nLab
Dirac-Born-Infeld action

Contents

Context

Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

String theory

Contents

Idea

What is known as Born-Infeld theory (Born-Infeld 34, often also attributed to Dirac 62 and abbreviated “DBI theory”) is a deformation of the theory of electromagnetism which coincides with ordinary electromagnetism for small excitations of the electromagnetic field but is such that there is a maximal value for the field strength which can never be reached in a physical process.

Just this theory happens to describe the Chan-Paton gauge field on single D-branes at low energy, as deduced from open string scattering amplitudes (Fradkin-Tseytlin 85, Abouelsaood-Callan-Nappi-Yost 87, Leigh 89).

In this context the action functional corresponding to Born-Infeld theory arises as the low-energy effective action on the D-branes, and this is referred to as the DBI-action. This is part of the full Green-Schwarz action functional for super D-branes, being a deformation of the Nambu-Goto action-summand by the field strength of the Chan-Paton gauge fields.

On coincident D-branes, where one expects gauge enhancement of the Chan-Paton gauge field to a non-abelian gauge group, a further generalization of the DBI-action to non-abelian gauge fields is expected to be an analogous deformation of that of non-abelian Yang-Mills theory. A widely used proposal is due to Tseytlin 97, Myers 99, but a derivation from string theory of this non-abelian DBI action is lacking; and it is in fact known to be in conflict, beyond the first few orders of correction terms, with effects argued elsewhere in the string theory literature (Hashimoto-Taylor 97, Bain 99, Bergshoeff-Bilal-Roo-Sevrin 01). The issue remains open.

When the D-branes in question are interpreted as flavor branes, then the maximal/critical value of the electric field which arises from the DBI-action has been argued (Semenoff-Zarembo 11) to reflect, via holographic QCD, the Schwinger limit beyond which the vacuum polarization caused by the electromagnetic field leads to deconfinement of quarks.

Definition

On 44-dimensional Minkowski spacetime

In the simplest situation of flat 4-dimensional Minkowski spacetime 3,1\mathbb{R}^{3,1} and no other fields besides that of electromagnetism, encoded in a Faraday tensor differential 2-form

F=F abdx adx b F \;=\; F_{a b} d x^a \wedge d x^b

the Lagrangian density of the Born-Infeld action functional is

(1)L BI=det((η μν)+1T(F μν))dvol. \mathbf{L}_{BI} \;=\; \sqrt{ - det \big( (\eta_{\mu\nu}) + \tfrac{1}{T} (F_{\mu\nu}) \big) } \, dvol \,.

Here

In the following, for ω 4\omega_4 any differential 4-form on 3,1\mathbb{R}^{3,1} we write ω 4/dvol\omega_4 / dvol for the unique smooth function 3,1\mathbb{R}^{3,1} \to \mathbb{R} such that

(ω 4/dvol)dvol=ω 4. (\omega_4 / dvol) \cdot dvol \;=\; \omega_4 \,.
Lemma

The determinant in (1) evaluates to

(3)det(η+F)=112(FF)Lagrangian ofelectromagnetism/dvol+(12(FF)/dvol) 2correctionterm, det( \eta + F ) \;=\; - 1 - \tfrac{1}{2} \underset{ \mathclap{ {\color{blue}\text{Lagrangian of}} \atop {\color{blue}\text{electromagnetism}} } }{ \underbrace{ (F \wedge \star F) } } / dvol + \underset{ {\color{blue}\text{correction}} \atop {\color{blue}\text{term}} }{ \underbrace{ \big( \tfrac{1}{2} (F\wedge F) / \mathrm{dvol} \big)^2 } } \,,

where

Proof

We compute as follows:

