I-Brane Inflow and Anomalous Couplings on D-Branes

Gregory Moore; Jeffrey A. Harvey; Michael Green

REVIEW

I-Brane Inflow and Anomalous Couplings on D-Branes

Michael Green, Jeffrey A. Harvey, Gregory Moore

High Energy Physics - Theory · Submitted 6 May 1996 · 24 references

Pith's verdict: The sqrt(A-hat) D-brane coupling paper: the eight-form gravitational term and the inflow method are the real contributions, and the derivation is a consistency argument that holds up as long as you accept the factorized ansatz.

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arXiv IDhep-th/9605033 (version 1)

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CitationGregory Moore, Jeffrey A. Harvey, Michael Green. "I-Brane Inflow and Anomalous Couplings on D-Branes." arXiv:hep-th/9605033 (1996). https://pith.science/paper/NGEUNSKO

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Editor's note

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This is the paper that pins down the D-brane Wess-Zumino coupling as C ∧ ch_N(F) ∧ sqrt(Â(R)), and the genuinely new piece is the degree-eight gravitational term plus the inflow method itself. The gauge Chern character was already in Li [16], and the four-form curvature term came from Bershadsky-Vafa-Sadov [21] by duality chasing. What Green, Harvey and Moore add is the square-root structure and the observation that you can get all of it, including the eight-form term, from anomaly inflow on an intersecting brane pair. That is a clean idea and it is executed cleanly. The bonus is that the same matching reproduces Polchinski's mu^2/alpha = 2 pi, which is a nice independent check on the charge quantum rather than a new assumption. The I-brane spectrum is imported from Chaudhuri-Johnson-Polchinski [22], which is fine; that is a cited result, not a gap. The one step I would flag to a student is the split. The I-brane anomaly matching (2.3), (2.9), (2.11) only fixes the product Y1 Y2 = ch1 ch2 Â(R). Getting individual Y's requires the counterterm choice (2.10), justified by 'symmetry between the two branes.' The stress-test is right that this is the soft step, and right that it is mild: the provisional ansatz (2.4) already treats Y as a property of a single brane depending only on its own F and the bulk R, so universality is built in and the product constraint then forces Y = ch sqrt(Â) up to the sign fixed by the leading term. The paper does not spell that reasoning out, and a reader could reasonably ask whether Y_i = ch_i Â^{1/2} e^{±g(R)} is excluded. It is excluded by the ansatz, not by the matching alone. That is worth one explicit sentence in a revision, not more. The other acknowledged limitations (B set to zero, Romans mass omitted) are honestly stated and do not touch the central claim. The remark that (2.4) is not well-defined because RR potentials are not single-valued, and the fix by integrating by parts to (2.5), is a real subtlety that the authors handle rather than paper over. Who should read it: anyone computing brane charges on curved cycles, anyone doing anomaly inflow, and anyone who wants to see why the sqrt(Â) is there rather than just quoting it. It deserves a serious referee, and I would cite it without hesitation.

Referee report

Significance

If correct, the result is a clean and economical derivation of the D-brane anomalous couplings, including the eight-form gravitational term, from a physical input (the I-brane fermion spectrum) rather than from a string amplitude computation. The derivation is parameter-free: the only inputs are the anomaly polynomial of the I-brane spectrum and the provisional form of the brane coupling, and the output includes both the functional form of Y and the numerical RR charge quantum µ²/α = 2π, independently obtained by Polchinski. The appearance of sqrt(Â) rather than Â is a genuine output of the symmetric split of the two-brane matching and is a nontrivial check against later direct computations. The paper is also methodologically valuable in making precise how consistent versus covariant anomalies are distributed between the brane and the bulk, and it correctly observes that because open strings have only two ends, all charge violation is captured by pairs of branes.

