Generalized Fuzzy Torus and its Modular Properties

We consider a generalization of the basic fuzzy torus to a fuzzy torus with non-trivial modular parameter, based on a finite matrix algebra. We discuss the modular properties of this fuzzy torus, and compute the matrix Laplacian for a scalar field. In the semi-classical limit, the generalized fuzzy torus can be used to approximate a generic commutative torus represented by two generic vectors in the complex plane, with generic modular parameter $\tau$. The effective classical geometry and the spectrum of the Laplacian are correctly reproduced in the limit. The spectrum of a matrix Dirac operator is also computed.


Introduction
In recent years, Matrix models of Yang-Mills type have become a promising tool to address fundamental questions such as the unification of interacting forces and gravity in physics. The fundamental degrees of freedom are operators acting on a finite or infinite-dimensional Hilbertspace, represented by matrices X A . Specific Yang-Mills matrix models appear naturally in string theory [1], and provide a description of branes, as well as strings stretching between the branes.
It is well-known how to realize certain basic compact branes in the framework of matrix models. A simple and well-known example is the rectangular torus or fuzzy torus, realized in terms of finite-dimensional clock-and shift matrices. It is important to distinguish here the fuzzy torus form the noncommutative tori as studied by Connes etal. [2,3]. In the latter case the algebra of functions and the corresponding Hilbert space are infinite-dimensional, with non-trivial functional-analytic structure. In contrast, we are interested here only in fuzzy tori defined in terms of finite-dimensional matrices, with algebra of functions given by a finite-dimensional matrix algebra A N = M N (C). They are quantizations of classical tori with quantizable symplectic form with finite symplectic volume, without any additional sector. These fuzzy tori have an intrinsic UV cutoff, and are therefore excellent candidates for fuzzy extra dimensions, along the lines of [4].
As quantized symplectic manifolds, the fuzzy tori have a priori no metric structure. However if realized in matrix models, they inherit an effective metric as discussed in [5]. Again this should be contrasted with the case of the infinite-dimensional noncommutative torus, which is usually equipped with a differentiable calculus given by outer derivations, and subsequently with a metric structure e.g. via a Laplace operator. The fuzzy torus admits only inner derivations, and the natural Laplace operator is provided by the matrix model.
In this work, we study in detail the most general fuzzy torus embedded in the matrix model first considered in [6], and study in detail its effective geometry. We demonstrate that the embedding provides a fuzzy analogue for a general torus with non-trivial modular parameter. It turns out that non-trivial tori are obtained only if certain divisibility conditions for relevant integers hold, in particular N should not be prime. In the limit of large matrices, our construction allows to approximate any generic classical torus with generic modular parameter τ . Moreover, we obtain a finite analogue of modular invariance, with modular group SL(2, Z N ). The effective Riemannian and complex structure are determined using the general results in [5]. In addition we determine the spectrum of the associated Laplace operator, and verify that the spectral geometry is consistent with the effective geometry as determined before.
The origin for the non-trivial geometries of tori is somewhat surprising, since the embedding in the matrix model is in a sense always rectangular. A non-rectangular effective geometry arises due to different winding numbers along the two cycles in the apparent embedding. This finite winding feature leads to a non-trivial effective metric, due to the non-commutative nature of the branes. This paper is organized as follows. We first review the classical results on the flat torus, as well as the quantization of the basic rectangular fuzzy torus in the matrix model. We then give the construction of the general fuzzy torus embedding, and determine its effective geometry. Its modular properties are studied, and the modular group SL(2, Z N ) is identified. We also compute the spectrum of the corresponding Laplace operator, and determine its first Brillouin zone. Finally we also discuss the Dirac operator in the rectangular case and obtain its spectrum.

The classical torus
Before discussing the fuzzy torus, we review in detail the geometric structure of the classical torus.