det((η ab)+(F ab)) =+14!ϵ a 1a 2a 3a 4(η a 1b 1+F a 1b 1)(η a 2b 2+F a 2b 2)(η a 3b 3+F a 3b 3)(η a 4b 4+F a 4b 4)ϵ b 1b 2b 3b 4 =+14!ϵ a 1a 2a 3a 4η a 1b 1η a 2b 2η a 3b 3η a 4b 4ϵ b 1b 2b 3b 4=det(η)=1 =+34!ϵ a 1a 2a 3a 4η a 1b 1η a 2b 2η a 3b 3F a 4b 4ϵ b 1b 2b 3b 4=0 =+64!ϵ a 1a 2a 3a 4η a 1b 1η a 2b 2F a 3b 3F a 4b 4ϵ b 1b 2b 3b 4=264!F abF ab =+34!ϵ a 1a 2a 3a 4η a 1b 1F a 2b 2F a 3b 3F a 4b 4ϵ b 1b 2b 3b 4=0 =+14!ϵ a 1a 2a 3a 4F a 1b 1F a 2b 2F a 3b 3F a 4b 4ϵ b 1b 2b 3b 4=(12(FF)/dvol) 2 =+112(FF)/dvol+(12(FF)/dvol) 2 \begin{aligned} \mathrm{det} \big( (\eta_{a b}) + (F_{a b}) \big) & \; = \phantom{+} \tfrac{1}{4!} \epsilon^{a_1 a_2 a_3 a_4} (\eta_{a_1 b_1} + F_{a_1 b_1}) (\eta_{a_2 b_2} + F_{a_2 b_2}) (\eta_{a_3 b_3} + F_{a_3 b_3}) (\eta_{a_4 b_4} + F_{a_4 b_4}) \epsilon^{b_1 b_2 b_3 b_4} \\ & \; = \phantom{+} \underset{ = \mathrm{det}(\eta) = -1 }{ \underbrace{ \tfrac{1}{4!} \epsilon^{a_1 a_2 a_3 a_4} \eta_{a_1 b_1} \eta_{a_2 b_2} \eta_{a_3 b_3} \eta_{a_4 b_4} \epsilon^{b_1 b_2 b_3 b_4} } } \\ & \phantom{\; =} + \underset{ = 0 }{ \underbrace{ \tfrac{3}{4!} \epsilon^{a_1 a_2 a_3 a_4} \eta_{a_1 b_1} \eta_{a_2 b_2} \eta_{a_3 b_3} F_{a_4 b_4} \epsilon^{b_1 b_2 b_3 b_4} } } \\ & \phantom{\; =} + \underset{ = -\tfrac{2\cdot 6}{4!} F_{a b} F^{a b} }{ \underbrace{ \tfrac{6}{4!} \epsilon^{a_1 a_2 a_3 a_4} \eta_{a_1 b_1} \eta_{a_2 b_2} F_{a_3 b_3} F_{a_4 b_4} \epsilon^{b_1 b_2 b_3 b_4} } } \\ & \phantom{\; =} + \underset{ = 0 }{ \underbrace{ \tfrac{3}{4!} \epsilon^{a_1 a_2 a_3 a_4} \eta_{a_1 b_1} F_{a_2 b_2} F_{a_3 b_3} F_{a_4 b_4} \epsilon^{b_1 b_2 b_3 b_4} } } \\ & \phantom{\; =} + \underset{ = \big( \tfrac{1}{2} (F \wedge F) / \mathrm{dvol} \big)^2 }{ \underbrace{ \tfrac{1}{4!} \epsilon^{a_1 a_2 a_3 a_4} F_{a_1 b_1} F_{a_2 b_2} F_{a_3 b_3} F_{a_4 b_4} \epsilon^{b_1 b_2 b_3 b_4} } } \\ & \; = \phantom{+} -1 - \tfrac{1}{2} (F \wedge \star F) / \mathrm{dvol} + \big( \tfrac{1}{2} (F\wedge F) / \mathrm{dvol} \big)^2 \end{aligned}

In the first line we used the expression of the determinant via the Levi-Civita symbol (here) with the Einstein summation convention being understood throughout. Then we multiplied out the terms, collecting those with the same number of factors of η\eta (of FF), using that under exchange of the order of factors both Levi-Civita symbols give a sign, which hence cancel. Of the five terms that appear, the first and the last are themselves the plain determinants of η\eta and of FF, respectively (again by that formula).

We discuss the identifications of the resulting four summands shown under the braces:

  • (first summand) The determinant of η\eta equals -1 by definition.