Major objections (1)

[§2, Eqs. (2.3), (2.8)–(2.11)] The I-brane anomaly matching determines only the product Y1(F1,R) Y2(F2,R) = ch_{N1}(F1) ch_{N2}(F2) Â(R); the step from this product to the individual coupling Y(F,R) = ch_N(F) sqrt(Â(R)) is carried by the counterterm choice Eq. (2.10), justified in one sentence as 'symmetry between the two branes.' This is the one place where the logic of the central derivation is compressed, and it deserves to be made explicit. The needed assumption is that Y(F,R) is a universal property of a single brane, depending only on that brane's own field strength and on the bulk curvature, independent of which other brane is present; the ansatz Eq. (2.4) already presupposes this, but the text does not say so. The argument would be materially tightened by noting that the matching must hold for arbitrary multiplicities N1, N2, so that the coefficients of N1 and of N2 in the degree-four (and degree-eight) terms must match separately; this constrains the split well beyond the product constraint and excludes alternatives of the form Y = ch_N(F) sqrt(Â(R)) e^{g(R)} with g a curvature polynomial of degree at least four. As written, a reader cannot tell from the text alone whether such alternatives are excluded by the matching or by the ansatz. Relatedly, the fact that the symmetric split produces sqrt(Â) rather than Â is a nontrivial prediction that distinguishes this result from the fivebrane case, and the paper would benefit from saying so explicitly, since it is the strongest internal check on the new eight-form term.

Minor objections (9)

[§2, Eqs. (2.8)–(2.11)] The passage from the variation (2.8) through the descent polynomial (2.9) to the numerical relation µ²/α = 2π in (2.11) skips the intermediate algebra, including the tracking of the factor of 2 in (2.3) and the 2 in (2.9). Since the charge quantum is one of the two headline results, one or two displayed intermediate lines would let the reader verify the normalization without reconstructing it.
[§1, Eq. (1.1) and Eq. (1.6)] The descent formula (1.1) carries an explicit factor of 2πi, while the inflow cancellation statement (1.6) does not. It would help to state once whether the 2πi is absorbed into the normalization of the fermion determinant or into the definition of the descent chain, so that the two equations can be compared directly.
[§2, footnote 3] The convention of working in Minkowski space for IIB and Euclidean space for IIA is confined to a footnote and is not obviously consistent with the use of Â(R) and of chirality throughout §2. A sentence in the main text explaining how the Â genus and the chirality of the I-brane fermions are defined in each signature would remove a possible source of confusion.
[§2, Eq. (2.2)] The table of boundary conditions would be considerably easier to read if the rows were labelled by which brane each endpoint lies on (brane 1 or brane 2) rather than only by the boundary-condition pattern, since the identification of the first two sectors with U(N_i) Yang–Mills on B_i depends on that assignment.
[§2, paragraph after Eq. (2.2)] The text first states that chiral zero modes occur 'when there are chiral unbroken supersymmetries on the I-brane' and then states that the Ramond ground states are chiral if and only if p12 is odd, regardless of supersymmetry. These two statements should be reconciled explicitly, since the second is the one actually used and is the stronger one.
[§2, Eq. (2.6)] Calling dH = µα Σ δ(B_p → M10) Y(F,R) 'the equation of motion (Bianchi identity)' for the RR field strength is potentially confusing: it is a modified Bianchi identity for H, and the genuine equation of motion is the one displayed in Eq. (1.4). A brief clarification of which equation is being used where would help.
[§1, reference [12]] The fivebrane eight-form coupling is attributed jointly to reference [12], which is listed as unpublished, and to reference [13], which is published. Since [13] appears to contain the result, this is not a problem, but the reliance on an unpublished note could be made explicit or the citation trimmed to the published source.
[§2, final paragraph] The observation that multiple intersecting branes require no new inflow analysis because open strings have only two ends is correct and worth stating, but as written it is asserted rather than argued. One sentence on why a three-brane configuration adds no independent constraint would make the closure of the argument apparent.
[Throughout] There are several typographical errors that should be corrected: 'ambigous' and 'discused' in §1, 'Evidentally' and 'interesection' in §2, and 'Poincare' for 'Poincaré' in the caption to Eq. (1.4). The hyphenation artifacts in the running text ('descr ibed', 't o', 'a ccounted', 'gravitatio nal') should also be cleaned up in the final version.