The most general flat 2-dimensional torus can be considered as a parallelogram in the complex plane C, with opposite edges identified. The torus naturally inherits the metric and the complex structure of the complex plane. The shape of the parallelogram is given by two complex numbers ω 1 and ω 2 , as illustrated in fig.1. One can think of the vectors ω 1 and ω 2 as generators of a lattice in the complex plane C. Denoting this lattice by L(ω 1 , ω 2 ) = {nω 1 + mω 2 , n, m ∈ Z} a point z on the torus is given by with coordinates σ 1 , σ 2 ∈ [0, 2π]. These points are identified according to the lattice L(ω 1 , ω 2 ). Such coordinates σ 1 , σ 2 with periodicity 2π will be called standard coordinates. In these standard coordinates, the line element is We can read off the metric components Furthermore, we introduce the modular parameter where H is the complex upper half-plane H = {z ∈ C|z > 0}. We identify conformally related metrics on the torus. Using a Weyl scaling g → e φ g of the metric as well as a diffeomorphism (a rotation), the lattice vectors of the torus can be brought in the standard form ω 1 = τ and ω 2 = 1, see ( fig.2). Then z = σ 1 + τ σ 2 for (σ 1 , σ 2 ) (σ 1 , σ 2 ) + 2π(n, m). The line element in these standard coordinates then simplifies as with metric components In these coordinates z = σ 1 + τ σ 2 , one can express the modular parameter through the metric components (6) as follows where g = det(g ab ). Now on any oriented two-dimensional Riemann surface, there is a covariantly constant antisymmetric tensor 3 1 √ g ab with 10 = −1. Together with the metric and the antisymmetric tensor, we can build the tensor 3 this corresponds to the inverse of the volume form.
In the above standard coordinates, this tensor is explicitly (9) and the square of J is J 2 = −1. It is therefore an almost complex structure. In fact it is a complex structure, since it is constant and thus trivially integrable. It is instructive to choose Euclidian coordinates z = x+iy on the same torus, with metric ds 2 = dx 2 + dy 2 . Then the periodicity becomes z z + 2π(m + τ n). In these coordinates, the almost complex structure takes the standard form which is Now J 2 = −1 is obvious. Now we can discuss modular invariance. Note that two tori are always diffeomorphic as real manifolds, but not necessarily biholomorphic as complex manifolds. This can be illustrated e.g. with two tori T 1 and T 2 defined by the lattice L(ω 1 , ω 2 ) = ((1, 0), (0, 1)) and L(u 1 , u 2 ) = ((1, 0), (0, 2)), see fig.3. On T 1 we choose coordinates (x 1 , y 1 ), and on T 2 we Let us introduce complex coordinates on tori z = x 1 + iy 1 and w = x 2 + iy 2 . Using the above diffeomorphism, we obtain w = x 1 + 2iy 1 , and together with we find This is clearly not a holomorphic function of z.
Clearly two tori are equal as complex manifolds if their modular parameters τ ω = ω 1 /ω 2 and τ u = u 1 /u 2 coincide. Moreover, two tori are also equivalent if they are related by a modular transformation To see this, it suffices to note that the two lattices L(ω 1 , ω 2 ) and L(u 1 , u 2 ) are equivalent if they are related by a P SL(2, Z) transformation This leads to fractional transformation of their modular parameters This modular group is in fact generated by two generators which obey the relations S 2 = (ST ) 3 = 1. The moduli space of τ is the fundamental domain F, which is the complex upper half-plane H modulo the group P SL(2, Z) A standard choice for this fundamental domain is −1/2 ≤ τ 1 ≤ 1/2 and 1 ≤ |τ |, see fig,4. The fundamental domain is topologically equal to the complex plane F C. Adding the point τ = i∞ we obtain the compactified moduli space, which is topological equivalent to the Riemann sphere. The action of the modular transformations T : τ → τ + 1 and S : τ → −1/τ on the torus is illustrated in fig.5.   a quantization map Q, which is an isomorphism of two vector spaces. It maps the space of function to a space of operators

Poisson manifolds and quantization
In the present context the space of operators will be the simple matrix algebra A N = M N (C). The quantization map Q depends on the Poisson structure, and should satisfy the conditions for θ → 0. The algebra A is interpreted as quantized algebra of functions C(M) on M.
