We perform a systematic study of integrated four-point functions of half-BPS operators in four-dimensional N = 4 super Yang-Mills theory with gauge group SU(N). These observables, defined by a certain spacetime integral of (O2O2OPOP) where O-P is a superconformal primary of charge p, are known to be computable by supersymmetric localization, yet are non-trivial functions of the complexified gauge coupling tau. We find explicit and remarkably simple results for several classes of these observables, exactly as a function of N and tau. Their physical and formal properties are greatly illuminated upon employing the SL(2, Z) spectral decomposition: in this S-duality-invariant eigenbasis, the integrated correlators are fixed simply by polynomials in the spectral parameter. These polynomials are determined recursively by linear algebraic equations relating different N and p, such that all integrated correlators are ultimately fixed in terms of the integrated stress tensor multiplets in the SU(2) theory. Our computations include the full matrix of integrated correlators at low values of p, and a certain infinite class involving operators of arbitrary p. The latter satisfy an open lattice chain equation for all N, reminiscent of the Toda equation obeyed by extremal correlators in N = 2 superconformal theories. We compute ensemble averages of these observables and analyze our solutions at large N, confirming and predicting features of semiclassical AdS(5) x S-5 supergravity amplitudes.
