Duality in Scalar Field Theory on Noncommutative Phase Spaces

Edwin Langmann; Richard J. Szabo

REVIEW

Duality in Scalar Field Theory on Noncommutative Phase Spaces

Edwin Langmann, Richard J. Szabo

High Energy Physics - Theory · Submitted 6 Feb 2002 · 11 references

Pith's verdict: A clean classical duality with a plausible but schematic quantum extension; the stress-test objection about the mass term does not hold up.

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arXiv IDhep-th/0202039 (version 2)

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CitationEdwin Langmann, Richard J. Szabo. "Duality in Scalar Field Theory on Noncommutative Phase Spaces." arXiv:hep-th/0202039 (2002). https://pith.science/paper/VEWRSI25

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Langmann and Szabo have a genuine result here: the Fourier-transform duality of the free charged scalar in a constant B field extends to the interacting star-product theory. The classical statement (Proposition 1) is a direct computation — the vertex identity in Eq. (27) follows from the Gaussian integral in Appendix A, and the S_beta case is handled by the sign relabeling in Eq. (28). The free part, including the mass term, is the harmonic-oscillator identity. That checks out. The identification of the self-dual points Bθ = ±I is the interesting structural payoff, and the connection to Morita duality is stated honestly as prior work in [1]–[3]. The new content is the interacting extension, and it is real. The quantum extension (Proposition 2) is the part that needs care. The regulator C_Λ = (P²+m²)^{-1}f(Λ^{-2}[P²+Q²]) is duality invariant, and the paper's argument for this is essentially right. I checked the stress-test concern that the mass term breaks the invariance: it does not. Under the duality P² maps to P² (the free action with mass is invariant, as Section 2 shows), not to Q². Q² is separately invariant. So the full propagator is invariant, and the stress test's objection does not land. The real soft spot is the finiteness argument in Appendix B. The claim that all Feynman diagrams are finite sums at finite Λ rests on the observation that only finitely many Landau levels satisfy E(n)+F(n) < 2Λ². That is plausible, but the argument is schematic — it does not spell out bounds on the vertices v(n1,n2,n3,n4) or the sum over internal labels. The paper acknowledges the Λ→∞ limit and general G are conjectural in Section 4, but the finite-Λ finiteness claim itself is not fully written out. Minor: the abstract says 'to all orders of perturbation theory' without the regulator and G=I qualifications. The claim in Section 3 that Q_μ ↦ -Q_μ looks off to me (I get Q_μ ↦ Q_μ), though it does not affect the regulator since Q² is invariant either way. The citation [10] is in preparation. Who should read it: people working on noncommutative field theory, UV/IR mixing, and Morita duality. The self-dual points Bθ = ±I are the interesting special case. Recommendation: send it out. The classical result is solid, the quantum claim is plausible and the limitations are stated. A referee should push on Appendix B.

Referee report

Significance

If correct, the result identifies a genuine strong-weak-type duality in a class of noncommutative scalar field theories, extending the harmonic-oscillator Fourier duality to star-product interactions. The observation that the interaction vertex in position space, Eq. (27), has the same non-local form as the momentum-space vertex with θ → θ^{-1} is a clean and useful technical fact, and the identification of the self-dual points Bθ = ±I connects to known special points in noncommutative gauge theory and the noncommutative Landau problem. The paper is explicit that the quantum statement is perturbative, restricted to G = I, and made at finite cutoff, and it flags the Λ → ∞ limit and general G as open; that candour is a strength. The main value is the classical duality and the vertex identity; the quantum extension is the part that needs the most scrutiny.

Major objections (3)