The quantization map Q is not unique, since higher order terms in θ are not unique. The natural integration on symplectic manifolds is related to its operator version in the semiclassical limit, as I(Q(f )) → I(f ). Here and in the following, semiclassical limit means taking the inverse of the quantization map Q −1 (F ) = f in the limit θ → 0, keeping only the leading contribution [., .] → i{., .} and dropping higher-order corrections in θ. Sometimes this semi-classical limit is indicated by F → f . We are interested here in manifolds which can be realized as Poisson manifold M embedded in the Euclidean space R D , with Cartesian coordinates x A , A = 1, ..., D. The embedding is a map where A quantization of such a Poisson manifold provides in particular quantized embedding functions x A via Now consider the action for a scalar field Φ on such a quantized Poisson manifold in the matrix model, given by In the semiclassical limit Φ ∼ φ, the action becomes where ρ = det θ −1 ab . Thus G ab = θ ac θ bd g cd is identified as effective metric. In dimensions 4 or higher, this can be cast in the standard form for a scalar field coupled to a (conformally rescaled) metric [9]. In the present case of 2 dimensions this is not possible in general due to Weyl invariance, cf. [7]. However we are only considering tori with constant ρ and G ab here, where this problem is irrelevant. Then G ab = e σ g ab as above is indeed the effective metric, up to possible conformal rescaling. Moreover, the matrix Laplace operator defined by reduces in the semi-classical limit to where G is the standard Laplacian on manifold with metric G ab . Thus the equation of motion for the scalar field reduces to or equivalently g φ = 0.

The rectangular fuzzy torus in the matrix model
The rectangular fuzzy torus can be defined in terms of two N × N unitary matrices, clock C and shift S Here we introduce the deformation parameter q = e i2πθ , with phase θ = 1/N and positive integer N ∈ N. The clock and shift matrices satisfy the relation and thus These matrices are traceless and obey C N = S N = 1 N . The fuzzy torus has a Z N × Z N symmetry, which acts on the algebra A N as and similar for the other Z N replacing C by S. Here ω denotes the generator of Z N . Thus we have a decomposition of the algebra of function A N over the torus into harmonics or irreducible representations of Z N × Z N , An element in A N can thus be written uniquely as The corresponding basis of functions on the classical torus is e inσ 1 e imσ 2 , for n, m ∈ Z and coordinates σ 1 , σ 2 ∈ [0, 2π]. Thus we obtain a quantization map from the functions on the torus to a matrix algebra which is one-to-one below the UV cutoff n max , m max = N/2. This defines the fuzzy torus T 2 N . Now we consider the fuzzy torus embedded in R 4 , via the quantized embedding functions The hermitian matrices X 1 , X 2 , X 3 and X 4 satisfy the algebraic relations which tells us that R 1 , R 2 are the radii of the torus. This embedding defines derivations given by the adjoint action [X i , f ] on A N . Now consider the semi-classical limit. Then the clock and shift operators become plane waves, C → c = e iσ 1 and S → s = e iσ 2 , where σ a ∈ [0, 2π]. Observe that due to this periodicity, these σ a are standard coordinates on the torus as discussed before. We have then the embedding functions x A (σ 1 , σ 2 ) which again satisfy the algebraic relations Using these embedding functions, we can compute the embedding (induced) metric in standard coordinates. The Poisson structure is obtained from the semiclassical limit of the commutator where we expanded q to first order of 1/N . On the other hand, classically we can write for the Poisson bracket We can read off the Poisson tensor The corresponding symplectic structure is ω = N π dσ 1 ∧ σ 2 . Given the embedding metric g ab and the Poisson tensor θ cd , we can compute the effective metric and the Laplacian. It is easy to see that in 2 dimensions, the effective metric G ab = θ ac θ bd g cd is always proportional to the embedding metric g ab by a conformal rescaling For the Laplacian in 2 dimensions such conformal factors drop out, and indeed we have always identified conformally equivalent metrics on the torus. It is therefore sufficient here to work only with the embedding metric g ab . With these tensors at hand, we can build the complex structure according to (8) Since these are standard torus coordinates, we can read off the modular parameter which is purely imaginary, Recalling that τ = ω 1 /ω 2 , this corresponds to a rectangular torus with lattice vectors ω 1 = iR 2 and ω 2 = R 1 .

