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Rademacher Expansions and the Spectrum of 2d CFT

Luis F. Alday, Jin-Beom Bae

TL;DR

The paper extends the Rademacher circle method to non-holomorphic 2d CFT partition functions with $c>1$ and no extended chiral algebra, constructing a convergent spectral density $\rho_R(e,j)$ that encodes the light spectrum while respecting modular invariance. It shows that $\rho_R(e,j)$ converges for all nonzero spin $|j|$ and reproduces Cardy-like asymptotics at large $e$, while matching a Maloney–Witten–Keller–type Poincaré density at finite spin up to a modular-ambiguity term arising from uncensored seeds. The work clarifies the relation between Rademacher and MWK constructions, highlighting a physically meaningful ambiguity that corresponds to extra uncensored operators and oscillatory contributions, and discusses the problematic negative-norm densities that arise in the pure gravity dual and potential remedies. Overall, the approach provides a modularly consistent framework to constrain the spectrum of 2d CFTs from first principles and suggests avenues to regulate negativity and pursue a discrete spectrum at finite central charge and spin.

Abstract

A classical result from analytic number theory by Rademacher gives an exact formula for the Fourier coefficients of modular forms of non-positive weight. We apply similar techniques to study the spectrum of two-dimensional unitary conformal field theories, with no extended chiral algebra and $c>1$. By exploiting the full modular constraints of the partition function we propose an expression for the spectral density in terms of the light spectrum of the theory. The expression is given in terms of a Rademacher expansion, which converges for spin $j \neq 0$. For a finite number of light operators the expression agrees with a variant of the Poincare construction developed by Maloney, Witten and Keller. With this framework we study the presence of negative density of states in the partition function dual to pure gravity, and propose a scenario to cure this negativity.

Rademacher Expansions and the Spectrum of 2d CFT

TL;DR

The paper extends the Rademacher circle method to non-holomorphic 2d CFT partition functions with and no extended chiral algebra, constructing a convergent spectral density that encodes the light spectrum while respecting modular invariance. It shows that converges for all nonzero spin and reproduces Cardy-like asymptotics at large , while matching a Maloney–Witten–Keller–type Poincaré density at finite spin up to a modular-ambiguity term arising from uncensored seeds. The work clarifies the relation between Rademacher and MWK constructions, highlighting a physically meaningful ambiguity that corresponds to extra uncensored operators and oscillatory contributions, and discusses the problematic negative-norm densities that arise in the pure gravity dual and potential remedies. Overall, the approach provides a modularly consistent framework to constrain the spectrum of 2d CFTs from first principles and suggests avenues to regulate negativity and pursue a discrete spectrum at finite central charge and spin.

Abstract

A classical result from analytic number theory by Rademacher gives an exact formula for the Fourier coefficients of modular forms of non-positive weight. We apply similar techniques to study the spectrum of two-dimensional unitary conformal field theories, with no extended chiral algebra and . By exploiting the full modular constraints of the partition function we propose an expression for the spectral density in terms of the light spectrum of the theory. The expression is given in terms of a Rademacher expansion, which converges for spin . For a finite number of light operators the expression agrees with a variant of the Poincare construction developed by Maloney, Witten and Keller. With this framework we study the presence of negative density of states in the partition function dual to pure gravity, and propose a scenario to cure this negativity.

Paper Structure

This paper contains 15 sections, 1 theorem, 117 equations, 5 figures.

Table of Contents

  1. Introduction
  2. From asymptotic to convergent expansions
  3. A toy model
  4. From asymptotic to convergent series
  5. Modular transformations
  6. Rademacher's circle method
  7. Examples
  8. Application to 2d CFT
  9. Constraints on the spectrum
  10. Ambiguities, comparison to MWK and negative norms
  11. Ambiguities
  12. Comparison to MWK
  13. Negative densities
  14. Conclusions
  15. MWK density of states

Key Result

Theorem 1

Suppose $Z(q) = \sum_{n=0}^\infty a_n q^n$ is a power series, analytic for $|q|<1$ and $q \not\in \mathbb{R}_{\leq 0}$, such that it satisfies the following two conditions. First as $z=x+i y \to 0$ in the arc $Arg[z]< \delta$, with $\delta<\pi/2$, and $\alpha,\kappa,\alpha_s$ real coefficients. Here the dots denote contributions exponentially suppressed with respect to the leading one. Second fo

Figures (5)

  • Figure 1: Ford circles corresponding to the Farey sequences of order three (left) and five (right)
  • Figure 2: For $N=3$ we deform the contour as shown in the figure and write $\Gamma = \Gamma_{01} \cup \Gamma_{13} \cup \Gamma_{12} \cup \Gamma_{23} \cup \Gamma_{11}$.
  • Figure 3: $\Gamma_{01} \cup \Gamma_{01}$ tend to the circle $C'_{01}$, shown in the figure, as $N \to \infty$.
  • Figure 4: On the left we see the structure of essential singularities in the $x-$plane, at all points of the form $x = \pm i y + \mathbb{Q}$. On the right we have performed a Rademacher deformation as to isolate only two of those, at $x = \pm i y$.
  • Figure 5: The Rademacher contour for a single pole (left) is mapped to a straight in the $w-$plane (right).

Theorems & Definitions (1)

  • Theorem 1