added pointer to today’s

- Dmitri Sorokin,
*Introductory Notes on Non-linear Electrodynamics and its Applications*(arXiv:2112.12118)

added pointer to today’s

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

and this one:

- Gary Gibbons,
*Born-Infeld particles and Dirichlet p-branes*, Nucl. Phys. B514:603-639, 1998 (arXiv:hep-th/9709027)

added pointer to

- Konstantin G. Savvidy,
*Born-Infeld Action in String Theory*, 1999 (arXiv:hep-th/9906075, spire:501510)

```
<p>added pointer to more of the precursor proposals for the non-abelian DBI-action:</p>
<p>Via a plane <a href="https://ncatlab.org/nlab/show/trace">trace</a>:</p>
<ul>
<li>T. Hagiwara, <em>A non-abelian Born-Infeld Lagrangian</em>, J. Phys., A14:3059, 1981 (<a href="https://iopscience.iop.org/article/10.1088/0305-4470/14/11/027">doi:10.1088/0305-4470/14/11/027</a>)</li>
</ul>
<p>Via an antisymmetrized <a href="https://ncatlab.org/nlab/show/trace">trace</a>:</p>
<ul>
<li><a href="https://ncatlab.org/nlab/show/Philip Argyres">Philip Argyres</a>, <a href="https://ncatlab.org/nlab/show/Chiara Nappi">Chiara Nappi</a>, <em>Spin-1 effective actions from open strings</em>, Nuclear Physics B Volume 330, Issue 1, 22 January 1990, Pages 151-173 Nuclear Physics B (< ahref=”https://doi.org/10.1016/0550-3213(90)90305-W”>doi:10.1016/0550-3213(90)90305-W</a>)</li>
</ul>
<p><a href="https://ncatlab.org/nlab/revision/diff/Dirac-Born-Infeld+action/37">diff</a>, <a href="https://ncatlab.org/nlab/revision/Dirac-Born-Infeld+action/37">v37</a>, <a href="https://ncatlab.org/nlab/show/Dirac-Born-Infeld+action">current</a></p>
```

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have instantiated the string-tension factor (previously suppressed) and made more explicit the cross-link between the DBI critical field strength and the Schwinger limit

]]>added expression for the Born-Infeld Lagrangian on 4d Minkowski spacetime in terms of the electric and magnetic field strenghts:

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 for the DBI-Lagrangian reduces to (Born-Infeld 34, p. 437, review in Nastase 15, 9.4):

$\mathbf{L}_{BI} \;=\; \sqrt{ - det( \eta + F ) } \, dvol_4 \;=\; \sqrt{ 1 - ( \vec E \cdot \vec E - \vec B \cdot \vec B ) - (\vec B \cdot \vec E)^2 } \, dvol_4$Notice that this being well-defined, in that the square root is a real number, hence its argument a non-negative number, means that

$\begin{aligned} & - \mathrm{det} \big( (\eta_{\mu \nu}) + (F_{\mu \nu}) \big) \geq 0 \\ & \Leftrightarrow \; 1 \;-\; (E \cdot E - B \cdot B) \;-\; (B \cdot E)^2 \;\geq\; 0 \\ & \Leftrightarrow \; E^2 - B^2 + E^2 B_{\parallel}^2 \;\leq 1\; \\ & \Leftrightarrow \; E^2 \;\leq\; \frac{ 1 + B^2 }{ 1 + B_{\parallel}^2 } \end{aligned}$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

$E_{crit} \;\coloneqq\; \sqrt{ \frac{ 1 + B^2 }{ 1 + B_{\parallel}^2 } }$turns out to be the Schwinger limit beyond which the electric field would cause deconfining quark-pair creation (Hashimoto-Oka-Sonoda 14b, (2.17)).

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and added pointer to the original:

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

added this sentence to the end of the Idea-section:

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 to reflect, via holographic QCD, the Schwinger limit beyond which the vacuum polarization caused by the electromagnetic field leads to deconfinement of quarks.

And added pointer to relevant references:

Koji Hashimoto, Takashi Oka,

*Vacuum Instability in Electric Fields via AdS/CFT: Euler-Heisenberg Lagrangian and Planckian Thermalization*, JHEP 10 (2013) 116 (arXiv:1307.7423)Koji Hashimoto, Takashi Oka, Akihiko Sonoda,

*Magnetic instability in AdS/CFT : Schwinger effect and Euler-Heisenberg Lagrangian of Supersymmetric QCD*, J. High Energ. Phys. 2014, 85 (2014) (arXiv:1403.6336)Koji Hashimoto, Takashi Oka, Akihiko Sonoda,

*Electromagnetic instability in holographic QCD*, J. High Energ. Phys. 2015, 1 (2015) (arXiv:1412.4254)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)

Thanks, fixed now. I had fixed it in the proof while writing the proof, forgetting to fix also in the statement.

I have now also fixed the factor of $4!$ in front of $F \wedge F$ to $\tfrac{1}{2}$. (This comes from the formula for the Pfaffian, here)

]]>Haven’t any time to look at the moment, but the coefficient of the second term on RHS is 1/6 in Lemma 2.1, but then 1/2 in the proof.

]]>Yes, I think so.

Now to compute the first generalization, to field strengths that are not necessarily constant, but linear functions of the coordinates. This will pick up a “higher derivative correction”. So to check now if that is also as expected (here).

]]>Still true?

]]>I think I have now proof – at least for the special case of constant field strength – that the super-exceptional correction term to the M5-Lagrangian (second but last of the open issues listed at the end of arxiv:1908.00042) is indeed proportional to the first DBI-correction term.

I put the computation in the *Sandbox*. If this is still true tomorrow morning, I’ll polish it up and expand.

Thanks.

And I just fixed a coefficient prefactor.

]]>Put ’t’ in ’electromagnetism’. Just mentioning in case you have copied this formula in a paper.

]]>Have spelled out detailed proof/computation (here) that the determinant in the DBI action comes out as

$det( \eta + F ) \;=\; - 1 + \tfrac{1}{6} \underset{ \mathclap{ {\color{blue}\text{Lagrangian of}} \atop {\color{blue}\text{elecromagnetism}} } }{ \underbrace{ (F \wedge \star F) } } / dvol + \underset{ {\color{blue}\text{correction}} \atop {\color{blue}\text{term}} }{ \underbrace{ \big( 4! (F\wedge F) / \mathrm{dvol} \big)^2 } } \,,$(I have not yet found a single reference that would bother to go through this derivation. If anyone has a pointer to a reference that does, let’s add it.)

]]>I have given the entry more of an actual Idea-section. Now it reads as follows:

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.

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Have added more original references, in particular the very first

- 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)

I have also added pointer to

- Paul Dirac,
*An Extensible Model of the Electron*, Proc. Roy. Soc. A268, (1962) 57-67 (jstor:2414316)

which everyone cites. But looking through this I don’t see anything like the DBI action in there (?)

]]>added pointer to today’s

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

added pointer to

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

added also pointer to

- Eric Bergshoeff, Adel Bilal, M. de Roo and A. Sevrin,
*Supersymmetric non-abelian Born-Infeld revisited*, JHEP 0107, 029 (2001) (arXiv:hep-th/0105274)

for more problems with the non-abelian DBI action

]]>added pointer to

- Akikazu Hashimoto, Washington Taylor,
*Fluctuation Spectra of Tilted and Intersecting D-branes from the Born-Infeld Action*, Nucl. Phys. B503: 193-219, 1997 (arXiv:hep-th/9703217)

on potential issues with the non-abelian DBI action

]]>added a few more references and a tad more text

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