algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
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.
In the simplest situation of flat 4-dimensional Minkowski spacetime and no other fields besides that of electromagnetism, encoded in a Faraday tensor differential 2-form
the Lagrangian density of the Born-Infeld action functional is
Here
denotes the metric tensor of Minkowski spacetime, regarded as a matrix
is the Faraday tensor, regarded as a matrix
is the string tension,
denotes the determinant of the sum of these matrices,
denotes the positive square root,
denotes a volume form on Minkowski spacetime, which after a choice of global coordinates may be taken to be
In the following, for any differential 4-form on we write for the unique smooth function such that
The determinant in (1) evaluates to
where
is the 4-form wedge product of with its Hodge dual, hence is the Lagrangian density of ordinary electromagnetism,
is the wedge product of with itself.
We compute as follows:
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 (of ), 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 and of , respectively (again by that formula).
We discuss the identifications of the resulting four summands shown under the braces:
(first summand) The determinant of equals -1 by definition.
(second summand) If we exchange indices the form of this summand remains unchanged, also the factors do not change, since is a symmetric matrix, by definition. But the single factor of 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 relaced by the identity matrix (to be indicated by a Kronecker delta ). Observe then that the contraction not involving any factor of yields
where the symbol on the right is defined to be
Hence the full expression (with still replaced by ) is
where we used that due to the skew-symmetry of 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 instead of : 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 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 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 , which is (by this formula)
This is proportial to the coefficient of the wedge product of with itself, relative to the volume form:
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 . 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 expressed in terms of the electric field and magnetic field as
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):
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
where
is the component of the magnetic field which is parallel to the electric field.
The resulting maximal electric field strength
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)).
As a proposal for a modification of electromagnetism in spacetime, the (Dirac-)Born-Infeld (DBI) action originates in
The article by Dirac which came to be commonly cited in this context is
Paul Dirac, An Extensible Model of the Electron, Proc. Roy. Soc. A268, (1962) 57-67 (jstor:2414316)
(which proposes a membrane-model as a unification of the electron and the muon)
Broad review and further developments:
As the low energy action functional for single D-branes the DBI action is due to
Efim Fradkin, Arkady Tseytlin, Non-linear electrodynamics from quantized strings, Physics Letters B Volume 163, Issues 1–4, 21 November 1985 (doi:10.1016/0370-2693(85)90205-9)
A. Abouelsaood, Curtis Callan, Chiara Nappi, S. A. Yost, Open strings in background gauge fields, Nuclear Physics B Volume 280, 1987, Pages 599-624 (doi:10.1016/0550-3213(87)90164-7)
Robert Leigh, Dirac-Born-Infeld Action from Dirichlet Sigma Model, Mod. Phys. Lett. A4 (1989) 2767 (spire:26399, doi:10.1142/S0217732389003099)
and a full -symmetric Green-Schwarz sigma-model for D-branes:
Martin Cederwall, Alexander von Gussich, Bengt Nilsson, Per Sundell, Anders Westerberg, The Dirichlet Super-p-Branes in Ten-Dimensional Type IIA and IIB Supergravity, Nucl.Phys. B490 (1997) 179-201 (arXiv:hep-th/9611159)
Mina Aganagic, Jaemo Park, Costin Popescu, John Schwarz, Dual D-Brane Actions, Nucl. Phys. B496 (1997) 215-230 (arXiv:hep-th/9702133)
Review:
Gary Gibbons, Born-Infeld particles and Dirichlet p-branes, Nucl. Phys. B514:603-639, 1998 (arXiv:hep-th/9709027)
Konstantin G. Savvidy, Born-Infeld Action in String Theory, 1999 (arXiv:hep-th/9906075, spire:501510)
Arkady Tseytlin, Born-Infeld action, supersymmetry and string theory, in: Mikhail Shifman (ed.) The many faces of the superworld, pp. 417-452, World Scientific (2000) (arXiv:hep-th/9908105, doi:10.1142/9789812793850_0025)
John Schwarz, Comments on Born-Infeld Theory, in: Atish Dabholkar, Sunil Mukhi, Spenta R. Wadia (eds.) Strings 2001: Proceedings, Strings 2001 Conference, Tata Institute of Fundamental Research, Mumbai, India, January 5-10, 2001 (arXiv:hep-th/0103165, spire:554347)
Paul Koerber, Abelian and Non-abelian D-brane Effective Actions, Fortsch. Phys. 52 (2004) 871-960 (arXiv:hep-th/0405227)
Horatiu Nastase, Section 9.2 and 9.4 of: Introduction to AdS/CFT correspondence, Cambridge University Press, 2015 (cds:1984145, doi:10.1017/CBO9781316090954)
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:
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)
Discussion of one-loop corrections:
Derivation of the first DBI-correction from an M5-brane model via super-exceptional geometry:
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:
The now widely accepted proposal via a symmetrized trace is due to
followed by
Robert Myers, Dielectric-Branes, JHEP 9912 (1999) 022 (arXiv:hep-th/9910053)
(introducing the Myers effect)
The symmetrized trace proposal has become widely accepted.