Laplacian of a scalar field
Now consider a scalar field Φ ∈ A N on the basic fuzzy torus, with action and equation of motion Φ = 0. The matrix Laplacian operator (28) can be evaluated explicitly on the torus as where we have introduced the q-number In the semiclassical limit, the spectrum 4 reduces to the spectrum of the commutative Laplacian 4 The fuzzy torus on a general lattice and fuzzy modular invariance To construct more general fuzzy tori, we define two unitary operators where C and S are the clock and shift matrix, and k x , l x , k y , l y ∈ Z. The operators V x and V y generalize the clock and shift matrices, and satisfy V N x = V N y = 1. Note that the k x , l x , k y , l y should be considered more properly as elements of Z N , due to C N = S N = 1. We combine these k x , l x , k y , l y in two discrete complex vectors which define a lattice L N (k, l) = {nk + ml, n, m ∈ Z N } .
This is the fuzzy analogue of the lattice L(ω 1 , ω 2 ) which defines a commutative torus. The operators V x (k x , l x ) and V y (k y , l y ) satisfy the commutations relations where is the area of the parallelogram spanned by k and l. Note that the operators V x (k x , l x ) and V y (k y , l y ) commute if and only if k ∧ l = 0 mod N , corresponding to collinear vectors spanning a degenerate torus, or tori whose area is a multiple of N . Let us transform the lattice L N (k, l) with a P SL(2, Z N ) = SL(2, Z N )/Z 2 transformation to another lattice L N (k , l ): Clearly the entries of the matrix should be elements of Z N , so that the transformed lattice vectors k and l are in Z N . On the P SL(2, Z N ) transformed lattice L N (k , l ) the commutation relations are Since the area k ∧ l is invariant under a P SL(2, Z N ) transformation it follows that this commutation relation is the same as for the original lattice under the transformations (58). Thus we have established fuzzy modular invariance at the algebraic level, and we will consider noncommutative tori whose lattices are related by P SL(2, Z N ) as equal. Later we will see that the spectrum of the Laplacian and the equation of motion for the noncommutative tori are also invariant under P SL(2, Z N ). The moduli space of the lattice L N (k, l) or the fuzzy fundamental domain F N is defined accordingly as To obtain a metric structure, we define an embedding of these fuzzy tori into the R 4 via the operators V x and V y as follows (cf. [6]) This embedding satisfies the algebraic relations X 2 1 + X 2 2 = R 2 1 and X 2 3 + X 2 4 = R 2 2 corresponding to two orthogonal S 1 ×S 1 . Nevertheless, the non-trivial ansatz for the V x,y will lead to a non-trivial effective geometry on the tori. As usual, this embedding defines derivations on the algebra A N given by [X i , .], and the integral is defined by the trace I(Φ) = 1 N T r(Φ), where Φ denotes a scalar field on the torus Here the momentum space is Z 2 N ∼ = [−N/2 + 1, N/2] 2 if N is even, to be specific. We are now ready to compute the spectrum of the Laplacian for a scalar field on the fuzzy torus, It is easy to see that this spectrum is invariant under the SL(2, Z N ) modular transformations acting on the defining lattice L N (k, l) as in (58), and simultaneously on the momenta as follows Therefore fuzzy modular invariance is indeed a symmetry of fuzzy tori and their the scalar field spectrum.

Spectrum and Brillouin zone.
