fˆ = f(nˆ) as follows:Lˆifˆ = [Lˆi, fˆ ] , Lˆi = ρ−1nˆi , i = 1, 2, 3 . (8)In the fuzzy case, the integral over S2 is replaced by the normalized trace∫dn f(~n) →4πL+ 1Trf(nˆ) . (9)In the commutative limit ρ → 0, i.e. L → ∞, eqs. (8) and (9) reduce,respectively, to the standard definitions of generators of rotations and theusual integral over S2.The modifications indicated in (8) and (9) lead to the na¨ıve fuzzy fieldaction proposed in [1]:SL[Φˆ] =4πL+ 1Tr[12(LˆiΦˆ)2 +r2Φˆ2 +λ4!Φˆ4] . (10)Of course, this gives a tadpole contributions to the pertubation series, whichindeed, are finite since the number of field modes is finite.However, in the noncommutative case the permutation symmetry of thevertex contributions, originated from∫dnΦ4, is reduced the cyclic symme-try of the fuzzy vertex 4π(L + 1)−1 TrΦˆ4. This yealds two independenttadpole contributions: instead of 12 tadpole subtractions in (5), we havenow 8 subtractions of the planar tadpole diagram and 4 subtractions ofthe non-planar one. The planar diagram gives the same contribution asthe tadpole in the commutative case, whereas the nonplanar possesses acorrection. Therefore, the correct fuzzy interaction is, [8]:Sint[Φˆ] =4πL+ 1Tr : Φˆ4 :] , (11)with the normal ordering defined as follows:Tr : Φˆ4 : = Tr[Φˆ4 − 12∑lmΦˆYˆlmYˆlmΦˆl(l + 1) + t+ 2∑lm[Φˆ, Yˆlm]†[Φˆ, Yˆlm]l(l + 1) + t]. (12)Here, Yˆlm are fuzzy analogs of spherical functions determined by the con-ditions:Lˆ2i Yˆlm = l(l+1)Yˆlm , Lˆ3Yˆlm = mYˆlm ,4πL+ 1TrYˆ †lmYˆl′m′ = δll′δmm′ . (13)5The middle term in (12) is the usual tadpole subtraction and correspondsto a normal ordering in the commutative vertex. The last term is the addi-tional noncommutative subtraction that is necessary to obtain the correctcommutative limit.4. ConclusionsThe correct matrix model that represents a lattice regularization of thecommutative theory is given by the field action with the interaction termgiven by (11) and (12). It can be put into the formS˜L[Φˆ] =4πL+ 1Tr[12Φˆ(Lˆ2 +λ2QL(Lˆ2) + t− gL(t))Φˆ +λ4!Φˆ4] . (14)The term (λ/2)gL(t), given by the middle term in (12), corresponds to theusual tadpole subtraction (the parameter r is replaced by t − (λ/2)gL(t),exactly, like in the commutative theory). The term (λ/2)QL(Lˆ2) is given bythe manifestly positive last term in (12). It interpreted as the momentumdependent wave-function renormalization since QL(Lˆ2) is a power series inLˆ2 which starts at Lˆ2, and therefore we could have written (see, [7], [8]):Lˆ2 +λ2QL(Lˆ2) = Lˆ2ZL(Lˆ2) .This term exactly cancels the unwanted momentum dependent quadraticterms in the effective action arising from non-planar diagrams in the fuzzytheory. This cancellation then guarantees that the continuum limit of thistheory is the standard scalar φ4 on the sphere.The modified action (14) can serve for the non-perturbative definitionof the fuzzy quantum averages〈F [Φˆ]〉L =∫DLΦˆ e−S˜L[Φˆ]F [Φˆ] . (15)Here, DLΦˆ = NLΠlmdclm is a finite-dimensional measure associated toarbitrary field configurationsΦˆ =L∑l=0+L∑m=−LclmYˆlm , cl,−m = c∗l,m − complex , (16)with the normalization constant NL fixed by 〈1〉L = 1. The action S˜L[Φˆ] in(15) guarantees the correct commutative limit L→∞ of quantum averages〈F [Φˆ]〉L for any fixed field functional F [Φˆ] not depending explicitely on L.6It would be desirable to investigate in the same spirit the φ4 theoryin four-dimensional fuzzy spaces, e.g. as a suitable canditate could servethe fuzzy space S2F × S2F . However, in this case, the problem will be moresevere since there will be additional residual non-local differences for two-and four-point functions. It will be therefore more difficult to establish themodel which will reproduce the commutative limit.References1. Grosse, H, Klimcˇ´ık, C. and Presˇnajder, P. (1996) Int. J. Theor. Phys., 35, 231;Grosse, H. and Strohmaier, A. (1999) Lett. Math. Phys., 48, 163.2. Grosse, H., Klimcˇ´ık, C. and Presˇnajder, P. (1996) Commun. Math. Phys., 178,507; Grosse, H. and Presˇnajder, P. (1998) Lett. Math. Phys., 46, 61; Presˇnajder, P.(2000) J. Math. Phys., 41, 2789.3. Balachandran, A.P. and Vaidya, S. (2001) Int. J. Mod. Phys., A16, 17; Balachan-dran, A.P., Govindaradjan, T.R. and Ydri, B. (2000) Int. J. Mod. Phys., A15, 1279;Balachandran, A.P., Martin, X. and O’Connor, D. (2001) Int. J. Mod. Phys., A16,2577; Balachandran, A.P., Vaidya, S. and Ydri, B. (2000) Commun. Math. Phys.,208, 787.4. Madore, J. (1992) it Class. Quant. Grav. 9, 69.5. Minwalla, S., Van Raamsdonk, M. and Seiberg, N. (2000)JHEP 0002 020.6. Vaidya, S. (2001) Phys. Lett. , B512, 403.7. Chong-Sun Chu, Madore, J. and Steinacker, H. (2001) JHEP 0108 038.8. Dolan, B.P., O’Connor, Denjoe and Presˇnajder, P. (2002) JHEP 0203 013.

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