Contents

# 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 $4$-dimensional Minkowski spacetime

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

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

the Lagrangian density of the Born-Infeld action functional is

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

Here

• $\eta$ denotes the metric tensor of Minkowski spacetime, regarded as a $4 \times 4$ matrix

$\eta \;=\; (\eta_{a b}) \;=\; diag(-1,1,1,1) \,,$
• $F$ is the Faraday tensor, regarded as a $4 \times 4$ matrix

$F = (F_{a b}) \,,$
• $T = \frac{1}{2\pi \, \alpha'} = \frac{1}{2\pi \, \ell_2^2}$ is the string tension,

• $det(\cdots)$ denotes the determinant of the sum of these matrices,

• $\sqrt{}$ denotes the positive square root,

• $dvol$ denotes a volume form on Minkowski spacetime, which after a choice of global coordinates may be taken to be

(2)$dvol \;=\; d t \wedge d x^1 \wedge d x^2 \wedge d x^3$

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

$(\omega_4 / dvol) \cdot dvol \;=\; \omega_4 \,.$
###### Lemma

The determinant in (1) evaluates to

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

where

###### Proof

We compute as follows:

\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 $\eta$ (of $F$), 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 $\eta$ and of $F$, respectively (again by that formula).

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

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

• (second summand) If we exchange indices $(a_i \leftrightarrow b_i)$ the form of this summand remains unchanged, also the factors $\eta_{a_i b_i}$ do not change, since $\eta$ is a symmetric matrix, by definition. But the single factor of $F$ 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 $\eta$ relaced by the identity matrix (to be indicated by a Kronecker delta $(\delta_{a b})$). Observe then that the contraction not involving any factor of $F$ yields

$\epsilon^{a_1 a_2 a_3 a_4} \delta_{a_1 b_1} \delta_{a_2 b_2} \epsilon^{b_1 b_2 b_3 b_4} \;=\; 2 \delta^{a_3 a_4}_{b_3 b_4} \,,$

where the symbol on the right is defined to be

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

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

$\tfrac{2\cdot 2}{4!} \delta^{a_3 a_4}_{b_3 b_4} F_{a_3 b_3} F_{a_4 b_4} \;=\; - \sum_{a, b} F_{a b} F_{a b}$

where we used that due to the skew-symmetry of $F$ 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 $\eta$ instead of $\delta$: 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 $F$ 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 $F$ 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 $F$, which is (by this formula)

$Pf(F) \;=\; \tfrac{1}{4 \cdot 4!} \epsilon^{a_1 b_1 a_2 b_2} F_{a_1 b_1} F_{a_2 b_2}$

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

\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 $(\partial F)^4$. 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 $F$ expressed in terms of the electric field $\vec E$ and magnetic field $\vec B$ as

\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)$\mathbf{L}_{BI} \;=\; \sqrt{ - det\left( \eta + \tfrac{1}{T} F \right) } \, dvol_4 \;=\; \sqrt{ 1 - \tfrac{1}{T^2} ( \vec E \cdot \vec E - \vec B \cdot \vec B ) - \tfrac{1}{T^4} (\vec B \cdot \vec E)^2 } \, dvol_4$

### 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

\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

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

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

The resulting maximal electric field strength

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

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 $\kappa$-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)

• 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)

Discussion of one-loop corrections:

• Garrett Goon, Scott Melville, Johannes Noller, Quantum Corrections to Generic Branes: DBI, NLSM, and More (arXiv:2010.05913)

Derivation of the first DBI-correction from an M5-brane model via super-exceptional geometry:

### 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 plain 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

• T. Daniel Brennan, Christian Ferko, Savdeep Sethi, A Non-Abelian Analogue of DBI from $T \bar T$ (arXiv:1912.12389)

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

On KK-compactification of the non-abelian DBI-action from 10d to 4d:

• Yoshihiko Abe, Tetsutaro Higaki, Tatsuo Kobayashi, Shintaro Takada, Rei Takahashi, 4D effective action from non-Abelian DBI action with magnetic flux background (arXiv:2107.11961)

### 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:

• 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 $O(\alpha'{}^2)$, Phys. Lett. B701:465-470, 2011 (arXiv:1103.3121)

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

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

• Komeil Babaei Velni, Ali Jalali, Higher derivative corrections to DBI action at $\alpha'{}^2$ 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 July 28, 2021 at 03:11:50. See the history of this page for a list of all contributions to it.