The above spectrum of L N has a complicated periodicity structure, and typically some degeneracy in momentum space C N . In order to correctly identify the irreducible spectrum and the spectral geometry of the torus, we have to find the unit cell, or the first Brillouin zone B( s, r). This unit cell is spanned by two vectors in momentum space which characterize the basic periodicity of the spectrum. We can associate to them two elements W r = C r 1 S r 2 and W s = C s 1 S s 2 in A N . Then the shift in momentum space n → n + r of the field Φ along r is realized by ΦW r , and the shift n → n + s is realized by ΦW s . In order to compute these s and r, we rewrite the spectrum in factorized form using trigonometric identities, setting R 1 = R 2 = 1 for simplicity. This allows to identify r as primitive periodicity of the first cos factor while leaving the second unchanged, and s as primitive periodicity of the second cos factor leaving the first unchanged. Explicitly, This leads to the equations and These four equations are equivalent to and which amount to [V x,y , W r,s ] = 0. In complex notation, these 4 equations can be written as or in matrix form In particular, this implies reflecting the decomposition of the momentum space Z 2 N into Brillouin zones. Alternatively, these equations can be written as introducing the following complex combinations Inverting (74) gives However, all quantities in these equations must be integers in [− N 2 , N 2 ], to be specific. Therefore non-trivial Brillouin zones B( s, r) are typically possible only if their area | r ∧ s| = r 1 s 2 − r 2 s 1 divides 5 N . Similarly, inverting (75) gives and again |k ∧ l| = k y l x − k x l y must typically divide N . The above analysis leads to a very important point. The equations (77) which determine the first Brillouin zone are Diophantic equations, so that their naive solutions in R 2 may not be admissible in C N . This follows also from (75), which is very restrictive e.g. if N is a prime number. If (77) gives non-integer (r, s) for given (k, l), then these naive Brillouin zones and their apparent spectral geometry are not physical; in that case, the full spectrum obtained by properly organizing all physical modes in momentum space (n 1 , n 2 ) may look very different. To see this, consider N prime and k, l relatively prime. Then there are unitary operators C = V n x ,S = V m y which generate A N with V xC = qCV x and V yS = q −1C V y , leading to the spectral geometry (51) of a rectangular torus; this is in contrast to (76) which falsely suggests a non-trivial lattice and Brillouin zone. On the other hand, if N is divisible by (k y l x − k x l y ), then the above equations (77) can be solved for a, b ∈ C N , for any given non-trivial lattice L N (k, l). In that case, we obtain indeed a fuzzy version of the desired non-trivial torus as discussed below, with periodic spectrum decomposing into several isomorphic Brillouin zones B( s, r).
To illustrate this, we choose a lattice L N (k, l) with vectors l = 2 + i and k = 2 + 4i, with area k ∧ l = 6. The smallest matrix size to accomodate this is N = 6, and in this case the corresponding Brillouin zone B( r, s) is spanned by r = −2 + i, s = −6 − 3i with r ∧ s = 12, see ( fig.6). Thus momentum space decomposes into 3 copies of the Brillouin zone.

Effective geometry.
Now we want to understand the effective geometry of the torus L N (k, l) in the semi-classical limit. We will discuss both the spectral geometry as well as the effective geometry in the sense of section 3, which should of course agree. In the semi-classical limit, we would like that the integers k x , l x , k y , l y approach in some sense the real numbers ω 1x , ω 2x , ω 1y , ω 2y corresponding to some generic classical torus. More precisely, the lattice L N (k, l) should approach some given lattice L(ω 1 , ω 2 ). This can be achieved via a sequence of rational numbers approximating these real numbers. Explicitly, we require where ρ N is some increasing function of N . Now consider the spectrum λ n 1 n 2 = 4 sin 2 ( π N (k x n 2 − l x n 1 )) + 4 sin 2 ( π N (k y n 2 − l y n 1 )) setting R 1 = R 2 = 1. This approximation is valid as long as the argument of the sin() terms are smaller than one, i.e. in the interior of the first Brillouin zone. As we will verify below, this spectrum indeed reproduces the spectrum of the classical Laplace operator on the torus L(ω 1 , ω 2 ) in the semi-classical limit N → ∞, as long as |ω 1 n 2 − ω 2 n 1 | < N ρ N . Now consider the effective geometry in the semi-classical limit, as discussed in section 3. Since C ∼ e iσ 1 and S ∼ e iσ 2 , the defining matrices V x and V y (55) of the fuzzy torus (80) The Poisson brackets can be obtained from The semi-classical approximation makes sense as long as k ∧ l < N , which holds for at least one equivalent torus L N (k , l ) if C N decomposes into at least N fundamental domains F N (62). We can then identify this with the Poisson bracket and read off the Poisson tensor for theσ i coordinates The embedding functions in R 4 become and satisfy again the algebraic relations The embedding metric is computed via (42), This reproduces indeed the metric of the general torus L(ω 1 , ω 2 ) (3) for R 1 = R 2 = 1, which is recovered here from a series of fuzzy tori L N (k N , l N ). As a consistency check, we compute the spectrum of the commutative Laplacian and compare it with the semiclassical limit (79). Since G ab ∼ g ab in 2 dimensions as discussed before, the Laplacian is proportional to setting R 1 = R 2 = 1 and dropping the tilde on σ i . Evaluating this on e inσ 1 e imσ 2 we obtain This agrees (up to an irrelevant factor) with the semiclassical spectrum (79) of the matrix Laplacian.
