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On the relation between non-commutative field theories at theta = infinity and large N matrix field theories

It is well-known that non-commutative (NC) field theories at theta = infinity are ``equivalent'' to large N matrix field theories to all orders in perturbation theory, due to the dominance of planar diagrams. By formulating a NC field theory on the lattice non-perturbatively and mapping it onto a twisted reduced model, we point out that the above equivalence does not hold if the translational symmetry of the NC field theory is broken spontaneously. As an example we discuss NC scalar field theory, where such a spontaneous symmetry breakdown has been confirmed by Monte Carlo simulations.


Introduction
Field theories on non-commutative (NC) spaces, which are characterized by the commutation relation among the coordinate operatorsx µ (µ = 1, · · · , D), have recently attracted much attention in the context of string theory and quantum gravity.
In particular perturbative aspects of such theories have been discussed extensively in the literature. Diagrammatically the difference from ordinary field theories in the commutative space is represented by a momentum dependent phase factor of the form assigned to each vertex, where Θ µν is the non-commutativity tensor in eq. (1.1), and p (i) µ (i = 1, · · · , n) represent the momenta flowing into the vertex. The phase factors (except those depending only on the external momenta) cancel in the planar diagrams, whereas in the non-planar diagrams, at least one phase factor depending on an internal momentum remains. Let θ be the typical magnitude of the non-zero elements of the non-commutativity tensor Θ µν . If we take the θ → ∞ limit (after removing the phase factor depending only on the external momenta), the non-planar diagrams vanish due to the oscillating phase, and we are left with the planar diagrams. In this way the θ → ∞ limit of NC field theories is "equivalent" to the large N matrix field theory in the commutative space to all orders in perturbation theory.
In this letter we point out that this well-known "equivalence" does not always hold non-perturbatively. In order to discuss this issue, we definitely need a non-perturbative formulation of NC field theory. As in ordinary field theories, we may regularize a NC field theory non-perturbatively by putting it on a lattice [1]. This is achieved by imposing the operator identity e 2 π ixµ/a = 1 1 , where a is the lattice spacing, and the lattice sites are "fuzzy" due to the non-commutativity (1.1). Using a correspondence between a field on a finite lattice and a matrix, which is exact in the sense that the finite degrees of freedom on one side are mapped onto the other, one to one, we can map a lattice NC field theory onto the twisted reduced model [1]. This is a refinement of the earlier work [2], where continuum NC theory is mapped to the continuum version of the twisted reduced model [3], which is formally defined in terms of infinite-dimensional matrices. Reduced models appeared in history first as an equivalent description of large N gauge theory in the early 80s [4]. The original Eguchi-Kawai (EK) model is proved to be "equivalent" to SU(∞) lattice gauge theory under the assumption that the U(1) D symmetry of the model is not spontaneously broken. It was found, however, that this symmetry is actually broken in the weak coupling regime at D ≥ 3 [5]. The twisted EK model is one of the proposals to overcome this problem [6] 1 , and the equivalence was confirmed by the weak coupling expansion as well as the strong coupling expansion. Soon later it was noticed that this idea of "twisted reduction" can be applied also to non-gauge theories written in terms of matrix fields [8,9]. As in the case of gauge theory [4,6], the proof based on the Schwinger-Dyson (SD) equations reveals that there are certain symmetries which should not be spontaneously broken in order for the equivalence to hold [9].
In the new interpretation of the twisted reduced model as a lattice regularization of NC field theories, the non-commutativity parameter θ is identified as [1] θ ∝ N 2/D a 2 , (1.4) where N is the size of the matrices that appear in twisted reduced models. In the context of EK equivalence, one has to take the large N limit at a fixed lattice spacing a (hence at fixed bare parameters with appropriate normalization with respect to N ) 2 , which corresponds to the θ = ∞ limit of NC field theories as is clear from eq. (1.4). Thus the EK equivalence provides a non-perturbative account for the aforementioned "equivalence" between the NC field theories at θ = ∞ and the large N matrix field theories. We will see that the condition for this "equivalence" is -in the language of NC field theories -that the translational symmetry is not spontaneously broken.
In fact in NC scalar field theories the spontaneous breakdown of translational invariance was conjectured in Refs. [13,14], and confirmed recently by Monte Carlo simulations in Refs. [11,12,15]. Before we discuss the NC scalar field theory, we start in the next Section with a brief comment on the gauge theory case. For a comprehensive review on the EK equivalence we refer the reader to Ref. [16], and for reviews on the novel connection between reduced models and NC field theories to Refs. [17,18].

