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Constraining gravitational interactions in the M theory effective action
High Energy Physics - Theory · Submitted 12 Aug 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This procedure yields partial contributions to several multi--loop type IIB string amplitudes, from which we extract the transcendental nature of the corresponding M theory couplings.
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We consider purely gravitational interactions of the type D^{6n} R^4 in the effective action of M theory which are related to the type IIA interactions of the form e^{2n\phi_A} D^{6n} R^4 where \phi_A is the type IIA dilaton. The coefficients of the M theory interactions are determined by the strongly coupled type IIA theory. Given the nature of the dilaton dependence, it is plausible that for low values of n, the coefficient has a similar structure as the genus (n+1) string amplitude of the type IIA D^{6n} R^4 interaction, namely the transcendental nature. Assuming this, and focussing on the even--even spin structure part of the type IIA string amplitude, this coefficient is given by the type IIB genus (n+1) amplitude, which we constrain using supersymmetry, S--duality and maximal supergravity. The source terms of the Poisson equations satisfied by the S--duality invariant IIB couplings play a central role in the analysis. This procedure yields partial contributions to several multi--loop type IIB string amplitudes, from which we extract the transcendental nature of the corresponding M theory couplings. For n \leq 2, all possible source terms involve only BPS couplings. While the R^4 and D^6 R^4 M theory couplings agree with known results, the coefficient of the D^{12} R^4 interaction takes the form \zeta (2)^3 (\Omega_1+\Omega_2 \zeta (3)). We also analyze the D^{18} R^4 and D^{24} R^4 interactions, and show that their coefficients have at least the terms \zeta (2)^4 (\tilde\Omega_1 +\tilde\Omega_2\zeta (3) +\tilde\Omega_3\zeta (5)) and \zeta (2)^5 (\underline\Omega_1 +\underline\Omega_2\zeta (3) +\underline\Omega_3\zeta (5) + \underline\Omega_4\zeta (3)^2 + \underline\Omega_5\zeta (7) +\underline\Omega_6\zeta (3) \zeta (5) +\underline\Omega_7 \zeta (3)^3) respectively. The various undetermined constants have vanishing transcendentality.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider purely gravitational interactions of the type D^{6n} R^4 in the effective action of M theory which are related to the type IIA interactions of the form e^{2n\phi_A} D^{6n} R^4 where \phi_A is the type IIA dilaton. | background 0.69 | 0.29 | 0.04 | asserted 0.99 |
| s1 | The coefficients of the M theory interactions are determined by the strongly coupled type IIA theory. | background 0.70 | 0.22 | 0.07 | asserted 1.00 |
| s2 | Given the nature of the dilaton dependence, it is plausible that for low values of n, the coefficient has a similar structure as the genus (n+1) string amplitude of the type IIA D^{6n} R^4 interaction, namely the transcendental nature. | method 0.37 | 0.18 | 0.08 | hedged possibility 0.95 |
| s3 | Assuming this, and focussing on the even--even spin structure part of the type IIA string amplitude, this coefficient is given by the type IIB genus (n+1) amplitude, which we constrain using supersymmetry, S--duality and maximal supergravity. | method 0.95 | 0.77 | 0.11 | asserted 0.75 |
| s4 | The source terms of the Poisson equations satisfied by the S--duality invariant IIB couplings play a central role in the analysis. | method 0.96 | 0.49 | 0.04 | asserted 1.00 |
| s5 | This procedure yields partial contributions to several multi--loop type IIB string amplitudes, from which we extract the transcendental nature of the corresponding M theory couplings. | result 0.63 | 0.92 | 0.26 | asserted 1.00 |
| s6 | For n \leq 2, all possible source terms involve only BPS couplings. | result 0.97 | 0.75 | 0.06 | asserted 1.00 |