  • (second summand) If we exchange indices (a ib i)(a_i \leftrightarrow b_i) the form of this summand remains unchanged, also the factors η a ib i\eta_{a_i b_i} do not change, since η\eta is a symmetric matrix, by definition. But the single factor of FF changes by a sign, since the components of a differential 2-form constitute a skew-symmetric matrix. In summary this says that the second term is equal to minus itself, and hence has to be zero.

  • (third summand) Consider this term first with η\eta relaced by the identity matrix (to be indicated by a Kronecker delta (δ ab)(\delta_{a b})). Observe then that the contraction not involving any factor of FF yields

    ϵ a 1a 2a 3a 4δ a 1b 1δ a 2b 2ϵ b 1b 2b 3b 4=2δ b 3b 4 a 3a 4, \epsilon^{a_1 a_2 a_3 a_4} \delta_{a_1 b_1} \delta_{a_2 b_2} \epsilon^{b_1 b_2 b_3 b_4} \;=\; 2 \delta^{a_3 a_4}_{b_3 b_4} \,,

    where the symbol on the right is defined to be

    (4)δ b 3b 4 a 3a 4{+1 | a 3a 4anda 3=b 3anda 4=b 4 1 | a 3a 4anda 3=b 4anda 4=b 3 +0 | otherwise \delta^{a_3 a_4}_{b_3 b_4} \;\coloneqq\; \left\lbrace \array{ +1 &\vert& a_3 \neq a_4 \;\text{and}\; a_3 = b_3 \;\text{and}\; a_4 = b_4 \\ -1 &\vert& a_3 \neq a_4 \;\text{and}\; a_3 = b_4 \;\text{and}\; a_4 = b_3 \\ \phantom{+}0 &\vert& \text{otherwise} } \right.

    Hence the full expression (with η\eta still replaced by δ\delta) is

    224!δ b 3b 4 a 3a 4F a 3b 3F a 4b 4= a,bF abF ab \tfrac{2\cdot 2}{4!} \delta^{a_3 a_4}_{b_3 b_4} F_{a_3 b_3} F_{a_4 b_4} \;=\; - \sum_{a, b} F_{a b} F_{a b}

    where we used that due to the skew-symmetry of FF the first case in (4) does not contribute, only the second case does.

    Now it just remains to translate this back to the situation at hand where we use η\eta instead of δ\delta: This just differs by a minus sign in the component with both indices corresponding to the temporal direction, while this is also the case for which raising an index on FF picks up a minus sign. Since either of these cases contributes in each summand, there is a global minus sign.

  • (fourth summand) Since this involves three factors of FF which jointly pick up one minus sign when the indices on each of them are exchanged simultaneously, this vanishes by the same kind of skew-symmetry argument as for the second term.

  • (fifth summand) Since this is the determinant of a skew-symmetric matrix, the Pfaffian-theorem (here) says that this term equals the square of the Pfaffian of FF, which is (by this formula)

    Pf(F)=144!ϵ a 1b 1a 2b 2F a 1b 1F a 2b 2 Pf(F) \;=\; \tfrac{1}{4 \cdot 4!} \epsilon^{a_1 b_1 a_2 b_2} F_{a_1 b_1} F_{a_2 b_2}

    This is proportial to the coefficient of the wedge product of FF with itself, relative to the volume form:

    FF =(12F a 1b 1dx a 1dx b 1)(12F a 2b 2dx a 2dx b 2) =14F a 1b 1F a 2b 2dx a 1dx b 1dx a 2dx b 2 =14F a 1b 1F a 2b 2ϵ a 1b 1a 2b 2dvol =4!Pf(F)dvol \begin{aligned} F \wedge F & = \; \big( \tfrac{1}{2}F_{a_1 b_1} d x^{a_1} \wedge d x^{b_1} \big) \wedge \big( \tfrac{1}{2}F_{a_2 b_2} d x^{a_2} \wedge d x^{b_2} \big) \\ & = \; \tfrac{1}{4} F_{a_1 b_1} F_{a_2 b_2} d x^{a_1} \wedge d x^{b_1} \wedge d x^{a_2} \wedge d x^{b_2} \\ & = \; \tfrac{1}{4} F_{a_1 b_1} F_{a_2 b_2} \epsilon^{a_1 b_1 a_2 b_2} \, dvol \\ & = 4! Pf(F) \, dvol \end{aligned}

The expression (1) is supposed to be exact for constant field strength (e..g. Bachas-Bain-Green 99, above (1.9)), and to pick up higher curvature corrections for non-constant field strength. The first derivative correction to (1) is supposed to arise at order (F) 4(\partial F)^4. The explicit expression is given in Garousi 15 (7) (argued there by appealing to T-duality and S-duality applied to earlier results on higher curvature corrections in other fields involved).


