Aspects of Born-Infeld Theory and String/M-Theory

G W Gibbons

REVIEW

Aspects of Born-Infeld Theory and String/M-Theory

G W Gibbons

High Energy Physics - Theory · Submitted 7 Jun 2001 · 22 references

Pith's verdict: A well-informed set of lecture notes whose real contribution is the geometric framing of open-string causality, not new results.

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arXiv IDhep-th/0106059 (version 1)

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CitationG W Gibbons. "Aspects of Born-Infeld Theory and String/M-Theory." arXiv:hep-th/0106059 (2001). https://pith.science/paper/RCTGKTAA

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Editor's note

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Bottom line: this is Gibbons' lecture notes, not a new result, and you should read it as a tour of Born-Infeld theory by someone who has thought hard about the geometry. The one genuinely useful synthesis is the framing of open-string causality through the Boillat cone and the identity C = g + T (his Hooke's law), which then makes the finite-temperature Hagedorn redshift a short consequence of G00 = 1 - E^2. That framing is his, and it is a good one: it turns a scattered set of facts about the open-string metric into a single statement about characteristic cones. The strong-coupling limits, the string-fluid reading of the Weyl-invariant Bialynicki-Birula limit, and the M5-brane extension with the sign-flipped Hooke's law are all cleanly presented. The pulse/Monge-Ampere section is the nicest piece of physics in the notes and is worth reading on its own. Credit where due: the derivations are standard and internally consistent, and the pedagogical aim is met. On the stress-test note: it is right that the reader's WEC worry does not land. Section 8.1 cites [18] for the energy-condition analysis, and the tachyon-condensate limit of Section 10.5 gives a null-fluid stress tensor T = H l l with H >= 0, which satisfies the null energy condition. The worry that the WEC might fail in a condensate is answered by the paper's own treatment. The real soft spot is different and smaller: the characteristic analysis in Section 8.1 is explicitly linearized ('small disturbances'), and the global causality claim would need symmetric hyperbolicity of the full nonlinear BI system with respect to the Boillat metric. The notes gesture at this via exceptionality and no-shocks in Section 9.1 but do not close the loop. That is a scope limitation, not an error, and is normal for lecture notes. The missing bibliography entries are a real gap for a review: Boillat, Perry-Schwarz, Howe-Sezgin-West, and Sen are all invoked by name with no reference. Minor but worth fixing. The abstract's 'new insights' is a mild overclaim given that the finite-temperature result is attributed to [13] and [14]. Typos are cosmetic ('insighst', 'banes', 'strong states' for 'strings', a duplicated cone label in Section 8.1, and a missing zero in 'T down arrow' in Section 11.4). Who should read it: students entering Born-Infeld and nonlinear electrodynamics, and anyone who wants the Hooke's law identity in one place. It deserves a serious referee as lecture notes, and I would cite it for the cone framing.

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Major objections (2)

[§11.3, Eqs. (136)–(137)] Eq. (137) defines Q = −3H²/(1 + (2/3)H²), which tends to zero as H → 0. The prefactor Q(2−Q) in Eq. (136) then also tends to zero, so the proposed Boillat co-metric C_{αγ} vanishes in the weak-field limit. Hooke's law in Eq. (138), T^{αβ} = g^{αβ} − C^{αβ}, requires instead that C → g (equivalently T → 0) as H → 0. Either the normalization of Q or the placement of Q(2−Q) in Eq. (136) needs correction; please check against the source and state the H → 0 limit explicitly.
[§9.1] The sentence 'Boillat has shown that the only form of non-linear electrodynamics with a sensible weak field limit is that of Born-Infeld' is stronger than the result usually attributed to Boillat. The established statement is that Born-Infeld is the unique exceptional (shock-free) theory with the standard weak-field limit; general Lagrangians L(x,y) certainly have sensible weak-field limits and are not Born-Infeld. As written the sentence would also exclude the Weyl-invariant limit of §10.3. Please state the theorem precisely and give the citation.