| s7 | While the R^4 and D^6 R^4 M theory couplings agree with known results, the coefficient of the D^{12} R^4 interaction takes the form \zeta (2)^3 (\Omega_1+\Omega_2 \zeta (3)). | result 1.00 | 0.88 | 0.22 | asserted 1.00 |
| s8 | We also analyze the D^{18} R^4 and D^{24} R^4 interactions, and show that their coefficients have at least the terms \zeta (2)^4 (\tilde\Omega_1 +\tilde\Omega_2\zeta (3) +\tilde\Omega_3\zeta (5)) and \zeta (2)^5 (\underline\Omega_1 +\underline\Omega_2\zeta (3) +\underline\Omega_3\zeta (5) + \underline\Omega_4\zeta (3)^2 + \underline\Omega_5\zeta (7) +\underline\Omega_6\zeta (3) \zeta (5) +\underline\Omega_7 \zeta (3)^3) respectively. | result 0.99 | 0.94 | 0.21 | asserted 0.99 |
| s9 | The various undetermined constants have vanishing transcendentality. | result 0.98 | 0.79 | 0.13 | asserted 1.00 |
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The paper
Original abstract
We consider purely gravitational interactions of the type D^{6n} R^4 in the effective action of M theory which are related to the type IIA interactions of the form e^{2n\phi_A} D^{6n} R^4 where \phi_A is the type IIA dilaton. The coefficients of the M theory interactions are determined by the strongly coupled type IIA theory. Given the nature of the dilaton dependence, it is plausible that for low values of n, the coefficient has a similar structure as the genus (n+1) string amplitude of the type IIA D^{6n} R^4 interaction, namely the transcendental nature. Assuming this, and focussing on the even--even spin structure part of the type IIA string amplitude, this coefficient is given by the type IIB genus (n+1) amplitude, which we constrain using supersymmetry, S--duality and maximal supergravity. The source terms of the Poisson equations satisfied by the S--duality invariant IIB couplings play a central role in the analysis. This procedure yields partial contributions to several multi--loop type IIB string amplitudes, from which we extract the transcendental nature of the corresponding M theory couplings. For n \leq 2, all possible source terms involve only BPS couplings. While the R^4 and D^6 R^4 M theory couplings agree with known results, the coefficient of the D^{12} R^4 interaction takes the form \zeta (2)^3 (\Omega_1+\Omega_2 \zeta (3)). We also analyze the D^{18} R^4 and D^{24} R^4 interactions, and show that their coefficients have at least the terms \zeta (2)^4 (\tilde\Omega_1 +\tilde\Omega_2\zeta (3) +\tilde\Omega_3\zeta (5)) and \zeta (2)^5 (\underline\Omega_1 +\underline\Omega_2\zeta (3) +\underline\Omega_3\zeta (5) + \underline\Omega_4\zeta (3)^2 + \underline\Omega_5\zeta (7) +\underline\Omega_6\zeta (3) \zeta (5) +\underline\Omega_7 \zeta (3)^3) respectively. The various undetermined constants have vanishing transcendentality.
Human review
How the literature uses it
7 arXiv papers cite this in 7 sentences
Papers citing it · 2 sentences without a confident label
Cites it as background 5 papers
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Such interactions have not been studied in detail[See [ref] for some analysis of non–BPS interactions.], and are crucial in understanding the effective action beyond the first few orders in the low momentum expansion.
Constraining non--BPS interactions from counterterms in three loop maximal supergravity · Introduction · 0.90
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These BPS interactions have been studied in considerable detail, though not much is known about the non–BPS interactions [ref] .
Some finite terms from ladder diagrams in three and four loop maximal supergravity · Introduction · 0.99
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Such interactions are not well understood, apart from some analysis based on constraints due to supersymmetry, and multi–loop supergravity [ref] .
Poisson equation for the Mercedes diagram in string theory at genus one · Discussion · 0.99
What it cites · 48 sentences without a confident label
Builds on 6 papers
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We describe the method below, which is a generalization of [ref] .
