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Fuzzy Torus via q-Parafermion

We note that the recently introduced fuzzy torus can be regarded as a q-deformed parafermion. Based on this picture, classification of the Hermitian representations of the fuzzy torus is carried out. The result involves Fock-type representations and new finite dimensional representations for q being a root of unity as well as already known finite dimensional ones.


Introduction
Fuzzy spaces have been widely accepted as models of noncommutative manifolds in the context of quantum field theory (see for example [1]). The fuzzy 2-sphere [2] has been studied extensively. Regarding the ordinary 2-sphere S 2 as an one-dimensional complex projective space CP 1 , construction of the fuzzy S 2 has been extended to higher dimensional fuzzy projective spaces [3]. Many of the fuzzy spaces obtained so far are related to the complex projective space CP n . Supersymmetric extensions of the fuzzy S 2 [4] and the fuzzy CP n [5] embody such relations. Exploiting the observation that CP 3 is an S 2 bundle over S 4 , the fuzzy S 4 has been constructed [6,7]. Fuzzy S 5 has been established by utilizing a U(1) fibration over a fuzzy CP 2 [8]. Further examples are found in the investigations of fuzzy unitary Grassmannian spaces [9,10] and fuzzy orthogonal Grassmannian spaces [11]. Furthermore, fuzzy versions of various toric varieties have been considered in [12] by embedding such varieties in CP n .
The sphere S 2 is a Riemann surface of genus zero that may be embedded in R 3 . Using the Cartesian coordinates (x, y, z) in R 3 it is described by a constrained polynomial C(x, y, z) = 0, where C(x, y, z) = x 2 + y 2 + z 2 − 1. Similar polynomial description of compact connected Riemann surfaces of arbitrary genus has been introduced in [13]. These authors have developed a general recipe for constructing fuzzy Riemann surfaces. Explicit realization for the torus (genus one) is completed by defining the fuzzy torus as a nonlinear C-algebra with three generators [13]. The fuzzy torus algebra T 2 F has been recast as a q-deformed Lie algebra in [14]. This may be understood as a linearization of the nonlinear T 2 F algebra by using the q-deformed commutators. In the present work, we are looking at the fuzzy torus algebra from a different viewpoint. We regard T 2 F as a q-deformed parafermion algebra which is different from the q-deformations of parafermion or paraboson discussed so far [15][16][17][18][19][20]. Using this approach, we classify the Hermitian representations of T 2 F . In addition to recovering the known finite dimensional representations [13,14], some new and nontrivial representations are found. A Fock-type infinite dimensional representation, and a finite dimensional representation for q being a root of unity are produced. The classification of the Hermitian representations will facilitate deeper understanding of T 2 F and model building of quantum field theory on the fuzzy torus. In the next section, we recall the definition of T 2 F and explain the interpretation as q-deformed parafermion. Representations of T 2 F are systematically constructed under simple assumptions in §3 so as to achieve the classification. Our concluding remarks are given in §4.

Fuzzy torus and deformed parafermion
The polynomial description [13] of the torus comprising of three real commuting variables reads where µ and c (> 0) are real parameters. In [13], the parameter c is set equal to unity. Following [14], we, however, keep its value arbitrary. The relation C(x, y, z) = 0 describes a surface in R 3 , which is a squashed sphere for the domain − √ c < µ < √ c, and a torus for √ c < µ. Fuzzy analog of the surface is introduced by replacing the commuting variables (x, y, z) with the generating elements (X, Y, Z) that satisfy the commutation relations [13] [ and the constraint The parameter imparts noncommutativity. Its commutative limit is given by → 0. The associative algebra T 2 F , generated by the variables (X, Y, Z) obeying the relations (2.2, 2.3), satisfy the Jacobi identity. The element C F is the center of the algebra.
Introducing the complex variable W = X + iY and its Hermitian adjoint W † , the defining relations (2.2, 2.3) are reexpressed as Eliminating Z via the first equation in (2.5), it is observed [13] that the operators W, W † obey the following trilinear relation and its Hermitian adjoint: (2.7) The deformation parameter q is introduced [14] by the relation It follows that the complex parameter q is of unit magnitude: |q| = 1. The commutative → 0 limit now corresponds to q → 1. We now scale the conjugate variables W and W † as follows For a fixed µ above scaling is well-defined in the noncommutative q = 1 i.e. = 0 regime. The trilinear relation (2.7) now assumes two distinct forms depending on the sign of the parameter µ. For the choice µ < 0, it reads where the q-number is defined as usual For the alternate µ > 0 case, the relation (2.7) may be recast as The trilinear relation (2.10) may be regarded as q-deformation of the parafermion [21]. The relation (2.12) has the same form albeit with an opposite sign on the right hand side. One may regard this as a variant of the q-deformed parafermion. It becomes evident in the next section that the q-parafermionic picture observed in (2.10) and (2.12) facilitates a systematic study of the representations. The transformation (2.9) is singular for a fixed µ in the limit q → 1. In this scenario the classical q = 1 parafermions do not arise. There is, however, an interesting possibility which allows taking the q → 1 limit in (2.9). This limit is pertinent in understanding some of the representations discussed below.