Minor objections (10)

[References and in-text attributions] Several results that carry weight in the text are attributed by name but have no bibliography entry: Boillat (§8.1, §8.2, §11.3), Perry–Schwarz and Howe–Sezgin–West (§11.2), and Sen (§10.5). For notes whose stated purpose is to give newcomers a rapid survey, the reader cannot follow these up. In addition, [19] and [20] give only 'hep-th/' with no number, [21] and [23] both carry hep-th/0007019 although they are listed as different papers, and [15] omits its arXiv number. [13] should read 'de la Incera'.
[§6, Eq. (39)] Eq. (39) as printed reads 'D = D/√(1−D²)', which is not a relation between two different quantities. The intended inverse of Eq. (38) is D = E/√(1−E²); please correct.
[§10.2, Eq. (105)] Eq. (105) gives ρ = 1/√(1+s²) − 1. Tracing the Legendre transform from Eq. (104) gives instead ρ = √(1+s²) − 1; with the printed form, Eq. (106) and Eq. (107) do not follow. Please check.
[§10.5, Eq. (131)] Tracing Eq. (130) directly gives T^μ_μ = −K²/√(−½K²) = +√(−2K²), and the explicit rest-frame form in Eq. (133) gives T^μ_μ = 2τ > 0. Eq. (131) as printed has the opposite sign. The conclusion that the limit is not conformally invariant is unaffected, but the displayed trace should be corrected.
[§11.4, Eq. (145)] In Eq. (145) the factor 'B₂ dx² ∧ dx²' vanishes identically, and x⁵ appears in both factors of the wedge product. Presumably the intended expression is (dt − dx⁵) ∧ (B₂ dx¹ ∧ dx² + B₁ dx³ ∧ dx⁴) or similar; please correct.
[§7.2, §7.3, §8.3] Eqs. (46) and (48) are missing '= 0'. In §8.3 the sentence 'Let v +t +z,u = t −z,ξ = Z −T,η = Z −T' defines ξ and η identically and should presumably read 'v = t + z, u = t − z, ξ = Z − T, η = Z + T'; please check against [16].
[§2.1, §2.3, §8.1] In §8.1 the Einstein co-cone is written 'C^⋆_Einstein : C^{μν}p_μp_ν ≥ 0', which repeats the Boillat expression; it should use g^{μν}. In §2.1 the dilation factor is R, which should be R⁺. In §2.3 the conclusion 'if T^{μν} vanishes on S then it vanishes everywhere in S' should read 'everywhere in D⁺(S)'.
[§8, §9] The text refers to 'closed strong states' (§8) and to the 'open strong metric' and 'open strong states' (§9); these should read 'string'. The same section writes '√(−G_{ij})' for the volume factor, which should be √(−det G_{ij}) or equivalent.
[§8.4, §10.4] The statement in §8.4 that the fermion characteristics are 'given by the Open String metric or equivalently the Boillat metric' is correct only up to the conformal factor displayed in Eq. (63); since conformal factors do not affect null cones this is harmless, but the equivalence should be stated as such. In Eq. (119) the coefficients 'µ4' and 'ν4' should presumably be µ/4 and ν/4, and the symbol T is used for the brane tension in §10.4 and for temperature in §10.2, which should be distinguished.
[Abstract and throughout] The abstract contains 'some-one' and 'insighst'. Other typographical slips include 'it's equation', 'in addition and closed 3-form' and 'be using' (§11), 'T ↓' for 'T ↓ 0' (§11.4), 'signature indepbedent' and 'Kmuν' (§10.1), 'by the calculating' (§10.2), 'It may be checked that is Lorentz-invariant' (§10.3), 'they two pulses' and 'This follows form another useful identity is' (§8.2–8.3), and 'The quantities corresponds' (§10.2). Eq. (10) omits the √(−det) and the overall sign, and the sentence in §7.2 'There is a branch 2-surface at a finite radius' is garbled.

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The paper

The paper claims

Born-Infeld theory gives open-string fields one shared light cone, and Hooke's law keeps it inside the gravitational cone.

