Coarse-graining in time, emergent temperature in 2d CFT at large central charge

TL;DR

This work analyzes two light operators on time-dependent heavy states in large-$c$ 2d CFTs using the monodromy method to obtain a compact identity-block contribution $Q(z_1,z_2)=\big(\rho_+(z_1)\rho_-(z_2)-\rho_+(z_2)\rho_-(z_1)\big)^{-2h}$. Analytic continuation to Lorentzian time reveals a branch structure tied to lightcone crossings, which, through Floquet theory for Hill’s equation with a real periodic potential, yields late-time exponential decay and an emergent effective temperature $T_{|V\rangle}=|\alpha|/\pi$. The analysis shows a coarse-graining of the heavy-state data to a single Virasoro coadjoint orbit, linking the Floquet exponent to an orbit invariant and to a high-weight CFT representation, with the projection supported by conformal bootstrap results for specific heavy states. These results illuminate a 2d-CFT no-hair-like behavior at large $c$, where long-time dynamics are governed by a minimal set of heavy-state data and a universal thermal-like response in the light sector. The framework provides a concrete bridge between monodromy methods, Lorentzian evolution, Floquet dynamics, and Virasoro representation theory with bootstrap consistency checks, offering a pathway to analyze thermalization and chaos diagnostics in holographic and non-holographic CFTs.

Abstract

Using the monodromy method, a compact expression is obtained for the identity block contribution to the expectation value of two low-energy probe operators on a broad class of time-dependent heavy pure states in large-$c$ 2d CFTs. It will be shown that the size of the compact spatial dimension sets a coarse-graining time-scale above which the thermal two-point function naturally emerges. A phenomena will be highlighted where the information of the pure state appears to get projected down to a single conformal representation. We will see that this projection is corroborated by known conformal bootstrap results.

Figures (10)

• Figure 1: Restricting the heavy operator to lie within a thin annulus of width $2\sigma$ centered around the unit circle gives us a controllable parameter with which the identity block dominance of the Euclidean correlator can be controlled.
• Figure 2: Deforming the contour around the two light operators to the dumbbell contour.
• Figure 3: Visual representation of the time-ordering of the Euclidean correlator $\langle V|O_LO_L|V\rangle$ on the Euclidean cylinder.
• Figure 4: The Lorentzian time evolution of the operators $O(t_1)$ and $O(t_2)$ that takes the Eucldian correlator to the Lorentzian expectation value $\Phi(t_1,t_2)$.
• Figure 5: Left: The Lorentzian cylinder is obtained by identifying the two boundaries. The lightcone structure of the initial state is represented by the dashed lines. The red dashed lines represent the rightmoving lightcones and the blue dashed lines the leftmoving lightcones. Right: Turning on Lorentzian time-evolution by analytically continuing $z$ while keeping $\bar{z}$ fixed results in the following trajectories on the Minkowski diagram. Every time the light operator crosses a lightcone of the heavy sector results in a half-monodromy picked up by continuing the Euclidean correlator across a branch cut.