Given the metric and the Poisson structure, we can compute the complex structure which satisfies J 2 = −1. Hereθ −1 = det(θ −1 ab ). The effective modular parameter in the commutative case is given by τ = ω 1 /ω 2 ∈ F. In the fuzzy case, we can choose a sequence of moduli parameter depending on N which for N → ∞ approximates the complex number τ to arbitrary precision. Finally let us discuss the quantization map. There is a natural map Q : where n 1 , n 2 ∈ Z, and σ 1 , σ 2 ∈ [0, 2π] are coordinates on T 2 , which respects the harmonic decomposition with respect to the classical and matrix Laplacians. In particular, Q(e i(ω 1yσ1 +ω 2yσ2 ) ) = C ky S ly = V y (93) (up to phase factors) with Now assume that (77) is solved by integers r i , s i , defining the Brillouin zone B( r, s). Then the spectrum of is n-fold degenerate, and (91) describes the quantization of an n-fold covering of the basic torus. Indeed the elements W r , W s generate a discrete group G W ⊂ U (N ) acting on A N from the right, which leaves invariant and permutes the different tori resp. Brillouin zones. Accordingly, the space of functions on a single fuzzy torus L N (k, l) is given by the quotientÃ N = M N (C)/G W , which is a vector space rather than an algebra. Nevertheless, it is natural to consider the map Q : as quantization of the torus L(ω 1 , ω 2 ) under consideration.

Partition function
The partition function is defined via the functional approach as Z N (k, l) = dφ nm dφ n m e −c N nm;n m φnmΩ nn ;mm φ n m (96) 2 and Ω nn ;mm = δ nn δ mm (Q nm + ). Here is a small number introdued to regularize the divergence due to the zero modes. The Gaussian integral gives We renormalize the partition function by multiplying with 1/2 , and after taking the limit → 0 we find This is completely well-defined, and invariant under the fuzzy modular group SL(2, Z N ) using the above results. For example, the partition function for the rectangular fuzzy torus corresponds to the lattice k y = l x = 1 and k x = l y = 0, In the limit N → ∞, the partition function (98) looks very similar to the partition function of the commutative torus L(ω 1 , ω 2 ) ∼ = L(τ, 1), which up to a factor takes the form However Z N provides a regularization which is not equivalent to a simple cutoff or zeta function regularization (see for example [8]), because the spectrum of the fuzzy torus significantly differs from the commutative one near the boundary of the Brillouin zone, thus regularizing the theory. Moreover, there may be some multiplicity due to the periodic structure of Brillouin zones. The fuzzy torus regularization should provide a useful new tool in this context, taking advantage of the bounded spectrum. Consider e.g. the free energy in terms of the partition function ln[sin 2 ( π N (k x n 2 − l x n 1 )) + sin 2 ( π N (k y n 2 − l y n 1 ))] (102) using the identity (68). In the semi-classical approximation we can replace the sum by an integral over the appropriate Brillouin zone, where N denotes its multiplicity. This integral is invariant under SL(2, R) transformation of the lattice vectors ω 1 and ω 2 . However we have not been able to evaluate it in closed form.