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The paper

The paper claims

Anomaly inflow fixes how D-branes couple to Ramond-Ramond potentials, including a new eight-form gravitational term.

The reading

The paper asks how the Ramond-Ramond potentials of type II string theory couple to the gauge and gravitational fields that live on a D-brane. Instead of computing string scattering amplitudes order by order, the authors place two D-branes at right angles so that they intersect on a lower-dimensional brane whose light spectrum contains chiral fermions, and then require that the charge these fermions anomalously lose be supplied by the surrounding bulk. Matching the anomaly on the intersection against the variation of the bulk coupling fixes the polynomial of gauge and gravitational curvatures that multiplies the Ramond-Ramond potential, and simultaneously fixes the constant relating brane charge to the string coupling. The answer reproduces the previously known gauge-field couplings, supplies a new degree-eight gravitational coupling, and recovers the quantum of Ramond-Ramond charge a D-brane carries. If the argument holds, an entire tower of brane couplings follows from one anomaly-matching condition rather than from a term-by-term string computation.

What carries the argument

The mechanism is anomaly inflow applied to an intersecting-brane configuration. Two orthogonal D-branes of dimensions $p_1$ and $p_2$ meet on an I-brane $B_{12} = B_1 cap B_2$; the open-string sectors with one end on each brane give, after the GSO projection, one Weyl fermion in the $(N_1, overline{N}_2)$ and one in $(overline{N}_1, N_2)$, chiral precisely when $p_{12}$ is odd. Descent from the polynomial $2, mathrm{ch}_{N_1}(F_1), mathrm{ch}_{N_2}(F_2), widehat{A}(R)$ gives the anomaly on the intersection, while the Bianchi identity $dH = mu alpha sum_p delta(B_p to M_{10}), Y(F,R)$ converts the variation of the bulk coupling into a term localized on the same intersection. Matching the two, with the descent counterterm fixed by symmetry between the two branes, yields $Y(F,R) = mathrm{ch}_N(F) sqrt{widehat{A}(R)}$ and $mu^2/alpha = 2pi$.

Extended reading notes

Core claim

The central claim is that the anomalous couplings of a stack of $N$ D-branes take the form $I_{CS} = -T_p - T_p + T_p + T_p$ ... more precisely $I_{CS} = T_p$ ...

Load-bearing premise

The derivation assumes in advance that each brane couples through a gauge-invariant polynomial $Y(F,R)$ of the form $N + cdots$, and it fixes the descent counterterm ambiguity by demanding symmetry between the two intersecting branes; uniqueness of the square-root A-roof answer rests on that factorized starting point together with that counterterm choice, neither of which is derived.

Editorial extensions

If this is right

The gauge-field couplings $mathrm{ch}_N(F)$ of brane fields to Ramond-Ramond potentials are confirmed by an independent argument rather than by direct string amplitude computation.
A new degree-eight gravitational coupling appears, entering through $sqrt{widehat{A}(R)} = 1 + tfrac{1}{2}widehat{A}_4 + (tfrac{1}{2}widehat{A}_8 - tfrac{1}{8}widehat{A}_4^2) + cdots$, which the paper states is determined here for the first time.
The relation $mu^2/alpha = 2pi$ between the Ramond-Ramond charge normalization and the string coupling follows from the same matching, verifying the quantum of charge a D-brane carries.
The argument explains why consistent and covariant anomalies both appear: the anomaly on the intersecting brane is cancelled by inflow from the bulk, so the two descriptions are two faces of one mechanism.

