A Fuzzy Three Sphere and Fuzzy Tori

DIAS-STP-03-5 A Fuzzy Three Sphere and Fuzzy Tori Brian P. Dolana),b)∗and Denjoe O’Connorb)† a)Dept. of Mathematical Physics, NUI, Maynooth, Ireland b)School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Rd., Dublin 8, Ireland September 23, 2005 Abstract A fuzzy circle and a fuzzy 3-sphere are constructed as subspaces of fuzzy complex projective spaces, of complex dimension one and three, by modifying the Laplacians on the latter so as to give unwanted states large eigenvalues. This leaves only states corresponding to fuzzy spheres in the low energy spectrum (this allows the commutative algebra of functions on the continuous sphere to be approximated to any required degree of accuracy). The construction of a fuzzy circle opens the way to fuzzy tori of any dimension, thus circumventing the problem of power law corrections in possible numerical simulations on these spaces. 1 Introduction One of the principal goals of the study of field theories on fuzzy spaces is to develop an alternative non-perturbative technique to the familiar lattice one [1]. To date, this new approach in the case of four dimensional field theories has been limited to studies of Euclidean field theory on S2×S2 [2], CP2[3] and S4[4]. All but S2×S2have additional complications. For ∗[email protected] †[email protected] 1 example, CP2is not spin but spincand S4is really a squashed CP3and includes many unwanted massive Kaluza-Klein type modes. Even S2×S2 is not ideal since it has curvature effects that drop off as power corrections rather than exponentially as in the case of toroidal geometries. The fuzzy approach does, however, have the advantage of preserving continuous symmetries such as the SU(2) symmetry of a round S2and does not suffer from fermion doubling [5]. The advantages are gained at the cost of introducing a non-locality associated with the non-commutativity of the fuzzy sphere. There is therefore a balance of advantages and disadvantages associated with the fuzzy approach. The final decision on whether the approach has real advantages over the standard lattice approach should be determined by doing genuine simulations. For this reason Monte Carlo simulations of the fuzzy approach are now in progress. In the lattice approach non-locality is also a problem when fermions are included. So our expectation is that as far as Monte Carlo simulations are concerned the fuzzy approach will not be competitive with the lattice one until fermions are included. The approach will gain further advantages in situations where symmetries are more important. It also extends naturally to allow for supersymmetry. (see [6] where a fuzzy supersphere was constructed). So we expect the true power of the approach to emerge when supersymmetry and chiral symmetry are present in a model. A radically different alternative to the Euclidean Monte Carlo approach becomes available once one has a fuzzy three-dimensional space. Such a space has the advantage that it allows one to develop very different nonperturbative methods, since now one can address the non-perturbative questions from a Hamiltonian point of view. The purpose of this article is to introduce precisely such fuzzy threedimensional spaces. We will begin by presenting a fuzzy version of the circle S1 F, from which one can obtain tori of arbitrary dimension. We will then present a fuzzy approximation to the three-sphere, S3 F. Unfortunately, both of these spaces are still not ideal in that they involve many unwanted additional degrees of freedom which we suppress so that they do not contribute to the low energy physics. The presence of additional degrees of freedom is probably unavoidable as it seems to be the price one pays for the classical space not being a phase space. The three-sphere is also curved and hence the results obtained from studies of field theories on this space will approach those of a flat three-dimensional space with polynomial corrections. It has, however, the advantages of preserving the full SO(4) symmetry of a round S3. From the construction it seems clear that both of these spaces will also be free of fermion doubling problems. We will restrict our focus here to scalar field theories and demonstrate how 2 the unwanted degrees of freedom can be suppressed so that the limiting large matrix theory of a scalar field theory recovers field theory on the commutative spaces. We will argue that the data