T-duality and Actions for Non-BPS D-branes

E.A. Bergshoeff; E. Eyras; M. de Roo; S. Panda; T.C. de Wit

REVIEW

T-duality and Actions for Non-BPS D-branes

E.A. Bergshoeff, M. de Roo, T.C. de Wit, E. Eyras, S. Panda

High Energy Physics - Theory · Submitted 24 Mar 2000 · 31 references

Pith's verdict: A short, honest structural paper: the tachyon-extended Born-Infeld form is Garousi's, but the T-duality argument and the Wess-Zumino coefficient fixing are theirs and they hold up.

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arXiv IDhep-th/0003221 (version 2)

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CitationE.A. Bergshoeff, E. Eyras, M. de Roo, S. Panda, T.C. de Wit. "T-duality and Actions for Non-BPS D-branes." arXiv:hep-th/0003221 (2000). https://pith.science/paper/KUUWNO7K

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Editor's note

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Short version: this is a modest, candid structural paper, and the modesty is the point. The Born-Infeld form in Eq. (1.3) — tachyon derivative sitting inside the determinant — is not theirs; Garousi proposed it independently in ref. [14], and they say so plainly in Section 4. What they add is the argument: reduce a IIA Dp and a IIB D(p+1) to nine dimensions, show the tachyon-extended determinants agree (2.4)–(2.10), then repeat for the Wess-Zumino term, where T-duality pins the coefficients to (3.6). That last piece is the cleanest result here — a two-sided comparison that lands on exactly the coefficients ordinary D-branes require. The exclusion of the coupling (2.12) is a nice concrete negative: it exists in IIA, has no IIB counterpart, so it fails T-duality and should not appear in the action. That is a useful thing to have on record, since people had been writing it down [21]. They are also unusually straight about scope: the conjecture is not proven, the relative coefficients of tachyon couplings are not fixed by these two symmetries, and g(T) is untouched. Soft spots, in proportion. The T-duality check assumes T is constant along the duality circle and inert under supersymmetry — both stated explicitly in Section 2, so this is a declared subsector check, not a concealed gap. It does mean the abstract's 'satisfies T-duality' is narrower than it reads. The one genuinely unshown claim is in Section 4: the assertion that L0 + Lm breaks Lorentz invariance is reported as 'we have checked' with no details. That is a discussion remark, not load-bearing, but a referee should ask them to write it out or drop it. Citations look right; [14] is credited, [11] and [12,13] are the correct anchors. Who should read it: anyone working on non-BPS effective actions or tachyon condensation who wants the symmetry constraints written down cleanly. It will not change your picture of the subject, but it saves you rederiving the Wess-Zumino coefficients. Recommendation: send it out. Short, checkable, a real if narrow result, and the report should be quick.

Referee report

Significance

If the result holds, it gives a compact packaging of the tachyon-derivative dependence of the non-BPS Born-Infeld action that is stable under T-duality at all orders in ∂T, rather than order by order, and it identifies which tachyon-fermion couplings are permitted. The negative result in Section 2, that a coupling of the form ∂_i θ̄ ∂_j θ G^{ij} times a function of T has no IIB counterpart and should therefore be absent, is a useful constraint on a proposal that has appeared in the literature. The Wess-Zumino coefficient matching in Eq. (3.6) is a clean two-sided comparison and is the most concrete check in the paper. The paper is also candid about its scope: it presents Eq. (1.3) as a conjecture supported by symmetry criteria, not as a derivation, and it does not overclaim for the unrestricted potential g(T). The work is a short letter with a modest but genuine technical content, and its value lies in the symmetry constraints it isolates rather than in a first-principles derivation.

