arXiv subjects

Michael B. Green

Publications and source records attributed to Michael B. Green.

At least 19 recordsLinked to original sources

arXiv2025

Modular Features of Superstring Scattering Amplitudes: Generalised Eisenstein Series and Theta Lifts

In previous papers it has been shown that the coefficients of terms in the large-$N$ expansion of a certain integrated four-point correlator of superconformal primary operators in $\mathcal{N}=4$ supersymmetric Yang-Mills theory are rational sums of real-analytic Eisenstein series and "generalised Eisenstein series''. The latter are novel modular functions first encountered in the context of graviton amplitudes in type IIB superstring theory. Similar modular functions, known as two-loop modular graph functions, are also encountered in the low-energy expansion of the integrand of genus-one closed superstring amplitudes. In this paper we further develop the mathematical structure of such generalised Eisenstein series emphasising, in particular, the occurrence of $L$-values of holomorphic cusp forms in their Fourier mode decomposition. We show that both the coefficients in the large-$N$ expansion of the integrated correlator and two-loop modular graph functions admit a unifying description in terms of four-dimensional lattice sums generated by theta lifts of local Maass functions, which generalise the structure of real-analytic Eisenstein series. Through the theta lift representation, we demonstrate that elements belonging to these two families of non-holomorphic modular functions can be expressed as rational linear combinations of generalised Eisenstein series for which all the $L$-values of holomorphic cusp forms precisely cancel.

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arXiv2023

Relations between integrated correlators in $\mathcal{N}=4$ Supersymmetric Yang--Mills Theory

Integrated correlation functions in $\mathcal{N}=4$ supersymmetric Yang--Mills theory with gauge group $SU(N)$ can be expressed in terms of the localised $S^4$ partition function, $Z_N$, deformed by a mass $m$. Two such cases are $\mathcal{C}_N=(\text{Im} τ)^2 \partial_τ\partial_{\barτ} \partial_m^2\log Z_N\vert_{m=0}$ and $\mathcal{H}_N=\partial_m^4\log Z_N\vert_{m=0}$, which are modular invariant functions of the complex coupling $τ$. While $\mathcal{C}_N$ was recently written in terms of a two-dimensional lattice sum for any $N$ and $τ$, $\mathcal{H}_N$ has only been evaluated up to order $1/N^3$ in a large-$N$ expansion in terms of modular invariant functions with no known lattice sum realisation. Here we develop methods for evaluating $\mathcal{H}_N$ to any desired order in $1/N$ and finite $τ$. We use this new data to constrain higher loop corrections to the stress tensor correlator, and give evidence for several intriguing relations between $\mathcal{H}_N$ and $\mathcal{C}_N$ to all orders in $1/N$. We also give evidence that the coefficients of the $1/N$ expansion of $\mathcal{H}_N$ can be written as lattice sums to all orders. Lastly, these large $N$ and finite $τ$ results are used to accurately estimate the integrated correlators at finite $N$ and finite $τ$.

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arXiv2023

Modular-invariant large-$N$ completion of an integrated correlator in $\mathcal{N}=4$ supersymmetric Yang-Mills theory

The use of supersymmetric localisation has recently led to modular covariant expressions for certain integrated correlators of half-BPS operators in $\mathcal{N} = 4$ supersymmetric Yang-Mills theory with a general classical gauge group $G_N$. Here we determine generating functions that encode such integrated correlators for any classical gauge group and provide a proof of previous conjectured formulae. This gives a systematic understanding of the relation between properties of these correlators at finite $N$ and their expansions at large $N$. In particular, it determines a duality-invariant non-perturbative completion of the large-$N$ expansion in terms of a sum of novel non-holomorphic modular functions. These functions are exponentially suppressed at large $N$ and have the form of a sum of contributions from coincident $(p, q)$-string world-sheet instantons.

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arXiv2023

The SAGEX Review on Scattering Amplitudes, Chapter 10: Selected topics on modular covariance of type IIB string amplitudes and their $\mathcal{N}=4$ supersymmetric Yang-Mills duals

This article reviews some results of the SAGEX programme that have developed in the understanding of the interplay of supersymmetry and modular covariance of scattering amplitudes in type IIB superstring theory and its holographic image in $\mathcal{N}=4$ supersymmetric Yang-Mills theory (SYM). The first section includes the determination of exact expressions for BPS interactions in the low-energy expansion of type IIB superstring amplitudes. The second section concerns properties of a certain class of integrated correlators in $\mathcal{N}=4$ SYM with arbitrary classical gauge group that are exactly determined by supersymmetric localisation. Not only do these reproduce known features of perturbative and non-perturbative $\mathcal{N}=4$ SYM for any classical gauge group, but they have large-$N$ expansions that are in accord with expectations based on the holographic correspondence with superstring theory. The final section focusses on modular graph functions. These are modular functions that are closely associated with coefficients in the low-energy expansion of superstring perturbation theory and have recently received quite a lot of interest in both the physics and mathematics literature.