Duality and higher derivative terms in M theory · Transcendentality from integration by parts · 0.81
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Note that the various non–analytic terms in 18more that lead to hg are of the form such that their total contribution can be non–perturbatively completed to give the non–local term in the type IIB effective action in the Einstein frame of the form Y( f^(10) + E_3/2 E_5/2 + Y^2 ) s^9 ℛ^4,which follows from generalizing [ref] .
Non-renormalisation Conditions in Type II String Theory and Maximal Supergravity · Schematics of the $D^18 ℛ^4$ interaction · 0.97
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So far as analyzing the structure of higher genus amplitudes is concerned, one should be able to see this structure and much beyond, coming out directly from the four graviton amplitude in maximal supergravity beyond 2 loops, generalizing the results of [ref] and using [ref] .
The Ultraviolet Behavior of N=8 Supergravity at Four Loops · The genus five $D^12 ℛ^4$ amplitude · 0.93
Uses the method or tool of 3 papers
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The reason we want to do the calculation in the type IIB theory is because we want to impose the constraints implied by S–duality which allow us to obtain explicit expressions for these amplitudes at high genera, along the lines of [ref] .
Recursion Relations from Space-time Supersymmetry · The leading terms in the M theory effective action · 0.90
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The reason we want to do the calculation in the type IIB theory is because we want to impose the constraints implied by S–duality which allow us to obtain explicit expressions for these amplitudes at high genera, along the lines of [ref] .
Supersymmetry Constraints on Type IIB Supergravity · The leading terms in the M theory effective action · 0.90
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For example, this was used to obtain the genus 3 amplitude for the $D^6 ℛ^4$ interaction in [ref] .].
Duality and higher derivative terms in M theory · Transcendentality from integration by parts · 0.80
Uses data or values from 1 paper
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Now apart from $ζ(11)τ_2^11/2$ and $ζ(3)^2 ζ(5)τ_2^11/2$ at genus 0, $f^(14;5)$ has additional terms given by [ref] T(11) τ_2^11/2≡( 9ζ(2) ζ(9)+6/25ζ(2)^2 ζ(7)-4/35ζ(2)^3 ζ(5)+1/5 ζ(3,3,5) ) τ_2^11/2where $ζ(3,3,5)$ is the Multiple Zeta Value (MZV) of depth 3 defined by ζ(3,3,5) = ∑_0<m<n<p 1/m^3 n^3 p^5.Apart from the MZV, the $D^14 ℛ^5$ is the simplest multi–graviton amplitude where zeta functions of even transcendentality arise as genus 0.
Motivic Multiple Zeta Values and Superstring Amplitudes · Some generalities · 0.86
Compares with 2 papers
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Now let us consider the analytic parts of the genus 3 and 4 amplitudes obtained from ans6loop, and compare with the 1 and 2 loop maximal supergravity calculations in [ref] .
Modular properties of two-loop maximal supergravity and connections with string theory · The analysis for generic $λ_i$ · 1.00
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The genus 0 contribution matches the $ζ(3)^3$ part of univ12 , while the structure of the genus 1 contribution agrees with the string calculation [ref] .
Low energy expansion of the four-particle genus-one amplitude in type II superstring theory · The analysis for generic $λ_i$ · 0.96
Cites as support 1 paper
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For the genus 1 contribution, our argument follows from the string calculation of [ref] .
Low energy expansion of the four-particle genus-one amplitude in type II superstring theory · The analysis for generic $λ_i$ · 0.98
Cites as background 14 papers
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We refer the reader to [ref] for the details.
Recursion Relations from Space-time Supersymmetry · Constraints on the type IIB effective action from supersymmetry and S–duality, and M theory interactions · 0.80
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At the eight derivative level, the $ℛ^4$ and the $C_3 ∧X_8$ interactions have been analyzed in [ref] .
Eleven Dimensional Origin of String/String Duality: A One Loop Test · Introduction · 0.95
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At the eight derivative level, the $ℛ^4$ and the $C_3 ∧X_8$ interactions have been analyzed in [ref] .
D-instantons, Strings and M-theory · Introduction · 0.95
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