Born-Infeld action, supersymmetry and string theory

A.A. Tseytlin

REVIEW

Born-Infeld action, supersymmetry and string theory

A.A. Tseytlin

High Energy Physics - Theory · Submitted 16 Aug 1999 · 89 references

Pith's verdict: A solid, honest review of Born-Infeld from string theory; the only new derivation is a small SU(3) matching in §4.3 that deserves a careful look but doesn't carry the paper.

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arXiv IDhep-th/9908105 (version 5)

Pith IDRRUX2LTO, Pith's permanent record of this paper and its review

CitationA.A. Tseytlin. "Born-Infeld action, supersymmetry and string theory." arXiv:hep-th/9908105 (1999). https://pith.science/paper/RRUX2LTO

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Editor's note

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This is the review you want on the shelf if you need the Born-Infeld-from-string-theory story in one place. Tseytlin wrote it for the Golfand volume, and it does what a good review should: it puts the disc path integral, the constrained-superfield construction of the N=1 and N=2 actions, the D-brane/DBI reduction, and the non-abelian symmetrized-trace proposal side by side and shows where they agree. The Appendix is the nicest part for teaching: the M-regulator that interpolates between the Born-Infeld and Schwinger partition functions is a clean way to see why the string case gives a square root and the particle case gives F/sinh F, and the superstring version showing the linear divergence cancels is a genuinely useful check to have written down explicitly. The section on deformed supersymmetry from gauge-fixed D-brane actions is also a good map of a confusing corner of the literature, and the comparison with the manifest-supersymmetry actions is stated plainly rather than smoothed over. Credit where due: the paper is candid about what it does not have. The N=4 action is stated as unknown, the non-abelian action is stated as a truncation that should not be expected to match string theory past α′², and the field-redefinition argument in §4.2 is flagged as likely rather than proved. Those are the right hedges, and they line up with the actual gaps. The one place I would actually sit down and check is §4.3, the only new derivation in the paper. The claim that P1, P2, P3 are the only SU(3)-invariant generalizations of the Φ⁴ terms is asserted rather than argued, and the matching to the (∂X)⁴ structures of (4.29) is done by dropping ∂²Φ terms and total derivatives, which is fine in spirit but leaves the uniqueness claim doing more work than the text supports. It is peripheral to the central Born-Infeld claim, so it does not sink anything, but if someone is going to build on the N=4 four-derivative term they should redo that matching themselves. Minor editorial note: §3 has a leftover fragment, 'duality-symmetric actions was explained', that should have been cut. Who is this for: anyone entering D-brane effective actions or the open-string derivative expansion, and anyone who wants the BI/supersymmetry dictionary in one place. It is a reference, not a source of new results. It deserves a serious referee, and I would cite it.

Referee report

Significance

The review is an authoritative and useful synthesis by a leading practitioner, and it is explicit about what is established and what is conjectural. The derivation of the Born-Infeld action from the disc partition function and the constrained-superfield constructions of the N=1 and N=2 actions are standard results, correctly attributed to the cited literature. The paper gives credit where due and states its limitations plainly: §4.3 says the N=4 action is not known, §4.2 says the elimination of higher-derivative scalar terms by field redefinitions is likely rather than proved, and §6.1 says the symmetrized-trace non-abelian action should not be expected to reproduce the full string effective action beyond order α′². The new §4.3 result, if correct, supplies the N=4 generalization of the F^4 invariant and is a genuine, if modest, addition to the literature.

Major objections (2)

[§4.3, Eq. (4.26)] As printed, Eq. (4.26) does not follow from Eq. (4.25). Squaring the matrix under the determinant gives (η+t+F)² = η + 2t + t² + 2F + F² + 2tF, so the equivalent form should be [−det(η + 2t + t² + 2F + F² + 2tF)]^{1/2}, not [−det(η + 2t + t² + F² + 2tF)]^{1/4}. A concrete check: set t=0 and take a single magnetic component F_12 = B in Euclidean signature; Eq. (4.25) gives sqrt(det(δ+F)) = 1+B², while Eq. (4.26) gives [det(δ+F²)]^{1/4} = (1−B²)^{1/2}, which differs already at order B². Since Eq. (4.26) is presented as the starting point for the §4.3 discussion, the exponent and the missing 2F term should be corrected.
[§4.3, Eq. (4.33)] The statement that P1, P2, P3 are 'the only possible SU(3) invariant generalizations' of the Φ⁴ terms in Eq. (4.21) is asserted without justification. A structure such as (Φ̄_a∂_mΦ_a)(Φ_b∂_mΦ̄_b) is not obviously reducible to P1, P2, P3 modulo ∂²Φ and total derivatives, even if it may be after integration by parts; the manuscript should either show the reduction or soften the claim. Relatedly, the matching that fixes the combination P1+P2−3P3 against the (∂X)⁴ structures of Eq. (4.29) is reported as 'we find' without intermediate steps; since this is the paper's only new derivation, the reader cannot check it. Please show the component expansion or state the result as a conjecture.

