REVIEW
Duality and higher derivative terms in M theory
High Energy Physics - Theory · Submitted 4 Oct 2005 · 29 references
Pith's verdict: The two-loop D^6R^4 computation and its Laplace equation are solid and well cross-checked; the word 'exact' in the abstract is doing more work than Section 6 supports.
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Referee report
The paper
The paper claims
A two-loop gravity calculation fixes the exact coupling dependence of a four-graviton correction in string theory, D-instantons included.
The reading
What carries the argument
The load-bearing device is a change of variables that turns the three Schwinger parameters of the two-loop scalar integrals into the volume $V$ and complex structure $tau$ of an auxiliary two-torus, after Poisson resummation converts Kaluza-Klein charges into winding numbers. The integration domain becomes the fundamental domain of the congruence subgroup $Gamma_0(2)$, which covers three copies of the standard $SL(2,Z)$ domain, so the amplitude reduces to an integral over $V$ and $tau$ weighted by a kernel $A(tau_1,tau_2) = |tau|^2 - |tau_1| + 1$ over $tau_2$ plus a correction. The whole argument then rides on one identity for that kernel, $Delta_tau A = 12A - 12tau_2delta(tau_1)$, read as a weak equality on the fundamental domain: because the exponent depends on $tau$ and $Omega$ in the same way, the same operator acting on $Omega$ produces the inhomogeneous Laplace equation for $E^{(3/2,3/2)}$, with the delta-function term becoming the $Z_{3/2}^2$ source. A recursion relation in an integer $p$ labelling generalised integrals then separates the answer into an infinite series over generalised Eisenstein series $Z^{(p+3/2,p+3/2)}$ plus a remainder evaluated by a saddle point at $tau_0 = -(hat m_1 + hat m_2Omega)/(hat n_1 + hat n_2Omega)$.
Extended reading notes
Core claim
The central claim is that the coefficient of the $l_s^4 D^6R^4$ interaction in the type IIB effective action is a modular function $E^{(3/2,3/2)}(Omega,bar Omega)$ satisfying $$Delta_Omega $E^{{(3/2,3/2)}}$ = 12, $E^{{(3/2,3/2)}}$ - 6, Z_{3/2}, Z_{3/2}$$ on the fundamental domain of $SL(2,Z)$, where $Z_{3/2}$ is the non-holomorphic Eisenstein series that carries the $R^4$ coupling dependence and $Delta_Omega = 4Omega_2^2 partial_Omega partial_{bar Omega}$. The authors obtain this by expanding the two-loop eleven-dimensional supergravity four-graviton amplitude to first order in the Mandelstam invariants and compactifying on $T^2$; the $V to 0$ limit of that compactification is finite and yields the ten-dimensional IIB answer. The perturbative part of the solution is $$4zeta(3)^2 $Omega_2^{3}$ + 8zeta(3)zeta(2)Omega_2 + tfrac{48}{5}zeta(2)^2 $Omega_2^{{-1}}$ + $\beta$, $Omega_2^{{-3}}$,$$ where the first two coefficients match explicit string calculations of the tree-level and one-loop terms, the third is a new two-loop prediction, and the fourth is a three-loop coefficient $beta = tfrac{16}{189}pi^2zeta(4)$ fixed by requiring that the solution be no more singular than the tree-level term. That same $beta$ is independently recovered from the type IIA three-loop contribution extracted from earlier work, a match the authors emphasise because the IIA and IIB four-graviton amplitudes need not agree beyond two loops. Compactification on a circle instead gives a $zeta(3)^2/R_{11}^6$ term whose normalisation matches the tree-level string amplitude, and lifting the two-loop IIB coefficient back to eleven dimensions leaves a finite $D^6R^4$ term in the M-theory action.
Load-bearing premise
The claim that this is the exact coupling dependence rests on the assumption that eleven-dimensional supergravity diagrams with three or more loops contribute nothing further to $D^6R^4$, an assumption the authors explicitly say they cannot verify because the systematics of the theory beyond two loops are not understood.
Editorial extensions
If this is right
- The two-loop coefficient $tfrac{48}{5}zeta(2)^2$ for $D^6R^4$ becomes a concrete target for a two-loop string calculation extended to the first non-leading order in external momenta.
- The three-loop coefficient $tfrac{16}{189}pi^2zeta(4)$ is predicted, and its agreement with the independently extracted type IIA value says the four-graviton amplitudes of the two theories coincide at three loops for this interaction.
- The eleven-dimensional limit leaves a finite $D^6R^4$ term, in contrast to $D^4R^4$ which vanishes there, matching the expectation that the first curvature power beyond $R^4$ surviving in eleven dimensions is $R^7$.