Reading between the lines

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

Because the coupling is a square root of the standard anomaly polynomial, the natural object being integrated is a bispinor Ramond-Ramond potential rather than a plain form; this suggests the coupling is most compactly organized in a spinor language that the paper does not develop.
The new $widehat{A}_8$ term should be checkable in the same duality settings where the $widehat{A}_4$ term was already used, since a duality that tests the four-form part also constrains its eight-form partner.
The authors note that a nonzero Neveu-Schwarz potential and the Romans mass can be included by the same bookkeeping; if so, the inflow argument extends to massive type IIA without new input, which would be a sharp test of the mechanism.
Since open strings have only two ends, the paper's remark that all charge violation is captured by pairs of branes suggests the same pairwise logic should suffice for configurations with many intersecting branes, though the authors do not carry that out.

Original abstract

We show that the anomalous couplings of $D$-brane gauge and gravitational fields to Ramond-Ramond tensor potentials can be deduced by a simple anomaly inflow argument applied to intersecting $D$-branes and use this to determine the eight-form gravitational coupling.

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We show that the anomalous couplings of $D$-brane gauge and gravitational fields to Ramond-Ramond tensor potentials can be deduced by a simple anomaly inflow argument applied to intersecting $D$-branes and use this to determine the eight-form gravitational coupling.

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s0	We show that the anomalous couplings of $D$-brane gauge and gravitational fields to Ramond-Ramond tensor potentials can be deduced by a simple anomaly inflow argument applied to intersecting $D$-branes and use this to determine the eight-form gravitational coupling.	contribution 0.92	0.95	0.92	asserted 1.00

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References

Works cited (24)

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S. Chaudhuri, C. Johnson, and J. Polchinski, “Notes on D -branes,” hep-th/9602052. 7

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J. Blum and J. A. Harvey, unpublished

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C. Vafa and E. Witten, “A One-Loop Test of String Duality ,” Nucl. Phys. B447 (1995) 261, hep-th/9505053

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Analyzes TsT deformations and Penrose limits on fibered I-branes from prior work, finding preserved solvability when TsT precedes the limit and new asymptotically free or parallelizable sectors in the reverse order.
TASI Lectures on D-Branes
hep-th 1996-11 accept novelty 3.0 of 10

D-branes arise from T-duality of open strings and supply the charged objects needed for string duality, black hole state counting, and short-distance probes.
TASI 2003 Lectures on Anomalies
hep-th 2005-09 accept novelty 2.0 of 10

A pedagogical survey that derives chiral and gauge anomalies, develops the descent and inflow machinery, and applies it to axion strings, intersecting D-branes and M5-brane anomaly cancellation.
D-Branes on Calabi-Yau Manifolds
hep-th 2004-03 accept novelty 2.0 of 10

The review argues that B-branes on Calabi-Yau threefolds are best described by the derived category of coherent sheaves, with physical stability captured by Pi-stability and mirror symmetry relating this to the Fukaya...
Four-dimensional String Compactifications with D-Branes, Orientifolds and Fluxes
hep-th 2006-10 unverdicted

A comprehensive pedagogical review of four-dimensional orientifold and flux compactifications with D-branes, assembling model-building rules, effective-action methods, and statistical vacuum counts.

How later papers used it

103 later papers cite it

Evidence: substantial. 101 of the citing papers share no author with it, and 9 of those carry a confident label from Pith's citation classifier.

Pith read 16 of the 140 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

1 later paper by other authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 4 papers with this label

Used the usual descent procedure from this paper to derive the cancelling form for the Ramond-Ramond gauge anomaly inflow due to chiral fermions on the intersection worldvolume for families of branes.

Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory (2007) · 0.93 · paraphrased: the licence does not allow quoting

How the literature uses it

99 arXiv papers cite this in 133 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 · 100 sentences without a confident label

Contests it 1 paper

This is, however, indeed too naive generically: While the supergravity calculation performed in section 4 of [ref] actually admits the cycle which is wrapped by the D7-brane to be compact (and therefore captures the corrections $∼^2 ∫∧$), curvature-induced D3-brane charge [ref] was neglected.

