REVIEW
Curvature terms in D-brane actions and their M-theory origin
High Energy Physics - Theory · Submitted 24 Mar 1999 · 62 references
Pith's verdict: A solid, load-bearing paper: the curvature-squared brane action and its D-instanton completion have held up, with the one real gap being that the nine-coefficient matching is asserted rather than shown.
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Referee report
The paper
The paper claims
The curvature-squared terms of a D-brane's world-volume action are fixed, and on a D3-brane they sum an infinite series of D-instantons into a single modular function of the string coupling.
The reading
What carries the argument
Three pieces of machinery carry the argument. First, the disk-level four-point open-string amplitude with its $t_8$ kinematic factor, converted into graviton–brane, graviton–ripple and ripple–ripple scattering by the substitutions $2k_1 to p$, $2k_2 to Dcdot p$, $zeta_1 zeta_2 to epsilon D$; expanding the gamma-function ratio to subleading order isolates the $O(alpha'^2)$ contact terms, since the bulk vertices and graviton/ripple mixing are protected by supersymmetry. Second, the Gauss–Codazzi decomposition, which rewrites the ambient curvature pull-backs in terms of the intrinsic world-volume curvature, the normal-bundle field strength and the second fundamental form $Omega^a_{alphabeta}$, so that a single Lagrangian covers geodesic and non-geodesic embeddings. Third, the Dedekind eta function: its modular transformation laws, $log eta(tau+1) = log eta(tau) + i pi/12$ and $log eta(-1/tau) = log eta(tau) - i pi/4 + frac{1}{2}log tau$, are exactly what is needed for the anomalous variation of $log eta$ to cancel the anomalous phases of the world-volume fermion determinant and of the Abelian Maxwell partition function under an $S$ transformation, leaving only a $tau$-independent piece proportional to the first Pontryagin class. On the M-theory side the corresponding tool is a light-cone superparticle loop with two graviton vertex operators, a Kaluza–Klein sum over momenta on $T^2$, and Poisson resummation that separates the D-instanton sum, the tree-level disk term and the open-string one-loop piece.
Extended reading notes
Core claim
The paper's central claim is that the $(curvature)^2$ terms of the D-brane effective action are now known completely at disk level. For the parity-conserving sector the result is Eq. (2.14), built out of the world-volume curvature $(R_T)_{alphabeta gammadelta}$, the $SO(9-p)$ field strength $(R_N)^{ab}{}_{alphabeta}$ of the normal bundle, and the combination $bar{R}_{ab} = hat{R}_{ab} + Omega_{a}cdot Omega_{b}$ of the second fundamental form, with the Gauss–Bonnet coefficient fixed to zero by a K3 compactification argument. The parity-violating partner, Eq. (2.15), pairs the Ramond–Ramond potential with $mathrm{tr}(R_T wedge R_T) - mathrm{tr}(R_N wedge R_N)$, the natural closed form needed for gauge invariance of the Wess–Zumino coupling. Specialising to a D3-brane with trivial normal bundle, the authors observe that these disk-level terms, written as $-tau_2 mathrm{tr}(R wedge *R) + tau_1 mathrm{tr}(R wedge R)$ up to a normalisation, are not invariant under $SL(2,mathbb{Z})$ acting on $tau = lambda_B^{-1}(C^{(0)} + i e^{-phi})$. Requiring that the full action transform so as to cancel the modular anomaly of the massless $N=4$ vector multiplet, and cross-checking against the known exact one-loop $R^2$ amplitude of the dual type IIA compactification, leads to Eq. (1.5): the complete answer is $frac{1}{16pi^2} int mathrm{Re}[log eta(tau), mathrm{tr}(R wedge *R - i R wedge R)]$. Its weak-coupling expansion reproduces the disk result as the leading $tau_2$ term and then generates the whole tower $q + 3q^2/2 + 4q^3/3 + dots$ with $q = e^{2pi i tau}$, i.e. a sum over D-instantons. The authors then show that the parity-conserving part of this same series follows from a one-loop diagram in the six-dimensional $(2,0)$ tensor multiplet of the M5-brane compactified on $T^2$, where the Kaluza–Klein momentum along the hidden eleventh direction plays the role of instanton number and the sum over windings reproduces the disk term.
Load-bearing premise
The nine independent coefficients of Eq. (2.14) are asserted rather than displayed — section 2 says only that 'this straightforward but tedious comparison leads to the following result' — so everything downstream, including the D3-brane specialisation and the normalisation of $log eta(tau)$, inherits a matching that the reader cannot check from the text.
Editorial extensions
If this is right
- The curvature-squared couplings of a D3-brane are a modular function of $tau$, not a constant, so any computation of graviton scattering off a D3-brane at order $alpha'^2$ must include the whole D-instanton tower, weighted by the divisor sum $sum_{n|N} 1/n$.