- The same bookkeeping suggests a homogeneous Laplace equation with solution $Z_{7/2}$ for the $D^8R^4$ coefficient and an inhomogeneous equation with source $Z_{3/2}Z_{5/2}$ for $D^{10}R^4$, and at $D^{12}R^4$ a branching into two distinct modular functions because two independent tree-level coefficients, $zeta(9)$ and $zeta(3)^3$, appear.
Reading between the lines
- If the pattern holds, the coupling dependence of each four-graviton interaction is fixed by a Laplace-type equation whose source is a product of lower-order solutions, so the whole tower could be organised as a recursive system rather than computed diagram by diagram.
- The equality of the three-loop IIA and IIB coefficients is a check that could be sharpened: the two theories are expected to differ first through odd-odd spin structures, so a three-loop string computation of $D^6R^4$ would test whether those structures really cancel here.
- The double-D-instanton terms with $k_1 = -k_2$ sit in the zero Fourier mode alongside the perturbative powers, which means a weak-coupling expansion that keeps only power-law terms is systematically incomplete even at large $Omega_2$.
- The authors note that the inhomogeneous source should follow from supersymmetry mixing $D^6R^4$ with both the classical action and the $R^4$ term; making that argument precise would give an independent derivation not tied to the two-loop amplitude.
Original abstract
Dualities of M-theory are used to determine the exact dependence on the coupling constant of the D^6R^4 interaction of the IIA and IIB superstring effective action. Upon lifting to eleven dimensions this determines the coefficient of the D^6R^4 interaction in eleven-dimensional M-theory. These results are obtained by considering the four-graviton two-loop scattering amplitude in eleven-dimensional supergravity compactified on a circle and on a two-torus -- extending earlier results concerning lower-derivative interactions. The torus compactification leads to an interesting SL(2,Z)-invariant function of the complex structure of the torus (the IIB string coupling) that satisfies a Laplace equation with a source term on the fundamental domain of moduli space. The structure of this equation is in accord with general supersymmetry considerations and immediately determines tree-level and one-loop contributions to D^6R^4 in perturbative IIB string theory that agree with explicit string calculations, and two-loop and three-loop contributions that have yet to be obtained in string theory. The complete solution of the Laplace equation contains infinite series' of single D-instanton and double D-instanton contributions, in addition to the perturbative terms. General considerations of the higher loop diagrams of eleven-dimensional supergravity suggest extensions of these results to interactions of higher order in the low energy expansion.
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
Dualities of M-theory are used to determine the exact dependence on the coupling constant of the D^6R^4 interaction of the IIA and IIB superstring effective action. Upon lifting to eleven dimensions this determines the coefficient of the D^6R^4 interaction in eleven-dimensional M-theory. These results are obtained by considering the four-graviton two-loop scattering amplitude in eleven-dimensional supergravity compactified on a circle and on a two-torus -- extending earlier results concerning lower-derivative interactions. The torus compactification leads to an interesting SL(2,Z)-invariant function of the complex structure of the torus (the IIB string coupling) that satisfies a Laplace equation with a source term on the fundamental domain of moduli space. The structure of this equation is in accord with general supersymmetry considerations and immediately determines tree-level and one-loop contributions to D^6R^4 in perturbative IIB string theory that agree with explicit string calculations, and two-loop and three-loop contributions that have yet to be obtained in string theory. The complete solution of the Laplace equation contains infinite series' of single D-instanton and double D-instanton contributions, in addition to the perturbative terms. General considerations of the higher loop diagrams of eleven-dimensional supergravity suggest extensions of these results to interactions of higher order in the low energy expansion.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Dualities of M-theory are used to determine the exact dependence on the coupling constant of the D^6R^4 interaction of the IIA and IIB superstring effective action. | contribution 0.51 | 0.80 | 0.69 | asserted 1.00 |
| s1 | Upon lifting to eleven dimensions this determines the coefficient of the D^6R^4 interaction in eleven-dimensional M-theory. | result 0.91 | 0.81 | 0.23 | asserted 1.00 |
| s2 | These results are obtained by considering the four-graviton two-loop scattering amplitude in eleven-dimensional supergravity compactified on a circle and on a two-torus -- extending earlier results concerning lower-derivative interactions. | method 0.96 | 0.87 | 0.10 | asserted 1.00 |
| s3 | The torus compactification leads to an interesting SL(2,Z)-invariant function of the complex structure of the torus (the IIB string coupling) that satisfies a Laplace equation with a source term on the fundamental domain of moduli space. | result 0.89 | 0.83 | 0.10 | asserted 1.00 |
| s4 | The structure of this equation is in accord with general supersymmetry considerations and immediately determines tree-level and one-loop contributions to D^6R^4 in perturbative IIB string theory that agree with explicit string calculations, and two-loop and three-loop contributions that have yet to be obtained in string theory. | result 0.71 | 0.81 | 0.30 | asserted 0.91 |
| s5 | The complete solution of the Laplace equation contains infinite series' of single D-instanton and double D-instanton contributions, in addition to the perturbative terms. | result 1.00 | 0.76 | 0.15 | asserted 1.00 |
| s6 | General considerations of the higher loop diagrams of eleven-dimensional supergravity suggest extensions of these results to interactions of higher order in the low energy expansion. | future work 0.71 | 0.24 | 0.07 | hedged possibility 1.00 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:hep-th/0510027 by the authors.