specifying the geometries can be cleanly specified by giving a suitable Laplace-type operator for the scalar field, which together with the matrix algebra and its Hilbert space structure gives a spectral triple. Aside from our personal motivations, non-commutative geometry has recently become a very popular area of research from both the point of view of possible new physics in string theory and D-brane theory, [7, 8], and as a new regularisation technique in ordinary quantum field theory, [2]-[4] and [9]-[12]. In both these endeavours “fuzzy” spaces play an important rˆole. Roughly speaking a fuzzy space is a finite matrix approximation to the algebra of functions on a continuous manifold, the seminal example being the fuzzy two-sphere, [13]. It has the important property of preserving the isometries of the space that it is approximating. As such the idea can serve as a source of examples related to matrix models in string theory and as a regularisation technique for ordinary quantum field theory. As a regularisation method it provides one that preserves the underlying space-time symmetries and is amenable to numerical computation. Fuzzy spheres in dimensions other than two were analysed in [14]-[17], but the construction there was incomplete. They also advocate projecting out the unwanted modes and working with a non-associative algebra which we consider unsatisfactory. Also the case of odd spheres works very differently to that of even spheres. An alternative approach for the fuzzy four-sphere, S4 F, was given in [4], based on the fact that fuzzy CP3and CP1∼ =S2are well understood [18], and, in the continuum limit, CP3is an S2bundle over S4. In this paper we show how the odd-dimensional fuzzy spheres S1 Fand S3 F can be extracted from the matrix algebras associated with the fuzzy complex projective spaces CP1 Fand CP3 F. An alternative approach to obtaining a finite approximation to S3∼ =SU(2), based on conformal field theory, was presented in [19], however, in this approach it is unclear how the unwanted modes are to be suppressed. Our method uses a similar suppression mechanism to that used for S4 Fin [4]. Although there is no closed finite dimensional matrix algebra for SN Funless N= 2, the relevant degrees of freedom when N= 1 and N= 3 are contained in the matrix algebras for CP1and CP3 respectively. One can therefore obtain functional integrals for field theories over S1 Fand S3 Fby starting with functional integrals over CP1 Fand CP3 F and then suppressing the unwanted modes so that they do not contribute to the functional integral. Because of the high degree of symmetry inherent in the construction, the unwanted modes can be suppressed simply by using appropriate quadratic Casimirs in the Laplacian. In this way we by-pass 3 the problems associated with the fact that the algebra of matrices associated with functions on the sphere does not close on the sphere, but necessarily lifts into the enveloping complex projective space. In a similar fashion we expect that when a Hamiltonian approach to field theory is developed using these spaces the unwanted modes will cause no difficulties since they can be made arbitrarily difficult to excite. The paper is organized as follows. In section 2 we summarize how a given geometry is captured in the fuzzy approach. Section 3 then gives our construction of a fuzzy circle, S1 F. Section 4 summarises the construction of S4 Fpresented in [4] and in section 5 we present our fuzzy three-sphere, S3 F. Section 6 gives an alternative construction of S3 Fwhich lends itself to a generalisation to SN Ffor any N[20]. 2 Encoding the geometry of a fuzzy space Fr¨ohlich and Gaw¸edzki [19] (following Connes, [21]) have demonstrated that the abstract triple (H, A,∆γ), where His the Hilbert space of square integrable functions on the manifold M, with Laplace-Beltrami operator ∆γ,γ being the metric, and A=C∞(M) is the algebra of smooth bounded functions on M, captures a topological space together with its metrical geometry. In a similar fashion one can specify a fuzzy space, MF, as the sequence of triples MF:= (HL,MatdL,∆L) (1) parameterized by L, where HL=Cd2 Lis the Hilbert space acted of the complete matrix algebra MatdLof dimension d2 Lwith inner product < M, N >= 1 dLTr(M†N) and ∆Lis a suitable Laplacian acting on matrices. One can readily extract information such as the dimension of the space from these data. The Laplacian comes with a cutoff