Hermitian representations
In this section, we obtain all Hermitian representations of the algebra defined by (2.2). Our strategy is as follows: we start with the smallest number of assumptions and construct the most general representation. Then, further assumptions are imposed to obtain possible subclasses of the representations. As observed in [13], the operators aa † and a † a commute, and, therefore, they are simultaneously diagonalizable. We thus assume the existence of a normalized state |0 such that Other states may be obtained by repeated actions of a or a † on |0 . The states are required to be normalizable: where N ±n are the normalization constants. Normalization of the |±1 states puts constraints on the values of p and r: It follows that p, r > 0, and we choose For the states of |±n ∀n ≥ 2, we use the relations (2.10) or (2.12) appropriately for evaluating the norm. For the regime µ < 0, the following relations are verified inductively: and The diagonal operators read For the choice µ > 0 the construction of states proceeds as before: while the relations (3.5) and (3.6) hold with the replacement of the normalization constant A n by A n . The Casimir operator is quartic in the variables W, W † . Its eigenvalues may be computed via the results obtained above: (3.9) We have formally obtained a double-sided infinite dimensional representation with two parameters: p, r. Our representation has close kinship with the symplecton realization [22] of the boson calculus. Reflecting the symmetry of (2.10) (or (2.12)) and its adjoint under the exchange of a and a † , the representation has the symmetry structure A n+1 (p, r) = A −n (r, p). The usefulness of the above representations becomes evident below where we impose restrictions for obtaining the subclasses of the above infinite dimensional representation. These restrictions terminate the infinite series of state vectors.
Assuming that the state |0 is annihilated by the operator a, we have, r = 0. All states |−n labeled by negative integers are eliminated while the semi-infinite series of states {|n |n = 0, 1, 2, . . . ∞} are retained. This is a Fock-type representation constructed on the vacuum |0 . The number of parameters is reduced to one.
A key requirement for the existence for the classes of representations discussed below is that their Hermiticity needs to be maintained. This, in turn, requires the constants A 2 ±n , A 2 ±n ∀n > 0 to be nonnegative. Keeping this in mind, we study all the possibilities in order.
(2) Cyclic representations The cyclic finite dimensional representations may be ensured by identifying the states |N and |0 . In particular, this makes the eigenvalues of the operators aa † and a † a on the states |N and |0 same. For the µ > 0 case, we obtain Eliminating r from the relations in (3.18), we obtain It then turns out that r = p, and A n = √ p ∀ n. This leads to Z |n = 0 [13]. This case is uninteresting. Alternate choice [N] √ q = 0, i.e. q N = 1 yields a class of two-parametric (p, r) N-dimensional representations: These representations corresponds to the 'loop solutions' in [13,14], and possess the symmetry A n (p, r) = A N +1−n (r, p). (3.20) To examine their nonnegativity the elements {A 2 n |n ∈ (1, . . . , N)} may be recast by isolating their symmetric part S n (p + r) as follows (3.21) Due to the symmetry (3.20) we only need to check the nonnegativity of the elements A 2 n for n = 1, . . . , N/2 ((N + 1)/2) for an even (odd) N. In this domain we have P n ≥ 0, and S n (p + r) = sin 2 ((n − 1)π/N) sin 2 (π/N) (1 + (1 − p − r) f n−1 ), (3.22) where the sequence {f n } has been defined in (3.17). In the present context the entries of the sequence {f n−1 }, where n = 2, . . . , N/2 ((N + 1)/2) for an even (odd) N, is bounded as 0 < f n−1 ≤ 1. This yields the parametric values 0 < p, 0 < r, p + r < 2 for the required nonnegativity. For the odd-N case the symmetric element reads It may be noticed that the solutions obtained in (3.16) may be obtained as the r → 0 limiting case of the present cyclic representations. The corresponding matrix structures are, however, of different ranks, and in that sense inequivalent. Moreover the cyclic property of the solutions (3.19) is lost when r = 0 is substituted in them. Hermitian cyclic representations do not exist for the µ < 0 case as the elements {A 2 n |n = 1, . . . , N} are not nonnegative. For instance, we have (3.24)

