Bootstrapping 2D CFTs in the Semiclassical Limit

TL;DR

This work develops a analytic framework for 2D CFTs in the semiclassical (large c) limit, where crossing symmetry reduces to a saddle-dominated Virasoro-block expansion. It proves universality relations for heavy-primary structure constants when the light spectrum is sparse, and demonstrates these ideas in Liouville theory, product orbifolds, and meromorphic CFTs, yielding a semiclassical Hellerman bound and a logarithmically corrected Cardy formula for h ≥ c/12. The Z2 twist-field analysis in symmetric product orbifolds ties twist correlators to torus partition functions, clarifying Renyi entropies and phase transitions in holographic settings. The authors further connect these CFT results to bulk AdS3 gravity via worldline actions, discuss heavy-light and light-limit bulk duals, and explore implications for entanglement and extremal CFTs, suggesting a broad holographic interpretation for classical Virasoro blocks beyond probe limits.

Abstract

We study two dimensional conformal field theories in the semiclassical limit. In this limit, the four-point function is dominated by intermediate primaries of particular weights along with their descendants, and the crossing equations simplify drastically. For a four-point function receiving sufficiently small contributions from the light primaries, the structure constants involving heavy primaries follow a universal formula. Applying our results to the four-point function of the $\mathbb Z_2$ twist field in the symmetric product orbifold, we produce the Hellerman bound and the logarithmically corrected Cardy formula that is valid for $h \geq c/12$.

Figures (7)

• Figure 1: The ratios $\widehat{m}(m_{ext}) \over m_{ext}$ and $\widehat{m}_{\overline{vac}}(m_{ext}|1/2) \over 2m_{ext}$ as functions of the external weight $m_{ext}$. See \ref{['hatmm']} and \ref{['hatmvac']} for definitions.
• Figure 2: The weight $\widehat{m}_{\overline{vac}}(m_{ext}, x)$ as a function of the cross ratio $x$ for external weights $m_{ext} = {\alpha}/24$. See \ref{['hatmvac']} for a definition. The curves from top to bottom are for ${\alpha} = 1/100, 1/10, 1/2, 1, 2, 12$.
• Figure 3: The universal classical $p_{\sigma_{ext}}(m)$ and one-loop $q^{cont}_{\sigma_{ext}}(m)$ parts of the structure constants as functions of the internal weight $m$, for external weights $m_{ext} = {\alpha}/24$. See \ref{['UniversalC']} for definitions. The curves from top to bottom in both (a) and (b) are for ${\alpha} = 1/100, 1/10, 1/2, 1, 2, 12$.
• Figure 4: The ground state $m_{ext} = 1/24$ four-point function in Liouville theory. The dashed lines plot the classical branching ratio $S_{\sigma_{ext}}(m|x)$ (defined in \ref{['BranchingRatio']}) as a function of the internal weight $m$, for cross ratios $x = 10^{({\alpha}-5)/10}/2$ with ${\alpha} = 0, 1, \dotsc, 5$ from bottom to top. The solid line traces the dominant saddle as the cross ratio is varied. The dominant saddle is at $m = 1.32 \, m_{ext}$ (semiclassical: $\widehat{m}(m_{ext}) = 1.32 \, m_{ext}$) at the crossing symmetric point.
• Figure 5: The $\sigma^{64}$ ($m_{ext} = 1/8$) four-point function in the product Ising model. The dashed lines plot the classical branching ratio $S_{\sigma_{ext}}(m|x)$ (defined in \ref{['branch']}) for scalars as a function of the internal weight $m$, for cross ratios $x = 10^{({\alpha}-5)/10}/2$ with ${\alpha} = 0, 1, \dotsc, 5$ from bottom to top. The solid line traces the dominant saddle as the cross ratio is varied. At the crossing symmetric point, the dominant weight is at $m = 1.12 \, m_{ext}$. It further approaches the semiclassical value $\widehat{m}(m_{ext}) = 1.24 \,m_{ext}$ as the number of copies is increased.

Theorems & Definitions (7)

• Proposition 1
• proof
• Lemma 1
• proof
• Proposition 2
• Proposition 3
• Proposition 4