The reading

These lecture notes ask what geometry governs causal propagation on a D-brane carrying a Born-Infeld electromagnetic field. The answer they develop is that the open-string fields all see one characteristic cone, the Boillat cone, and that for Born-Infeld theory this cone is fixed by a Hooke-like identity, $C^{μν}=g^{μν}+T^{μν}$, relating the effective co-metric to the background metric and the brane stress tensor. Because the Born-Infeld stress tensor satisfies the weak energy condition, the Boillat cone lies inside the gravitational cone, so small disturbances do not outrun gravity. The same effective metric controls the thermal redshift of open-string states at finite temperature, lowering the Hagedorn temperature by $√{1-E^2}$ in an electric field. The notes then use this picture to describe strong-coupling limits in which Born-Infeld and M5-brane theories dissolve into fluids of strings.

What carries the argument

The load-bearing object is the Boillat co-metric $C^{μν}$, the characteristic co-metric for small disturbances in a nonlinear electrodynamic background. For Born-Infeld it takes the special form $C^{μν}=g^{μν}+T^{μν}$, which the paper calls Hooke's law because stress is being measured by a difference of metrics. This identity does three jobs: it removes birefringence, it makes the characteristic cone coincide up to a conformal factor with the open-string metric $G^{μν}$ defined by $(g+F)^{-1}=G+θ$, and it turns the weak energy condition into an inclusion of cones, with the Boillat cone inside the cone of the background metric. The same construction is repeated for the M5-brane with a 3-form field strength and a nonlinear self-duality relation.

Extended reading notes

Core claim

The central claim is that Born-Infeld theory is special among nonlinear electrodynamic theories because its causal structure is carried by a single effective metric rather than a pair of birefringent cones. The paper states this as Hooke's law, $C^{μν}=g^{μν}+T^{μν}$, where $C$ is the Boillat co-metric and $T$ is the Born-Infeld energy-momentum tensor; the same identity implies $det(δ^μ{}_ν+T^μ{}_ν)=1$ and $det C=det g$. From this and the weak energy condition the paper argues that the Boillat cone lies inside or touches the gravitational cone, so fluctuations of the vector, scalars and spinors on the brane travel no faster than gravitational waves. The effective metric is conformal to the open-string metric, $C^{-1 μν}=(1/√{1+2x-y^2})G^{μν}$, which is why a single open-string equivalence principle governs both causality and the finite-temperature Hagedorn redshift. The notes extend the same pattern to the M5-brane, where a nonlinear self-duality condition and an analogous Hooke law $T^{αβ}=g^{αβ}-C^{αβ}$ lead to a dominant-energy-condition argument and a scale-invariant strong-coupling limit.

Load-bearing premise

The causality conclusion depends on the Born-Infeld energy-momentum tensor satisfying the weak energy condition, which is what places the Boillat cone inside the gravitational cone; if that condition fails, the no-superluminal-signal argument no longer follows.

Editorial extensions

If this is right

All open-string fields on a D-brane, including gauge vectors, scalars and fermions, share the same maximum propagation speed, set by the Boillat/open-string metric rather than by the closed-string metric.
An electric field lowers the Hagedorn temperature to $T_{Hagedorn}√{1-E^2}$, so the critical field $E=1$ is also the point where the open-string temperature scale collapses.
At the critical strong-coupling limit the Born-Infeld theory becomes a scale-invariant theory of null magnetic strings, with $T^{μν}=H l^μ l^ν$ and infinitely many conserved symmetric tensors.
When a tachyon potential vanishes, the Born-Infeld vector describes a massive string fluid with a simple timelike 2-form $K$ and energy density proportional to electric flux.
The M5-brane has the same causal structure: its Boillat cone lies inside the gravitational cone, and its strong-coupling limit is a scale-invariant null 3-form fluid.

Reading between the lines

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

If the single Boillat cone really is universal, a D-brane in a strong background should show no species-dependent horizon for open-string signals; comparing arrival times of gauge, scalar and fermion modes would test the open-string equivalence principle.
The conformal relation between the Boillat and open-string metrics suggests that the finite-temperature redshift and the causal cone are two readings of one effective geometry, so any correction that breaks the conformal factor should also break the universal Hagedorn shift.
The strong-coupling string-fluid limits offer a route to connecting Born-Infeld dynamics with tachyon condensation; a natural extension is to ask whether the integrable 2-plane distribution survives quantum or higher-derivative corrections.
The M5-brane analogy suggests that the same cone-inclusion argument could constrain higher-form-brane effective actions whenever a Hooke-type identity can be written.

