Holographic Equidistribution

Hecke operators acting on modular functions arise naturally in the context of 2d conformal field theory, but in seemingly disparate areas, including permutation orbifold theories, ensembles of code CFTs, and more recently in the context of the AdS$_3$/RMT$_2$ program. We use an equidistribution theorem for Hecke operators to show that in each of these large $N$ limits, an entire heavy sector of the partition function gets integrated out, leaving only contributions from Poincaré series of light states. This gives an immediate holographic interpretation as a sum over semiclassical handlebody geometries. We speculate on further physical interpretations for equidistribution, including a potential ergodicity statement.

💡 Research Summary

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The paper “Holographic Equidistribution” investigates the role of SL(2,ℤ) Hecke operators in two‑dimensional conformal field theories (CFTs) and their holographic duals in AdS₃. The authors start by recalling that low‑dimensional holography often requires an ensemble of boundary theories rather than a single CFT, a phenomenon observed in JT gravity and recent AdS₃/RMT₂ studies. In many constructions—permutation orbifolds, code CFTs built from Narain lattices, and other large‑N models—Hecke operators naturally appear when expressing the orbifolded partition function.

Section 2 reviews the geometry of the upper‑half plane H², the fundamental domain F = H²/SL(2,ℤ), and the spectral decomposition of L²(F) in terms of Maass cusp forms νₙ(τ) and real‑analytic Eisenstein series Eₛ(τ). Both families are simultaneous eigenfunctions of the Laplacian and the Hecke operators T_N. The eigenvalues are given by explicit divisor‑sum formulas for Eisenstein series, a(s)N = σ{2s‑1}(N) N^{-s}, while the cusp‑form eigenvalues bₙ(N) have no closed form but are known numerically to follow a Sato‑Tate (Wigner semicircle) distribution for large N.

The crucial mathematical input is the equidistribution theorem for Hecke points: for any square‑integrable modular function f∈L²(F) and any ε>0, \