Construction of non-Abelian gauge theories on noncommutative spaces

Branislav Jurco; Julius Wess; Lutz M\"oller; Peter Schupp; Stefan Schraml

REVIEW

Construction of non-Abelian gauge theories on noncommutative spaces

Branislav Jurco, Lutz Möller, Stefan Schraml, Peter Schupp, Julius Wess

High Energy Physics - Theory · Submitted 18 Apr 2001 · 15 references

Pith's verdict: A solid construction paper: the first-order solutions check by hand, the second-order ones are the real new content and the real risk, and the authors are honest about what they leave open.

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arXiv IDhep-th/0104153 (version 4)

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CitationBranislav Jurco, Julius Wess, Lutz M\"oller, Peter Schupp, Stefan Schraml. "Construction of non-Abelian gauge theories on noncommutative spaces." arXiv:hep-th/0104153 (2001). https://pith.science/paper/W6RT723B

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Bottom line: this is the paper that made theta-expanded non-Abelian Seiberg-Witten maps a computation anyone could actually run, and the construction holds up. The first-order solutions (2.18), (3.3), (4.4) you can verify by substitution in a few lines. The second-order results are the genuinely new content, and they are the part you have to take on trust as printed. What is new: ref [4] supplied the consistency relation and the enveloping-algebra framework; this paper solves it explicitly to second order for arbitrary gauge groups and carries the matter sector, the potential and the field strength along with it, in (2.27), (3.5), (4.7) and (4.10), then writes the invariant actions (5.4) and (5.5). That last step is what everyone downstream uses. Section 6's all-orders Abelian expressions are quoted from the authors' own published work and independently rederived by Okuyama, so that section is on firm ground. Soft spots, in order of weight. The second-order formulas are long, introduced with 'Finally we obtain a solution' and no intermediate algebra, and the acknowledgment records that (2.27) was wrong in an earlier version and that (5.4) carried a misprint. Both were caught and corrected, which is to the authors' credit, but it tells you where errors live and that the paper contains no independent check of that sector. Second, the c = 0 choice in (2.22) and (2.25): the text says outright that the extra homogeneous terms are essential for renormalizability and defers the choice to ref [6]. So the actions as written are one representative of a family, not the physical one. That is a scope limit rather than an error, since the construction works for any c, but a reader who takes (5.5) as 'the' noncommutative Yang-Mills action is overreading it. Third, Section 7 concedes that the expansion in powers of the ordinary potential fails beyond lowest order for non-Abelian groups because of the Bianchi identity. The classifier record reports no stated limitations; that is wrong, Section 7 states one plainly. Who it is for: anyone computing theta corrections in noncommutative gauge theory or building on the Seiberg-Witten map. It is a working reference, not a conceptual paper. Send it to referees; it would have passed.

Referee report

Significance

The paper supplies what was at the time a genuinely useful piece of technology: an explicit, group-independent algorithm for the Seiberg–Witten map of an arbitrary non-Abelian gauge theory, together with the enveloping-algebra-valued gauge parameter needed to close the algebra. The first-order results are compact, checkable by direct substitution, and have been widely used; the second-order potential and field strength are the kind of output that later work on noncommutative Yang–Mills at order θ² draws on. The paper is honest about the freedom in the Seiberg–Witten map and about the physical ambiguity that freedom induces, and it correctly defers the choice of representative to the renormalizability analysis of ref. [6]. The construction is parameter-free in the sense that no quantity is fitted; the only inputs are the Lie algebra, the Poisson tensor, and the consistency relation itself.

Major objections (3)

