Laplace-difference equation for integrated correlators of operators with general charges in $\mathcal{N}=4$ SYM

Abstract

We consider the integrated correlators associated with four-point correlation functions $⟨{\mathcal{O}}_2{\mathcal{O}}_2{\mathcal{O}}_p^{(i)}{\mathcal{O}}_p^{(j)}⟩$ in four-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills theory (SYM) with $SU(N)$ gauge group, where ${\mathcal{O}}_p^{(i)}$ is a superconformal primary with charge (or dimension) $p$ and the superscript $i$ represents possible degeneracy. These integrated correlators are defined by integrating out spacetime dependence with a certain integration measure, and they can be computed via supersymmetric localisation. They are modular functions of complexified Yang-Mills coupling $\tau$. We show that the localisation computation is systematised by appropriately reorganising the operators. After this reorganisation of the operators, we prove that all the integrated correlators for any $N$, with some crucial normalisation factor, satisfy a universal Laplace-difference equation (with the laplacian defined on the $\tau$-plane) that relates integrated correlators of operators with different charges. This Laplace-difference equation is a recursion relation that completely determines all the integrated correlators, once the initial conditions are given.

arXiv comment30 pages; v2: references added, typos corrected; v3: typos corrected, version published in JHEP

References in corpus (46)

• Localization of gauge theory on a four-sphere and supersymmetric Wilson loops
• Mellin amplitudes for $Ad{S}_5\times S^5$
• How to Succeed at Holographic Correlators Without Really Trying
• Superconformal Ward Identities and their Solution
• Gravitational S-matrix from CFT dispersion relations
• Loop Corrections to Supergravity on $Ad{S}_5\times S^5$
• $\mathcal{N}=4$ Super-Yang-Mills Correlators at Strong Coupling from String Theory and Localization
• Partial non-renormalisation of the stress-tensor four-point function in N=4 SYM and AdS/CFT
• Correlation Functions of Coulomb Branch Operators
• All Tree-Level Correlators in AdS${}_5\times$S${}_5$ Supergravity: Hidden Ten-Dimensional Conformal Symmetry
• Quantum Gravity from Conformal Field Theory
• Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N=4 SYM
• A non-renormalization theorem for chiral primary 3-point functions
• Amplitudes and Correlators to Ten Loops Using Simple, Graphical Bootstraps
• Genus-One String Amplitudes from Conformal Field Theory
• Far Beyond the Planar Limit in Strongly-Coupled $\mathcal{N}=4$ SYM
• $t{t}^{\ast }$ equations, localization and exact chiral rings in 4d N=2 SCFTs
• Massive N=2 Gauge Theories at Large N
• Loop corrections for Kaluza-Klein AdS amplitudes
• Genus-2 Holographic Correlator on $Ad{S}_5\times S^5$ from Localization
• Modular Invariance in Superstring Theory From $\mathcal{N}=4$ Super-Yang-Mills
• Exact properties of an integrated correlator in $\mathcal{N}=4$ $SU(N)$ SYM
• On exact correlation functions in SU(N) $\mathcal{N}=2$ superconformal QCD
• New Modular Invariants in $\mathcal{N}=4$ Super-Yang-Mills Theory
• One-loop amplitudes in $Ad{S}_5\times S^5$ supergravity from $\mathcal{N}=4$ SYM at strong coupling
• All three-loop four-point correlators of half-BPS operators in planar N=4 SYM
• A novel representation of an integrated correlator in $\mathcal{N}=4$ SYM theory
• Near-extremal correlators and vanishing supergravity couplings in AdS/CFT
• Harnessing S-Duality in $\mathcal{N}=4$ SYM & Supergravity as $SL(2,\mathbb{Z})$-Averaged Strings
• AdS Virasoro-Shapiro from dispersive sum rules
• Single Particle Operators and their Correlators in Free $\mathcal{N}=4$ SYM
• String corrections to AdS amplitudes and the double-trace spectrum of N=4 SYM
• Towards the Virasoro-Shapiro Amplitude in AdS5xS5
• Maximal $U(1{)}_Y$-violating $n$-point correlators in $\mathcal{N}=4$ super-Yang-Mills theory
• Wilson Loops and Chiral Correlators on Squashed Sphere
• The Virasoro-Shapiro amplitude in AdS${}_5\times$S${}^5$ and level splitting of 10d conformal symmetry
• Modular invariant holographic correlators for $\mathcal{N}=4$ SYM with general gauge group
• Four-point functions of 1/2-BPS operators of any weights in the supergravity approximation
• All five-loop planar four-point functions of half-BPS operators in $\mathcal{N}=4$ SYM
• Modular-invariant large-$N$ completion of an integrated correlator in $\mathcal{N}=4$ supersymmetric Yang-Mills theory
• Integrated correlators in $\mathcal{N}=4$ super Yang-Mills and periods
• 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
• Non-planar data of $\mathcal{N}=4$ SYM
• One-loop string corrections for AdS Kaluza-Klein amplitudes
• A wide class of four point functions of BPS operators in N=4 SYM at order g^4
• Large $N$ expansion of an integrated correlator in $\mathcal{N}=4$ SYM

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• Integrated correlators at strong coupling in an orbifold of $\mathcal{N}=4$ SYM
• Integrated correlators with a Wilson line in a $\mathcal{N}=2$ quiver gauge theory at strong coupling
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