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arXiv2023

The SAGEX Review on Scattering Amplitudes

This is an introduction to, and invitation to read, a series of review articles on scattering amplitudes in gauge theory, gravity, and superstring theory. Our aim is to provide an overview of the field, from basic aspects to a selection of current (2022) research and developments.

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arXiv2022

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

We present exact expressions for certain integrated correlators of four superconformal primary operators in the stress tensor multiplet of $\mathcal{N}=4$ supersymmetric Yang--Mills (SYM) theory with classical gauge group, $G_N$ $= SO(2N)$, $SO(2N+1)$, $USp(2N)$. 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 $τ=θ/(2π) + 4πi /g^2_{_{YM}}$ on any integrated correlator for gauge group $G_N$ relates it to a linear combination of correlators with gauge groups $G_{N+1}$, $G_N$ and $G_{N-1}$. These "Laplace-difference equation" 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 $AdS_5\times S^5/\mathbb{Z}_2$ background.

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arXiv2022

Snowmass White Paper: String Perturbation Theory

The purpose of this White Paper is to review recent progress towards elucidating and evaluating string amplitudes, relating them to quantum field theory amplitudes, applying their predictions to string dualities, exploring their connection with gravitational physics, and deepening our understanding of their mathematical structure. We also present a selection of targets for future research.

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arXiv2021

Exact expressions for $n$-point maximal $U(1)_Y$-violating integrated correlators in $SU(N)$ $\mathcal{N}=4$ SYM

The exact expressions for integrated maximal $U(1)_Y$ violating (MUV) $n$-point correlators in $SU(N)$ ${\mathcal N}=4$ supersymmetric Yang--Mills theory are determined. The analysis generalises previous results on the integrated correlator of four superconformal primaries and is based on supersymmetric localisation. The integrated correlators are functions of $N$ and $τ=θ/(2π)+4πi/g_{_{YM}}^2$, and are expressed as two-dimensional lattice sums that are modular forms with holomorphic and anti-holomorphic weights $(w,-w)$ where $w=n-4$. The correlators satisfy Laplace-difference equations that relate the $SU(N+1)$, $SU(N)$ and $SU(N-1)$ expressions and generalise the equations previously found in the $w=0$ case. The correlators can be expressed as infinite sums of Eisenstein modular forms of weight $(w,-w)$. For any fixed value of $N$ the perturbation expansion of this correlator is found to start at order $( g_{_{YM}}^2 N)^w$. The contributions of Yang--Mills instantons of charge $k>0$ are of the form $q^k\, f(g_{_{YM}})$, where $q=e^{2πi τ}$ and $f(g_{_{YM}})= O(g_{_{YM}}^{-2w})$ when $g_{_{YM}}^2 \ll 1$. Anti-instanton contributions have charge $k<0$ and are of the form $\bar q^{|k|} \, \hat f(g_{_{YM}})$, where $\hat f(g_{_{YM}}) = O(g_{_{YM}}^{2w})$ when $g_{_{YM}}^2 \ll 1$. Properties of the large-$N$ expansion are in agreement with expectations based on the low energy expansion of flat-space type IIB superstring amplitudes. We also comment on the identification of $n$-point free-field MUV correlators with the integrands of $(n-4)$-loop perturbative contributions to the four-point correlator. In particular, we emphasise the important rôle of $SL(2, \mathbb{Z})$-covariance in the construction.