[Section 3, Eqs. (37)–(39), Proposition 2] The invariance argument for the regulator appears to check only the cutoff function, not the full propagator. Eq. (37) defines C_Λ = (P²+m²)^{-1} f(Λ^{-2}[P²+Q²]). Under the duality the paper states that P_μ ↦ -P_μ and Q_μ ↦ -Q_μ (the sentence preceding Eq. (39)), which would leave P² and hence (P²+m²)^{-1} invariant. But the actual transformation induced by Eq. (12) on the covariant operators is P_μ ↦ -Q_μ and Q_μ ↦ -P_μ, as is implicit in the text's own remark that the duality maps the differential operator P_μ to k_μ - iB_{μν}∂̂_ν and then back to -P_μ after k ↦ B^{-1}k. Under P_μ ↦ -Q_μ the mass term is not invariant: (P²+m²)^{-1} ↦ (Q²+m²)^{-1}. The regulated free action ⟨Φ, f^{-1}(P²+m²)Φ⟩ would then map to ⟨Φ, f^{-1}(Q²+m²)Φ⟩, which differs from the original unless P² = Q². The paper verifies invariance of f alone (Eq. (39)) and infers invariance of the regularization procedure, but the mass factor in the propagator is not covered by that argument. This is a gap in the finite-Λ invariance claim itself, distinct from the acknowledged open issues of the Λ → ∞ limit and the finiteness argument of Appendix B. The authors should either exhibit the correct action of the duality on P_μ and Q_μ and show that the full propagator, mass term included, is mapped to the propagator of the dual theory, or state precisely in what sense Eq. (40) holds.
[Section 3, Eq. (35) and Proposition 2] The derivation of Eq. (35) from Eq. (33) uses the change of variables Φ ↦ Φ̃ in the path integral and asserts that the measure is invariant. This is plausible for the Gaussian measure defined by the free action, but the argument as written does not display the Jacobian of the field rescaling in Eq. (12) or show that it cancels against the |det B| factors from the source terms in Eq. (32). Since the central quantum claim is Eq. (35), the Jacobian bookkeeping should be made explicit, at least at the level of the free Gaussian measure, so that the reader can verify the normalization.
[Appendix B, Eq. (54)] The finiteness argument rests on the claim that c_Λ(n) is nonzero only for finitely many n at finite Λ because E(n)+F(n) = Σ_j B_j(m_j+ℓ_j+1) < 2Λ². This controls the sum over Landau-level labels, but the interaction vertices v(n₁,n₂,n₃,n₄) of Eq. (53) are only asserted to be well-defined, not shown to decay in the labels. For the claim that all Feynman diagrams are finite sums to be convincing, the authors should state the bound they use on the vertices, or explain why the sum over internal labels is finite independently of vertex growth. As written, the argument establishes that the propagator sum is truncated but not that the full diagram is finite.

Minor objections (7)

[Abstract] There are typographical errors: 'regualarization' and the doubled 'that that' in the sentence corresponding to abstract:s4. These should be corrected.
[Section 1, Eq. (5) and Proposition 1] The transformation rules in Eq. (5) list G ↦ G and B ↦ B as if they were part of the duality action, but Proposition 1 keeps G, B, m, α, β fixed. It would be clearer to say explicitly that these parameters are inert under the duality rather than displaying them as transformation rules.
[Section 1, discussion of θB = ±I] The text says the self-dual points are 'essentially but not quite self-dual since the duality transformation changes the sign of θ'. This is correct, but the phrase 'for each coupling constant g, the duality maps the field theory with θ(B) = +B^{-1} to that with θ(B) = -B^{-1}' would read more clearly if the dependence on B were made explicit, since B is held fixed.
[Section 4, Eq. (41)] The second duality under Fourier transformation alone is stated without proof. Since it is presented as an easy consequence, a one-line verification or a pointer to the analogous steps in Section 2 would help the reader.
[Section 4, citation [10]] Reference [10] (Langmann and Szabo, in preparation) is cited for the conjecture that the models are exactly solvable in 2n = 2 and for the matrix-model interpretation. Since this reference is unpublished, the claims that depend on it should be flagged as preliminary in the text, which they partly are; making that explicit at each point of use would be preferable.
[Appendix A, Eq. (45)] The step 'det(A_θ) = det(θ)⁴' and the use of (A_θ)^{-1} = A_{θ^{-1}} are correct but compressed. A brief indication of how the 8n×8n determinant reduces to the 2n×2n one would improve readability.
[Section 2, Eq. (24)] The vertex function V̂ in Eq. (24) is written with the star-product evaluated on plane waves; the intermediate step (e^{-ik₁·x} ⋆ e^{ik₂·x}) could be spelled out once, since the sign conventions for θ in Eq. (9) are easy to get wrong and the rest of the paper depends on them.

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The paper

The paper claims

A scalar field theory with star-product interactions and noncommuting momenta is shown to have a Fourier-transform duality that survives, inside a particular cutoff, to all orders of perturbation theory.

