Noncommutative Sugawara Construction

The noncommutative extension of the Sugawara construction for free massless fermionic fields in two dimensions is studied. We prove that the equivalence of the noncommutative Sugawara energy-momentum tensor and symmetric energy-momentum tensor persists in the noncommutative extension. Some relevant physical results of this equivalence are also discussed.


I. INTRODUCTION
One of the outstanding features of two dimensional field theories is bosonization where a free massless fermionic field can be written as a bosonic filed. This property is rooted in the work of Jordan and Wigner [1] where it was shown that the fermionic creation and annihilation operators may be represented as the bosonic counterparts. On the other hand, the idea of describing strong interaction process in terms of currents was proposed in [2][3][4]. In this approach, the dynamical variables are taken to be the currents and the canonical formalism is abandoned. In other words, each particle does not correspond to a field which satisfies the canonical commutation relation but Hilbert space is built upon current operators. Accordingly, it was shown that the energy momentum-tensor of these theories can be expressed as a quadratic function of the currents known as Sugawara construction [4]. Later on, it was proved that the symmetric energy-momentum tensor of the two-dimensional free massless fermionic theory is exactly equivalent to the Sugawara energy-momentum tensor which is bilinear in fermionic currents [5]. Indeed, this equivalence confirmed the result of [1], equivalence of free massless fermions and bosons, in an elegant way. Then generalization of this famous equivalence to the curved space-time was performed in [6] where boson-fermion correspondence was shown for a general metric in two dimensions. Our purpose is to study whether this equivalence is satisfied for noncommutative space, where the nature of the space-time changes at very short distances [7][8][9], which is not a trivial extension. The authors in [10] considered the noncommutative generalization of the Sugawara energy-momentum tensor and then used the Seiberg-Witten map. While in the present work, the correspondence between the noncommutative Sugawara construction and the symmetric energy-momentum tensor for two dimensional free fermionic theory is addressed, without employing the Seiberg-Witten map. Applying the techniques described in [5], we demonstrate that the noncommutative Sugawara energy-momentum tensor exactly leads to the symmetric energy-momentum tensor. An interesting physical consequence where g indicates the determinant of the metric with signature (+, +). The variation of the action corresponding to the Lagrangian density (II.1) can be written as Using the relation (II.3) and the cyclic property of the star product under the integral, we have Applying the equation of motion for free massless fermions, we find the energy-momentum tensor as which is completely symmetric under µ ↔ ν.

B. Sugawara energy-momentum tensor
The equivalence of the Sugawara construction and the symmetric energy-momentum tensor in commutative space has been shown in [5] and is reviewed in appendix A. In the present section, we construct the noncommutative version of the Sugawara energy-momentum tensor to demonstrate that it is precisely equivalent to (II.6). The Lagrangian (II.1) is invariant under global U(1) transformation which yields two different Noether currents [12] J µ (x) =: where : : denotes normal ordering. Now, we extend the commutative Sugawara construction to the noncommutative one as a bilinear function of J µ (x) with inserting star product instead of ordinary product where c is the Schwinger constant which appears in the equal-time commutator of currents. Since the mass dimension of energy-momentum tensor and of the currents in two dimensions is equal to two and one respectively, the coefficient c should be dimensionless while in four dimensions, it is a dimensionful quantity with dimension of a mass square. The detailed analysis of the current algebra in two dimensions shows that the value of c in noncommutative case is the same as the commutative one, c = 1 π [13]. To prove that (II.8) is exactly equivalent to (II.6), we need to regularize the operator products in (II.8). To this end, we use the point-splitting technique [14] and replace J µ (x) ⋆ J ν (x) with which leads to To perform some algebraic manipulations on (II.10), we employ the star product definition (II.2) where F ab is an abbreviated notation for the exponential operator appearing in (II.2). Accordingly, the first term of the equation (II.10) can be written as : . (II.12) Using Wick's theorem, (II.12) changes into : Rewriting the other terms of (II.10) similar to (II.13) and substituting them again into (II.10), we obtain : The field ordering appearing in the vacuum expectation value of the relation (II.15) is not time ordering and is defined as 1 We follow the convention used in [5].
In view of the above definition, the quantities R µν (x, ǫ) and S νµ (x, −ǫ) can be represented as : Converting equation (II.17) to the star product form, we obtain Note that the minus sign in R µν (x, ǫ) comes from the odd permutation of the fermionic fields.
Expanding ψ andψ up to the first order in ǫ yields and using the following symmetric limits Inserting the result (II.22) in (II.14), we arrive at The product of gamma matrices in (II.23) can be simplified using the Clifford algebra Substituting (II.24) into (II.23) and applying the equation of motion γ ξ ∂ ξ ψ = 0 and ∂ ξψ γ ξ = 0, we obtain the Sugawara energy-momentum tensor We notice that the last term of (II.23) vanishes in two dimensions as a result of the identity γ λ γ ξ γ λ = (2 − d)γ ξ . In order to show that T ⋆ µν = T s ⋆ µν , it is enough to demonstrate Q µν = 0. For simplicity, we carry out computations in the light-cone coordinate system, x ± = x 1 ± ix 2 . The representation of the Euclidean gamma matrices in the light-cone coordinates is given by (II.28) The equation of motion for two-dimensional massless fermions, which is described by iγ µ ∂ µ ψ = 0 with ψ = ψ 1 ψ 2 , in the light-cone coordinate system reduces to ∂ + ψ 1 = ∂ − ψ 2 = 0. Thus Using [x + , x − ] = 2θ, the expression ψ β (x) ⋆ψ α (x)(γ µ ) αβ with on-shell Dirac fermions is rewritten as One may then readily show that Performing some straightforward permutations, we get Q ++ = 0. Similarly Also for off-diagonal components Q ±∓ , we have Consequently Q µν = 0. This means that the equivalence of the Sugawara energy-momentum tensor and energy-momentum tensor T s ⋆ µν = T ⋆ µν in two-dimensional noncommutative space for free massless fermions is still satisfied. Also this equivalence occurs for the Sugawara form in terms of the current J µ (x), as defined in (II.7). We have To show this, let us write first all the components of the currents J µ (x) and J µ (x) using the representation of the gamma matrices (II.26) as follows (II.38) as well as