D7-Brane Moduli Space in Axion Monodromy and Fluxbrane Inflation · Relevance of Loop Corrections · contest 0.79

Builds on it 1 paper

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

Uses its method or tool 4 papers

To extract the full anomaly inflow counterterm cancelling the 7-brane gauge anomalies from the 3-7 sector as well as the tangent bundle anomalies along the D3-brane, we follow the standard procedure [ref] and rewrite the non-kinetic terms in the action ([ref]) as S|_C_4 ⊃ S_1 + S_2 S_1 = 2 π/4 ∫C_4 ∧∑_i ([C_i] + [C_i'] ) S_2 = 2 π∫F_5 ∧( 1/8 ∑_a (𝐰_3a ∧[D_a] + 𝐰_3a' ∧[D_a'] ) + 1/32 𝐰_3Y ∧[O7] ) .

The Green-Schwarz Mechanism and Geometric Anomaly Relations in 2d (0,2) F-theory Vacua · 3-7 anomaly from gauging in Type IIB · 0.97
We derive the cancelling form for the Ramond-Ramond gauge anomaly inflow due to chiral fermions on the intersection worldvolume for families of branes using the usual descent procedure [ref] , which is due to curvature of the spacetime manifold $X$ itself.

Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory · Gravitational pairings · 0.93
It could be computed by performing a gauge variation $δA = dλ$ of the Wess-Zumino action for a D7-brane, along the lines of [ref] and more recently [ref] .

A chiral cascade via backreacting D7-branes with flux · Brane Engineering · 0.96

Compares with it 2 papers

Moreover, this root is not unique and $Γ̂_X$ is just a particular choice [ref] , however this choice is different from the one corresponding to the Ramond-Ramond (RR) charge computed in [ref] .

GLSM monodromy on quantum period lattice of Calabi-Yau fourfold flops · A-period lattice for CY4 · 0.84
Moreover, this root is not unique and $Γ̂_X$ is just a particular choice [ref] , however this choice is different from the one corresponding to the Ramond-Ramond (RR) charge computed in [ref] .

Monodromy of Calabi-Yau threefold flops via grade restriction rule and their quantum Kahler moduli · Hemisphere partition function and A-periods · 0.84

Cites it as support 1 paper

The stringy choice for the $X_4^α$ is justified by the anomalous couplings of D-branes [ref] and O-planes [ref] to the tensors $C_2^α$ in the Wess-Zumino term.

Rigid Vacua with Brane Supersymmetry Breaking · The anomaly polynomial and the R-R tadpoles · 0.97

Cites it as background 23 papers

This action for a single D-brane is given by [ref] , S_CS = T_p∫_M^p+1C∧(e^2πα'F+B)∧(A(4π^2α'R_T)/A(4π^2α'R_N))^1/2CS where $M^p+1$ represents the world volume of the D$_p$-brane, $C$ is meant to represent a sum over all appropriate RR potential and $A(R_T,N)$ is the Dirac roof genus of the tangent and normal bundle curvatures respectively, √(A(4π^2α'R_T)/A(4π^2α'R_N)) = 1+(4π^2α')^2/384π^2(R_T^2-R_N^2)+⋯roof For totally-geodesic embeddings of world-volume in the ambient spacetime, $R_T,N$ are the pulled back curvature 2-forms of the tangent and normal bundles respectively (see the appendix in ref.

T-duality of curvature terms in D-brane actions · Introduction · 0.85
The curvature corrections to the WS action have been found in [ref] by requiring that the chiral anomaly on the world volume of intersecting D-branes (I-brane) cancels with the anomalous variation of the WS action.

Higher derivative corrections to WZ action: One RR, one NSNS and one NS couplings · Introduction and Results · 0.96
The so-called I-brane background [ref] , [ref] rises as the 1+1 dimensional intersection of two orthogonal stacks of NS5-branes, with one set of branes lying along $(x^0, x^1,⋯, x^5)$, and other set lying along $(x^0, x^6,⋯, x^9)$ directions.

On spinning strings in I-brane background · Introduction · 0.95

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.

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