References
How later papers used it
109 later papers cite it
2 later papers by other authors are listed below: 2 used its method or tool. 2 more papers by the same authors are listed: 1 used its method or tool and 1 used its data or values.
Uses its method or tool 3 of 12 papers with this label · 1 by the same authors
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Calculated lambda-hat_2 along the lines of this paper.
The D^6 R^4 term in type IIB string theory on T^2 and U-duality (2007) · 0.99 · paraphrased: the licence does not allow quoting
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Proceeded exactly along the lines of this paper and mentioned only the results.
Recursion Relations from Space-time Supersymmetry (2008) · 0.98 · paraphrased: the licence does not allow quoting
By the same authors
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Determined the value of beta as in section 5.4 of this paper by integrating over the cutoff fundamental domain for SL(2,Z).
Modular properties of two-loop maximal supergravity and connections with string theory (2008) (same authors) · 1.00 · paraphrased: the licence does not allow quoting
Uses its data or values 1 of 10 papers with this label · 1 by the same authors
By the same authors
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Used the value of the genus-three coefficient found in this paper to fix a relation upon decompactification to ten dimensions.
Automorphic properties of low energy string amplitudes in various dimensions (2010) (same authors) · 0.97 · paraphrased: the licence does not allow quoting
How the literature uses it
100 arXiv papers cite this in 287 sentences
Papers citing it · 188 sentences without a confident label
Contests it 1 paper
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The combination of terms in ([ref]), which was pointed out to us by Don Zagier, has asymptotic behaviour that guarantees convergence of the sums that arise in ([ref]), whereas the expression used in [ref] leads to a divergent result.] h(u) = 7+44u^2+40u^43√(1 + u^2) - 16/3π(4/3 + 5 u^2 + u (3 + 5 u^2) tan^-1(u))
SL(2,Z)-invariance and D-instanton contributions to the $D^6 R^4$ interaction · Sum over translates of one-dimensional solution · contest 0.85
Builds on it 4 papers
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The explicit $^(D)_(0,0)$ coefficients in dimensions $D≥6$ will be obtained in terms of Eisenstein series that satisfy Laplace eigenvalue equations on moduli space space, building on the work of [ref] .
Automorphic properties of low energy string amplitudes in various dimensions · Introduction · 0.99
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Instead, we will study the appropriate constant terms of this coefficient in two of the parabolic subgroups, $P_α_1$ and $P_α_d+1$, of relevance in this paper, as was done for the cases $6≤D≤10$ (i.e., for duality groups of rank $r≤5$) in [ref] .
Eisenstein series for higher-rank groups and string theory amplitudes · Constant terms for the solution of the $E_8$ inhomogeneous Laplace eigenvalue equation · 0.81
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The method used to evaluate these integrals is an extension of that used for the $(0,1)$ case in [ref] , which we will review in the appendix [ref] ($I_(1,0)^(d=10)$ was evaluated in [ref] ).
Modular properties of two-loop maximal supergravity and connections with string theory · Interactions from circle compactification to ten dimensions · 0.92
Uses its method or tool 12 papers
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Nevertheless, the full solution, $(Δ^(3)+198)^-1 (^(3)_(0,0))^2$, does contain such a contribution in its zero mode expansion that can be extracted by projecting this solution onto a solution of the homogeneous equation, as in section 5.4 of [ref] .