(II.39)
We notice that the star product appearing in the noncommutative currents is removed. Applying the permutation on the fermionic fields of the relation (II.38), we obtain This is an interesting result in two dimensions. Unlike the four-dimensional case, where J µ (x) and J µ (x) correspond to each other by the charge conjugation transformation [15], which is not conserved, in two dimensions the charge conjugation, as well as the Lorentz invariance, remain the symmetry of the theory as in their commutative case. Inserting (II.40) in (II.37) then leads to One of the physical consequences of this equivalence is noncommutative bosonization, which is obtained by writing the transformation of the field ψ under the spatial translation where T s ⋆ 21 is the conserved current arising from translational invariance 2 . We have and substituting the value of T s ⋆ 21 from (II.8) in (II.43) yields (II.44) To simplify (II.44), we insert dx ′ 2 δ(x 2 − x ′ 2 ) = 1 to use the trace property of the star product, which is given by Inserting (II.45) in (II.44) and applying the operator definition of the star product (II.11), we have .
The bracket terms appearing in the right-hand side of (II.46) are derived by considering the quantization condition {ψ α (x), ψ † β (x ′ )} = δ αβ δ(x 1 − x ′ 1 ) as follows (II.47) with γ 5 = iγ 1 γ 2 . Substituting (II.47) into (II.46) and then converting the result into the star product form, we have The solution of this equation is represented by where P denotes the path-ordering operator and ψ 0 is a constant spinor in two dimensions. Now, as a result of (II.38), we can use the bosonized form of the commutative current, which is introduced in appendix A. Hence, we conclude

III. DISCUSSION
In this paper, we established the noncommutative extension of the Sugawara construction in bilinear form of the currents for free massless fermions in two dimensions. It was shown that this construction is precisely equivalent to the symmetric energy-momentum tensor.
To prove the correctness of this equivalence, we determined the energy-momentum tensor in two separate methods. The first was the direct calculation using the symmetric definition of the energymomentum tensor for on-shell Dirac fermions and the second contained a detailed analysis of noncommutative Sugawara construction by applying the point-splitting regularization. Furthermore, for simplification in our calculation, we considered the light-cone system. In this coordinate, we realized that the currents J µ and J µ , apart from a minus sign, are actually the same in two dimensions which leads to the charge conjugation symmetry restoration. Eventually, we presented a physical consequence of this equivalence, named as noncommutative bosonization (e.g. see [16][17][18]), that relates a fermionic field to a bosonic field through an exponential function and demonstrated that a free massless fermion theory with a global U (1) symmetry in noncommutative space corresponds to a free massless boson theory. Also, the bosonized version of a theory with local U (1) symmetry such as two-dimensional noncommutative QED (NC-QED 2 ) was addressed in [19], where it was proven that the bosonized action contains a noncommutative Wess-Zumino-Witten (WZW) part, a gauge kinetic part and an interaction part between the WZW and gauge field.
The physical significance of the bosonization procedure is that it specifies a duality between the strong and weak couplings for particular interacting quantum field theories. The most famous example of this duality is the equivalence of the massive Thirring model and sine-Gordon model [20,21], where the weak coupling β of the bosonic theory, that is the sine-Gordon model, is related to the strong coupling g of the fermionic theory, the massive Thirring model, through the bosonization rule described by 4π β 2 = 1 + g π . Moreover, the duality between the noncommutative version of these models was studied in [16][17][18] where it was shown that the strong-weak duality is also preserved. However, it is notable that the strong-weak duality does not appear in the case of NC-QED 2 and its bosonized version, because of the appearance of the same couplings in two theories.