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arXiv2021

Exploring transcendentality in superstring amplitudes

It is well known that the low energy expansion of tree-level superstring scattering amplitudes satisfies a suitably defined version of maximum transcendentality. In this paper it is argued that there is a natural extension of this definition that applies to the genus-one four-graviton Type II superstring amplitude to all orders in the low-energy expansion. To obtain this result, the integral over the genus-one moduli space is partitioned into a region ${\cal M}_R$ surrounding the cusp and its complement ${\cal M}_L$, and an exact expression is obtained for the contribution to the amplitude from ${\cal M}_R$. The low-energy expansion of the ${\cal M}_R$ contribution is proven to be free of irreducible multiple zeta-values to all orders. The contribution to the amplitude from ${\cal M}_L$ is computed in terms of modular graph functions up to order $D^{12} {\cal R}^4$ in the low-energy expansion, and general arguments are used beyond this order to conjecture the transcendentality properties of the ${\cal M}_L$ contributions. Maximal transcendentality of the full amplitude holds provided we assign a non-zero weight to certain harmonic sums, an assumption which is familiar from transcendentality assignments in quantum field theory amplitudes.

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arXiv2021

Exact properties of an integrated correlator in $\mathcal{N}=4$ $SU(N)$ SYM

We present a novel expression for an integrated correlation function of four superconformal primaries in $SU(N)$ $\mathcal{N}=4$ SYM. This integrated correlator, which is based on supersymmetric localisation, has been the subject of several recent developments. The correlator is re-expressed as a sum over a two dimensional lattice that is valid for all $N$ and all values of the complex Yang-Mills coupling $τ$. In this form it is manifestly invariant under $SL(2,\mathbb{Z})$ Montonen-Olive duality. Furthermore, it satisfies a remarkable Laplace-difference equation that relates the $SU(N)$ to the $SU(N+1)$ and $SU(N-1)$ correlators. For any fixed value of $N$ the correlator is an infinite series of non-holomorphic Eisenstein series, $E(s;τ,\barτ)$ with $s\in \mathbb{Z}$, and rational coefficients. The perturbative expansion of the integrated correlator is asymptotic and the $n$-loop coefficient is a rational multiple of $ζ(2n+1)$. The $n=1$ and $n=2$ terms agree precisely with results determined directly by integrating the expressions in one- and two-loop perturbative SYM. Likewise, the charge-$k$ instanton contributions have an asymptotic, but Borel summable, series of perturbative corrections. The large-$N$ expansion of the correlator with fixed $τ$ is a series in powers of $N^{1/2-\ell}$ ($\ell\in \mathbb{Z}$) with coefficients that are rational sums of $E_s$ with $s\in \mathbb{Z}+1/2$. This gives an all orders derivation of the form of the recently conjectured expansion. We further consider 't Hooft large-$N$ Yang-Mills theory. The coefficient of each order can be expanded as a convergent series in $λ$. For large $λ$ this becomes an asymptotic series with coefficients that are again rational multiples of odd zeta values. The large-$λ$ series is not Borel summable, and its resurgent non-perturbative completion is $O(\exp(-2\sqrtλ))$.

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arXiv2021

New Modular Invariants in $\mathcal{N}=4$ Super-Yang-Mills Theory

We study modular invariants arising in the four-point functions of the stress tensor multiplet operators of the ${\cal N} = 4$ $SU(N)$ super-Yang-Mills theory, in the limit where $N$ is taken to be large while the complexified Yang-Mills coupling $τ$ is held fixed. The specific four-point functions we consider are integrated correlators obtained by taking various combinations of four derivatives of the squashed sphere partition function of the ${\cal N} = 2^*$ theory with respect to the squashing parameter $b$ and mass parameter $m$, evaluated at the values $b=1$ and $m=0$ that correspond to the ${\cal N} = 4$ theory on a round sphere. At each order in the $1/N$ expansion, these fourth derivatives are modular invariant functions of $(τ, \bar τ)$. We present evidence that at half-integer orders in $1/N$, these modular invariants are linear combinations of non-holomorphic Eisenstein series, while at integer orders in $1/N$, they are certain "generalized Eisenstein series" which satisfy inhomogeneous Laplace eigenvalue equations on the hyperbolic plane. These results reproduce known features of the low-energy expansion of the four-graviton amplitude in type IIB superstring theory in ten-dimensional flat space and have interesting implications for the structure of the analogous expansion in $AdS_5\times S^5$.

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arXiv2021

A novel representation of an integrated correlator in $\mathcal{N}=4$ SYM theory

An integrated correlator of four superconformal stress-tensor primaries of $\mathcal{N}=4$ supersymmetric $SU(N)$ Yang-Mills theory (SYM), originally obtained by localisation, is re-expressed as a two-dimensional lattice sum that is manifestly invariant under $SL(2,\mathbb{Z})$ S-duality. This expression is shown to satisfy a novel Laplace equation in the complex coupling constant $τ$ that relates the $SU(N)$ integrated correlator to those of the $SU(N+1)$ and $SU(N-1)$ theories. The lattice sum is shown to precisely reproduce known perturbative and non-perturbative properties of $\mathcal{N}=4$ SYM for any finite $N$, as well as extending previously conjectured properties of the large-$N$ expansion.