Consider now the Faraday tensor FF expressed in terms of the electric field E\vec E and magnetic field B\vec B as

F 0i =+E i F i0 =E i F ij =ϵ ijkB k \begin{aligned} F_{0 i} & = \phantom{+} E_i \\ F_{i 0} & = - E_i \\ F_{i j} & = \epsilon_{i j k} B^k \end{aligned}

Then the general expression (3) for the DBI-Lagrangian reduces to (Born-Infeld 34, p. 437, review in Gibbons 97, (56), Savvidy 99, (22), Nastase 15, 9.4):

(5)L BI=det(η+1TF)dvol 4=11T 2(EEBB)1T 4(BE) 2dvol 4 \mathbf{L}_{BI} \;=\; \sqrt{ - det\left( \eta + \tfrac{1}{T} F \right) } \, dvol_4 \;=\; \sqrt{ 1 - \tfrac{1}{T^2} ( \vec E \cdot \vec E - \vec B \cdot \vec B ) - \tfrac{1}{T^4} (\vec B \cdot \vec E)^2 } \, dvol_4

Critical field strength

For the DBI-action (5) to be well-defined, in that the square root is a real number, hence its argument a non-negative number, requires that

det((η μν)+1T(F μν))0 11T 2(EEBB)1T 4(BE) 20 1T 2E 21T 2B 2+1T 4E 2B 21 1T 2E 21+1T 2B 21+1T 2B 2 ETT 2+B 2T 2+B 2 \begin{aligned} & - \mathrm{det} \big( (\eta_{\mu \nu}) + \tfrac{1}{T} (F_{\mu \nu}) \big) \geq 0 \\ & \Leftrightarrow \; 1 \;-\; \tfrac{1}{T^2} (E \cdot E - B \cdot B) \;-\; \tfrac{1}{T^4} (B \cdot E)^2 \;\geq\; 0 \\ & \Leftrightarrow \; \tfrac{1}{T^2} E^2 - \tfrac{1}{T^2} B^2 + \tfrac{1}{T^4} E^2 B_{\parallel}^2 \;\leq 1\; \\ & \Leftrightarrow \; \tfrac{1}{T^2} E^2 \;\leq\; \frac{ 1 + \tfrac{1}{T^2} B^2 }{ 1 + \tfrac{1}{T^2} B_{\parallel}^2 } \\ & \Leftrightarrow \; E \;\leq\; T \sqrt{ \frac{ T^2 + B^2 }{ T^2 + B_{\parallel}^2 } } \end{aligned}

where

B 1EEBE B_{\parallel} \coloneqq \tfrac{1}{\sqrt{E\cdot E}} B \cdot E

is the component of the magnetic field which is parallel to the electric field.

The resulting maximal electric field strength

E critTT 2+B 2T 2+B 2 E_{crit} \;\coloneqq\; T \sqrt{ \frac{ T^2 + B^2 }{ T^2 + B_{\parallel}^2 } }

turns out to be the Schwinger limit (see there) beyond which the electric field would cause deconfining quark-pair creation (Hashimoto-Oka-Sonoda 14b, (2.17)).