Minor objections (11)

[§3, after Eq. (3.9)] The sentence 'There exists, however, an equivalent action containing one extra vector field variable which is manifestly duality-symmetric [37]. duality-symmetric actions was explained' is garbled; the second sentence is incomplete and should be repaired.
[§3, paragraph after Eq. (3.10)] The phrase 'the scalar sϕn being linear combinations of λ1, λ2, a1, a2' contains a typo; it should read 'the scalars ϕ_i being linear combinations'.
[§4.1, first paragraph] The sentence 'The action depends in general on dimensional scale parameter which is set to 1 as in (3.1)' is missing an article; it should read 'on a dimensional scale parameter'.
[§4.2, after Eq. (4.14)] The phrase 'the two auxiliary complex scalar fields fields a and λ' duplicates the word 'fields'.
[§6.2, first paragraph] The phrase 'the usial cyclic symmetry' contains a typo for 'usual'.
[§2, Eq. (2.2)] Eq. (2.2) should be checked against Eq. (A.6): the factor [Γ(1+T/M)]² appears in the denominator in (A.6), and the argument '1 + T+iF/M' should be read as 1 + (T+iF)/M. The printed form is ambiguous.
[§6.1] The paper acknowledges that the symmetrized-trace action is a truncation and states that the first expected deviation from the full string effective action is at order α′⁴F⁶. It would help to repeat this caveat in the introduction, so that a reader who stops before §6.1 does not mistake the non-abelian action for a derived result.
[Appendix A] The statement that in the Green-Schwarz light-cone case 'the M-regulator does not seem to preserve the world-volume supersymmetry' is a negative result stated without derivation. Please clarify whether this is a regulator artifact or a statement about the light-cone Green-Schwarz action, and whether the ζ-function prescription is preferred.
[§5, after Eq. (5.2)] The sentence 'The two formulations are presumably related by a field redefinition' is a conjecture; it should be flagged as such rather than stated as an expectation.
[§7, after Eq. (7.2)] The claim that the function F(F) 'can be, in principle, computed exactly' is a conjecture; it should be flagged as such.
[§2, footnote 4] The claim that the Born-Infeld action is 'unambiguous' because field redefinitions produce terms with derivatives of F is specific to the abelian case; in the non-abelian case commutators of F also appear, so the statement should be qualified.

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The paper

The paper claims

The Born-Infeld square root is the derivative-independent part of the abelian open-string effective action, and supersymmetry fixes its $N=1$ and $N=2$ completions.

The reading

This review argues that the Born-Infeld Lagrangian — the square root of a determinant built from a metric plus a field strength — is not one nonlinear electrodynamics among many but the derivative-independent piece of the abelian open-string effective action, and it traces what supersymmetry does to that object. The author shows how a disc path integral in a constant field strength produces exactly the square root, how the four-dimensional case interpolates between Maxwell at weak field and a total-derivative topological density at strong field, and how requiring partially broken extended supersymmetry pins down the $N=1$ and $N=2$ supersymmetric versions. If the organizing claim holds, the Born-Infeld form is forced by string theory rather than postulated, and the flat-space D-brane actions are simply its dimensional reductions. The review also carries a proposal for the non-abelian case, where the field strength becomes a matrix and the action is defined with a symmetrized trace.

What carries the argument

Two devices carry the argument. The first is the boundary path integral on the disc in a constant field strength, where the boundary kinetic operator is regularized and the resulting product of gamma functions reduces to the Born-Infeld square root in the limit of vanishing regulator, with the linearly divergent piece cancelling in the superstring case. The second is the constrained superfield construction: one starts from an $N=2$ vector multiplet, gives its field strength a constant condensate to break $N=2$ to $N=1$, imposes $W^2 = 0$, and finds that the surviving $N=1$ chiral superfield obeys $�� = ��$ $��^2 ��$ + $W^��W_��$. Solving that constraint yields the $N=1$ Born-Infeld action together with the nonlinear transformation law of the broken supersymmetry, which is what makes the construction more than a guess at an invariant.