and so the dimension can be read from the growth of the number of eigenvalues. The data contained in the triple (H, A,∆γ) are precisely the data that go into the Euclidean action for a scalar field theory on the space Mand hence specifying the scalar action is a convenient method of prescribing these data. In the fuzzy approach the algebra will always be a matrix algebra and we will retain the Hilbert space inner product specified above so the only data from the triple, (HL,MatdL,∆L), remaining to be supplied are the permitted matrix dimensions, dLand a realization of the Laplacian, ∆L. Once this information is given the fuzzy geometry is specified. Though it may be convenient to give a map to functions this is not necessary. Once the Laplacian is given its eigenmatrices and spectrum can be used to provide such a map if needed. Suppose for example that the spectrum of 4 ∆Lis identical to that of ∆γup to some cutoff and a complete set of eigenmatrices is given by ˆ Ψλwith the corresponding commutative eigenfunctions being Ψλ, then the symmetric symbol-map D given by D = d2 L X λ Ψλˆ Ψλ(2) provides a map to functions with fM=1 dL Tr(DM) (3) the function corresponding to the matrix M. By construction the map has no kernel and the symbol-map induces a ∗product on functions given by fM∗DfN=1 dL Tr(DMN) (4) which represents matrix multiplication in terms of an operation on the image functions. The ∗product depends on D, a different but equivalent one could be obtained by giving a nonzero weighting cλ(L) to the different terms in the sum (2). In the case of CPNa particular choice of the cλ(L) will give the diagonal coherent state prescription1as discussed in [18]. If the symbol-map (2) has the property that ∆γfM=1 dL Tr(D∆LM) (5) where ∆γis a natural Laplacian for the space to be approximated, then the spectrum of the fuzzy space will be precisely a cutoff version of that of the commutative space M. This is precisely what happens in the case of CPN F, see [18]. However, it is convenient to extend the definition of fuzzy space to the case where the spectrum coincides for low-lying eigenvalues, but deviates for a family of eigenvalues that can be given arbitrarily high value and which correspond to degrees of freedom that have no counterpart in the commutative space M. This allows us to obtain fuzzy approximations to additional spaces — in particular, as we will see, to tori and the three sphere. 1In the case where the symbol-map is the projector of coherent states the function fM is referred to as the covariant symbol of the matrix Mand since the coefficients cλ(L) are not one it will differ from the corresponding contravariant symbol, see Berezin [22]. The symbol-map is referred to as symmetric when its covariant and contravariant symbols are equal and coincides with the case of cλ(L) = 1. 5 If one takes the Euclidean quantum field theory point of view then the desired geometry appears as that associated with the accessible configurations of the field theory and the deviations are suppressed in a probabilistic fashion. A successful method of suppressing the unwanted modes would be to add to the scalar action a term SI[Φ] which is non-negative for any Φ, zero only for matrices that correspond to functions on M, and positive for those that do not. The modified action would therefore be of the form S[Φ] + hSI[Φ]. The parameter hshould be chosen to be large and positive. The probability of any given matrix configuration then takes the form P[Φ] = e−S[Φ]−hSI[Φ] Z(6) where Z=Zd[Φ]e−S[Φ]−hSI[Φ] (7) is the partition function of the model. If the prescription is to work for free field theories, then SI[Φ] should be at most quadratic in Φ. This can then be thought of as a modification of the Laplacian in the triple (1). Furthermore the problem of UV/IR mixing in scalar theories can be removed by including a higher derivative operator in the quadratic term of the field theory such that it renders all diagrams finite when the matrix size is sent to infinity. With such a prescription since each diagram has a limiting commutative value in the large matrix limit each diagram must take this value and hence no UV/IR mixing can occur. The prescription of sending the matrix size to infinity and sending the coefficient of the irrelevant higher derivative operator to zero do not commute. This prescription of adding an irrelevant operator to the action is simpler than the normal ordering prescription proposed in [23] and works for any dimension. From the above discussion it should be clear that the entire problem of constructing a fuzzy approximation to a space is the problem of giving a suitable prescription for the matrix Laplacian. 