The reading

The paper asks whether a scalar field theory whose quartic interactions are built with the nonlocal Groenewold-Moyal star product, and whose momenta fail to commute because of a constant background field $B$, keeps the same form under Fourier transformation combined with a rescaling of the fields. It proves that it does at the classical level: the transformed theory has the same free action and the same two quartic interaction types, but the noncommutativity matrix $theta$ is replaced by $-B^{-1}theta^{-1}B^{-1}$ and the coupling $g$ by $|det(Btheta)|^{-1/2}g$, and applying the transformation twice returns the original parameters. The paper then introduces a cutoff that depends on the combination $P^2+Q^2$ and argues that it is invariant under the same transformation, so the identity for connected Green's functions persists to all orders of perturbation theory at finite cutoff. The interest is that this is an interacting version of the Fourier duality of a charged particle in a constant magnetic field, and it gives a precise exchange between short-distance and long-distance cutoffs in a class of noncommutative theories.

What carries the argument

The load-bearing mechanism is the Fourier transform of the Groenewold-Moyal star-product interaction. For the vertex $V$ of Eq. (27), position space and momentum space have the same nonlocal form, with $theta$ inverted and a factor $|det theta|^{-1}$; this is what makes the interacting action map back to the same interaction type. The regulator is built from the operator combination $P^2+Q^2$, where $P_mu=-i partial_mu-B_{mu nu}x^nu$ and $Q_mu=-i partial_mu+B_{mu nu}x^nu$, and the duality sends both $P$ and $Q$ to minus themselves, so any cutoff function of $P^2+Q^2$ is invariant. In the Landau-level basis the free propagator is diagonal and the regulated propagator is nonzero only for finitely many levels at finite cutoff, which is how the paper argues that every regulated Feynman diagram is a finite sum.

Extended reading notes

Core claim

The paper's central claim is that the action $S=S_0+g^2S_{int}$ for a charged scalar field with star-product quartic interactions obeys $S[Phi;B,g,theta]=S[tilde Phi;B,tilde g,tilde theta]$ under the field transformation $tilde Phi(x)=sqrt{|det B|},hat Phi(Bx)$, with $tilde theta=-B^{-1}theta^{-1}B^{-1}$ and $tilde g=|det(Btheta)|^{-1/2}g$, and that this map is an involution. The proof reduces to a Fourier-transform identity for the quartic star-product vertex: in position space the vertex has the same form as the momentum-space vertex with $theta$ replaced by $theta^{-1}$, up to a factor $|det theta|^{-1}$. The quantum statement is Proposition 2: with the regulator $C_Lambda=(P^2+m^2)^{-1}f(Lambda^{-2}(P^2+Q^2))$, the regulated generating functional of connected Green's functions satisfies $G_Lambda(J;B,g,theta)=G_Lambda(tilde J;B,tilde g,tilde theta)$ to all orders of perturbation theory for the trivial Euclidean metric $G=I$. At the special points $Btheta=pm I$ the coupling is unchanged and $theta$ only changes sign, so the regulated correlation functions are Fourier-self-similar up to rescaling and that sign flip.

Load-bearing premise

The load-bearing premise is that the specific regulator $C_Lambda=(P^2+m^2)^{-1}f(Lambda^{-2}(P^2+Q^2))$ is both invariant under the duality and strong enough to make every regulated Feynman diagram finite, since the all-orders quantum statement is proved only inside that regularization and the removal of the cutoff is left open.

Editorial extensions

If this is right

At $Btheta=pm I$ the coupling is unchanged and $theta$ only changes sign, so the regulated correlation functions are invariant under Fourier transformation up to rescaling and the sign of $theta$.
If the ultraviolet and infrared cutoffs are split as $Lambda_{UV}$ and $Lambda_{IR}$, the duality exchanges them, so a divergence that looks ultraviolet on one side appears infrared on the other.
The duality makes explicit why the correlation functions have no continuous $theta to 0$ limit: it relates finite-$theta$ data to data at transformed parameters rather than to a smooth limiting theory.
Dropping the field rescaling gives a second Fourier duality, with $B to B^{-1}$, $G to -B^{-1}GB^{-1}$, $theta to theta^{-1}$ and the same coupling scaling, which also changes the free part and at $B=pmtheta^{-1}$ matches the zero-rank limit of gauge Morita duality.
A two-flavor model with $theta$ and $-theta$ can be made fully self-dual at $Btheta=pm I$ if the duality also exchanges the two flavors.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the duality survives cutoff removal, ultraviolet/infrared mixing should be understood as an artifact of using separate position- and momentum-space cutoffs; an exact formulation would treat them as one duality-paired scale.
The self-dual points $Btheta=pm I$, where the paper conjectures exact solvability in two dimensions, are a natural place to test whether the duality is nonperturbative rather than only a finite-cutoff statement.
The transformation has the shape of an electromagnetic duality in $2n$ dimensions, with $B$ and $theta$ playing inverse roles; this suggests looking for the same structure in quantum Hall effective actions, where $B$ is a real magnetic field.

