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Super D-branes

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E. Bergshoeff, P.K. Townsend

High Energy Physics - Theory · Submitted 21 Nov 1996 · 21 references

Pith's verdict: The standard reference for the general super D-brane action and its kappa-symmetry; the p ≤ 6 core is solid, but the p ≥ 7 verification is genuinely incomplete and the 'embedding' argument does not cover the new n=4 term.

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arXiv IDhep-th/9611173 (version 3)
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CitationE. Bergshoeff, P.K. Townsend. "Super D-branes." arXiv:hep-th/9611173 (1996). https://pith.science/paper/CPQOOZXQ
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The paper

The paper claims

One worldvolume action is proposed for every type II D-brane, from the point particle to the 9-brane, in an arbitrary supergravity background.

The reading

The paper sets out to write a single worldvolume action for every type II Dirichlet $p$-brane — the objects on which open strings end — that keeps ten-dimensional Lorentz invariance and spacetime supersymmetry manifest and carries the fermionic gauge symmetry called $`$kappa$-symmetry', in a general type II supergravity background rather than only in flat space. Earlier work had handled the D-2-brane and D-3-brane case by case, or treated general $p$ only in a Minkowski background. Here the action is the sum of a Dirac-Born-Infeld determinant term, built from the induced metric and the modified two-form $F = dV - B$, and a Wess-Zumino term built from superspace Ramond-Ramond potentials. The claim that carries weight is that one algebraic structure covers all $p$ up to 9, and that the $p = 9$ case amounts to a supersymmetrization of the ten-dimensional Born-Infeld action, a problem previously solved only in pieces. The reason a reader should care is that $`$kappa$-symmetry' is the working criterion for a brane action to be the right supersymmetric completion of the bosonic one, so a uniform construction is what lets D-branes be used as probes of general supergravity backgrounds, including the massive IIA theory.

What carries the argument

The load-bearing object is the matrix $X^{i}{}_{j} = g^{ik}F_{kj}$ and the resulting $Gamma$ in Eq. (4.21), a sum of totally antisymmetrized gamma matrices contracted with powers of $X$. Two ingredients make the cancellation work. The first is the identity $(1 + Gamma)K^{i}(p) equiv 0$ of Eq. (4.19): once it holds, the $`$kappa$-variation' of the whole action collapses to a term proportional to $(1 - Gamma)$T^{{i}}$(p)$, which vanishes for the projected parameter. The second is that the surviving condition reduces to a polynomial identity in the $(p+1) times (p+1)$ matrix $X$, namely $$A^{{(0)}}$(X) = det(1 + X) mathbb{1}$, and such identities are generated recursively: a $p$-th order identity for $X$ implies a $(p+1)$-th order one, so everything descends from the $p = 1$ relation $$X^{2}$ - frac{1}{2}mathrm{tr},$X^{2}$ equiv 0$. The paper records the expansion of $det(1 + X)$ through tenth order in $X$, which is as far as $p leq 9$ requires. A separate piece of machinery is the eleven-form superspace field strength $$R^{{(11)}}$$, which is what allows the $p = 9$ Wess-Zumino term to be $`$kappa$-invariant' at all.

Extended reading notes

Core claim

The paper proposes the action $I = I_{DBI} + I_{WZ}$, with $I_{DBI} = -int d^{p+1}xi, e^{-varphi} sqrt{-det(g_{ij} + F_{ij})}$ and $I_{WZ} = int_W C e^{F} + m I_{CS}$, where all forms are superspace forms and the top form in the $C$ expansion is a ten-form potential whose eleven-form field strength is nonzero on superspace even though it vanishes on spacetime. The $`$kappa$-variations' are $delta_kappa E = barkappa(1 + Gamma)$, with $Gamma$ given explicitly as a sum over powers of $$X^{{i}}${}_{j} = $g^{{ik}}$F_{kj}$, weighted by $Gamma_{(0)} = frac{1}{(p+1)!sqrt{-det g}} $varepsilon^{{i_1 dots i_{p+1}}$} gamma_{i_1 dots i_{p+1}}$ and by $Gamma_{11}$ or $sigma_3$ according to the type II theory. The authors report that $$Gamma^{2}$ = 1$ and that the vanishing of $(1 + Gamma)$K^{{i}}$(p)$ — the condition that makes the variation of the action cancel — has been checked by explicit gamma-matrix algebra for $p leq 6$ in bosonic backgrounds, and that the $p = 0$ and $p = 2$ cases hold in general backgrounds because they follow by dimensional reduction from the D=11 massless superparticle and supermembrane. For $p = 7, 8, 9$ the check is partial: the authors argue that the algebra for the D-$p$-brane is contained in that for the D-$(p+1)$-brane and state that they have no doubt the symmetry holds there too.

Load-bearing premise

The whole construction rests on the matrix identity $(1 + Gamma)K^{i}(p) equiv 0$, which the authors verify by explicit gamma-matrix algebra only up to $p = 6$ and extend to $p = 7, 8, 9$ by the assertion that the D-$p$-brane algebra is contained in the D-$(p+1)$-brane algebra.