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arXiv2021

Asymptotics of the $D^8 R^4$ genus-two string invariant

We continue our investigation of the modular graph functions and string invariants that arise at genus-two as coefficients of low energy effective interactions in Type II superstring theory. In previous work, the non-separating degeneration of a genus-two modular graph function of weight $w$ was shown to be given by a Laurent polynomial in the degeneration parameter $t$ of degree $(w,w)$. The coefficients of this polynomial generalize genus-one modular graph functions, up to terms which are exponentially suppressed in $t$ as $t \to \infty$. In this paper, we evaluate this expansion explicitly for the modular graph functions associated with the $D^8 {\cal R}^4$ effective interaction for which the Laurent polynomial has degree $(2,2)$. We also prove that the separating degeneration is given by a polynomial in the degeneration parameter $\ln (|v|)$ up to contributions which are power-behaved in $v$ as $v \to 0$. We further extract the complete, or tropical, degeneration and compare it with the independent calculation of the integrand of the sum of Feynman diagrams that contributes to two-loop type II supergravity expanded to the same order in the low energy expansion. We find that the tropical limit of the string theory integrand reproduces the supergravity integrand as its leading term, but also includes sub-leading terms proportional to odd zeta values that are absent in supergravity and can be ascribed to higher-derivative stringy interactions.

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arXiv2021

Maximal $U(1)_Y$-violating $n$-point correlators in $\mathcal{N}=4$ super-Yang-Mills theory

This paper concerns a special class of $n$-point correlation functions of operators in the stress tensor supermultiplet of $\mathcal{N}=4$ supersymmetric $SU(N)$ Yang-Mills theory. These are "maximal $U(1)_Y$-violating" correlators that violate the bonus $U(1)_Y$ charge by a maximum of $2(n-4)$ units. We will demonstrate that such correlators satisfy $SL(2,\mathbb{Z})$-covariant recursion relations that relate $n$-point correlators to $(n-1)$-point correlators in a manner analogous to the soft dilaton relations that relate the corresponding amplitudes in flat-space type IIB superstring theory. These recursion relations are used to determine terms in the large-$N$ expansion of $n$-point maximal $U(1)_Y$-violating correlators in the chiral sector, including correlators with four superconformal stress tensor primaries and $(n-4)$ chiral Lagrangian operators, starting from known properties of the $n=4$ case. We concentrate on the first three orders in $1/N$ beyond the supergravity limit. The Mellin representations of the correlators are polynomials in Mellin variables, which correspond to higher derivative contact terms in the low-energy expansion of type IIB superstring theory in $AdS_5 \times S^5$ at the same orders as $R^4, d^4R^4$ and $d^6R^4$. The coupling constant dependence of these terms is found to be described by non-holomorphic modular forms with holomorphic and anti-holomorphic weights $(n-4,4-n)$ that are $SL(2, \mathbb{Z})$-covariant derivatives of Eisenstein series and certain generalisations. This determines a number of non-leading contributions to $U(1)_Y$-violating $n$-particle interactions ($n>4$) in the low-energy expansion of type IIB superstring amplitudes in $AdS_5\times S^5$.

