Poincaré series for modular graph forms at depth two. Part I. Seeds and Laplace systems

Dorigoni, Daniele; Kleinschmidt, Axel; Schlotterer, Oliver

doi:10.1007/jhep01(2022)133

Cited by 17 publications

(44 citation statements)

References 113 publications

(334 reference statements)

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“…In that case the coefficients of integer powers of 1/N are generalised Eisenstein series that satisfy inhomogeneous Laplace eigenvalue equations with sources terms that are quadratic in nonholomorphic Eisenstein series. It would be of interest to discover the structure of such correlators at finite values of N , perhaps using the recent results of [37,38], and for more general gauge groups.…”

Section: Discussionmentioning

confidence: 99%

Exact results for duality-covariant integrated correlators in $\mathcal{N}=4$ SYM with general classical gauge groups

Dorigoni¹,

Green²,

Wen³

2022

Preprint

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We present exact expressions for certain integrated correlators of four superconformal primary operators in the stress tensor multiplet of N = 4 supersymmetric Yang-Mills (SYM) theory with classical gauge group, GN = SO(2N ), SO(2N + 1), U Sp(2N ). These integrated correlators are expressed as two-dimensional lattice sums by considering derivatives of the localised partition functions, generalising the expression obtained for SU (N ) in our previous works. These expressions are manifestly covariant under Goddard-Nuyts-Olive duality. The integrated correlators can also be formally written as infinite sums of non-holomorphic Eisenstein series with integer indices and rational coefficients. Furthermore, the action of the hyperbolic Laplace operator with respect to the complex coupling τ = θ/(2π)on any integrated correlator for gauge group GN relates it to a linear combination of correlators with gauge groups GN+1, GN and GN−1. These "Laplace-difference equations" determine the expressions of integrated correlators for all classical gauge groups for any value of N in terms of the correlator for the gauge group SU (2). The perturbation expansions of these integrated correlators for any finite value of N agree with properties obtained from perturbative Yang-Mills quantum field theory, together with various multi-instanton calculations which are also shown to agree with those determined by supersymmetric localisation. The coefficients of terms in the large-N expansion are sums of non-holomorphic Eisenstein series with half-integer indices, which extend recent results and make contact with low order terms in the low energy expansion of type IIB superstring theory in an AdS5 × S 5 /Z2 background.

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Section: Discussionmentioning

confidence: 99%

Exact results for duality-covariant integrated correlators in $\mathcal{N}=4$ SYM with general classical gauge groups

Dorigoni¹,

Green²,

Wen³

2022

Preprint

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“…As explained in Part I [1], as we increase the total transcendental weight w = m+k we encounter higher and higher eigenvalues s ≤ k+m−1 in the spectrum, see (1.4). This in turn means that the obstructions to finding modular solutions to the Laplace systems (1.1) are related to iterated integrals (2.29) of cusp forms ∆ 2s of higher and higher modular weight 2s.…”

Section: An Example Involving the Two Weight 24 Cusp Formsmentioning

confidence: 89%

“…In many cases, the Poincaré-series representations in this work may be viewed as interpolating between double sums over lattice momenta and double integrals over holomorphic 1 Absolute convergence is guaranteed for m < k and for m = k a suitable regularisation was described in Part I. 2 A more detailed review of the construction and properties of the β sv can be found in section 2.…”

Section: Jhep01(2022)134mentioning

confidence: 99%

Poincaré series for modular graph forms at depth two. Part II. Iterated integrals of cusp forms

Dorigoni

Kleinschmidt

Schlotterer

2022

J. High Energ. Phys.

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We continue the analysis of modular invariant functions, subject to inhomogeneous Laplace eigenvalue equations, that were determined in terms of Poincaré series in a companion paper. The source term of the Laplace equation is a product of (derivatives of) two non-holomorphic Eisenstein series whence the modular invariants are assigned depth two. These modular invariant functions can sometimes be expressed in terms of single-valued iterated integrals of holomorphic Eisenstein series as they appear in generating series of modular graph forms. We show that the set of iterated integrals of Eisenstein series has to be extended to include also iterated integrals of holomorphic cusp forms to find expressions for all modular invariant functions of depth two. The coefficients of these cusp forms are identified as ratios of their L-values inside and outside the critical strip.

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“…In particular, the formulas of Appendix B make it possible to express the modular graph functions C a,b,c as a linear combination of the modular functions F +(s) m,k which were introduced in [40,41] and obey the inhomogeneous Laplace equation (∆ − s(s − 1))F…”

Section: A Useful Byproductmentioning

confidence: 99%

“…In Appendix B, we derive explicit expressions for the coefficients d w;m;p a,b,c and d a,b,c w;m;p as well as for the h coefficients which appear in the inhomogeneous Laplace eigenvalue equation (2.44), thereby proving the following proposition. m,k = E m E k were recently studied in [40,41]. There it was shown that the solutions to these equations include the modular graph functions C w;m;p as well as modular functions which are not modular graph function.…”

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confidence: 99%

Integrating three-loop modular graph functions and transcendentality of string amplitudes

D'Hoker,

Geiser

2021

Preprint

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Modular graph functions (MGFs) are SL(2, Z)-invariant functions on the Poincaré upper half-plane associated with Feynman graphs of a conformal scalar field on a torus. The low-energy expansion of genus-one superstring amplitudes involves suitably regularized integrals of MGFs over the fundamental domain for SL(2, Z). In earlier work, these integrals were evaluated for all MGFs up to two loops and for higher loops up to weight six. These results led to the conjectured uniform transcendentality of the genus-one four-graviton amplitude in Type II superstring theory. In this paper, we explicitly evaluate the integrals of several infinite families of three-loop MGFs and investigate their transcendental structure. Up to weight seven, the structure of the integral of each individual MGF is consistent with the uniform transcendentality of string amplitudes. Starting at weight eight, the transcendental weights obtained for the integrals of individual MGFs are no longer consistent with the uniform transcendentality of string amplitudes. However, in all the cases we examine, the violations of uniform transcendentality take on a special form given by the integrals of triple products of non-holomorphic Eisenstein series. If Type II superstring amplitudes do exhibit uniform transcendentality, then the special combinations of MGFs which enter the amplitudes must be such that these integrals of triple products of Eisenstein series precisely cancel one another. Whether this indeed is the case poses a novel challenge to the conjectured uniform transcendentality of genus-one string amplitudes.

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Poincaré series for modular graph forms at depth two. Part I. Seeds and Laplace systems

Cited by 17 publications

References 113 publications

Exact results for duality-covariant integrated correlators in $\mathcal{N}=4$ SYM with general classical gauge groups

Exact results for duality-covariant integrated correlators in $\mathcal{N}=4$ SYM with general classical gauge groups

Poincaré series for modular graph forms at depth two. Part II. Iterated integrals of cusp forms

Integrating three-loop modular graph functions and transcendentality of string amplitudes

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