[§2, Eqs. (2.20)–(2.23)] There is a θ-counting inconsistency in the discussion of the homogeneous solution. Eq. (2.20) defines F_i = θ^{jk} D_j F^0_{ki}, which already carries one power of θ, so the additional term ~Λ1α[a] = c θ^{ij}{∂_iα, F_j} in (2.21) is second order in θ and cannot be a homogeneous solution of the first-order equation (2.17); it belongs with the second-order solutions. Either (2.20) should read F_i = D^j F^0_{ji} (no explicit θ), or (2.21) should not carry the extra θ^{ij}. Relatedly, the symbol F^α_i in the potential redefinition (2.23) is never defined; presumably it is the F_i of (2.20). Since the paper's claim that the first-order solution set is classified by (2.22) and (2.25) rests on this passage, the counting should be made unambiguous.
[§2, Eq. (2.27); §3, Eq. (3.5); §4, Eq. (4.7)] The second-order solutions are the load-bearing explicit output of the paper, but they are printed without any intermediate algebra: the text goes from (2.26) directly to 'Finally we obtain a solution of (2.26)', and similarly for (3.5) and (4.7). The acknowledgement records that (2.27) previously contained an error corrected in this version, and that (5.4) contained a misprint. Given that history, the reader has no way to distinguish a verified formula from a transcribed one. I would ask the authors to state explicitly that each of (2.27), (3.5), (4.7) has been checked by substitution into (2.26), (3.4), (4.6) respectively, and ideally to add a short appendix recording the verification or the intermediate steps for at least (2.27). A comparison of (2.27) with the U(n) results of refs. [7] and [8] in the specialization to that group would also serve as an independent cross-check and would strengthen the claim that the method 'works for arbitrary gauge groups'.
[§5, Eqs. (5.4)–(5.5)] The actions (5.4) and (5.5) are presented as the expansion of the noncommutative theory, but they are computed with the specific representative (2.18) and with the homogeneous terms of (2.22) and (2.25) set to zero. The text acknowledges the freedom ('To define a physical theory, we have to make use of the full freedom of the Seiberg-Witten map') and cites [6] for the statement that extra terms are essential for renormalizability, but it does not state that the coefficients in (5.4) and (5.5) would change under a different choice of representative. A sentence in §5 making that dependence explicit — that these are the actions for the c = 0 representative, and that the physical (e.g. renormalizable) theory may correspond to a different point in the solution space — would prevent the results from being read as unique predictions of the construction.

Minor objections (8)

[Abstract] The abstract states that the construction is for 'a star product with aconstant Poisson tensor', but §2 explicitly says the methods work for other star products and §6 quotes results valid for arbitrary Poisson structures θ(x). The abstract should be brought into line with the body, and the typo 'aconstant' corrected.
[§2, Eq. (2.10)] The index structure of the expansion coefficients Λ^{n-1 a_1...a_n} is hard to parse as typeset; a short sentence stating that the superscripts are the symmetrized Lie-algebra indices of the basis (2.11) would help.
[§2, Eq. (2.15)] The expansion in h is formal. The paper never states whether the series is expected to converge or is to be treated as an asymptotic/perturbative expansion, nor whether the tower Λ_α[a] terminates for a given algebra. A remark on the status of the expansion would be appropriate, particularly since §6 contrasts the h-expansion with the expansion in powers of a.
[§3, after Eq. (3.5)] The statement that the adjoint field is 'easily obtained' from (3.3) and (3.5) 'keeping in mind that a_i is supposed to be self-adjoint for a unitary gauge group' is the only place hermiticity enters. The hermiticity properties of Λ_α[a] and A_i[a] themselves are not stated, and they matter for the unitarity of the resulting action. A short explicit statement would be useful.
[§5, Eqs. (5.4)–(5.5)] The potentials and field strength are constructed to second order in h, but the actions are expanded only to first order. The text says so ('We only do this here to first order in h'), but a reader may expect the second-order results of §4 to feed into a second-order Lagrangian. A sentence explaining why the action expansion is truncated earlier than the field expansion would remove the ambiguity.
[§6, Eq. (6.6)] The index structure of (6.6) should be checked: the free index of the bracket multiplying θ^{ij} appears to be j, but the second term contracts a_l with θ^{kl} and differentiates a_j with ∂_k, which does not obviously reproduce the standard Mehen–Wise expression. Please verify the placement of indices and the normalization of the ⋆_2 term.
[§7, title and Eq. (7.18)] The section title 'Expansion of non-Abelian fields in a' is truncated and should read 'in powers of a'. The product (7.18) is introduced with a bullet notation that is not used again in the paper; either use it in (7.20) or drop it.
[References] Reference [5] is missing a comma between 'Wulkenhaar' and 'Perturbative', and reference [6] similarly runs the title into the author list. A consistent reference style would improve readability.

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The paper

The paper claims

Non-Abelian gauge theories can be built order by order on a noncommutative space, with the gauge parameter, potential, matter fields, and field strength all fixed by one consistency relation.

