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Dielectric-Branes

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R.C. Myers

High Energy Physics - Theory · Submitted 6 Oct 1999 · 66 references

Pith's verdict: A landmark construction — the T-duality-derived nonabelian D-brane action and the dielectric/fuzzy-sphere effect — with the quantitative fuzzy-sphere numbers explicitly a leading-order toy, as the author says.

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arXiv IDhep-th/9910053 (version 2)
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CitationR.C. Myers. "Dielectric-Branes." arXiv:hep-th/9910053 (1999). https://pith.science/paper/VK4DATDH
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The paper

The paper claims

A T-duality-consistent action for many coincident D-branes predicts that D0-branes placed in a constant four-form field polarize into a fuzzy two-sphere, a spherical D2-D0 bound state.

The reading

The paper asks what the world-volume theory of $N$ coincident D-branes looks like once one insists that it be consistent with T-duality. Starting from the D9-brane, where there are no transverse scalars and hence no commutators to get wrong, the author T-dualizes down to a D$p$-brane and finds that the resulting Born-Infeld and Chern-Simons actions contain commutator terms that a naive 'nonabelianize and take a trace' procedure would miss. The most consequential of these is a new coupling: through interior products with the nonabelian scalars, a D$p$-brane couples to Ramond-Ramond potentials of form degree larger than $p+1$, not just smaller. The payoff is a concrete dynamical prediction: $N$ D0-branes in a constant Ramond-Ramond four-form field have a lower-energy noncommutative ground state than the separated, commuting configuration, and that state is naturally read as a spherical D2-brane carrying $N$ units of flux. If the construction is right, the same commutator terms are what let a collection of branes act as a multipole source for fields to which each individual brane is neutral.

What carries the argument

The mechanism is the interior-product exponential $e^{i2i, i_{2i}i_{2i}}$ sitting inside the Chern-Simons trace, combined with the matrix $Q^{ij}=2i^{ij}+i2i[2i^i,2i^k]E_{kj}$ inside the Born-Infeld determinant. The interior product with a nonabelian scalar is an anticommuting, form-degree-lowering operator, and because the scalars do not commute its square does not vanish; it produces the commutators $[2i^i,2i^j]$ that carry both the new higher-form Ramond-Ramond couplings and the scalar potential whose extremum is the fuzzy sphere. The symmetrized trace prescription, adopted from Tseytlin's proposal and from the matrix-theory comparison, is what makes the ordering of these noncommuting objects well defined.

Extended reading notes

Core claim

The central claim is that the nonabelian world-volume action for $N$ coincident D-branes is fixed by T-duality, and that the fixed action contains interactions invisible to the naive extension of the abelian Born-Infeld and Chern-Simons actions. Concretely, the Born-Infeld part becomes $-T_p$ times a symmetrized trace of $e^{-2i}$ times the square root of $e^{-2i}$ times the determinant of the pulled-back combination of $E_{ab}=G_{ab}+B_{ab}$ with $E_{ai}(Q^{-1}-2i)^{ij}E_{jb}$ plus $2i F_{ab}$, multiplied by $e^{-2i}$ times the square root of $e^{-2i}$ times the determinant of $Q^{ij}=2i^{ij}+i2i[2i^i,2i^k]E_{kj}$; the Chern-Simons part becomes $2i_p$ times a symmetrized trace of the pull-back of $e^{i2i, i_{2i}i_{2i}}$ acting on $e^{-2i}$ times the sum of the Ramond-Ramond potentials times $e^{B}$, all times $e^{2i F}$. The exponential of interior products is the load-bearing new ingredient: because the displacement vectors are noncommuting matrices, $i_{2i}i_{2i}C^{(2)}$ produces $2i C^{(2)}_{ij}[2i^j,2i^i]$, which is exactly the commutator structure T-duality demands and which gives D$p$-branes their couplings to higher-form Ramond-Ramond potentials. Applied to $N$ D0-branes in a constant $F^{(4)}_{tijk}=-2f2i_{ijk}$, the leading potential is $-2i^2T_0/4$ times the trace of $[2i^i,2i^j]^2$ minus $i2i^2 2i_0/3$ times the trace of $2i^i2i^j2i^k F^{(4)}_{tijk}$, whose extremum condition is solved by $[2i^i,2i^j]=2i f 2i_{ijk}2i^k$, i.e. $2i^i = (f/2)2i^i$ with $2i^i$ an $N$-dimensional representation of $SU(2)$. The irreducible representation has energy lower than the commuting solution by an amount proportional to $N(N^2-1)$, so the separated D0-branes are unstable toward the noncommutative configuration, which has radius $R=2i 2i_s^2 f N$ times the square root of $1-1/N^2$ and carries a dipole coupling to $F^{(4)}$. A dual abelian D2-brane calculation with $N$ units of world-volume flux reproduces both the radius and the energy shift up to corrections of order $1/N^2$.

Load-bearing premise

The noncommutative ground state and its energy are computed from only the leading commutator terms of the symmetrized-trace action, in a constant four-form background that the paper itself says is not a solution of the type IIa equations of motion.