Extended reading notes

Core claim

The paper's central claim is that the part of the open-string effective action that depends on the field strength but not on its derivatives is the Born-Infeld action $S = T_p �� d^{p+1}x$ $[ -$det$_{p+1}(1 + T^{-1}F)]^{1/2}$, and that this identification survives supersymmetrization in a specific and restrictive way. The author reports that the disc partition function in a constant abelian background reproduces the square root after renormalization, that the conformal-invariance condition of the open-string $\sigma$ model reproduces the same action at leading order in $��F$, and that the $F^2 + ��'^2F^4$ terms agree with the four-point superstring amplitude. On the supersymmetric side, the claim is that the requirement of partial breaking of $N=2$ to $N=1$ supersymmetry, implemented with constrained superfields, fixes the $N=1$ Born-Infeld action uniquely, with the second supersymmetry realized nonlinearly; the $N=2$ action arises the same way from $N=4$ to $N=2$ breaking. The $N=4$ case, written with manifest linear supersymmetry, is stated to be unknown, and the paper offers only the four-derivative term, deduced from the known bosonic action and an assumed $SU(3)$ symmetry of the chiral superfield sector.

Load-bearing premise

The non-abelian proposal replaces the field strength by a matrix and averages over all orderings with a symmetrized trace, which drops path ordering and commutators by hand; this is a prescription for a candidate action rather than a derivation, and the paper itself says there is no reason to expect it to reproduce the full string expression beyond order $��'^2$.

Editorial extensions

If this is right

The Born-Infeld square root is selected by string theory rather than chosen for its analytic properties, so its maximal-field-strength behaviour is a prediction of the open-string effective action.
Flat-space D-brane actions follow by dimensional reduction of the ten-dimensional Born-Infeld action, so their solutions and their velocity-type bounds are inherited rather than separately derived.
The $N=1$ supersymmetric Born-Infeld action is the unique action for a Goldstone multiplet of partially broken $N=2$ supersymmetry, which ties the nonlinear action to a symmetry-breaking pattern rather than to an arbitrary Lagrangian.
The four-derivative correction to the Born-Infeld action computed from the disc path integral has the same supersymmetric structure as the one-loop divergence of the quantized supersymmetric Born-Infeld action, connecting a tree-level string result to a field-theory loop calculation.

Reading between the lines

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

If the symmetrized-trace non-abelian action is only a truncation, then the D-brane cluster actions built from it should show measurable deviations in near-commuting or near-covariantly-constant configurations, which is where the truncation is hardest to distinguish from the full answer.
The clash the author notes between first-derivative scalar dependence and manifest extended supersymmetry suggests a general trade-off: any $N \geq 2$ supersymmetric Born-Infeld action with manifest linear supersymmetry will carry higher-derivative scalar terms removable only by field redefinitions that spoil the manifest symmetry.
The same constrained-superfield method should apply to D-branes in curved backgrounds, where the modulus of the scalar field replaces the explicit $\sqrt{\alpha'}$ scale and the nonlinearly realized symmetry becomes conformal rather than supersymmetric.
If the deformed-supersymmetry D-brane actions and the manifest-supersymmetry superfield actions are related by field redefinition, then the fermionic terms in the two formulations should match order by order in $\alpha'$, which is a checkable statement independent of the bosonic agreement.

Original abstract

We review and elaborate on some aspects of Born-Infeld action and its supersymmetric generalizations in connection with string theory. Contents: BI action from string theory; some properties of bosonic D=4 BI action; N=1 and N=2 supersymmetric BI actions with manifest linear D=4 supersymmetry; four-derivative terms in N=4 supersymmetric BI action; BI actions with `deformed' supersymmetry from D-brane actions; non-abelian generalization of BI action; derivative corrections to BI action in open superstring theory.

The compiled record

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.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,459 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.08 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v5. Results and the main claim are underlined; show every role.

We review and elaborate on some aspects of Born-Infeld action and its supersymmetric generalizations in connection with string theory. Contents: BI action from string theory; some properties of bosonic D=4 BI action; N=1 and N=2 supersymmetric BI actions with manifest linear D=4 supersymmetry; four-derivative terms in N=4 supersymmetric BI action; BI actions with `deformed' supersymmetry from D-brane actions; non-abelian generalization of BI action; derivative corrections to BI action in open superstring theory.

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 review and elaborate on some aspects of Born-Infeld action and its supersymmetric generalizations in connection with string theory.	contribution 0.91	0.57	0.18	asserted 0.65
s1	Contents: BI action from string theory; some properties of bosonic D=4 BI action; N=1 and N=2 supersymmetric BI actions with manifest linear D=4 supersymmetry; four-derivative terms in N=4 supersymmetric BI action; BI actions with `deformed' supersymmetry from D-brane actions; non-abelian generalization of BI action; derivative corrections to BI action in open superstring theory.	contribution 0.87	0.29	0.06	unclear 0.83

What kind of paper the classifier reports this is: review 1.00 · evidence from literature 0.89.