The case of gauge theory
The twisted EK model [6] is defined by where U µ (µ = 1, · · · , D) are N × N unitary matrices, and Z µν = (Z νµ ) * is a phase factor which is referred to as the "twist". Here and henceforth we assume D to be an even integer. The model has the SU(N ) symmetry As SU(N ) covariant quantities we define where the Greek indices may take the values ±1 · · · ± D, and we define U −µ = U † µ . For each term w(µ, ν, · · · , σ), we introduce the displacement vector v =μ +ν + · · · +σ , (2.5) whereμ is a unit vector in the µ direction. Under the U(1) D transformation (2.3) the quantity (2.4) transforms as and Tr w(µ, ν, · · · , σ) is SU(N ) invariant. The EK equivalence states that in the planar large N limit the vacuum expectation value (VEV) for v = 0 agrees with the corresponding Wilson loop in the SU(∞) lattice gauge theory, where P λρ denotes the number of plaquettes in the (λ, ρ) plane within the surface which spans the Wilson loop. The equivalence requires all the VEVs Tr w(µ, ν, · · · , σ) for v = 0 to vanish, which is satisfied if the U(1) D symmetry (2.3) is not spontaneously broken. Now we interpret the twisted reduced model (2.1) as a NC gauge theory [1]. Note that the configuration which minimizes the action (2.1) is given by U µ = Γ µ , where Γ µ are SU(N ) matrices satisfying the 't Hooft-Weyl algebra 3 This motivates us to make a substitution which brings the action (2.1) into the form Then we consider a one-to-one correspondence from the N × N matricesŨ µ to the lattice gauge field of rank 1 on a L D lattice, where N 2 = L D due to the matching of the degrees of freedom. In this correspondence, the matrix product becomes the so-called "star-product" of fields on the lattice, and the transformatioñ represents a shift of the lattice gauge field in the ν direction by one lattice unit. Thus the expression (2.10) can be regarded as the "plaquette action" for the NC gauge theory on the lattice. Note also that the SU(N ) symmetry (2.2) of the twisted reduced model can be re-written asŨ which represents the "gauge symmetry" of the lattice NC gauge theory. The action (2.10) is invariant under the transformation (2.11), which represents the translational invariance of the NC gauge theory. The whole symmetry group generated by such transformations is (Z L ) D , whose elements are given explicitly as (2.14) Note here that (Γ µ ) L = 1 1, so the L D lattice is periodic. In terms of original variables in the twisted EK model, the transformation (2.11) reads It is known that the U(1) D symmetry of the twisted EK model is unbroken in both, the weak coupling (large β) and the strong coupling (small β) regime [6]. However, there is a certain indication in D = 4 that the symmetry is broken at intermediate couplings [19], which might be related to the perturbative instabilities pointed out in Ref. [20]. These issues should be clarified further.

The case of scalar field theory
Let us move on to the case of scalar field theory. For concreteness we discuss the (D + 1)dimensional NC scalar field theory, where non-commutativity is introduced only in D directions (D even), but it is straightforward to generalize our argument to the case with an arbitrary number of commutative directions (including the case with no commutative direction). The corresponding twisted reduced model can be given as [11,12] S TRM = N Tr whereφ(t) represents a Hermitian N × N matrix living on one of the discrete time points t = 1, . . . , T . The coupling constant λ is assumed to be real positive, while the mass squared m 2 can take any real number. The action (3.1) has the (Z L ) D symmetrŷ which plays a crucial role. The EK equivalence states that, in the large N limit with fixed m 2 and λ, the twisted reduced model (3.1) is "equivalent" to the matrix field theory on the (D + 1)-dimensional hypercubic lattice, where Φ( x, t) represents a Hermitian N × N matrix living on each site. This matrix field theory has the global SU(N ) symmetry where g ∈ SU(N ). Let us consider a matrix field product

5) whose trace is invariant under a global SU(N ) transformation (3.4).
In the twisted reduced model (3.1), we may define the corresponding quantity as Let us denote the VEV with respect to the twisted reduced model (3.1) and the matrix field theory (3.3) by · · · TRM and · · · MFT , respectively. Then the statement is that Trw( x 1 , t 1 ; · · · ; x k , t k ) TRM = Tr w( x 1 , t 1 ; · · · ; x k , t k ) MFT (3.7) in the large N limit with fixed m 2 and λ, if the (Z L ) D symmetry (3.2) is not spontaneously broken. A proof based on the SD equations is presented in the Appendix. When we interpret the twisted reduced model (3.1) as a (D + 1)-dimensional NC scalar field theory, we map the matrix configuration {φ(t)} to a one-component real scalar field {φ( x, t)} on the L D × T lattice, where N 2 = L D . In this correspondence, the (Z L ) D symmetry (3.2) is nothing but the translational symmetry in the NC directions. Hence we conclude that the NC scalar field theory at θ = ∞ is equivalent to the large N matrix field theory, provided that the translational symmetry in the NC directions is not spontaneously broken.
The (2 + 1)-dimensional case (D = 2) has been studied by Monte Carlo simulation in Refs. [11,12]. In the planar large N limit with m 2 below some (negative) critical value m 2 c,TRM (λ), the scalar field φ( x, t) acquires a VEV, 4 which depends on the coordinates x in the NC directions. In this "striped phase", the translational symmetry is spontaneously broken 5 , so the equivalence to the large N matrix field theory is no longer valid.
On the other hand, in the planar large N limit of the matrix field theory (3.3) one expects a standard Ising-type phase transition at some critical point m 2 = m 2 c,MFT (λ), below which the matrix field Φ( x, t) acquires a uniform VEV proportional to the unit matrix. From the simulation results of the twisted reduced model, we may conclude that m 2 c,TRM (λ) ≥ m 2 c,MFT (λ). If we assume the opposite, i.e. m 2 c,TRM (λ) < m 2 c,MFT (λ), we should have observed a uniformly ordered phase for m 2 c,TRM (λ) < m 2 < m 2 c,MFT (λ), which is not the case. A priori there is no compelling reason to believe that m 2 c,TRM (λ) = m 2 c,MFT (λ). In Ref. [12] we have determined the effective mass M eff (or the renormalized mass) squared from the dispersion relation E 2 = M 2 eff + p 2 in the planar large N limit of the twisted reduced model (3.1). We have seen that it behaves linearly with respect to the bare mass squared as to a good accuracy as we approach the critical point m 2 c,TRM (λ) from above. Since the correlation length may be identified as ξ = (M eff ) −1 , our result (3.8) implies that the (standard) critical exponent ν, defined by agrees with the mean field value ν = 1/2. (More precisely, our data imply ν = 0.48(4).) On the other hand, the critical exponent for the Ising-like phase transition in the 3d matrix field theory (3.3) is expected to deviate from the mean field value generically. For instance, Refs. [22,23] obtain ν ≃ 0.67 from a Wilsonian renormalization group study with a number of drastic simplifications. If we rely on that result, then the fact that we obtain the mean field value ν ≃ 1/2 suggests that the phase transition in the NC scalar field theory at θ = ∞ has nothing to do with the phase transition in the large N matrix field theory.