- The modular anomaly of the massless $N=4$ vector multiplet on the D3-brane is cancelled by the anomalous variation of the curvature terms, provided the background satisfies $frac{1}{48}p_1(R) in mathbb{Z}$, a condition met in particular by K3.
- Because the instanton measure $q = e^{2pi i tau}$ has trivial monodromy around a D7-brane, the anomaly-inflow argument for intersecting brane configurations is unaffected by these non-perturbative corrections.
- In the T-dual D4-brane description, the same series is the contribution of D-particle threshold bound states, and the measure $sum_{n|N} 1/|n|$ is consistent with a Witten index of one for the D-particle–D4 system.
- The parity-conserving $R^2$ action of the D3-brane is reproduced by a one-loop diagram in the M5-brane world-volume, extending the same eleven-dimensional one-loop logic that previously produced the bulk $R^4$ terms.
Reading between the lines
- If the modular completion is right, then the same $log eta(tau)$ structure should govern curvature couplings of seven-branes and, by the same anomaly-cancellation logic, the $R^4$, $R^2F^2$ and $F^4$ terms of the type I / F-theory background, which the authors flag as an open extension rather than a result.
- The argument fixes the coefficient of $log eta$ by anomaly cancellation and by matching a dual one-loop amplitude, so an independent check could come from a direct multi-instanton computation of the D3-brane effective action at fixed instanton number, comparing the divisor sum term by term.
- The acknowledged restriction to trivial normal bundles suggests that the $SO(6)$ R-symmetry curvature of a D3-brane embedded in ten dimensions may require its own anomalous counterterm, in analogy with the tangent-bundle counterterm $int varphi, mathrm{tr}(R wedge R)$ used here.
- A probe D3-brane parallel to the $AdS_5$ boundary sees nonzero ambient curvature and nonzero self-dual five-form flux; whether the $R^2$ terms derived here are cancelled or completed by $F_5$-dependent terms is a question the authors raise but do not settle.
Original abstract
We derive the complete $(curvature)^2$ terms of effective D-brane actions, for arbitrary ambient geometries and world-volume embeddings, at lowest order (disk-level) in the string-loop expansion. These terms reproduce the $o(\alpha'^2)$ corrections to string scattering amplitudes, and are consistent with duality conjectures. In the particular case of the D3-brane with trivial normal bundle, considerations of $SL(2,\mathbb{Z})$ invariance lead to a complete sum of D-instanton corrections for both the parity-conserving and the parity-violating parts of the effective action. These corrections are required for the cancellation of the modular anomalies of massless modes, and are consistent with the absence of chiral anomalies in the intersection domain of pairs of D-branes. We also show that the parity-conserving part of the non-perturbative R^2 action follows from a one-loop quantum calculation in the six-dimensional world-volume of the M5-brane compactified on a two-torus.
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.
The abstract, sentence by sentence
We derive the complete $(curvature)^2$ terms of effective D-brane actions, for arbitrary ambient geometries and world-volume embeddings, at lowest order (disk-level) in the string-loop expansion. These terms reproduce the $o(\alpha'^2)$ corrections to string scattering amplitudes, and are consistent with duality conjectures. In the particular case of the D3-brane with trivial normal bundle, considerations of $SL(2,\mathbb{Z})$ invariance lead to a complete sum of D-instanton corrections for both the parity-conserving and the parity-violating parts of the effective action. These corrections are required for the cancellation of the modular anomalies of massless modes, and are consistent with the absence of chiral anomalies in the intersection domain of pairs of D-branes. We also show that the parity-conserving part of the non-perturbative R^2 action follows from a one-loop quantum calculation in the six-dimensional world-volume of the M5-brane compactified on a two-torus.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We derive the complete $(curvature)^2$ terms of effective D-brane actions, for arbitrary ambient geometries and world-volume embeddings, at lowest order (disk-level) in the string-loop expansion. | contribution 0.92 | 0.97 | 0.57 | asserted 1.00 |
| s1 | These terms reproduce the $o(\alpha'^2)$ corrections to string scattering amplitudes, and are consistent with duality conjectures. | result 0.52 | 0.72 | 0.23 | asserted 0.89 |
| s2 | In the particular case of the D3-brane with trivial normal bundle, considerations of $SL(2,\mathbb{Z})$ invariance lead to a complete sum of D-instanton corrections for both the parity-conserving and the parity-violating parts of the effective action. | result 0.86 | 0.87 | 0.18 | asserted 0.97 |
| s3 | These corrections are required for the cancellation of the modular anomalies of massless modes, and are consistent with the absence of chiral anomalies in the intersection domain of pairs of D-branes. | interpretation 0.79 | 0.55 | 0.09 | asserted 0.81 |
| s4 | We also show that the parity-conserving part of the non-perturbative R^2 action follows from a one-loop quantum calculation in the six-dimensional world-volume of the M5-brane compactified on a two-torus. | result 0.71 | 0.90 | 0.15 | asserted 1.00 |
What was examined, and what was not
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References
How later papers used it
77 later papers cite it
1 later paper by other authors is listed below: 1 built on it.