Editorial extensions

If this is right

  • D$p$-branes acquire couplings to Ramond-Ramond potentials of form degree $p+3, p+5, \ldots$, which vanish for a single brane and appear only through commutators of the nonabelian scalars.
  • The scalar potential of the nonabelian Born-Infeld action contains the standard super-Yang-Mills commutator potential as its leading term, plus an infinite series of higher commutator terms fixed by the determinant of $Q$.
  • A system of $N$ D0-branes in a constant $F^{(4)}$ background has a noncommutative ground state of lower energy than separated D0-branes, with radius $\pi\ell_s^2 f N\sqrt{1-1/N^2}$ and a dipole coupling to the four-form field.
  • The same mechanism gives the noncommutative ground state a source coupling to the Neveu-Schwarz three-form field strength $H$, so an analogous polarization effect should occur for D0-branes in a constant $H$ background.
  • Comparing the linear closed-string couplings of the D0-brane action with those derived in matrix theory requires the transverse five-brane charge operators to vanish, at least perturbatively.

Reading between the lines

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

  • The fuzzy-sphere ground state is a bound state whose size grows linearly with $N$ and with the applied field, so sufficiently large $N$ or $f$ pushes the configuration to radii where the truncated action and the fixed-background approximation are both dubious; the paper's own convergence condition $R \ll \sqrt{N}\ell_s$ is the boundary of the trustworthy regime.
  • Because the polarization radius scales as $f N$, the effect is a collective one: a single D0-brane cannot polarize, and the classical intuition that a neutral object stays put in a field it does not couple to fails only through nonabelian commutators.
  • The agreement between the noncommutative D0-brane and abelian D2-brane descriptions at large $N$ suggests that the residual $1/N^2$ discrepancies are a bookkeeping of higher-derivative versus higher-commutator corrections, which makes this system a useful testing ground for the Seiberg-Witten map between ordinary and noncommutative descriptions.
  • If the higher-form couplings survive the sixth-order commutator corrections the paper expects, they should be visible as multipole moments in the long-range fields sourced by a polarized brane cluster, which is in principle a sharper signature than the energy shift.
Original abstract

We extend the usual world-volume action for a Dp-brane to the case of N coincident Dp-branes where the world-volume theory involves a U(N) gauge theory. The guiding principle in our construction is that the action should be consistent with the familiar rules of T-duality. The resulting action involves a variety of potential terms, i.e., nonderivative interactions, for the nonabelian scalar fields. This action also shows that Dp-branes naturally couple to RR potentials of all form degrees, including both larger and smaller than p+1. We consider the dynamics resulting from this action for Dp-branes moving in nontrivial background fields, and illustrate how the Dp-branes are ``polarized'' by external fields. In a simple example, we show that a system of D0-branes in an external RR four-form field expands into a noncommutative two-sphere, which is interpreted as the formation of a spherical D2-D0 bound state.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,979 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.18 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We extend the usual world-volume action for a Dp-brane to the case of N coincident Dp-branes where the world-volume theory involves a U(N) gauge theory. The guiding principle in our construction is that the action should be consistent with the familiar rules of T-duality. The resulting action involves a variety of potential terms, i.e., nonderivative interactions, for the nonabelian scalar fields. This action also shows that Dp-branes naturally couple to RR potentials of all form degrees, including both larger and smaller than p+1. We consider the dynamics resulting from this action for Dp-branes moving in nontrivial background fields, and illustrate how the Dp-branes are ``polarized'' by external fields. In a simple example, we show that a system of D0-branes in an external RR four-form field expands into a noncommutative two-sphere, which is interpreted as the formation of a spherical D2-D0 bound state.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We extend the usual world-volume action for a Dp-brane to the case of N coincident Dp-branes where the world-volume theory involves a U(N) gauge theory. contribution 1.00 0.93 0.08 asserted 0.99
s1 The guiding principle in our construction is that the action should be consistent with the familiar rules of T-duality. method 0.98 0.44 0.04 asserted 0.96
s2 The resulting action involves a variety of potential terms, i.e., nonderivative interactions, for the nonabelian scalar fields. result 0.88 0.91 0.12 asserted 1.00
s3 This action also shows that Dp-branes naturally couple to RR potentials of all form degrees, including both larger and smaller than p+1. result 0.96 0.89 0.22 asserted 1.00
s4 We consider the dynamics resulting from this action for Dp-branes moving in nontrivial background fields, and illustrate how the Dp-branes are ``polarized'' by external fields. contribution 0.96 0.89 0.13 asserted 0.99
s5 In a simple example, we show that a system of D0-branes in an external RR four-form field expands into a noncommutative two-sphere, which is interpreted as the formation of a spherical D2-D0 bound state. result 0.89 0.94 0.29 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References
How later papers used it

357 later papers cite it

Evidence: substantial. 356 of the citing papers share no author with it, and 51 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 629 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Builds on it 1 of 4 papers with this label

  • Generalized the Chern-Simons action for multiple Dp-branes in this paper to include higher curvature terms.

