Aspects of the S transformation Bootstrap

TL;DR

This paper reviews and develops bootstrap techniques based on the S-modular transformation in 2D conformal field theories, deriving asymptotic formulas and bounds for OPE coefficients and spectrum.

Contribution

It introduces a systematic approach to bootstrap methods using the S-transformation, deriving new asymptotic formulas and bounds in 2D CFTs.

Findings

Derived asymptotic formulas for Zamolodchikov recursion coefficients

Established bounds on off-diagonal squared OPE coefficients

Improved bounds on the spectrum for non-negative diagonal OPE coefficients

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.