Eisenstein series for higher-rank groups and string theory amplitudes · Constant terms for the solution of the $E_8$ inhomogeneous Laplace eigenvalue equation · 0.92
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Mutiplying the equation involving $f̃^(0,0)_2$ by $E_7 (τ, τ̅)$, and integrating over the fundamental domain of $SL(2,ℤ)$ gives a_5 = 32/13π^4 ∑_n=1^∞1/n^4 μ(n,3/2) μ(n, 5/2),where we have used both expSL2 and altrep, the Rankin–Selberg formula RSf, and ∫_0^∞dx x^6 K_1 (x) K_2 (x) = 32/5.Similarly, multiplying the equation involving $f̃^(0,0)_1$ by $E_5 (τ, τ̅)$ and using ∫_0^∞dx x^4 K_1 (x) K_2 (x) = 2,we find a_4 = 4/27π^2 ∑_n=1^∞1/n^2 μ(n,3/2) μ(n, 5/2).Finally, we use Ramanujan's formula [ref] ∑_n=1^∞1/n^r μ(n,s) μ(n, s') = ζ(r) ζ(r + 2s - 1) ζ(r + 2s' -1) ζ(r + 2s + 2s' -2)/ζ(2r + 2s + 2s' -2)to obtain a_4 =11/4050 π^2 ζ(6), a_5 = 8/2835 π^2 ζ(8).Taking into account only the terms involving $μ_1$ and $μ_2$ in calcmore, the non–perturbative contributions to $f̃^(0,0)_i$ can also be evaluated along the lines of [ref] .
Recursion Relations from Space-time Supersymmetry · The $k ≥5$ case and aspects of the $D^10 ℛ^4$ interaction · 0.86
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The value of $β$ is determined as in section 5.4 of [ref] by integrating over the cutoff fundamental domain for $SL(2,)$ the product of $$ with the Eisenstein series $E_s$ –which is a solution of the homogeneous equation associated with ([ref]).
Modular properties of two-loop maximal supergravity and connections with string theory · General method for determining $β$ terms · 1.00
Uses its data or values 9 papers
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Using the ten dimensional values $λ^(10)_(0,0)=3/4$, $λ^(10)_(1,0)=15/4$ and $λ^(10)_(0,1)=12$ determined in [ref] , we deduce that the coefficients of the terms discussed in [ref] satisfy the following set of Laplace eigenvalue equations with source terms, [equation]
String theory dualities and supergravity divergences · Constraints on the coefficients · 0.99
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The finite piece has been evaluated in [ref] and is given by π^6/96 l_11^6 _3𝒱_2^-3ℰ (Ω,Ω̅)where $ℰ (Ω,Ω̅)$ satisfies the Poisson equation 4Ω_2^2 ^2 ℰ/ΩΩ̅ = 12 ℰ - 6 E_3/2^2,the structure of which follows from supersymmetry [ref] .
The D^6 R^4 term from three loop maximal supergravity · The two loop four graviton amplitude · 0.93
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Decompactification to ten dimensions and the value of the genus-three coefficient found in [ref] fixes [equation]
Automorphic properties of low energy string amplitudes in various dimensions · Determination of $^(8)_(0,1)$. · 0.97
Compares with it 10 papers
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In the absence of concrete answers at three and four loops, we consider in detail the analysis at two loops, where several amplitudes are known [ref] and we can use the results to compare with what we get based on our analysis in the main text.
Some finite terms from ladder diagrams in three and four loop maximal supergravity · Generalities and our analysis of the two loop amplitude · 0.91
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in agreement with the predictions from S-duality and supersymmetry [ref] .
Matching the $D^6 {\cal R}^4$ interaction at two-loops · Genus three · 0.88
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For $d=0$, our approach provides a powerful computational method, alternative to the one used in [ref] , which reproduces the results found in [ref] by integrating the differential equation.
1/8-BPS Couplings and Exceptional Automorphic Functions · Weak coupling expansion · 0.90
Cites it as background 40 papers
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Many D-instanton induced terms were predicted using S-duality invariances of various compactified string theories [ref] , but except for the early attempts [ref] , the direct systematic computation of these amplitudes from first principles has not been carried out.
Normalization of Type IIB D-instanton Amplitudes · Introduction · 0.97
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Although the coefficients, $^(D)_(p,q)$, have not been determined in generality, a significant amount of information has accumulated for the first three terms for the cases with $D≥6$ (duality groups $E_d+1$ with $d ≤4$) [ref] , and there are various conjectures concerning the coefficient of the $^4$ interaction, $^(D)_(0,0)$, for higher-rank groups [ref] .
Eisenstein series for higher-rank groups and string theory amplitudes · Introduction · 0.80
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The terms in the low momentum expansion of the two loop four graviton amplitude up to the $D^12ℛ^4$ term have been obtained in [ref] .
Supergravity limit of genus two modular graph functions in the worldline formalism · Introduction · 0.90
Discussion
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