Original abstract

We describe a novel duality symmetry of Phi(4)-theory defined on noncommutative Euclidean space and with noncommuting momentum coordinates. This duality acts on the fields by Fourier transformation and scaling. It is an extension, to interactions defined with a star-product, of that which arises in quantum field theories of non-interacting scalar particles coupled to a constant background electromagnetic field. The dual models are in general of the same original form but with transformed coupling parameters, while in certain special cases all parameters are essentially unchanged. Using a particular regualarization we show, to all orders of perturbation theory, that that this duality also persists at the quantum level. We also point out various other properties of this class of noncommutative field theories.

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The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We describe a novel duality symmetry of Phi(4)-theory defined on noncommutative Euclidean space and with noncommuting momentum coordinates. This duality acts on the fields by Fourier transformation and scaling. It is an extension, to interactions defined with a star-product, of that which arises in quantum field theories of non-interacting scalar particles coupled to a constant background electromagnetic field. The dual models are in general of the same original form but with transformed coupling parameters, while in certain special cases all parameters are essentially unchanged. Using a particular regualarization we show, to all orders of perturbation theory, that that this duality also persists at the quantum level. We also point out various other properties of this class of noncommutative field theories.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We describe a novel duality symmetry of Phi(4)-theory defined on noncommutative Euclidean space and with noncommuting momentum coordinates.	contribution 0.98	0.93	0.30	asserted 1.00
s1	This duality acts on the fields by Fourier transformation and scaling.	contribution 0.57	0.72	0.08	asserted 1.00
s2	It is an extension, to interactions defined with a star-product, of that which arises in quantum field theories of non-interacting scalar particles coupled to a constant background electromagnetic field.	background 0.44	0.29	0.08	asserted 1.00
s3	The dual models are in general of the same original form but with transformed coupling parameters, while in certain special cases all parameters are essentially unchanged.	result 0.99	0.80	0.19	asserted 1.00
s4	Using a particular regualarization we show, to all orders of perturbation theory, that that this duality also persists at the quantum level.	result 0.92	0.95	0.43	asserted 1.00
s5	We also point out various other properties of this class of noncommutative field theories.	contribution 0.93	0.84	0.04	asserted 0.63

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v2

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (11)

[1]

A. Konechny and A. Schwarz, [hep-th/0012145; 0107251]; M.R. Douglas and N.A. Nekrasov, Rev. Mod. Phys. 73 (2002) 977 [hep-th/0106048]; R.J. Szabo, [hep- th/0109162]

View on Pith Pith review arXiv 2002
[3]

B. Pioline and A. Schwarz, J. High Energy Phys. 9908 (1999) 021 [h ep-th/9908019]; J. Ambjørn, Y.M. Makeenko, J. Nishimura and R.J. Szabo, J. High Ene rgy Phys. 0005 (2000) 023 [hep-th/0004147]; N. Seiberg, J. High Energy Phy s. 0009 (2000) 003 [hep-th/0008013]

View on Pith Pith review arXiv 1999
[10]

E. Langmann and R.J. Szabo, in preparation

View on Pith
[2]

N. Seiberg and E. Witten, J. High Energy Phys. 9909 (1999) 032 [h ep-th/9908142]

View on Pith 1999
[4]

L. Susskind, [hep-th/0101029]; A.P. Polychronakos, J. High Ene rgy Phys. 0104 (2001) 011 [hep-th/0103013]; S. Hellerman and M. Van Raamsdonk, J. High E nergy Phys. 0110 (2001) 039 [hep-th/0103179]; D. Karabali and B. Sakita, [hep -th/0106016]; C.D. Fosco and A. Lopez, [hep-th/0106136]; T.H. Hansson and A. Ka rlhede, [cond- mat/0109413]; A. Gorsky, I.I. Ko...

View on Pith Pith review arXiv 2001
[5]

Y. Habara, [hep-th/0105002]

View on Pith Pith review arXiv
[6]

V.P. Nair and A.P. Polychronakos, Phys. Lett. B 505 (2001) 267 [h ep-th/0011172]; B. Morariu and A.P. Polychronakos, Nucl. Phys. B 610 (2001) 531 [he p-th/0102157]; J. Gamboa, M. Loewe, F. M´ endez and J.C. Rojas, [hep-th/010422 4]; A. Jel- lal, J. Phys. A 34 (2001) 10159 [hep-th/0105303]; S. Bellucci, A. Ner sessian and C. Sochichiu, Phys. Lett. B 522 (2...