We conclude with some remarks on possible applications of the above results. In the context of string theory, a natural problem is to integrate over the moduli space of all tori. This corresponds here to the sum of the partition function (98) over all fuzzy tori defined by k and l. This is certainly finite for any given N , since the moduli space Z 2 N is finite. There are two natural prescriptions to define the sum over all tori. First, one can consider This of course entails an over-counting of lattices L N (l, k) related by SL(2, Z N ), but it is still finite. On the other hand, one could compute which is analogous to the one-loop partition function for a closed string [8] . If all SL(2, Z N ) orbits on Z 2 N have the same cardinality, then the two definitions for Z and Z are related by a factor and hence equivalent. However this may not be true in general, and the two definitions may not be equivalent in the large N limit. We leave a more detailed study of these issues to future work.
Finally, the form of the spectrum of the Laplacian on L N (l, k) suggests to formulate a finite analog of the modular form E(1, ω 1 , ω 2 ) which could be replaced here by the fuzzy analog This is invariant under P SL(2, Z N ), and reduces to 2πρ N N 2 E(1, w1, w2) in the limit N → ∞. It would be interesting to construct fuzzy E q (p, l, k) which reduce to Eisenstein series E(p, ω 1 , ω 2 ) in the limit N → ∞.

The general fuzzy tori as solution of the massive matrix model
It is easy to see that the general torus corresopnding to the lattice L N as above is a solution of the massive matrix model with equations of motion as observed in [6]. Using the matrices (63) we find with i = 2 for A = 1, 2 and i = 1 for A = 3, 4. Thus the embedding function X a are solutions of (109) for R 1 = R 2 = R and where c N is defined in (51). The spectrum is invariant under SL(2, Z N ) transformation, as shown before. In the semiclassical limit, the equations of motion reduce to or G x A ∼ −τ 2 2 x A if the lattice vectors are chosen to be ω 1 = τ and ω 2 = 1.

Dirac operator on the fuzzy torus
In this final section we study the Dirac equation on the rectangular fuzzy torus generated by C and S. First, we introduce the following representation of the two-dimensional Euclidean Gamma matrices which satisfy the Clifford algebra {γ i , γ j } = 2δ ij . Then a 4-dimensional Clifford algebra can be constructed as follows Now we define Explicitly The Dirac equation reads or in terms of the C and S operators In matrix form, the Dirac operator becomes As an ansatz for a four component spinor we take where a nm , b nm , c nm , d nm ∈ C, and |n, m >= C n S m ∈ A N .
Note that this does not and should not agree with the spectrum of the Dirac operator on a noncommutative torus in the sense of [2,3] with infinite-dimensional algebra A, since the differential calculus here is based on inner derivations, while for the noncommutative torus it is based on exterior derivations.

Conclusion
We studied general fuzzy tori with algebra of functions A = M N (C) as realized in Yang-Mills matrix models, and discussed in detail their effective geometry. Our main result is that if certain divisibility conditions are satisfied, then the tori can have non-trivial effective geometry. The corresponding modular space of such fuzzy tori is studied, and characterized in terms of a "fuzzy" modular group P SL(2, Z N ). We determined the irreducible spectrum of the Laplace operator on these tori, and exhibit their invariance under P SL(2, Z N ). In the semiclassical limit, the general commutative torus represented by two generic vectors in the complex plane is recovered, with generic modular parameter τ . This is quite remarkable since the "apparent" embedding is always rectangular.
The results of this paper demonstrate the generality of the class of fuzzy embedded noncommutative spaces with quantized algebra of functions A = M N (C). Moreover, our results suggest applications of the fuzzy torus to regularize field-theoretical or string-theoretical models involving tori. A more detailed description of the moduli space (62) would be desirable, which requires a detailed understanding of the structure of P SL(2, Z N ) for non-prime integers N . Our results also suggest the possibility to define fuzzy analogs of modular forms. We leave an exploration of these topics to future work.