Review includes:
Issues with this proposal at higher order have been found in
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)
P. Bain, On the non-abelian Born-Infeld action, In: L. Baulieu , Michael Green, Picco M., Windey P. (eds.) Progress in String Theory and M-Theory. NATO Science Series (Series C: Mathematical and Physical Sciences), vol 564. Springer, Dordrecht (arXiv:hep-th/9909154, doi:10.1007/978-94-010-0852-5_12)
Eric Bergshoeff, Adel Bilal, Mees de Roo, A. Sevrin, Supersymmetric non-abelian Born-Infeld revisited, JHEP 0107, 029 (2001) (arXiv:hep-th/0105274)
Correction terms have been proposed in
A completely different approach, which defines a theory that is analogous to non-abelian BI via deformation of Yang-Mills theory, is proposed in
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:
Proposals for non-abelian and supersymmetric DBI-actions
for D0-branes:
Sudhakar Panda, Dmitri Sorokin, Supersymmetric and Kappa-invariant Coincident D0-Branes, JHEP 0302 (2003) 055 [arXivLhep-th/0301065, doi:10.1088/1126-6708/2003/02/055]
Igor Bandos, Unai D. M. Sarraga, Complete nonlinear action for supersymmetric multiple D0-brane system, Phys. Rev. D 106 (2022) 066004 [doi:10.1103/PhysRevD.106.066004, arXiv:2204.05973]
Igor Bandos, Unai D. M. Sarraga, Properties of multiple D0-brane system: 11D origin, equations of motion and their solutions [arXiv:2212.14829]
using the pure spinor superstring:
Discussion of the DBI-action for flavor branes in holographic QCD:
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:
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, Shunichiro Kinoshita, Keiju Murata, Takashi Oka, Electric Field Quench in AdS/CFT, J. High Energ. Phys. 2014 (arXiv:1407.0798)
Koji Hashimoto, Takashi Oka, Akihiko Sonoda, Electromagnetic instability in holographic QCD, J. High Energ. Phys. 2015, 1 (2015) (arXiv:1412.4254)
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)
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 , Phys. Lett. B701:465-470, 2011 (arXiv:1103.3121)
Ali Jalali, Mohammad Garousi, On D-brane action at order , Phys. Rev. D 92, 106004 (2015) (arXiv:1506.02130)
Mohammad Garousi, An off-shell D-brane action at order in flat spacetime, Phys. Rev. D 93, 066014 (2016) (arXiv:1511.01676)
Komeil Babaei Velni, Ali Jalali, Higher derivative corrections to DBI action at order, Phys. Rev. D 95, 086010 (2017) (arXiv:1612.05898)
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:
Sanyaje Ramgoolam, Bill Spence, S. Thomas, Section 3.2 of: Resolving brane collapse with corrections in non-Abelian DBI, Nucl. Phys. B703 (2004) 236-276 (arxiv:hep-th/0405256)
Simon McNamara, Constantinos Papageorgakis, Sanyaje Ramgoolam, Bill Spence, Appendix A of: Finite effects on the collapse of fuzzy spheres, JHEP 0605:060, 2006 (arxiv:hep-th/0512145)
Simon McNamara, Section 4 of: Twistor Inspired Methods in Perturbative FieldTheory and Fuzzy Funnels, 2006 (spire:1351861, pdf, pdf)
Constantinos Papageorgakis, p. 161-162 of: On matrix D-brane dynamics and fuzzy spheres, 2006 (pdf)
On D1-D3 brane intersections as spikes/BIons in the D3-brane DBI-theory:
Curtis Callan, Juan Maldacena, Brane Dynamics From the Born-Infeld Action, Nucl. Phys. B513 (1998) 198-212 (arXiv:hep-th/9708147)
Paul Howe, Neil Lambert, Peter West, The Self-Dual String Soliton, Nucl. Phys. B515 (1998) 203-216 (arXiv:hep-th/9709014)
Gary Gibbons, Born-Infeld particles and Dirichlet p-branes, Nucl. Phys. B514: 603-639, 1998 (arXiv:hep-th/9709027)
Neil Constable, Robert Myers, Oyvind Tafjord, The Noncommutative Bion Core, Phys. Rev. D61 (2000) 106009 (arXiv:hep-th/9911136)
From the M5-brane
Last revised on July 3, 2023 at 17:40:47. See the history of this page for a list of all contributions to it.