Major objections (2)

[Section 2, Eqs. (2.9)-(2.10) and the accompanying text; Abstract, sentence s1] The T-duality check is carried out only in the sector where the tachyon is constant along the compact direction used for the duality, and the supersymmetry statement rests on the separate assumption that T is inert. Both assumptions are stated explicitly in Section 2, which is to the authors' credit, but the abstract's unqualified statement that 'the criteria of T-duality and supersymmetry are satisfied' does not carry them, and a reader coming to the abstract alone would take the result to be general. The abstract and the closing paragraph of Section 2 should state the restriction. It would also help to say whether the restriction is technical or structural: specifically, whether a tachyon configuration with nonzero ∂_9 T (a winding or momentum mode on the duality circle) is expected to preserve the determinant structure of Eq. (1.3) at all, or whether the rank-one form of the tachyon insertion is tied to the zero-mode sector. A sentence on this point would let the reader judge how much of the conjectured action the check actually covers.
[Section 4, Eqs. (4.1)-(4.2)] The conclusion of Section 4, that a local Higgs mechanism for a single self-dual tensor is not possible, rests on the statement that 'we have checked that, although the equations of motion corresponding to L0 and Lm are Lorentz covariant, the ones corresponding to the combination L = L0 + Lm violate Lorentz symmetry.' No equations or intermediate steps are given, so the reader cannot verify the check or see which terms break the symmetry. Since this is the sole support for the section's conclusion, the offending terms in the equations of motion should be exhibited, or the claim should be softened to a reported observation. This concern is confined to Section 4 and is separable from the Born-Infeld and Wess-Zumino results, which do not depend on it.

Minor objections (8)

[Section 2, Eq. (2.11)] The displayed expansion is truncated with an ellipsis, and the text asserts that the terms shown 'satisfy the requirement of T-duality'. It would be useful to state explicitly which orders in ∂T and in the fermions are retained in Eq. (2.11), so that the reader knows what the ellipsis covers and what is being claimed for the omitted terms.
[Section 2, Eq. (2.12) and the following paragraph] The exclusion of the coupling ∂_i θ̄ ∂_j θ G^{ij} times a function of T is argued by the absence of a IIB counterpart. As written this is an assertion; giving the T-dual image of the IIA expression, or showing that the corresponding IIB expression vanishes identically by the chirality projection, would make the argument checkable.
[Section 2, paragraph after Eq. (2.11)] The statement that the (∂T)^2 terms couple only to the symmetric part of G^{-1} follows from the rank-one determinant lemma, det(A + vv^T) = det A (1 + v^T A^{-1} v). A one-line derivation would make the subsequent remark about the mixing with a nonvanishing NS-NS background easier to follow.
[Section 3, Eq. (3.6) and the following paragraph] The overall normalization is said to be fixed by requiring that the Dp-brane action arising from the kink solution be κ-symmetric, but this step is not carried out. A short indication of how the kink produces the δ-function from the dT insertion, as anticipated in Section 1, would close the argument.
[Section 1, first paragraph] The phrase 'played a crucial rule' should read 'played a crucial role', and 'enables us to identity the structure' should read 'identify'.
[References] Several reference identifiers appear to be mistyped and should be checked against the published versions: Ref. [9] gives hep-th/0810188, Ref. [11] gives hep-th/99090622, Ref. [15] gives hep-th/904119, and Ref. [17] gives hep-th/96100148. The author name in Refs. [10] and [21] should be Hořava and Klusoň respectively.
[Section 4, final paragraph] The sentence 'the source for the three transverse scalars that are needed to describe the M2-brane remain unclear' has a subject-verb disagreement and should read 'remains unclear'.
[Section 2, Eqs. (2.1)-(2.7)] The superscript convention distinguishing the ten-dimensional quantity G^{(p)10} from its nine-dimensional image G^{(p)9} is used before it is defined. A single sentence at the start of Section 2 fixing the notation would remove the ambiguity.

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The paper

The paper claims

The tachyon can be folded into the D-brane Born-Infeld determinant without breaking T-duality or supersymmetry, provided it is constant along the duality circle.