The reading

The paper asks how to put a non-Abelian gauge theory — the kind that describes quarks and gluons — on a space whose coordinates do not commute, so that ordinary multiplication is replaced by a star product. The obstacle is that on such a space the gauge parameter can no longer be a Lie-algebra element; two successive gauge transformations close only if the parameter is promoted to an element of the enveloping algebra, an infinite tower of symmetrized products of generators. The authors' proposal is to demand that this tower be generated entirely by the ordinary Lie-algebra parameter $[abstract:s0]$, and to impose the closure condition $(2.14)$ that makes two such transformations compose correctly. Solving that condition order by order in a noncommutativity parameter $h$ yields explicit formulas for the gauge parameter, the noncommutative potential, matter fields, and the field strength, all written in terms of the ordinary gauge field and its derivatives $[abstract:s1]$. With those in hand the standard Yang-Mills and matter actions can be expanded in $h$, giving ordinary Lagrangians with new $h$-dependent couplings $[abstract:s2]$.

What carries the argument

The load-bearing device is the closure, or consistency, relation $i(2.14)$, namely $i(2.14)$, which says that the failure of two star-commuting gauge transformations to commute must itself be a gauge transformation with parameter $i(2.4)$. Because the star product of two enveloping-algebra-valued parameters stays in the enveloping algebra, the relation can be expanded in $h$ and solved recursively: at each order the inhomogeneous piece is a known functional of lower-order solutions, and a finite ansatz linear in $i(2.15)$ suffices to cancel it. The symmetrized basis $i(2.11)$ keeps the tower in a canonical form, and the structure of the solutions ties the power of $i(2.15)$ to the power of the generators — order $n$ in $h$ carries at most $n+1$ generators. The same relation, applied to fields and to the potential, generates the Seiberg-Witten-type maps $i(3.3)$, $i(4.4)$, and $i(4.10)$ that make the expanded actions possible.

Extended reading notes

Core claim

The central claim is that the consistency relation $i(2.14)$, namely $i(2.14)$, holds, in the form $i(2.14)$, and admits a solution in which every noncommutative object is an explicit power series in the noncommutativity parameter built from the ordinary Lie-algebra potential. Concretely, the authors solve the relation to first order in $h$ with $i(2.18)$, namely $i(2.18)$, giving $i(2.18)$; and to second order with the long expression $i(2.27)$. The same procedure delivers the matter field to first and second order, $i(3.3)$ and $i(3.5)$, the enveloping-algebra-valued potential $i(4.4)$ and $i(4.7)$, and the field strength $i(4.10)$. Inserting these into the trace action $i(5.2)$ and the matter action $i(5.3)$ and expanding produces ordinary Lagrangians $i(5.4)$ and $i(5.5)$ in which $h$ plays the role of a coupling constant, with new terms such as $i(5.5)$ and $i(5.5)$. The authors emphasise that the construction works for arbitrary gauge groups, unlike earlier approaches restricted to particular ones.

Load-bearing premise

The construction fixes one representative among a family of solutions by discarding the homogeneous terms in equations (2.22) and (2.25), and nothing in the paper shows that this particular choice is the one with acceptable high-energy behaviour — the authors themselves note that such extra terms are essential for renormalizability.

Editorial extensions

If this is right

Any non-Abelian gauge group, not just the unitary ones handled by earlier constructions, can be given a noncommutative deformation with the gauge parameter, potential, and matter fields determined order by order in the noncommutativity parameter.
The deformed Yang-Mills and matter Lagrangians in equations (5.4) and (5.5) are ordinary field theories in which the noncommutativity parameter enters as a coupling, so existing perturbative machinery can be applied to them directly.
The expanded actions contain genuinely new vertices — for instance the $\theta^{kl}\mathrm{Tr}\,F^{0}_{ik}F^{0}_{jl}F^{0}_{ij}$ term — which modify the tree-level S-matrix relative to the commutative theory.
In the Abelian case the authors recover, to all orders in $\theta$ and first order in the potential, the known closed forms (6.6) and (6.7) built from the auxiliary product $\star_2$ of equation (6.8), confirming that the general construction reduces correctly.
The non-Abelian expansion in powers of the commutative potential is shown to be obstructed beyond zeroth order by the Bianchi identity, a limitation the Abelian case does not share.

Reading between the lines

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

Because the solution is fixed only up to homogeneous terms, the physical content of the expanded actions depends on which representative is chosen; different choices are related by field redefinitions of the potential, as in equation (2.23), and a reader who wants a definite prediction must specify the choice.
The obstruction to expanding non-Abelian fields in powers of the commutative potential — traced to the Bianchi identity — suggests that any all-orders closed form for the non-Abelian map must be organised by something other than a naive potential expansion, perhaps by the covariant coordinates of reference [2].
The first-order action corrections are all of the form field strength times a covariant derivative or a commutator, which is the shape one expects from a lowest-order derivative expansion; it would be worth checking whether this pattern persists at second order or whether genuinely new tensor structures appear.
Since the construction is presented for a constant Poisson tensor, the natural test of its scope is whether the same recursive solution of the closure relation survives when the Poisson tensor depends on position, as the authors' own earlier work suggests it should.