3 Approximating a circle from a fuzzy sphere Consider the finite matrix algebra representation of the fuzzy sphere S2 F[13]. The algebra of (L+ 1) ×(L+ 1) matrices, which will be denoted by MatL+1, has the same dimension as the number of degrees of freedom in a spherical 6 harmonic expansion of a function on S2, truncated at angular momentum L, fL(θ, φ) = L X l=0 l X m=−l flmYlm(θ, φ).(8) That is L X l=0 (2l+ 1) = (L+ 1)2.(9) The precise identification between a matrix Φ ∈MatL+1 and a cut-off function fL(θ, φ), as discussed in the preceding section, is not unique, but the possible maps can be given in terms of coherent states or the symmetric symbolmap D of (2), and the resulting product of functions is non-commutative for finite L. It is crucial to our construction that only maps for which the product of functions becomes commutative in the limit L→ ∞ be considered. The symbol-map (2) associates orthonormal (L+ 1) ×(L+ 1) polarisation tensors ˆ Ylm with spherical harmonics Ylm(θ, φ). The conventions used here will be that ˆ Ylm =1 √L+ 1 ˆ Tlm (10) where the polarisation tensors ˆ Tlm are those of [24]. The SO(3) symmetric Laplacian, L2, on the fuzzy sphere acts on matrices Φ and is represented by the second order Casimir corresponding to the adjoint action of the angular momentum generators Liin the (L+ 1) ×(L+ 1) representation: L2Φ = [Li,[Li,Φ]].(11) Hence the action can be taken to be S[Φ] = 1 L+ 1Tr 1 2Φ†L2Φ + V(Φ)(12) for some scalar potential V(Φ)†=V(Φ), which is assumed to be bounded below. This action can then be used in a partition function which involves ordinary integration over (L+ 1)2degrees of freedom Z=ZDΦe−S[Φ].(13) The probability distribution for field configurations is then P[Φ] = e−S[Φ] Z(14) 7 where S[Φ] given is by (12). This probability distribution is associated with the geometry (HL,MatL+1,L2) which specifies a round fuzzy sphere. The field theory with quadratic potential, however, suffers from UV/IR mixing problems [23, 25]. If we add the term aL4to the Laplacian and use the triple (HL,MatL+1,L2+aL4) the UV/IR mixing problem is removed and we recover a field theory on the commutative S2in the infinite matrix size limit. The parameter acan finally be sent to zero with the result that the critical value of the mass parameter will be sent to infinity. The process of taking the large matrix limit and sending ato zero do not commute. To obtain the commutative theory on the sphere the matrix size must be sent to infinity for non-zero a. There is no finite matrix approximation to the algebra of functions on S1. Nevertheless, the degrees of freedom relevant to a circle are certainly contained in MatL+1. Focusing on the top harmonic in (8), with l=L, the YLm contain all −L≤m≤Land thus reproduce functions on the circle as m→ ∞. This implies that the partition function and correlation functions for a field theory on a circle can be extracted from that of the fuzzy sphere by suppressing all the modes with l < L in (13). One way of achieving this is to penalise modes with l < L by giving them a large positive weight in the action. To this end we modify the action (12) to Sh[Φ] = 1 L+ 1Tr 1 2Φ†[L3,[L3,Φ]] + h 2Φ†L(L+ 1) −L2Φ + V(Φ). (15) All modes with l < L now have the wrong sign for L2and, when his very large, are heavily penalised in the partition function (13), contributing nothing as h→ ∞. In this limit only the modes with l=Lremain and these have the correct sign for their kinetic energy, because the term linear in hvanishes on these and only these modes. The ‘wrong sign’ for the L2 contribution to the kinetic energy here is analogous to an anti-ferromagnetic coupling in a lattice theory and just as in the lattice theory with an antiferromagnetic coupling the action here is also bounded below. That the action remains bounded from below is intimately related to the fact that there is an ultraviolet cutoff in the model and therefore a maximum eigenvalue for the Laplacian or equivalently a shortest wavelength. To see that the commutative algebra of functions on S1is recovered in the l=Lsector of the fuzzy sphere as L→ ∞, we first decompose the matrix Φ using the basis of polarisation tensors: Φ = L X l=0 l X m=−l Φlm ˆ Ylm.