View on Pith Pith review arXiv 2001
[7]

I.Ya. Aref’eva, D.M. Belov and A.S. Koshelev, [hep-th/0001215]; F . Ruiz Ruiz, [hep- th/0202011]

View on Pith Pith review arXiv
[8]

Y.-G. Miao, H.J.W. M¨ uller-Kirsten and D.K. Park, [hep-th/0105029 ]

View on Pith Pith review arXiv
[9]

S. Minwalla, M. Van Raamsdonk and N. Seiberg, J. High Energy Phys . 0002 (2000) 020 [hep-th/9912072]

View on Pith Pith review arXiv 2000
[11]

R. Gopakumar, S. Minwalla and A. Strominger, J. High Energy Phy s. 0005 (2000) 020 [hep-th/0003160]. 13

View on Pith Pith review arXiv 2000

Forward citations

Cited by 13 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Renormalisation of \phi^4-theory on noncommutative R^4 in the matrix base
hep-th 2004-01 conditional novelty 8.0 of 10

Adding a harmonic oscillator potential to the free action makes four-dimensional noncommutative phi^4 renormalisable to all orders in perturbation theory, proved via Polchinski flow equations in the matrix base.
IR Dynamics from UV Divergences: UV/IR Mixing, NCFT, and the Hierarchy Problem
hep-ph 2019-09 conditional novelty 7.0 of 10

In noncommutative Yukawa theory, the CPT-required pair of interaction orderings converts the one-loop ultraviolet quadratic divergence into an infrared pole that is reachable in the s-channel.
Solution of the self-dual $\Phi^4$ QFT-model on four-dimensional Moyal space
math-ph 2019-08 accept novelty 7.0 of 10

The planar sector of the self-dual Φ^4 model on 4D Moyal space is solved in closed form by a hypergeometric function, and the model's spectral dimension is proven to be 4 - (2/π) arcsin(λπ).
Self-dual noncommutative \phi^4-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory
math-ph 2012-05 conditional novelty 7.0 of 10

A quartic matrix model on four-dimensional Moyal space is solved exactly: all correlation functions follow from a single nonlinear integral equation whose solution is proved to exist.
A translation-invariant renormalizable non-commutative scalar model
math-ph 2008-02 conditional novelty 7.0 of 10

Adding a 1/p² term to the propagator of the noncommutative φ⁴ model yields a translation-invariant theory that is renormalizable to all orders in perturbation theory.

Show 8 more citing papers

Constructive Matrix Theory
hep-th 2007-06 conditional novelty 7.0 of 10

A new loop-vertex expansion bounds connected functions of quartic matrix models uniformly in N, providing a missing ingredient for a rigorous construction of noncommutative phi^4_4 theory.
Noncommutative Induced Gauge Theory
hep-th 2007-03 conditional novelty 6.0 of 10

The one-loop effective action induced by a scalar on four-dimensional Moyal space contains, beyond the noncommutative Yang-Mills term, gauge-invariant quadratic and quartic terms in the gauge potential that resemble t...
Vanishing of Beta Function of Non Commutative $\Phi^4_4$ Theory to all orders
hep-th 2006-12 conditional novelty 6.0 of 10

The beta function of the Grosse-Wulkenhaar noncommutative phi-fourth model vanishes at the self-dual point to all orders in perturbation theory, proved via Ward identities.
A nontrivial solvable noncommutative \phi^3 model in 4 dimensions
hep-th 2006-03 conditional novelty 6.0 of 10

The 4D noncommutative phi-cubed model, mapped to the Kontsevich model, is renormalizable at each genus after a new field-shift counterterm, with finite n-point functions and a critical coupling for instability.
The \beta-function in duality-covariant noncommutative \phi^4-theory
hep-th 2004-02 conditional novelty 6.0 of 10

The one-loop beta functions of the duality-covariant noncommutative phi^4 model vanish at Omega=1, and beta_Omega also vanishes as Omega approaches 0.
Noncommutative D=4 gravity coupled to fermions
hep-th 2009-02 conditional novelty 5.0 of 10

A noncommutative extension of first-order gravity coupled to fermions is constructed, with a theta-dependent charge conjugation condition that ensures the commutative limit is ordinary gravity.
Non-commutative Renormalization
hep-th 2007-05 conditional novelty 3.0 of 10

Renormalizable quantum field theory exists on the non-commutative Moyal space, and its renormalization group flow appears to stop at a self-dual fixed point rather than running into a Landau ghost.
Quantum Gravity, Field Theory and Signatures of Noncommutative Spacetime
hep-th 2009-06 accept novelty 1.0 of 10

The review argues that noncommutative field theories are a tractable laboratory in which UV/IR mixing, renormalization, and gravitational dynamics can be studied together, and surveys the experimental bounds on spacet...