Builds on it 1 of 5 papers with this label
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Extended the discussion in this paper by considering a non-trivial normal bundle and by including the contribution to the overall anomaly from three U(1) currents.
Theories of Class F and Anomalies (2018) · 1.00 · paraphrased: the licence does not allow quoting
How the literature uses it
70 arXiv papers cite this in 181 sentences
Papers citing it · 145 sentences without a confident label
Builds on it 5 papers
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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
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In particular we have extended the discussion in [ref] by considering a non-trivial normal bundle, implying that there is a triangle anomaly involving the local $U(1)$ currents and the R-symmetry currents, and by including the contribution to the overall anomaly from three $U(1)$ currents.
Theories of Class F and Anomalies · Modular Anomaly Revisited · 1.00
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We do so by extending the leading order calculation of the potential of a nonabelian $D3$-brane stack of [ref] to include the next-to-leading order commutator corrections as well as the same set of $α'$ corrections [ref] which has been encountered in [ref] but here as corrections to $D3$-branes.
Higher order corrections to KPV: The nonabelian brane stack perspective · Introduction · 1.00
Uses its data or values 1 paper
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Furthermore, $c_3i^(1) = 𝒪(1)/48·16π^2 = 𝒪(10^-4)$ for both D-branes and O-planes up to the usual field-redefinition ambiguities [ref] .
Almost classical de Sitter? · Adding $α^'$ corrections · 0.86
Compares with it 5 papers
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We note with satisfaction that the $R^2$ and $(∇H)^2$ terms are consistent with known results [ref] , [ref] (we would require $c_37=0$ and $c_40=c_1$ to match the $R^2$ terms, and $c_4=-2c_1$, $c_18=0$, $c_46=c_1/2$, and $c_47=c_1/6$ to match the $(∇H)^2$ terms[To carry out the match with [ref] , we need to rewrite the coupling $∇^iH^abc∇_iH_abc$ in our basis, using the null vectors from appendix [ref], ∫√(-g)e^-Φ∇^iH^abc∇_iH_abc=∫√(-g)e^-Φ3∇^aH^bci∇_aH_bci+6∇^aH_a^abi∇^cH_bci+RH^2+∇ΦH∇H. ]), but many other coefficients have now been fixed.
Higher Derivative Corrections to O-plane Actions: NS-NS Sector · Warped product · 1.00
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We have done this calculation and found the following couplings for $ΩΩΩΩ$ and $ΩΩΩΩFF$: [equation] Note that the $ΩΩΩΩ$ terms above are exactly the second fundamental form correction that have been found in [ref] up to terms that involve the trace of the second fundamental form, (see [ref] ), which are removed in our scheme.
Couplings of order six in the gauge field strength and the second fundamental form on a D$_p$-brane at order $\alpha'^2$ · Towards all gauge field couplings · 0.96
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Finally, let us compare the $α'^3$-order spacetime action Y6 with the string frame $α'^2$-order D-brane action found in [ref] S ∼ ∫d^p+1x √(-G)e^-ϕ_0[-2(R̂_ab+ Φ_;ab)(R̂^ab+ Φ^;ab)+2(R̂_ij+ Φ_;ij)(R̂^ij+ Φ^;ij) +R_abcdR^abcd-R_abijR^abij-2/3_aH_ijk^aH^ijk-4/3_iH_abc^iH^abc+2_aH_bci^aH^bci ]In this equation the indices $a,b,c,d$ are the world volume indices and $i,j,k$ are the transverse indices.
Ricci curvature corrections to type II supergravity · Discussion · 1.00
Cites it as support 2 papers
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The $ℛ^2$ interaction receives perturbative contributions only from disc and annulus amplitudes [ref] , and non-perturbative contributions from D–instantons, and hence is protected.
Constraining the D3-brane effective action · Introduction · 0.99
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In string theory, scattering amplitude calculations in heterotic or higher curvature corrections to the D-brane effective action both lead to positive coefficients [ref] .
On higher derivative gravity, c-theorems and cosmology · Discussion · 0.98
Cites it as background 19 papers
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Let us start by recalling the curvature corrections to the string-frame DBI action of a D$p$-brane [ref] .
Curvature corrections to KPV: Do we need deep throats? · Curvature corrections · 0.92
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The curvature corrections to this action has been found in [ref] by requiring the consistency of the effective action with the $O(α'^2)$ terms of the corresponding disk-level scattering amplitude [ref] .
Towards extending the Chern-Simons couplings at order $O(\alpha'^2)$ · Introduction and results · 0.84
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The curvature corrections to this action has been found in [ref] by requiring the consistency of the effective action with the $O(α'^2)$ terms of the corresponding disk-level scattering amplitude [ref] .
On RR couplings on D-branes at order $O(\alpha'^2)$ · Introduction · 0.88
Discussion
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