Original abstract

We present a formalism to explicitly construct non-Abelian gauge theories on noncommutative spaces (induced via a star product with aconstant Poisson tensor) from a consistency relation. This results in an expansion of the gauge parameter, the noncommutative gauge potential and fields in the fundamental representation, in powers of a parameter of the noncommutativity. This allows the explicit construction of actions for these gauge theories.

The compiled record

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The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; show every role.

We present a formalism to explicitly construct non-Abelian gauge theories on noncommutative spaces (induced via a star product with aconstant Poisson tensor) from a consistency relation. This results in an expansion of the gauge parameter, the noncommutative gauge potential and fields in the fundamental representation, in powers of a parameter of the noncommutativity. This allows the explicit construction of actions for these gauge 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 present a formalism to explicitly construct non-Abelian gauge theories on noncommutative spaces (induced via a star product with aconstant Poisson tensor) from a consistency relation.	contribution 0.97	0.94	0.45	asserted 1.00
s1	This results in an expansion of the gauge parameter, the noncommutative gauge potential and fields in the fundamental representation, in powers of a parameter of the noncommutativity.	result 0.71	0.90	0.20	asserted 1.00
s2	This allows the explicit construction of actions for these gauge theories.	result 0.76	0.86	0.31	asserted 0.90

What kind of paper the classifier reports this is: method or tool 0.92 · evidence from proof or derivation 1.00.

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sources read

arXiv listing abstract v4

not examined

full text
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References

Works cited (15)

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[2]

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K. R. Garousi, Non-commutative world-volume interactions on D-brane and Dirac-Born-Infeld-Action, Nucl.Phys. B579, 209 (2000) [hep-th/9909214]

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How later papers used it

50 later papers cite it

Evidence: substantial. 49 of the citing papers share no author with it, and 12 of those carry a confident label from Pith's citation classifier.

Pith read 11 of the 102 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 8 papers with this label

Followed this paper to give the general strategy to solve the noncommutative gauge consistency condition and Seiberg-Witten map for general order n.

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How the literature uses it

45 arXiv papers cite this in 89 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 · 57 sentences without a confident label

Contests it 1 paper

The importance of this systematic approach is stressed a posteriori by discovering ambiguities in the SWM that have been missed in earlier $𝒪(θ^2)$ constructions of SWM [ref] .

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Builds on it 2 papers

This idea was further developed in [ref] where it was shown that any commutative (but possibly nonabelian) gauge algebra can be extended to a noncommutative version by a generalization of the Seiberg-Witten map.

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[ref] for non-supersymmetric gauge theories: now the noncommutative vector superfields will be valued in the enveloping algebra of the Lie algebra of the ordinary gauge group.

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Uses its method or tool 7 papers

To construct the NC extension of the standard model (SM) [ref] , which uses the same gauge group and particle content, or for that matter any other gauge theory, be it abelian or non-abelian, one expands the NC gauge fields in non linear power series of $Θ$ [ref] : λ_α(x,Θ) = α(x) + Θ^μν λ^(1)_μν(x;α) + Θ^μν Θ^ησ λ^(2)_μνησ(x;α) + ⋯ A_ρ(x,Θ) = A_ρ(x) + Θ^μν A^(1)_μνρ(x) + Θ^μν Θ^ησ A^(2)_μνησρ(x) + ⋯.

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[ref] – Seiberg-Witten maps.

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Within the enveloping-algebra framework of Refs. [ref] , one introduces at least a noncommutative field for each ordinary field.

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Compares with it 2 papers

As pointed out in section [ref], this solution is related to the one presented in [ref] , but incompatible with the one in [ref] .

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Note that the action [ref] has the same form as the expanded action for the NC gauge field in the case of $θ$-constant deformation [ref] .

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Cites it as background 16 papers

A dozen years ago, it was put forward a formalism –the enveloping-algebra formalism– which can be used to construct noncommutative gauge theories that are deformations of ordinary theories with simple gauge groups in arbitrary unitary representations [ref] .

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Noncommutative gauge theories with simple groups can only be formulated with the so called enveloping algebra approach, which makes use of Seiberg-Witten maps to relate noncommutative gauge orbits to ordinary ones [ref] .

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The second order solutions of ([ref]) are also studied by several authors [ref] .

Seiberg--Witten Maps to All Orders · Fundamental Representation · 0.89

Discussion

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