Original abstract

These notes provide a hopefully pedagogic introduction to Born-Infeld theory and some of its uses in String/M-Theory from the perspective of some-one interested in spacetime geometry. Causality and the consequences of having both the open string and closed string metric are stressed. This leads to some new insighst into old problems, such as strings at finite temperature. Various strong coupling limits of both Born-Infeld and the M5-brane theories are described and their relevance for tachyon condensations are mentioned.

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The abstract, sentence by sentence

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These notes provide a hopefully pedagogic introduction to Born-Infeld theory and some of its uses in String/M-Theory from the perspective of some-one interested in spacetime geometry. Causality and the consequences of having both the open string and closed string metric are stressed. This leads to some new insighst into old problems, such as strings at finite temperature. Various strong coupling limits of both Born-Infeld and the M5-brane theories are described and their relevance for tachyon condensations are mentioned.

background contribution method result interpretation main claim

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s0	These notes provide a hopefully pedagogic introduction to Born-Infeld theory and some of its uses in String/M-Theory from the perspective of some-one interested in spacetime geometry.	contribution 0.77	0.46	0.05	asserted 0.64
s1	Causality and the consequences of having both the open string and closed string metric are stressed.	contribution 0.92	0.37	0.10	asserted 0.91
s2	This leads to some new insighst into old problems, such as strings at finite temperature.	interpretation 0.59	0.60	0.13	asserted 0.86
s3	Various strong coupling limits of both Born-Infeld and the M5-brane theories are described and their relevance for tachyon condensations are mentioned.	contribution 0.97	0.57	0.11	asserted 0.97

What kind of paper the classifier reports this is: review 0.97 · evidence from argument 0.55.

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References

Works cited (22)

[18]

G W Gibbons and C A R Herdeiro, Phys rev D 63 (2001) 064006-1 - 064006-17 hep-th/0008052

View on Pith Pith review arXiv 2001
[13]

E J Ferrer, E S Fradkin and V de la Incarra, Eﬀect of a backgrou nd electric ﬁeld on the Hagedorn temperature Phys Lett B 248 (1990) 281-287

View on Pith 1990
[14]

A A Tseytlin, Nucl Phys B 460 (1998) 69

View on Pith 1998
[1]

L de Broglie, Un Itin´ eraire Scientiﬁque, ´Editions Scientiﬁque (1987)

View on Pith 1987
[2]

JP Gauntlett, G W Gibbons, C M Hull and P Townsend, BPS Staes of D=4 N=1 Supersymmetry Comm Math Phys 216 (2001) 431-459

View on Pith 2001
[3]

G W Gibbons, Convex Cones in Physics, unpublished notes of lectur es delivered at the Yuakawa Insitute

View on Pith
[4]

S W Hawking, The conservation of matter in general relativity Comm Math Phys 18 (1970) 303-306

View on Pith 1970
[5]

S W Hawking and GFR Ellis, The Large Scale Structure of Spacetime (1973) , Cambridge University Press

View on Pith 1973
[6]

Ya B Zeldovich, The Creation of Particles and Anti-Particles in Elect ric and Gravitational Fields Magic Without Magic , ed J R Klauder (277-288)

View on Pith
[7]

Ya B Zeldovich and L P Pitaevsky, On the possibility of the Creation o f Particles by a Classical Gravitional Field Comm Math Phys 23 (1971) 185

View on Pith 1971
[8]

G Monge, Application de l’analyse a `la geom´ etrie (1807)

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W Rouse Ball, A Hypothesis relating to the Nature of the Ether & Gr avity Messenger of Mathematics 21 (1891) 20-24

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L Jardine Ingenious Pursuits Little, Brown and Company (1999)

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[11]

I Bialynicki-Birula, Non-linear Electrodynamics: Variations on a Th eme of Born and Infeld, in Quantum Theory of Fields and Particles eds B. Jancerwicz and J Lukierski, World Scientiﬁc, Singapore (1983) 28

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I Bialynicki-Birula, Field Theory of a Photon Dust, Acta Physica Polonica B 23 (1992) 553-559

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G W Gibbons Nucl Phys Born-Infeld particles nad Dirichlet p-branes 514 (1998) 603-639 9709027