IV. ACKNOWLEDGMENTS
I am grateful to M.M. Sheikh-Jabbari for numerous fruitful discussions and careful reading of the manuscript. I would also like to thank M. Khorrami for his valuable comments and I appreciate the insightful and constructive remarks of M. Chaichian and the referee, which led to improvement of the manuscript. Moreover, I acknowledge the School of Physics of Institute for research in fundamental sciences (IPM) for the research facilities and environment.

Appendix A: Commutative Sugawara Construction
In this appendix, we present a detailed analysis on the proof of the relation T s µν = T µν in two dimensional commutative space for free massless fermions, as argued in [5], and describe some interesting consequences of this equivalence [22][23][24]. The Lagrangian for the massless fermions is given by which is invariant under the global phase transformation ψ → e iα ψ andψ → e −iαψ that gives the conserved current For this theory, the symmetric energy-momentum tensor is written as follows The energy-momentum tensor in Sugawara form is described by a bilinear function of the currents as To show T s µν = T µν , we start with (A.4) and replace j µ (x)j ν (x) with Applying Wick's theorem on (A.5) j µ (x + ǫ)j ν (x) = :ψ(x + ǫ)γ µ ψ(x + ǫ) ::ψ(x)γ ν ψ(x) : = :ψ(x + ǫ)γ µ ψ(x + ǫ)ψ(x)γ ν ψ(x) : and implementing a similar analysis for the other terms of (A.4), we arrive at where M µν and N µν are defined as and we have First, we concentrate on determining the value of M µν (x).
M 00 = M 11 =: (j 0 ) 2 + (j 1 ) 2 :, M 01 = M 10 =: j 0 j 1 + j 1 j 0 : . (A.10) Choosing γ 0 = σ z and γ 1 = iσ y , we find Therefore, all the components of M µν in terms of the fermionic fields are given by Performing some permutations on fermionic fields yields M µν = 0. In the next step, our purpose is to obtain the value of N µν . To this end, let us start with expansion of the fermionic fields up to the first order in ǫ Taking the symmetric limits (II.21), putting (A.14) in (A.7) and using the identity (II.24) for on-shell fermions in two dimensions, we find T s µν = i 4 : (ψγ µ ∂ ν ψ +ψγ ν ∂ µ ψ − ∂ µψ γ ν ψ − ∂ νψ γ µ ψ) :, (A. 15) which is exactly equal to T µν mentioned in (A.3). This equivalence suggests the existence of a canonical massless pseudo scalar field, satisfying [φ(x, t),φ(y, t)] = iδ(x − y), which is related to the conserved current j µ (x) through the following equation [23] j µ (x) = 1 √ π ǫ µν ∂ ν φ(x). (A. 16) If we substitute (A.16) into the Sugawara form (A.4) and use the identity ǫ µα ǫ νβ = g µβ g αν − g µν g αβ , it is found that which describes the energy-momentum tensor for a free massless boson. Another interesting result of the Sugawara construction is that it is possible to find explicitly the fermionic filed in terms of the bosonic field, as mentioned in the introduction part. To show this, we consider the equation describing the transformation of the field ψ under the spatial translation [24] where T s 01 is the Noether current of translational symmetry. Inserting T s 01 from (A.4) into (A.18), we have Applying the equal-time commutation relations j 0 (x ′ ), ψ(x) with γ 5 = γ 0 γ 1 , we arrive at Solving this equation yields where ψ 0 is a constant spinor in space-time. In the final step, we put the bosonized form of the currents from (A.16) in (A.22) As we see, the spinor field ψ is mapped to the bosonic field φ.