References

General

As a proposal for a modification of electromagnetism in spacetime, the (Dirac-)Born-Infeld (DBI) action originates in

  • Max Born, Leopold Infeld, Foundations of the New Field Theory, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 144, No. 852 (Mar. 29, 1934), pp. 425-451 (jstor:2935568)

The article by Dirac which came to be commonly cited in this context is

For single (abelian) D-branes

As the low energy action functional for single D-branes the DBI action is due to

and a full κ\kappa-symmetric Green-Schwarz sigma-model for D-branes:

Review:

Detailed discussion of the relation to the Polyakov action and the Nambu-Goto action is in

Discussion in terms of D-branes as leaves of Dirac structures on Courant Lie 2-algebroids of type II geometry is in

  • Tsuguhiko Asakawa, Shuhei Sasa, Satoshi Watamura, D-branes in Generalized Geometry and Dirac-Born-Infeld Action (arXiv:1206.6964)

See also

  • Martin Cederwall, Alexander von Gussich, Aleksandar Mikovic, Bengt Nilsson, Anders Westerberg, On the Dirac-Born-Infeld Action for D-branes, Phys.Lett.B390:148-152, 1997 (arXiv:hep-th/9606173)

  • Ian I. Kogan, Dimitri Polyakov, DBI Action from Closed Strings and D-brane second Quantization, Int. J. Mod. Phys. A18 (2003) 1827 (arXiv:hep-th/0208036)

For coincident (non-abelian) D-branes

Proposals for the generalization of the DBI action to non-abelian Chan-Paton gauge fields (hence: for coincident D-branes) includes the following:

Via a plane trace:

Via an antisymmetrized trace:

Via a combination of spacetime and gauge indices:

  • Jeong-Hyuck Park, A Study of a Non-Abelian Generalization of the Born-Infeld Action, Phys. Lett. B458 (1999) 471-476 (arXiv:hep-th/9902081)

The now widely accepted proposal via a symmetrized trace is due to

followed by

The symmetrized trace proposal has become widely accepted.

Review includes:

  • W. Chemissany, On the way of finding the non-Abelian Born-Infeld theory, 2004 (spire:1286212 pdf)

Issues with this proposal at higher order have been found in

Correction terms have been proposed in

A completely different approach via TT deformation of the abelian DBI action is proposed in

For actual derivation of gauge enhancement on coincident D-branes see the references there.

On flavor branes for holographic QCD

Discussion of the DBI-action for flavor branes in holographic QCD:

Holographic Schwinger effect

Interpretation in holographic QCD of the Schwinger effect of vacuum polarization as exhibited by the DBI-action on flavor branes:

Precursor computation in open string theory:

Relation to the DBI-action of a probe brane in AdS/CFT:

  • Gordon Semenoff, Konstantin Zarembo, Holographic Schwinger Effect, Phys. Rev. Lett. 107, 171601 (2011) (arXiv:1109.2920, doi:10.1103/PhysRevLett.107.171601)

  • S. Bolognesi, F. Kiefer, E. Rabinovici, Comments on Critical Electric and Magnetic Fields from Holography, J. High Energ. Phys. 2013, 174 (2013) (arXiv:1210.4170)

  • Yoshiki Sato, Kentaroh Yoshida, Holographic description of the Schwinger effect in electric and magnetic fields, J. High Energ. Phys. 2013, 111 (2013) (arXiv:1303.0112)

  • Yoshiki Sato, Kentaroh Yoshida, Holographic Schwinger effect in confining phase, JHEP 09 (2013) 134 (arXiv:1306.5512

  • Yoshiki Sato, Kentaroh Yoshida, Universal aspects of holographic Schwinger effect in general backgrounds, JHEP 12 (2013) 051 (arXiv:1309.4629)

  • Daisuke Kawai, Yoshiki Sato, Kentaroh Yoshida, The Schwinger pair production rate in confining theories via holography, Phys. Rev. D 89, 101901 (2014) (arXiv:1312.4341)

  • Yue Ding, Zi-qiang Zhang, Holographic Schwinger effect in a soft wall AdS/QCD model (arXiv:2009.06179)

Relation to DBI-action of flavor branes in holographic QCD:

See also:

  • Xing Wu, Notes on holographic Schwinger effect, J. High Energ. Phys. 2015, 44 (2015) (arXiv:1507.03208, doi:10.1007/JHEP09(2015)044)

  • Kazuo Ghoroku, Masafumi Ishihara, Holographic Schwinger Effect and Chiral condensate in SYM Theory, J. High Energ. Phys. 2016, 11 (2016) (doi:10.1007/JHEP09(2016)011)

