Exact properties of an integrated correlator in N = 4 SU(N) SYM

• Published in: The journal of high energy physics Vol. 2021; no. 5
• Main Authors: Dorigoni, Daniele, Green, Michael B., Wen, Congkao
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 12.05.2021; Springer Nature B.V
• Subjects: Applied mathematics; Asymptotic series; Classical and Quantum Gravitation; Correlation; Correlators; Difference equations; Elementary Particles; Field theory; High energy physics; Infinite series; Instantons; Localization; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; String Theory; Supersymmetry; Symmetry; Yang-Mills theory; Supersymmetry and Duality; Conformal Field Theory; 1/N Expansion; Nonperturbative Effects
• ISSN: 1029-8479
• DOI: 10.1007/JHEP05(2021)089

A bstract We present a novel expression for an integrated correlation function of four superconformal primaries in SU( N ) N = 4 supersymmetric Yang-Mills ( N = 4 SYM) theory. This integrated correlator, which is based on supersymmetric localisation, has been the subject of several recent developments. In this paper 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 = θ / 2 π + 4 πi / g YM 2 . In this form it is manifestly invariant under SL(2 , ℤ) Montonen-Olive duality. Furthermore, it satisfies a remarkable Laplace-difference equation that relates the SU( N ) correlator to the SU( N + 1) and SU( N − 1) correlators. For any fixed value of N the correlator can be expressed as an infinite series of non-holomorphic Eisenstein series, E s τ τ ¯ with s ∈ ℤ, and rational coefficients that depend on the values of N and s . The perturbative expansion of the integrated correlator is an asymptotic but Borel summable series, in which the n -loop coefficient of order ( g YM / π ) 2 n is a rational multiple of ζ (2 n + 1). The n = 1 and n = 2 terms agree precisely with results determined directly by integrating the expressions in one-loop and two-loop perturbative N = 4 SYM field theory. Likewise, the charge- k instanton contributions (| k | = 1 , 2 , . . . ) 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 − ℓ ( ℓ ∈ ℤ) with coefficients that are rational sums of E s τ τ ¯ with s ∈ ℤ + 1/2. This gives an all orders derivation of the form of the recently conjectured expansion. We further consider the ’t Hooft topological expansion of large- N Yang-Mills theory in which λ = g YM 2 N is fixed. The coefficient of each order in the 1 /N expansion can be expanded as a series of powers of λ that converges for |λ| < π 2 . For large λ this becomes an asymptotic series when expanded in powers of 1 / λ with coefficients that are again rational multiples of odd zeta values, in agreement with earlier results and providing new ones. We demonstrate that the large- λ series is not Borel summable, and determine its resurgent non-perturbative completion, which is O exp − 2 λ .