The reading

Non-BPS D-branes are unstable objects whose effective action must contain a tachyon, and the tachyon's couplings are tightly constrained: the action has to be T-dual to the corresponding action one dimension up or down, and it has to be supersymmetric once the tachyon condenses into an ordinary D-brane. This paper asks how much of the tachyon's dependence can be pinned down by those two requirements alone. The authors propose that the Born-Infeld part takes the form $-∫d^{p+1}σ√{|det(G_{ij}+∂_iT∂_jT)|}g(T)$, i.e. the tachyon gradient enters the determinant exactly like one more symmetric tensor alongside the metric and the Born-Infeld field strength. They then check that this ansatz survives the standard T-duality map between a type IIA D$p$-brane and a type IIB D$(p+1)$-brane reduced to nine dimensions, and that the Wess-Zumino term $∫C∧dT∧e^{F}$ has its coefficients fixed by the same requirement to the values in their Eq. (3.6). If correct, the derivative of the tachyon is not a special ingredient at all but simply another rank-two tensor in the determinant, which sharply narrows which tachyon couplings are allowed and rules out at least one coupling previously proposed in the literature.

What carries the argument

The load-bearing device is the T-duality map between D-branes of adjacent dimensionality, applied to a Lagrangian in which the tachyon gradient sits inside the Born-Infeld determinant: $L=-√{|det(G^{(p)}_{10,ij}+∂_iT∂_jT)|}g(T)$. T-duality reduces a IIA D$p$-brane over a transverse direction to a nine-dimensional D$p$-brane, and a IIB D$(p+1)$-brane over a worldvolume direction to the same nine-dimensional theory; the two reductions must produce identical Lagrangians. Because the tachyon is assumed not to depend on the compact direction, the whole reduction goes through with the single replacement $G^{(p)}_{9,ij}→G^{(p)}_{9,ij}+∂_iT∂_jT$, so the determinant identity that proved T-duality for ordinary D-branes continues to hold. The second device is the Wess-Zumino term $∫d^{p+1}σ C∧dT∧e^{F}$, whose expansion in powers of $F$ produces one coefficient per Ramond-Ramond rank; matching the IIA and IIB reductions term by term is what pins those coefficients down.

Extended reading notes

Core claim

The central claim is that a single, compact ansatz for the non-BPS D-brane Born-Infeld action — the ordinary D-brane determinant with $G_{ij}=g_{ij}+F_{ij}$ replaced by $G_{ij}+∂_iT∂_jT$, multiplied by an undetermined potential $g(T)$ — passes both tests the authors impose on it. The T-duality test is explicit: reduce a IIA D$p$-brane on a transverse direction and a IIB D$(p+1)$-brane on a worldvolume direction to nine dimensions, and check that the two resulting determinants agree. The authors show the equality holds once the same substitution is made on both sides, using the block-determinant identity $det A = det(A-BD^{-1}C)det D$. The supersymmetry test is that the tachyon is taken to be inert under the worldvolume supersymmetry, so the supersymmetric combinations already known for D-branes carry over unchanged. The same T-duality comparison is then applied to the Wess-Zumino term $∫C∧dT∧e^{F}$, and it fixes the relative coefficients $a_{p,k}$ and $b_{p+1,k}$ of the individual Ramond-Ramond couplings up to an overall normalisation, which is in turn fixed by requiring that the D-brane produced by tachyon condensation be κ-symmetric. The authors are explicit that this is evidence for a conjecture, not a proof.

Load-bearing premise

The T-duality check only covers tachyons that do not vary along the compact direction used for the duality and that are taken to be inert under supersymmetry, so the argument establishes the ansatz for a restricted class of tachyon configurations rather than in general.

Editorial extensions

If this is right

Any candidate tachyon coupling in the non-BPS D-brane action can be screened by asking whether it has a T-dual partner of the opposite chirality structure; a coupling such as $∂_i\barθ∂_jθ G^{ij}$ times a function of $T$ has no IIB counterpart and is therefore excluded.
The expansion of the determinant generates a definite infinite series of couplings between tachyons and worldvolume fields with fixed relative coefficients, so the allowed interaction structure is determined even if the overall normalisation is not.
The Wess-Zumino coefficients required for non-BPS branes coincide with those already required for ordinary D-branes, so the Ramond-Ramond couplings of the unstable brane are not a new ingredient.
The potential $g(T)$ is left completely unconstrained by these arguments, so T-duality and supersymmetry alone cannot determine the tachyon condensation point or the endpoint energy.
The determinant form suggests the non-BPS brane has a higher-dimensional origin, which motivates looking for its analogue among eleven-dimensional M-branes.

