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Aspects of the S transformation Bootstrap

Enrico M. Brehm, Diptarka Das

TL;DR

This work develops two analytic bootstrap strategies in two-dimensional CFTs based on the $S$-modular transformation: an extreme-temperature approach that yields universal asymptotics for spectral data, OPE coefficients, and Zamolodchikov recursion coefficients, and an intermediate-temperature approach that constructs $S$-invariant quantities to derive bounds on the spectrum and on OPE data. The extreme-temperature analysis produces Cardy-type density of states, ETH-consistent off-diagonal matrix elements, and asymptotics for Zamolodchikov blocks, with explicit expressions and regime-of-validity criteria. The intermediate-temperature program leverages modular invariance to bound the spectrum (Hellerman-type bounds) and, under large central charge with non-negative diagonal OPEs, tightens bounds on gaps and certain OPE coefficients through linear functional methods and zero-variety analysis. Together, these techniques illuminate universal and model-dependent structures in 2D CFTs, with holographic interpretations via BTZ black hole physics and potential extensions to charged, higher-point, and larger symmetry algebras.

Abstract

We review and systematize two (analytic) bootstrap techniques in two-dimensional conformal field theories using the S-modular transformation. The first one gives universal results in asymptotic regimes by relating extreme temperatures. Along with the presentation of known results, we use this technique to also derive asymptotic formulae for the Zamolodchikov recursion coefficients which match previous conjectures from numerics and from Regge asymptotic analysis. The second technique focuses on intermediate temperatures. We use it to sketch a methodology to derive a bound on off-diagonal squared OPE coefficients, as well as to improve existing bounds on the spectrum in case of non-negative diagonal OPE coefficients.

Aspects of the S transformation Bootstrap

TL;DR

This work develops two analytic bootstrap strategies in two-dimensional CFTs based on the -modular transformation: an extreme-temperature approach that yields universal asymptotics for spectral data, OPE coefficients, and Zamolodchikov recursion coefficients, and an intermediate-temperature approach that constructs -invariant quantities to derive bounds on the spectrum and on OPE data. The extreme-temperature analysis produces Cardy-type density of states, ETH-consistent off-diagonal matrix elements, and asymptotics for Zamolodchikov blocks, with explicit expressions and regime-of-validity criteria. The intermediate-temperature program leverages modular invariance to bound the spectrum (Hellerman-type bounds) and, under large central charge with non-negative diagonal OPEs, tightens bounds on gaps and certain OPE coefficients through linear functional methods and zero-variety analysis. Together, these techniques illuminate universal and model-dependent structures in 2D CFTs, with holographic interpretations via BTZ black hole physics and potential extensions to charged, higher-point, and larger symmetry algebras.

Abstract

We review and systematize two (analytic) bootstrap techniques in two-dimensional conformal field theories using the S-modular transformation. The first one gives universal results in asymptotic regimes by relating extreme temperatures. Along with the presentation of known results, we use this technique to also derive asymptotic formulae for the Zamolodchikov recursion coefficients which match previous conjectures from numerics and from Regge asymptotic analysis. The second technique focuses on intermediate temperatures. We use it to sketch a methodology to derive a bound on off-diagonal squared OPE coefficients, as well as to improve existing bounds on the spectrum in case of non-negative diagonal OPE coefficients.

Paper Structure

This paper contains 31 sections, 105 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Basic setup
  3. Examples
  4. Partition function
  5. Grand canonical ensemble (GCE)
  6. Four-point function on the plane
  7. Conformal four-point blocks
  8. N-point functions on the torus
  9. Asymptotic results from extreme temperature
  10. Examples
  11. Partition function and density of states
  12. Thermal one-point function and averaged expectation values
  13. Thermal two-point function and averaged matrix elements
  14. GCE two-point function and averaged expectation values
  15. Four-point function on the plane and Zamolodchikov expansion
  16. ...and 16 more sections

Figures (3)

  • Figure 1: The $(h,\bar{h})$ plane. The yellow line corresponds to the boundary that separates possible positive and negative contributions to the one-point function of an operator $\mathbb{O}$ with $h_\mathbb{O}= \frac{c}{12}$ in a theory with only positive diagonal OPE coefficients at $c = 2000$. There must be some contribution from the region bellow it. The region is contained in the region bound by $\Delta = h+\bar{h} = \frac{c}{10.38}$.
  • Figure 2: Left: Zero lines of \ref{['eq:boundfrom3a1']} in the positive quadrant for $c=3, \Delta_\mathbb{O} = 1/3,$ and $a_1 = 6, a_3=-3,a_5 =1$. Right: The bounds (y value of the black lines) (from \ref{['eq:boundfrom-a']}) and zero lines in the positive quadrant for $c=3, \Delta_\mathbb{O} = 1/3,$ and $a$ as given in \ref{['eq:aFor2pt']}.
  • Figure 3: Plot of \ref{['eq:boundfrom-a']} for $c=1, \Delta_\mathbb{O} = 1/3,$ and $a = 1$. The function is $<0$ within the shaded region. This exemplary plot, in particular, serves to show the appearance of an "island". However, here it appears together with a non-compact region, s.t. the proposed analysis cannot be applied.