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arXiv2020

Modular Invariance in Superstring Theory From ${\cal N} = 4$ Super-Yang-Mills

We study the four-point function of the lowest-lying half-BPS operators in the ${\cal N} =4$ $SU(N)$ super-Yang-Mills theory and its relation to the flat-space four-graviton amplitude in type IIB superstring theory. We work in a large-$N$ expansion in which the complexified Yang-Mills coupling $τ$ is fixed. In this expansion, non-perturbative instanton contributions are present, and the $SL(2, \mathbb{Z})$ duality invariance of correlation functions is manifest. Our results are based on a detailed analysis of the sphere partition function of the mass-deformed SYM theory, which was previously computed using supersymmetric localization. This partition function determines a certain integrated correlator in the undeformed ${\cal N} = 4$ SYM theory, which in turn constrains the four-point correlator at separated points. In a normalization where the two-point functions are proportional to $N^2-1$ and are independent of $τ$ and $\bar τ$, we find that the terms of order $\sqrt{N}$ and $1/\sqrt{N}$ in the large $N$ expansion of the four-point correlator are proportional to the non-holomorphic Eisenstein series $E({\scriptstyle \frac{3}{2}},τ,\barτ)$ and $E({\scriptstyle \frac{5}{2}},τ,\barτ)$, respectively. In the flat space limit, these terms match the corresponding terms in the type IIB S-matrix arising from $R^4$ and $D^4 R^4$ contact interactions, which, for the $R^4$ case, represents a check of AdS/CFT at finite string coupling. Furthermore, we present striking evidence that these results generalize so that, at order $N^{\frac{1}{2}-m}$ with integer $m \ge 0$, the expansion of the integrated correlator we study is a linear sum of non-holomorphic Eisenstein series with half-integer index, which are manifestly $SL(2,\mathbb{Z})$ invariant.

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arXiv2019

Superstring Amplitudes, Unitarity, and Hankel Determinants of Multiple Zeta Values

The interplay of unitarity and analyticity has long been known to impose strong constraints on scattering amplitudes in quantum field theory and string theory. This has been highlighted in recent times in a number of papers and lecture notes. Here we examine such conditions in the context of superstring tree-level scattering amplitudes, leading to positivity constraints on determinants of Hankel matrices involving polynomials of multiple zeta values. These generalise certain constraints on polynomials of single zeta values in the mathematics literature.

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arXiv2019

Modular Forms and $SL(2, {\mathbb Z})$-covariance of type IIB superstring theory

The local higher-derivative interactions that enter into the low-energy expansion of the effective action of type IIB superstring theory with constant complex modulus generally violate the $U(1)$ R-symmetry of IIB supergravity by $q_U$ units. These interactions have coefficients that transform as non-holomorphic modular forms under $SL(2, {\mathbb Z})$ transformations with holomorphic and anti-holomorphic weights $(w,-w)$, where $q_U=-2w$. In this paper $SL(2, {\mathbb Z})$-covariance and supersymmetry are used to determine first-order differential equations on moduli space that relate the modular form coefficients of classes of BPS-protected maximal $U(1)$-violating interactions that arise at low orders in the low-energy expansion. These are the moduli-dependent coefficients of BPS interactions of the form $d^{2p} \mathcal{P}_n$ in linearised approximation, where $\mathcal{P}_n$ is the product of $n$ fields that has dimension $=8$ with $q_U=8-2n$, and $p=0$, $2$ or $3$. These first-order equations imply that the coefficients satisfy $SL(2, {\mathbb Z})$-covariant Laplace eigenvalue equations on moduli space with solutions that contain information concerning perturbative and non-perturbative contributions to superstring amplitudes. For $p=3$ and $n\ge 6$ there are two independent modular forms, one of which has a vanishing tree-level contribution. The analysis of super-amplitudes for $U(1)$-violating processes involving arbitrary numbers of external fluctuations of the complex modulus leads to a diagrammatic derivation of the first-order differential relations and Laplace equations satisfied by the coefficient modular forms. Combining this with a $SL(2, {\mathbb Z})$-covariant soft axio-dilaton limit that relates amplitudes with different values of $n$ determines most of the modular invariant coefficients, leaving a single undetermined constant.

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arXiv2018

Higher genus modular graph functions, string invariants, and their exact asymptotics

The concept and the construction of modular graph functions are generalized from genus-one to higher genus surfaces. The integrand of the four-graviton superstring amplitude at genus-two provides a generating function for a special class of such functions. A general method is developed for analyzing the behavior of modular graph functions under non-separating degenerations in terms of a natural real parameter $t$. For arbitrary genus, the Arakelov Green function and the Kawazumi-Zhang invariant degenerate to a Laurent polynomial in $t$ of degree $(1,1)$ in the limit $t\to\infty$. For genus two, each coefficient of the low energy expansion of the string amplitude degenerates to a Laurent polynomial of degree $(w,w)$ in $t$, where $w+2$ is the degree of homogeneity in the kinematic invariants. These results are exact to all orders in $t$, up to exponentially suppressed corrections. The non-separating degeneration of a general class of modular graph functions at arbitrary genus is sketched and similarly results in a Laurent polynomial in $t$ of bounded degree. The coefficients in the Laurent polynomial are generalized modular graph functions for a punctured Riemann surface of lower genus.

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