  • Le Zhang, De-Fu Hou, Jian Li, Holographic Schwinger effect with chemical potential at finite temperature, Eur. Phys. J. A54 (2018) no.6, 94 (spire:1677949, doi:10.1140/epja/i2018-12524-4)

  • Wenhe Cai, Kang-le Li, Si-wen Li, Electromagnetic instability and Schwinger effect in the Witten-Sakai-Sugimoto model with D0-D4 background, Eur. Phys. J. C 79, 904 (2019) (doi:10.1140/epjc/s10052-019-7404-1)

  • Zhou-Run Zhu, De-fu Hou, Xun Chen, Potential analysis of holographic Schwinger effect in the magnetized background (arXiv:1912.05806)

  • Zi-qiang Zhang, Xiangrong Zhu, De-fu Hou, Effect of gluon condensate on holographic Schwinger effect, Phys. Rev. D 101, 026017 (2020) (arXiv:2001.02321)

Review:

  • Daisuke Kawai, Yoshiki Sato, Kentaroh Yoshida, A holographic description of the Schwinger effect in a confining gauge theory, International Journal of Modern Physics A Vol. 30, No. 11, 1530026 (2015) (arXiv:1504.00459)

  • Akihiko Sonoda, Electromagnetic instability in AdS/CFT, 2016 (spire:1633963, pdf)

Higher curvature corrections to the DBI-action for D-branes

On higher curvature corrections to the (abelian) DBI-action for (single) D-branes:

  • Oleg Andreev, Arkady Tseytlin, Partition-function representation for the open superstring effective action:: Cancellation of Möbius infinites and derivative corrections to Born-Infeld lagrangian, Nuclear Physics B Volume 311, Issue 1, 19 December 1988, Pages 205-252 (doi:10.1016/0550-3213(88)90148-4)

  • Constantin Bachas, P. Bain, Michael Green, Curvature terms in D-brane actions and their M-theory origin, JHEP 9905:011, 1999 (arXiv:hep-th/9903210)

  • Niclas Wyllard, Derivative corrections to D-brane actions with constant background fields, Nucl. Phys. B598 (2001) 247-275 (arXiv:hep-th/0008125)

  • Oleg Andreev, More About Partition Function of Open Bosonic String in Background Fields and String Theory Effective Action, Phys. Lett. B513:207-212, 2001 (arXiv:hep-th/0104061)

  • Niclas Wyllard, Derivative corrections to the D-brane Born-Infeld action: non-geodesic embeddings and the Seiberg-Witten map, JHEP 0108 (2001) 027 (arXiv:hep-th/0107185)

  • Mohammad Garousi, T-duality of curvature terms in D-brane actions, JHEP 1002:002, 2010 (arXiv:0911.0255)

  • Mohammad Garousi, S-duality of D-brane action at order O(α 2)O(\alpha'{}^2), Phys. Lett. B701:465-470, 2011 (arXiv:1103.3121)

  • Ali Jalali, Mohammad Garousi, On D-brane action at order α 2\alpha'{}^2, Phys. Rev. D 92, 106004 (2015) (arXiv:1506.02130)

  • Mohammad Garousi, An off-shell D-brane action at order α 2\alpha'{}^2 in flat spacetime, Phys. Rev. D 93, 066014 (2016) (arXiv:1511.01676)

  • Komeil Babaei Velni, Ali Jalali, Higher derivative corrections to DBI action at α 2\alpha'{}^2 order, Phys. Rev. D 95, 086010 (2017) (arXiv:1612.05898)

Single trace observables as weight systems on chord diagrams

Relation of single trace observables in the non-abelian DBI action on Dp-D(p+2)-brane bound states (hence Yang-Mills monopoles) to su(2)-Lie algebra weight systems on chord diagrams computing radii averages of fuzzy spheres:

Brane intersections as DBI-spikes/BIons

On D1-D3 brane intersections as spikes/BIons in the D3-brane DBI-theory:

From the M5-brane

Last revised on June 1, 2020 at 14:03:16. See the history of this page for a list of all contributions to it.