{"arxiv_id":"hep-th/9910055","incoming":{"papers":72,"sentences":196,"groups":[{"kind":"extends","label":"Builds on it","papers":1,"sentences":1,"quotes":[{"arxiv_id":"0807.0389","title":"Modular properties of two-loop maximal supergravity and connections with string theory","url":"/paper/0807.0389","section":"Interactions from circle compactification to ten dimensions","sentence":"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] ).","intent_p":0.92,"contests_p":null}]},{"kind":"method_or_tool","label":"Uses its method or tool","papers":10,"sentences":21,"quotes":[{"arxiv_id":"1604.02667","title":"Non--analytic terms from nested divergences in maximal supergravity","url":"/paper/1604.02667","section":"The $ℛ^4$, $D^2ℛ^4$ and $D^4ℛ^4$ coefficient functions from supergravity amplitudes","sentence":"To evaluate eval, we further define [ref] τ_1 = ρ̂/ρ̂+λ̂, τ_2 = √(Δ̂_2(,ł̂,̊̂))/ρ̂+λ̂, V_2 = l_11^2 √(Δ̂_2(,ł̂,̊̂)),to get that 𝒜_D^4ℛ^4^(2) = π^11κ_11^4/(2π)^D l_11^8 _2ℛ^4∑_m̂^I, n̂^I∫_0^∞dV_2 V_2^3 ∫_ℱ_2 d^2 τ/τ_2^2 e^-π^2 G_IJ (m̂+n̂τ)^I (m̂+n̂τ̅)^J V_2/τ_2,where $d^2 τ= dτ_1 dτ_2$ and $ℱ_2$ is the fundamental domain of $SL(2,ℤ)$ defined by ℱ_2 = { -1/2 ≤τ_1 ≤1/2, τ_2 ≥0, |τ|^2 ≥1}.Thus $V_2$ and $τ$ parametrize the volume and complex structure of an auxiliary $T^2$.","intent_p":0.91,"contests_p":null},{"arxiv_id":"0807.0389","title":"Modular properties of two-loop maximal supergravity and connections with string theory","url":"/paper/0807.0389","section":"$h_(3,0)^0$, $h_(0,2)^0$","sentence":"The integral over $τ$ in the $(m̂,n̂) (0,0)$ piece can be `unfolded' to the infinite strip as in [ref] , giving h_(p,q)^0 (1) = π^5 b_(p,q)^0 ∫_chi^dVV ( ∫_d^2ττ_2^2 +∫_0^∞dtt^2∑_p≠0exp(-πp^2 v V/τ_2)) = π^6 b_(p,q)^03 ∫_chi^dVV + π^6 b_(p,q)^03v ∫_chi^dVV^2 .","intent_p":0.96,"contests_p":null},{"arxiv_id":"1407.0535","title":"The D^6 R^4 term from three loop maximal supergravity","url":"/paper/1407.0535","section":"The two loop four graviton amplitude","sentence":"Hence we only need to isolate the boundary contribution, which is done along the lines of [ref] and so we only mention the results.","intent_p":0.99,"contests_p":null},{"arxiv_id":"1001.2535","title":"Automorphic properties of low energy string amplitudes in various dimensions","url":"/paper/1001.2535","section":"Two-loop amplitudes in $D=11$ and Eisenstein series","sentence":"Using the method of orbits this integral has three kinds of pieces [ref] [equation]","intent_p":0.99,"contests_p":null},{"arxiv_id":"1510.02409","title":"Infrared divergences and harmonic anomalies in the two-loop superstring effective action","url":"/paper/1510.02409","section":"Two-loop renormalization, generalities","sentence":"To describe the contributions from the region $_2^II$, which is associated to primitive two-loop divergences, it is convenient to use yet a different set of variables for the imaginary part of the period matrix, introduced in [ref] ,","intent_p":0.99,"contests_p":null}]},{"kind":"data_used","label":"Uses its data or values","papers":5,"sentences":8,"quotes":[{"arxiv_id":"1002.3805","title":"String theory dualities and supergravity divergences","url":"/paper/1002.3805","section":"Constraints on the coefficients","sentence":"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]","intent_p":0.99,"contests_p":null},{"arxiv_id":"0807.0389","title":"Modular properties of two-loop maximal supergravity and connections with string theory","url":"/paper/0807.0389","section":"$(p,q)=(1,0)$","sentence":"The contribution to the coefficient of the $σ_2 ^4∼^4 ^4$ term was shown in [ref] to have the form I^(d=10)_(1,0) + I_▹ (1,0)^(d=10) + δ_2 I^(d=10)_(1,0) = ζ(5)4 R_11^5 ,","intent_p":0.81,"contests_p":null},{"arxiv_id":"1510.01667","title":"Perturbative type II amplitudes for BPS interactions","url":"/paper/1510.01667","section":"The genus one amplitudes","sentence":"For the $D^4ℛ^4$ interaction, from 1loop2 we get that ( λ^2 ^2/λ^2 +2 λ/λ-2)Î^(1)_D^4ℛ^4 =0,leading to I^(1)_D^4 ℛ^4 = 4/π^2ζ(3)ζ(4)(r^3 +1/r^3 ), where we have fixed the overall normalization using the answer from two loop supergravity [ref] .","intent_p":0.87,"contests_p":null},{"arxiv_id":"0712.1252","title":"The D^6 R^4 term in type IIB string theory on T^2 and U-duality","url":"/paper/0712.1252","section":"Constraints using U–duality and the decompactification limit","sentence":"This has been computed directly in ten dimensions in [ref] , where it was shown