    ABJM Baryon Stability and Myers effect (2011) · 0.99 · paraphrased: the licence does not allow quoting

Uses its method or tool 9 of 33 papers with this label
Compares with it 2 of 15 papers with this label
Cites it as support 1 of 5 papers with this label
How the literature uses it

340 arXiv papers cite this in 593 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 · 425 sentences without a confident label

Builds on it 3 papers
  • In this section we show that generalizing the microscopical Chern-Simons action in [ref] to include higher curvature terms [ref] we can predict the existence of a dielectric higher curvature coupling in the action for multiple D0-branes that exactly cancels the $k/8$ contribution to the D6-brane tadpole that we obtained above.

    ABJM Baryon Stability and Myers effect · Dielectric higher-curvature terms · 0.99

  • This non-linear Lagrangian is an extension of the non-Abelian DBI Lagrangian [ref] of $N$ coincident D2-branes and includes only terms with even number of the totally antisymmetric tensor $M^IJK$.

    Janus Field Theories from Non-Linear BF Theories for Multiple M2-Branes · Introduction · 0.81

  • Inspired by the non-abelian DBI action [ref] in which there are different transverse scalar fields, one may modify the usual tachyon DBI action to the following form: S_DBI = -T_p/2∫d^p+1σ(/V( T^iT^i)√(1+1/2[T^i,T^j][T^j,T^i])).nonab . ×√(-(η_ab +2πα'F_ab+2πα'D_a T^i(Q^-1)^ijD_b T^j)) ) ,where Q^ij = Iδ^ij-i[T^i,T^j]The superscripts $i,j=1,2$ and there is no sum over $i,j$.

    On effective actions of non-BPS branes and their higher derivative corrections · Discussion · 0.81

Uses its method or tool 29 papers
  • Here we will use the non-abelian D brane action derived in [ref] .

    New non-abelian effects on D branes · Introduction · 0.99

  • Using the standard trick for recombining the determinant [ref] D̃=( [ A B; C D ] ) =(A-BD^-1C)(D), one can rearrange the action in a compact form similar to [ref].

    Multiple D$_p$-branes as a D$_{p+2}$-brane · DBI theory · 0.91

  • Using the same steps as those in [ref] for finding the T-dual action nonab, one finds S_1 = -∫d^p+1σV(|τ|)√((Q))e^-Φ(√(-A^(1))+√(-A^(2))),action1where A^(n)_ab = P^(n)[E_ab-|τ|^2/2πα'(Q)E_aiℓ^iℓ^jE_jb]+ 2πα'F_ab^(n) +1/(Q)(/πα' [D_aτ(D_bτ)^*+D_bτ(D_aτ)^*]. +ℓ·ℓ/4[τ(D_aτ)^*+τ^*D_aτ][τ(D_bτ)^*+τ^*D_bτ] .+i/2[E_ai+_a X^(n)jE_ji]ℓ^i[τ(D_bτ)^*-τ^*D_bτ].A1 .-i/2[τ(D_aτ)^*-τ^* D_aτ] ℓ^i[E_ib+E_ij_bX^(n)j]) ,where $(Q)=1+|τ|^2ℓ·ℓ/2πα'$.

    On the effective action of D-brane-anti-D-brane system · $D_pD̅_p$ effective action · 0.80

Compares with it 14 papers
  • By comparing with the non-abelian D-brane action in arbitrary backgrounds [ref] , one can then derive the supergravity solution.

    Emergent D4-Brane Background from D-Particles · Introduction · 0.81

  • The first case with energy $E_(i)$ is the same as the result of [ref] .

    New non-abelian effects on D branes · Adding a RR magnetic field · 0.98

  • This is very different from the rich collection of supersymmetric vacua of the Lorentzian BMN model [ref] which can be interpreted in terms of polarised D-branes [ref] and lead to an equally rich set of supergravity solutions as explained in [ref] .

    Spherical Branes and the BMN Matrix Quantum Mechanics · Introduction · 0.95

Cites it as support 3 papers
Cites it as background 95 papers
  • On the other hand, the Chern-Simons action is given by [ref] S_CS=μ_p∫STr(P[e^iλi_Xi_X (ΣC^(n)e^B)]e^λF), where $C^(n)$ denotes $(p+1)$-forms and $μ_p$ is the RR charge of the brane and $i_X$ is the exterior derivative in $X$ direction which acts on RR form fields.

    Multiple D$_p$-branes as a D$_{p+2}$-brane · Chern-Simons term · 0.84

  • Whenever one considers $p$-branes in background with $F_6-p$-flux there is the possibility that the $p$-branes polarise into a spherical $(p+2)$-brane carrying $(p+2)$-brane dipole charge and the original $p$-brane monopole charges induced by gauge fluxes on the worldvolume [ref] .

    Smeared antibranes polarise in AdS · Spherical 5-branes · 0.97

  • Non-Abelian fields on D$p$-branes normally give rise to dielectric D$(p+2)$-brane charges [ref] , and this happens when three of their real worldvolume scalars have non-commuting expectation values.

    There and back again: A T-brane's tale · Introduction · 0.86

Discussion

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