(3) Infinite dimensional representations
Turning towards the infinite dimensional representations we discuss the special cases where we notice the existence of such Hermitian representations. The identity allows us to rewrite the elements A 2 n for the Fock states (r = 0) in the µ > 0 case as For the choice p = 1, therefore, the Hermiticity of the infinite set of basis states with n = 1, 2, . . . ∞ is guaranteed. To establish a range of values of the order parameter p around p = 1 that preserves the positivity A 2 n > 0, we use (3.13) to obtain The entries of the sequence {f n |n = 1, 2, . . . ∞} for any positive integral value of n remain finite for the choice θ = 2π/N , where N is an irrational number. Employing the finite maximum and minimum entries of the said sequence a range of values of p may now be easily established for which Fock-state representations are Hermitian and well-defined. Another possible scenario is to turn to the quasi-classical states mentioned in the context of (2.13). Using the classical limit q → 1 in the context of the (p, r) two-parametric states for the regime µ > 0 we obtain A 2 n = n 2 + (p − r − 1) n + r ∀ n = 1, 2, . . . ∞. (3.27) For the choice p ≥ r + 1, these two-parametric states maintain A 2 n > 0 ensuring the Hermiticity of the representations. These infinite dimensional quasi-classical states reflect, in some sense, noncommutative spaces bordering on the commutative regime. Hermitian infinite dimensional representations for the µ < 0 case do not exist.
We have studied so far the case of µ = 0. To complete the study of Hermitian representations, we next investigate the case of µ = 0. Although the algebra of this case does not correspond to a deformed parafermion, one can repeat the analysis used earlier. We use the trilinear relation and assume a state |0 subject to the conditions Using the same arguments applied earlier the positivity p, r > 0 follows, and the states are given by This leads to The eigenvalue of the Casimir operator reads Relations parallel to (3.30) -(3.33) also hold for the state |−n . Fock-type representation is obtained by requiring W |0 = 0 i.e. r = 0. Implementing w N = 0 we may truncate the Fock-type representation and obtain a finite N dimensional one. There is a slight difference from the case of µ = 0. The equation w N = [N] q p = 0, p > 0 may be solved only for the choice [N] q = 0. Namely, it is possible to obtain finite dimensional Hermitian representations from the Fock-type one for q 2N = 1 i.e. q = exp(iπ/N). The expression for w 2 n is found to be nonnegative: Such a truncation for a generic q does not exist. Now we investigate the possible realizations of the cyclic representations for the µ = 0 case. Conditions for cyclic representation are given by Eliminating r from (3.36) we obtain a relation for p: Therefore the cyclic representations may exist only if q N = 1. By computation, however, it may be seen that negative values of w 2 n exist: This eliminates the possibility of having Hermitian cyclic representations for the µ = 0 case. Lastly, paralleling the µ > 0 case the infinite dimensional representations for the choice µ = 0 exist for the said semi-classical states in the q → 1 limit as the elements are nonnegative for p ≥ r.

Concluding remarks
We introduced a viewpoint that regards fuzzy torus algebra as a q-deformed parafermion. Based on this picture, we investigated Hermitian representations of the fuzzy torus under the assumptions (3.1) and normalizability of the representation basis. All known representations in [13,14] were recovered and some new representations (both finite and infinite dimensional) were discovered. To summarize, the finite dimensional Hermitian representations of the algebra (2.10) and (2.12) may be classified as follows. (i) For a generic value of q, the representations exist with N-dependent order parameter p for both the µ < 0 and µ > 0 regime. (ii) For the root of unity values of q = exp(i2π/N) the Hermitian Fock-type representations given in (3.16) exist only for the choice µ > 0. (iii) Two parametric cyclic representations given in (3.19) exist for root of unity values of q = exp(i2π/N) for the µ > 0 case. In the r → 0 limit these representations reduce to the Fock-type solutions (3.16). Nevertheless, the matrix structures of the two classes of solutions are of different ranks, and the cyclic property of the solutions (3.19) is lost on substituting r = 0 in them. As these two categories of solutions reflect the allowed noncommutative structure of spaces, their geometric properties are likely to be quite distinct. Turning towards the existence of an infinite dimensional Hermitian Fock-type representation discussed here, we note that these solutions allow us to introduce coherent states of the fuzzy torus as eigenstates of the operator a. The coherent state will lead to developing a star-product on the fuzzy torus by the method of [23]. This will be discussed in a separate publication.
We have seen that the fuzzy torus algebra admits interpretations as a q-deformed Lie algebra [14], or a q-deformed parafermion. These interpretations may not be the only possibilities. One can derive the following set of relations from (2. This set looks like a variant of the Sklyanin algebra [24]. The implications of this viewpoint is not clear at present. It is also important to develop a supersymmetric extension of the fuzzy torus, and fuzzy surfaces of higher genus. Their relations with generalizations of parafermions are of interest.