Reading between the lines

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

The determinant ansatz makes a sharp, checkable prediction beyond the paper's scope: the coefficient of the $(∂T)^4$ term in the four-tachyon amplitude is fixed relative to the $(∂T)^2$ term, so a string-amplitude computation of that coefficient would either confirm the ansatz or measure its first failure.
The assumption that the tachyon is constant along the duality circle is doing real work; a tachyon that winds or varies along that direction would mix with the worldvolume scalar $S$ coming from the reduced gauge field, and the clean determinant substitution would presumably need to be replaced by a more entangled structure.
Because only the symmetric part of the inverse Born-Infeld tensor couples to $(∂T)^2$, a non-symmetric background such as a nonzero Neveu-Schwarz field should produce a distinctive, computable pattern of tachyon-fermion mixings that a more general ansatz would have to reproduce.
The paper's closing observation — that a local mass term added to a chiral two-form action appears to break Lorentz invariance even though the equations of motion separately do not — is a structural obstruction that any membrane-based account of the same condensation has to circumvent, and it is the kind of statement that could be settled by an explicit covariant formulation.

Original abstract

We employ T-duality to restrict the tachyon dependence of effective actions for non-BPS D-branes. For the Born-Infeld part the criteria of T-duality and supersymmetry are satisfied by a simple extension of the D-brane Born-Infeld action.

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We employ T-duality to restrict the tachyon dependence of effective actions for non-BPS D-branes. For the Born-Infeld part the criteria of T-duality and supersymmetry are satisfied by a simple extension of the D-brane Born-Infeld action.

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s0	We employ T-duality to restrict the tachyon dependence of effective actions for non-BPS D-branes.	method 0.74	0.92	0.29	asserted 1.00
s1	For the Born-Infeld part the criteria of T-duality and supersymmetry are satisfied by a simple extension of the D-brane Born-Infeld action.	result 0.96	0.75	0.31	asserted 0.99

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References

Works cited (31)

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A. Sen, Stable Non-BPS Bound States of BPS D-branes , JHEP 9808 (1998) 010, hep-th/9805019

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Forward citations

Cited by 9 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Dirac-Born-Infeld Action on the Tachyon Kink and Vortex
hep-th 2003-03 conditional novelty 7.0 of 10

The kink and vortex solitons of the tachyon effective field theory reproduce the exact Dirac-Born-Infeld world-volume action of BPS D-branes, with no higher-derivative corrections.
Deforming tachyon kinks and tachyon potentials
hep-th 2006-03 conditional novelty 6.0 of 10

A gauge-field-dependent deformation of tachyon DBI actions generates new tachyon potentials and resolves singular tachyon kinks into smooth kinks that localize electromagnetic fields.
Strings from Tachyons
hep-th 2003-04 conditional novelty 6.0 of 10

The c=1 matrix model is reinterpreted as the worldline quantum mechanics of N unstable D-particles, with a quantitative match between tachyon radiation and eigenvalue radiation, plus a proposed IIB embedding.
On Wess-Zumino terms of Brane-Antibrane systems
hep-th 2007-10 conditional novelty 5.0 of 10

The four-point amplitude of one RR field, two tachyons, and one gauge field is computed and shown to match the symmetric trace tachyon DBI and Wess-Zumino actions at leading order, fixing a normalization and yielding ...
Route to nonlocal cosmology
hep-th 2007-05 conditional novelty 5.0 of 10

Nonlocal scalar-field cosmologies can be localized via a diffusion equation in an auxiliary variable, yielding explicit hypergeometric solutions for power-law universes, though the full Friedmann equations are only sa...