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sources read

arXiv listing abstract v5

not examined

full text
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P abstract

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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

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Forward citations

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How later papers used it

110 later papers cite it

Evidence: substantial. 110 of the citing papers share no author with it, and 8 of those carry a confident label from Pith's citation classifier.

Pith read 18 of the 142 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

3 later papers by other authors are listed below: 3 used its method or tool.

Uses its method or tool 3 papers

Applied this paper's symmetrized trace prescription to the non-abelian DBI action in the presence of the isospin chemical potential.

Isospin asymmetry in holographic baryonic matter (2021) · 1.00 · paraphrased: the licence does not allow quoting
Expanded the square root and took the symmetrized trace in each term separately, following this paper.

Layers of deformed instantons in holographic baryonic matter (2016) · 0.97 · paraphrased: the licence does not allow quoting
Chose to work with this paper's symmetrized-trace prescription, denoted by STr.

Advances in the holographic description of hot and dense QCD matter (2026) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

109 arXiv papers cite this in 140 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 · 77 sentences without a confident label

Uses its method or tool 2 papers

Here we follow Tseytlin [ref] : we first compute the determinant over space-time indices as if the field strengths were numbers, det(g+2πα'F) = U^8(R/U)^3/2{f_T(2πα')^2F_iU^2 +[1+(U/R)^3 f_T (∂_U X_4)^2 +(2πα')^2F̂_0U^2].

Layers of deformed instantons in holographic baryonic matter · Dirac-Born-Infeld action · 0.99
We choose to work with the symmetrized-trace prescription [ref] , denoted by STr.

Advances in the holographic description of hot and dense QCD matter · Action – general form · 0.98

Uses its data or values 1 paper

In string theory, these constants are given by $Q=(N-1) g_s'^2/π$ and $T_3 = 1/2πg_s'^2$ [ref] .

On the 1-loop effective action for the IKKT model and non-commutative branes · Coulomb branch · 0.85

Compares with it 4 papers

A comparison of SDBI with the corresponding amplitudes in string theory was done in [ref] at the level of mass dimension-8 interactions involving four bosons, four fermions, two bosons and two fermions, i.e. $F^4$, $(λ∂)^2$ and $(λ∂)F^2$, schematically.

Note on supersymmetric Dirac-Born-Infeld action with Fayet-Iliopoulos term · Introduction · 0.80
This expression in the limiting case $Φ=0$ is in agreement with earlier perturbative calculations of the low-energy effective action of superstrings (see a review and references in [ref] and restrictions implied by supersymmetry in $6D$ [ref] ).

Leading low-energy effective action in the 6D hypermultiplet theory on a vector/tensor background · Leading low-energy contribution to effective action · 0.98
It is remarkable[This is essentially known in the case of flat branes [ref] .] that up to this order, after dropping the trace-$U(1)$ terms, the effective action $S_YM[X] + _IR[X]$ is consistent[The “would-be topological terms” $η(x) F ∧F$ presumably correspond to the Chern-Simons terms which should be added to DBI-action-AdS.] with the expansion of the Dirac-Born-Infeld (DBI) action for a D3-brane in the background of $N-1$ coinciding D3 branes [ref] [equation] for $Q= (N-1)/2π^2 ^-4$ and $T_3 = ^4$.

On the 1-loop effective action for the IKKT model and non-commutative branes · Coulomb branch · 0.83

Cites it as support 1 paper

Actually, the DBI action benefits from non-renormalization properties, being also protected against higher derivative corrections in the ultra-relativistic limit [ref] .

Weakly Broken Galileon Symmetry in Cosmology · The role of symmetries in the EFT of inflation · 0.97

Cites it as background 48 papers

We will work within the framework of supersymmetric DBI actions which have been investigated by a number of authors (see, e.g., [ref] .

Enhancement of the Axion Decay Constant in Inflation and the Weak Gravity Conjecture · Dirac-Born-Infeld (DBI) · 0.89
One of the distinctive features of string/brane–induced non-linear electrodynamics is that higher-order corrections in the Maxwell tensor can be resummed into a closed-form expression, namely the BI Lagrangian [ref] .

Self-Gravitating Magnetic Monopoles and Dyons in String-Inspired Models: Structure and Stability · Magnetic Monopoles from Global Monopoles with a Dilaton Charge · 0.88
The supersymmetric extensions of the BI theory exhibit the additional highly non-trivial and unexpected features, such as (viii) the extra hidden (non-linearly) realized supersymmetries allowing (ix) the interpretation of the supersymmertic BI actions as the Maxwell-Goldstone actions associated with partial spontaneous supersymmetry breaking [ref] and (x) adding Fayet-Iliopoulos (FI) terms [ref] .

Modified Born-Infeld-dilaton-axion coupling in supersymmetry · Introduction · 0.92

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.

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