Editorial extensions

If this is right

  • Every type II D-$p$-brane with $p leq 9$ gets a single Lorentz-covariant, spacetime-supersymmetric worldvolume action with an explicit candidate $`$kappa$-symmetry' transformation, rather than a case-by-case construction.
  • The $p = 9$ case supplies a supersymmetrization of the ten-dimensional Born-Infeld action, a problem previously known only in fragments.
  • The construction extends to massive IIA backgrounds: the mass parameter enters through a Chern-Simons term in the Wess-Zumino action and through $m$-dependent modifications of the Ramond-Ramond field strengths.
  • The D-0-brane and D-2-brane actions, including their $`$kappa$-transformations', follow from the D=11 massless superparticle and supermembrane, which upgrades those two cases from bosonic-background checks to general backgrounds.
  • The eleven-form superspace field strength, which is identically zero when restricted to spacetime, is shown to be a useful building block rather than a formal curiosity.

Reading between the lines

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

  • If the embedding argument for $p = 7, 8, 9$ is correct, the verification burden for all D-branes collapses to a finite algebraic computation at low $p$; that would make the whole family checkable by computer algebra rather than by hand.
  • The same $X$-matrix polynomial identities plausibly govern the D-4-brane, so this machinery may be the practical route to the still-missing D=11 super-fivebrane action the authors point to.
  • Because the eleven-form field strength is nonzero only off the spacetime slice, a natural test of the construction is whether truncating the superspace to its bosonic body reproduces the known bosonic Wess-Zumino couplings without leftover terms.
  • The freedom noted in the D-0-brane section — that $`$kappa$-symmetry' fixes the torsion spinor only up to conventional superspace choices — suggests the general-$p$ result may be similarly convention-dependent, which is worth pinning down before treating the action as a definition of the theory.
Original abstract

We present a manifestly Lorentz invariant, spacetime supersymmetric, and `$\kappa$-invariant' worldvolume action for all type II Dirichlet p-branes, $p\le9$, in a general type II supergravity background, including massive backgrounds in the IIA case. The $p=0,2$ cases are rederived from D=11. The $p=9$ case provides a supersymmetrization of the D=10 Born-Infeld action.

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, 2,998 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.14 · compares with prior results 0.57 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We present a manifestly Lorentz invariant, spacetime supersymmetric, and `$\kappa$-invariant' worldvolume action for all type II Dirichlet p-branes, $p\le9$, in a general type II supergravity background, including massive backgrounds in the IIA case. The $p=0,2$ cases are rederived from D=11. The $p=9$ case provides a supersymmetrization of the D=10 Born-Infeld action.

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 present a manifestly Lorentz invariant, spacetime supersymmetric, and `$\kappa$-invariant' worldvolume action for all type II Dirichlet p-branes, $p\le9$, in a general type II supergravity background, including massive backgrounds in the IIA case. contribution 0.99 0.95 0.85 asserted 1.00
s1 The $p=0,2$ cases are rederived from D=11. method 0.48 0.89 0.05 asserted 1.00
s2 The $p=9$ case provides a supersymmetrization of the D=10 Born-Infeld action. result 0.58 0.85 0.14 asserted 1.00

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

What was examined, and what was not

sources read
  • arXiv listing abstract v3
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

74 later papers cite it

Evidence: substantial. 69 of the citing papers share no author with it, and 15 of those carry a confident label from Pith's citation classifier.

Pith read 13 of the 119 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

Uses its method or tool 3 of 14 papers with this label
How the literature uses it

71 arXiv papers cite this in 114 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 · 72 sentences without a confident label

Builds on it 1 paper
Uses its method or tool 14 papers
  • Fortunately, nonabelian D-brane actions [ref] and $κ$-symmetry in curved backgrounds [ref] provide the tools needed to construct such actions directly.

    Nonabelian Probes In Holography · Introduction and Summary of Results · 0.99

  • The kappa symmetry projection [ref] plays a crucial role in our research.

    't Hooft Operators on an Interface and Bubbling D5-Branes · Kappa symmetry projection for D5-branes and D1-branes · 0.99

  • We determine the kinetic and mass-like terms for the fermions using the abelian action of [ref] , and infer the non-abelian gauge and Yukawa couplings via a simple ansatz consistent with gauge invariance and supersymmetry.

    Holography for field theory solitons · Summary of results · 0.94

Cites it as background 23 papers
  • The corresponding action constructed in [ref] has a local fermionic $κ$-symmetry.

    Gaugino Condensation and Geometry of the Perfect Square · Introduction · 0.93

  • We continue, following [ref] , with a review of the $κ$-invariant worldvolume actions in a curved superspace background and the role of DBI and WZ parts of the action: they are separately supersymmetric but cancel each other contributions to local $κ$-symmetry variations.

    $\overline{\rm D3}$ and dS · Introduction · 0.92

  • The relation between the M2-branes compactified on target spaces with isometries and the D2-branes under scalar/vector dualization is well-known [ref] .

    D2-brane with RR and NSNS fluxes and its dual M-theory origin · Introduction · 0.95

Discussion

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