Summary and discussions
In this letter we pointed out that the "equivalence" between non-commutative field theories at θ = ∞ and large N matrix field theories, which is commonly believed based on perturbation theory, does not hold when the translational symmetry is spontaneously broken. Our argument is based on the new interpretation of the twisted reduced models as the lattice NC field theories. In particular the equivalence does not hold in the striped phase of the NC scalar field theory. 6 Even in such a parameter region, if one formally expands the two theories around the classical vacuum with a uniform order, one would obtain the same perturbative series. What occurs here is that the NC theory actually chooses a different vacuum.
In the traditional context of EK equivalence, the spontaneous symmetry breaking (SSB) is problematic if it occurs at the points in the parameter space where the continuum limit should be taken. For instance, in the case of scalar field theory, one cannot study the continuum limit of the large N matrix field theories by using the twisted reduced models. In the case of gauge theory, the SSB might occur at some intermediate coupling, but this is harmless since the continuum limit is taken by extrapolating the bare coupling constant to zero along with a → 0.
On the other hand, from the view point of NC field theories, the SSB of translational symmetry provides a possibility to construct NC field theories with finite θ which do not reduce to the corresponding large N matrix field theories in the θ → ∞ limit. The 3d NC scalar field theory studied in Ref. [12] provides the first explicit example of this sort. This implies that the universality class of NC field theories can be different from that of large N matrix field theories.

Acknowledgments
It is our pleasure to thank Tomomi Ishikawa, Satoshi Iso, Hikaru Kawai, Yoshihisa Kitazawa and Yoshiaki Susaki for useful discussions. The work of J.N. was supported in part by Grant-in-Aid for Scientific Research (No. 14740163) from the Ministry of Education, Culture, Sports, Science and Technology.

A. A proof of the EK equivalence for scalar field theory
In this Appendix we show that the EK equivalence (3.7) between the twisted reduced model (3.1) and the matrix field theory (3.3) holds in the large N limit with fixed m 2 and λ, if the (Z L ) D symmetry (3.2) is not spontaneously broken. For that purpose, we consider the SD equations, which form a closed set of equations for the VEVs in eq. (3.7). The proof is analogous to the ones for other models [4,6,9].
In the matrix field theory (3.3), let us consider the quantity where λ a is a generator of U(N ), and apply the change of variable where ǫ is an infinitesimal parameter. Summing over the index a = 1, · · · , N 2 and using the formula and the large N factorization property, we obtain 0 = Tr w( x 2 , t 2 ; · · · ; x k , t k ) δ x 1 , xs δ t 1 ts Tr w( x 1 , t 1 ; · · · ; x s−1 , t s−1 ) MFT × Tr w( x s+1 , t s+1 ; · · · ; x k , t k ) Summing over the indices and using the large N factorization property, we obtain 0 = Trw( x 2 , t 2 ; · · · ; x k , t k ) TRM + Trw( x 1 , t 1 ; · · · ; x k , t k ) δ t 1 ,ts Tr w( x 1 , t 1 ; · · · ; x s−1 , t s−1 )Γ( x s )Γ † ( x 1 ) TRM × Tr w( x s+1 , t s+1 ; · · · ; x k , t k )Γ( Note that the operator on the left-hand-side transforms non-trivially under the (Z L ) D transformation (3.2), which is a symmetry of the twisted reduced model. Therefore the SD equations coincide in the large N limit with fixed m 2 and λ, if the (Z L ) D symmetry is not spontaneously broken. Assuming that the SD equations have a unique solution 7 , we obtain eq. (3.7).