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D1-brane with Overcritical Electric Field in AdS3 and S-brane
hep-th 2006-02 conditional novelty 5.0 of 10

Unphysical dS2-branes in AdS3 with imaginary electric flux are reinterpreted, via T-duality, as S0-branes from tachyon condensation, with S(p-1)-branes carrying imaginary NS-NS tension.
Quintom Cosmology: Theoretical implications and observations
hep-th 2009-09 accept novelty 2.0 of 10

Quintom dark energy, in which the equation of state crosses w = -1, is reviewed as an observationally motivated but theoretically challenging extension of standard cosmology.
Analytical Solutions of Open String Field Theory
hep-th 2008-07 accept novelty 2.0 of 10

The paper is a comprehensive review of Schnabl's analytical tachyon vacuum solution in open string field theory, its proofs of Sen's conjectures, marginal deformations, and superstring generalizations.
Tachyon Dynamics in Open String Theory
hep-th 2004-10 accept novelty 2.0 of 10

A comprehensive review of open string tachyon condensation argues that unstable D-brane dynamics is captured by an internally consistent open string field theory without explicit closed-string coupling.

How the literature uses it

41 arXiv papers cite this in 48 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 · 32 sentences without a confident label

Builds on it 1 paper

As the next step we extend this analysis to the case of non-BPS Dp-branes [ref] that includes tachyon field on its world-volume together with the gauge field.

Note About D-Branes in Carrollian Background · Introduction and Summary · 0.84

Compares with it 1 paper

It is clear that they cannot be identified but rather they should be related by some redefinition of the tachyon field in the similar way as the tachyon effective action given in [ref] is related to the DBI like tachyon effective action suggested in [ref] .

Proposal for D0-brane anti-D0-brane Action · Introduction and Summary · 0.95

Cites it as background 14 papers

The leading order non-BPS effective actions including tachyonic modes were proposed in [ref] , where some of their properties such as their decays and tachyon condensation have also been clarified in detail [ref] .

On Special Limit of Non-Supersymmetric Effective Actions of type II String theory · Introduction · 0.96
The effective action for describing the dynamics of one non-BPS D$_p$-brane in flat background in static gauge is given by [ref] : S = -T_p∫d^p+1σV(T^2) √(-(η_ab+_aX^i_bX^i+F_ab+_a T_b T)) ,dbiac2where $≡2πα'$ and $V=1-π/2T^2+O(T^4)$ is the tachyon potential[Our index convention is that $μ,ν,...=0,1,...,9$; $a,b,...=0,1,...,p$; $i,j,...=p+1,...,9$ and $I,J,...=3,4,..., 10$.].

A proposal for M2-brane-anti-M2-brane action · D2-brane-anti-D2-brane effective action · 0.94
The first proposal for the action that describes dynamics of Dp-brane anti-Dp-brane system was given in [ref] that was mainly motivated by the effective actions for non-BPS Dp-branes in string theory [ref] .

Proposal for D0-brane anti-D0-brane Action · Introduction and Summary · 0.92

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [14]

M.R. Garousi, Tachyon couplings on non-BPS D-branes and Dirac-Born-Infe ld action, hep-th/0003122

View on Pith Pith review arXiv

[2] [21]

J. Kluso˘ n,Remark about Non-BPS D-Brane in Type IIA Theory , hep-th/9909194

View on Pith Pith review arXiv

[3] [11]

A. Sen, Supersymmetric World-volume Action for Non-BPS D-Branes , JHEP 9910 (1999) 008, hep-th/99090622

View on Pith 1999

[4] [1]

J. Polchinski, Dirichlet-Branes and Ramond-Ramond Charges , Phys. Rev. Lett. 75 (1995) 4724, hep-th/9510017

View on Pith Pith review arXiv 1995

[5] [2]

For a review, see J. Polchinski, TASI lectures on D-branes , hep-th/9611050

View on Pith Pith review arXiv

[6] [3]

A. Sen, Stable Non-BPS States in String Theory , JHEP 9806 (1998) 007, hep-th/9803194

View on Pith Pith review arXiv 1998

[7] [4]

A. Sen, Stable Non-BPS Bound States of BPS D-branes , JHEP 9808 (1998) 010, hep-th/9805019

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