Abstract
We apply the theory of harmonic analysis on the fundamental domain of SL(2, ℤ) to partition functions of two-dimensional conformal field theories. We decompose the partition function of c free bosons on a Narain lattice into eigenfunctions of the Laplacian of worldsheet moduli space ℍ/SL(2, ℤ), and of target space moduli space O(c, c; ℤ)\\O(c, c; ℝ)/O(c)×O(c). This decomposition manifests certain properties of Narain theories and ensemble averages thereof. We extend the application of spectral theory to partition functions of general two-dimensional conformal field theories, and explore its meaning in connection to AdS3 gravity. An implication of harmonic analysis is that the local operator spectrum is fully determined by a certain subset of degeneracies.
Highlights
It is often illuminating to ask how a quantum field theory behaves when it is deformed
The first is to use the fact that conformal symmetry together with the operator product expansion (OPE) allow one to define conformal field theory (CFT) in terms of a discrete set of “data” — typically, operator dimensions and OPE coefficients — so that, roughly speaking, the space of CFTs is the space of such data
One of the main advantages of the spectral representation is that it naturally separates the partition function into a piece that is constant over the moduli space, and residual pieces whose averages on the moduli space vanish
Summary
It is often illuminating to ask how a quantum field theory behaves when it is deformed. While one might reasonably have expected the same to be true of all c, we show that at c = 2, as well as at c > 2 in regions of the moduli space where we can perform the computation, the overlap with the cusp forms does not vanish but instead has an essentially closed form: in a sense to be made precise, the resulting overlap is equal to the number 8 for all cusp forms This closed-form relation between the spectrum of free bosons on a Narain lattice, which is an integrable model, and cusp forms, which exhibit chaotic properties, deserves further study. The partition function Z(c) of a fixed Narain lattice CFT is a sum of the square-integrable terms, which average to zero, and the ensemble average with respect to the Zamolodchikov metric, Z(c) ; in harmony with the above paradigm, the known result [20, 21] for the average Z(c) turns out to be precisely equal to the modular completion, via Poincaré sum, of the vacuum state. We will denote the real and imaginary parts of the torus modular parameter τ as x and y, respectively
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