(16) 8 In our conventions (10) the commutator of the polarisation tensors is given by (see e.g. [24] page 191, equation (46)) [ˆ Yl1m1,ˆ Yl2m2] = r(2l1+ 1)(2l2+ 1) L+ 1 L X l=0 (−1)L−l1−(−1)l1+l2+l ×l1l2l L/2L/2L/2Clm l1m1,l2m2 ˆ Ylm, (17) where l1l2l L/2L/2L/2are 6j-symbols and Clm l1m1,l2m2are Clebsch-Gordon co-efficients. Now for large L l1l2l L/2L/2L/2≈1 √L+ 1Cl0 l10,l20(18) and Cl0 l10,l20= 0 when l1+l2+lis odd. Thus [ˆ Yl1m1,ˆ Yl2m2]→0 (19) and the algebra is commutative when L→0 as promised. In particular [ˆ YLm1,ˆ YLm2]→0 (20) and the top harmonic alone reproduces the commutative algebra of functions on S1in the continuum. To summarize we can encode the geometry specifying a fuzzy circle by the triple S1 F:= HL,MatL+1,L2 3+hL(L+ 1) −L2.(21) with h >> 1. This picks out the fuzzy circle from the top angular momentum polarization tensor ˆ YL,m. One could equally pick it out from a lower one, ˆ YL0,m by modifying the term proportional to hto (L0(L0+ 1) −L2)2. This latter choice may have advantages for the suppression of UV/IR mixing effects in the fuzzy context. It roughly corresponds to a mixture of ‘nearest neighbour’ and next nearest neighbour ferromagnetic and anti-ferromagnetic couplings. Having constructed a fuzzy circle it is now clear that there is no obstacle to constructing fuzzy tori of arbitrary dimension, simply by taking products of fuzzy circles. This has the obvious advantage for numerical simulation of avoiding power-law curvature effects. 9 If we can penalise all modes with 2l < L and m6= 0 for 2l=Lin a functional integral over SO(5)/SO(3) ×SO(2)then we will really be doing a functional integral over S3 F. This is easily achieved since 2l=Land m= 0 has the largest second order Casimir, C(5) 2(L, 0) = L(L+ 3),(51) of all the SO(5) representations in Matd0 L. In the now familiar manner the unwanted modes in the functional integral over SO(5)/SO(3)×SO(2)can be suppressed by using the Laplacian L2 h0=1 2[Jαβ,[Jαβ,·]] + h0L(L+ 3) −L2 (5),(52) which acts on fields Φ ∈Matd0 Land Leven. The unwanted modes are completely eliminated in the limit h0→ ∞, giving S3 Ftruncated at level L. The constraint that Lis even does not change the fact that we get the full continuum S3as L→ ∞. 7 Conclusions By starting with the known finite matrix algebras for CP3and CP1, the fuzzy CP3 Fand the fuzzy sphere CP1 F∼ =S2 F, finite functional integrals for scalar field theories on S3 Fand S1 Fhave been constructed. The geometry of a fuzzy space is specified by a triple (HL,MatdL,∆L) and, although there is no known closed associated algebra giving a fuzzy S1as a triple directly, CP1 Fnevertheless contains the states required for a S1 Fplus other unwanted states. The unwanted states are given large eigenvalues by modifying the Laplacian on CP1 F, as in equation (21), leaving only the states of S1 Fin the low energy spectrum of the Laplacian. In a similar way CP3 Fcontains the states necessary for a fuzzy description of S3(via S4 F) and the Laplacian on CP3 Fcan be modified, as in equation (42), so that states not related to S3 Fare given large eigenvalues, leaving only S3 Fstates in the low energy spectrum. An alternative construction of S3 F, based on suppressing modes on a fuzzy version of the orthogonal Grassmannian SO(5)/SO(3) ×SO(2), has been presented in section 6. This has the advantage of having a natural extension to SN Ffor any N, [20]. Thus S3 Fcan be obtained either in two steps, via the fuzzy S4 Fconstructed in [4], CP3 F→S4 F→S3 F, or alternatively in a single step from the fuzzy version of SO(5)/[SO(3)×SO(2)] as described in section 6. 16 The construction of the fuzzy circle allows fuzzy tori to be defined in an obvious way, by taking products of fuzzy circles, thus opening the way to numerical simulations on tori while preserving the full U(1) ×···×U(1) isometry group and avoiding the fermion doubling problem [5]. One first writes down a finite functional integral for a field theory on S2 F× ··· × S2 F, which contain the modes relevant for propagation S1 F× ··· × S1 Fin its spectrum, and then damps the unwanted modes. This can be done, in a manner that preserves the isometries of the torus, by introducing appropriate combinations of second order Casimirs into the propagators. It is pleasure to acknowledge A. P. Balachandran, Peter Preˇsnajder, Harald Grosse and Daniel Roggenkamp for helpful discussions. 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