How the literature uses it

62 arXiv papers cite this in 81 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 60 sentences without a confident label

Cites it as background 17 papers

We propose here to review the three principal mathematical interpretations of the harmonic term: the Langmann–Szabo duality [ref] and its group formulation (in [ref] ), the superalgebraic approach [ref] , and the interpretation in terms of a noncommutative scalar curvature [ref] .

On the Origin of the Harmonic Term in Noncommutative Quantum Field Theory · Introduction · 0.88
Recently non-commutative quantum field theories such as the $ϕ^⋆4$ and the Gross-Neveu$_2$ models have been shown renormalizable [ref] provided the propagator is modified to obey Langmann-Szabo duality [ref] .

Non-Commutative Complete Mellin Representation for Feynman Amplitudes · Introduction · 0.85
The oscillator term provides the symmetry between short and long distances or between coordinates and momenta which is referred to as Langmann-Szabo (LS) duality, [ref] .

Gauge fields on noncommutative geometries with curvature · Introduction · 0.91

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

A. Konechny and A. Schwarz, [hep-th/0012145; 0107251]; M.R. Douglas and N.A. Nekrasov, Rev. Mod. Phys. 73 (2002) 977 [hep-th/0106048]; R.J. Szabo, [hep- th/0109162]

View on Pith Pith review arXiv 2002

[2] [3]

B. Pioline and A. Schwarz, J. High Energy Phys. 9908 (1999) 021 [h ep-th/9908019]; J. Ambjørn, Y.M. Makeenko, J. Nishimura and R.J. Szabo, J. High Ene rgy Phys. 0005 (2000) 023 [hep-th/0004147]; N. Seiberg, J. High Energy Phy s. 0009 (2000) 003 [hep-th/0008013]

View on Pith Pith review arXiv 1999

[3] [10]

E. Langmann and R.J. Szabo, in preparation

View on Pith

[4] [2]

N. Seiberg and E. Witten, J. High Energy Phys. 9909 (1999) 032 [h ep-th/9908142]

View on Pith 1999

[5] [4]

L. Susskind, [hep-th/0101029]; A.P. Polychronakos, J. High Ene rgy Phys. 0104 (2001) 011 [hep-th/0103013]; S. Hellerman and M. Van Raamsdonk, J. High E nergy Phys. 0110 (2001) 039 [hep-th/0103179]; D. Karabali and B. Sakita, [hep -th/0106016]; C.D. Fosco and A. Lopez, [hep-th/0106136]; T.H. Hansson and A. Ka rlhede, [cond- mat/0109413]; A. Gorsky, I.I. Ko...

View on Pith Pith review arXiv 2001

[6] [5]

Y. Habara, [hep-th/0105002]

View on Pith Pith review arXiv

[7] [6]

V.P. Nair and A.P. Polychronakos, Phys. Lett. B 505 (2001) 267 [h ep-th/0011172]; B. Morariu and A.P. Polychronakos, Nucl. Phys. B 610 (2001) 531 [he p-th/0102157]; J. Gamboa, M. Loewe, F. M´ endez and J.C. Rojas, [hep-th/010422 4]; A. Jel- lal, J. Phys. A 34 (2001) 10159 [hep-th/0105303]; S. Bellucci, A. Ner sessian and C. Sochichiu, Phys. Lett. B 522 (2...

View on Pith Pith review arXiv 2001

[8] [7]

I.Ya. Aref’eva, D.M. Belov and A.S. Koshelev, [hep-th/0001215]; F . Ruiz Ruiz, [hep- th/0202011]

View on Pith Pith review arXiv

[9] [8]

Y.-G. Miao, H.J.W. M¨ uller-Kirsten and D.K. Park, [hep-th/0105029 ]

View on Pith Pith review arXiv

[10] [9]

S. Minwalla, M. Van Raamsdonk and N. Seiberg, J. High Energy Phys . 0002 (2000) 020 [hep-th/9912072]

View on Pith Pith review arXiv 2000

[11] [11]

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View on Pith Pith review arXiv 2000