that the divergent term and the genus two contribution together is proportional to 2/3 ζ(4) e^4ϕ^B + 1/2 ζ(5) r_B^4.","intent_p":0.86,"contests_p":null},{"arxiv_id":"2409.10003","title":"The Quantum Gravity Scale and the Swampland","url":"/paper/2409.10003","section":"Further quantitative tests","sentence":"The first one, which corresponds to a gravitational operator of mass dimension $n=12$, can be computed to be [ref]","intent_p":0.89,"contests_p":null}]},{"kind":"comparison","label":"Compares with it","papers":6,"sentences":7,"quotes":[{"arxiv_id":"1004.0163","title":"Eisenstein series for higher-rank groups and string theory amplitudes","url":"/paper/1004.0163","section":"Solutions for the coefficients $^(D)_(0,0)$ and $^(D)_(1,0)$.","sentence":"The various contributions in [ref] agree with the expressions obtained by evaluating the sum of one-loop and two-loop Feynman diagram contributions to the amplitude in eleven-dimensional supergravity compactified on a $(d+1)$-torus [ref] .","intent_p":0.98,"contests_p":null},{"arxiv_id":"1505.04384","title":"Some finite terms from ladder diagrams in three and four loop maximal supergravity","url":"/paper/1505.04384","section":"Generalities and our analysis of the two loop amplitude","sentence":"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.","intent_p":0.91,"contests_p":null},{"arxiv_id":"0801.0322","title":"Low energy expansion of the four-particle genus-one amplitude in type II superstring theory","url":"/paper/0801.0322","section":"Summary of the expansion of the nine-dimensional genus-one amplitude","sentence":"These coefficients follow from the methods described in appendix [ref], and the results agree with the type IIA expressions derived by taking the one-loop amplitude of eleven dimensional supergravity on $T^2$ [ref] .","intent_p":1.0,"contests_p":null},{"arxiv_id":"0708.2950","title":"The D^4 R^4 term in type IIB string theory on T^2 and U-duality","url":"/paper/0708.2950","section":"Decompactifying to nine dimensions","sentence":"Keeping only the terms that survive in the large $r_B$ limit, note that ninedim reduces to l_s^3 ∫d^9 x √(- g_9) r_B [ e^ϕ_B/2 E_5/2 (τ,τ̅)^SL(2,ℤ) + 8/π^2 ζ(3) ζ(4) r_B^2 ] D^4 R^4, which is precisely what has been obtained in [ref] , which is a consistency check of our modular form.","intent_p":0.91,"contests_p":null},{"arxiv_id":"0807.0389","title":"Modular properties of two-loop maximal supergravity and connections with string theory","url":"/paper/0807.0389","section":"World-line presentation of the two-loop amplitude","sentence":"which agrees with [ref] .","intent_p":0.95,"contests_p":null}]},{"kind":"background","label":"Cites it as background","papers":26,"sentences":30,"quotes":[{"arxiv_id":"2104.11109","title":"Normalization of Type IIB D-instanton Amplitudes","url":"/paper/2104.11109","section":"Introduction","sentence":"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.","intent_p":0.97,"contests_p":null},{"arxiv_id":"1803.08329","title":"Supergravity limit of genus two modular graph functions in the worldline formalism","url":"/paper/1803.08329","section":"Introduction","sentence":"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] .","intent_p":0.9,"contests_p":null},{"arxiv_id":"1407.0535","title":"The D^6 R^4 term from three loop maximal supergravity","url":"/paper/1407.0535","section":"The four graviton supergravity amplitude at one and two loops","sentence":"We review various details of the four graviton amplitude in $N=1$ supergravity in 11 dimensions compactified on $ℝ^8,1 ×T^2$ at one and two loops [ref] .","intent_p":0.91,"contests_p":null},{"arxiv_id":"0807.0389","title":"Modular properties of two-loop maximal supergravity and connections with string theory","url":"/paper/0807.0389","section":"Overview of low orders in the momentum expansion","sentence":"where the ellipsis stand for the non-analytic contributions [ref] and $h_k$ are simple constants and $^(k)$ is a polynomial in $σ̂_2$ and $σ̂_3$ of order $k=2p+3q$ in the Mandelstam invariants.","intent_p":0.82,"contests_p":null},{"arxiv_id":"1002.3805","title":"String theory dualities and supergravity divergences","url":"/paper/1002.3805","section":"Constraints on the coefficients","sentence":"A variety of calculations in compactified eleven-dimensional supergravity at one loop and two loops provide much information about this limit [ref] 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