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G W Gibbons, Pulse Propagation in Born-Infeld Theory hep-th/0104015

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G W Gibbons and P C West, The metric and strong coupling limit of the M5-brane J Math Phys hep-th/0011149

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C G Callan and J Maldacena, Brane Dynamics from the Born-Infeld action, Nucl Phys B 513 (1998) 198 hep-th/9708147

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G W Gibbons and K Hashimoto, Nonlinear Electrodynamics in Curved Backgrounds hep-th/0007019 29

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How the literature uses it

34 arXiv papers cite this in 39 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 · 20 sentences without a confident label

Builds on it 1 paper

Interestingly, we show that Weyl quadratic gravity is the leading order (in some expansion) of a more general Weyl gauge invariant theory (in $d$ dimensions!) that we construct and which is a version of Dirac-Born-Infeld (DBI) [ref] action but in Weyl conformal geometry.

Quantum gravity from Weyl conformal geometry · Motivation and brief history · 0.90

Cites it as background 17 papers

The Born-Infeld (BI) equations [ref] , which preserve the electromagnetic duality invariance of Maxwell's equations, is perhaps the best known example, for reasons reviewed in [ref] .

A non-linear duality-invariant conformal extension of Maxwell's equations · Introduction · 0.97
Originally proposed to regularize the divergent self-energy of point charges through a maximum field strength [ref] , BI electrodynamics emerges naturally as the low-energy effective theory of D-branes and open strings [ref] .

Effects of Born-Infeld Electrodynamics on Chiral Symmetry Restoration and Meson Susceptibilities in Holographic QCD · Introduction · 0.89
(See [ref] for a later review.) BIons were shown to have a point-like source in their core with strings, or flux tubes, emanating from the core and going all the way to infinity.

Frozen stars: Black hole mimickers sourced by a string fluid · Introduction · 0.84

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] [18]

G W Gibbons and C A R Herdeiro, Phys rev D 63 (2001) 064006-1 - 064006-17 hep-th/0008052

View on Pith Pith review arXiv 2001

[2] [13]

E J Ferrer, E S Fradkin and V de la Incarra, Eﬀect of a backgrou nd electric ﬁeld on the Hagedorn temperature Phys Lett B 248 (1990) 281-287

View on Pith 1990

[3] [14]

A A Tseytlin, Nucl Phys B 460 (1998) 69

View on Pith 1998

[4] [1]

L de Broglie, Un Itin´ eraire Scientiﬁque, ´Editions Scientiﬁque (1987)

View on Pith 1987

[5] [2]

JP Gauntlett, G W Gibbons, C M Hull and P Townsend, BPS Staes of D=4 N=1 Supersymmetry Comm Math Phys 216 (2001) 431-459

View on Pith 2001

[6] [3]

G W Gibbons, Convex Cones in Physics, unpublished notes of lectur es delivered at the Yuakawa Insitute

View on Pith

[7] [4]

S W Hawking, The conservation of matter in general relativity Comm Math Phys 18 (1970) 303-306

View on Pith 1970

[8] [5]

S W Hawking and GFR Ellis, The Large Scale Structure of Spacetime (1973) , Cambridge University Press

View on Pith 1973

[9] [6]

Ya B Zeldovich, The Creation of Particles and Anti-Particles in Elect ric and Gravitational Fields Magic Without Magic , ed J R Klauder (277-288)

View on Pith

[10] [7]

Ya B Zeldovich and L P Pitaevsky, On the possibility of the Creation o f Particles by a Classical Gravitional Field Comm Math Phys 23 (1971) 185

View on Pith 1971

[11] [8]

G Monge, Application de l’analyse a `la geom´ etrie (1807)

View on Pith

[12] [9]

W Rouse Ball, A Hypothesis relating to the Nature of the Ether & Gr avity Messenger of Mathematics 21 (1891) 20-24

View on Pith

[13] [10]

L Jardine Ingenious Pursuits Little, Brown and Company (1999)

View on Pith 1999

[14] [11]

I Bialynicki-Birula, Non-linear Electrodynamics: Variations on a Th eme of Born and Infeld, in Quantum Theory of Fields and Particles eds B. Jancerwicz and J Lukierski, World Scientiﬁc, Singapore (1983) 28

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