Late time Wilson lines

TL;DR

This work advances the understanding of Virasoro conformal blocks at large central charge in the late-time regime by employing the Virasoro Wilson line to resum all $t/c$ dependence for light-light blocks. The main result is a simple late-time block where each double-trace level $n$ acquires a shift $\gamma_n/2$ in the energy, with $\gamma_n = -{12\over c}[C_2(h+h'+n)-C_2(h)-C_2(h')]$, leading to a rich, decaying-and-recurring time dependence. Through the Lorentzian inversion formula, the authors connect this late-time behavior to gravitational contributions to anomalous dimensions of double-trace operators, obtaining $\gamma_{n,\overline{n}} = -{12\over c}[C_2(h+h'+\min(n,\overline{n}))-C_2(h)-C_2(h')]$ and showing how vacuum-block results compare to full correlator data. The work illustrates how Wilson-line techniques encapsulate stress-tensor exchanges and provides a concrete, pedagogical example for inversion-formula machinery, while outlining clear directions to extend to heavy-light blocks and non-vacuum sectors.

Abstract

In the AdS$_3$/CFT$_2$ correspondence, physical interest attaches to understanding Virasoro conformal blocks at large central charge and in a kinematical regime of large Lorentzian time separation, $t\sim c$. However, almost no analytical information about this regime is presently available. By employing the Wilson line representation we derive new results on conformal blocks at late times, effectively resumming all dependence on $t/c$. This is achieved in the context of "light-light" blocks, as opposed to the richer, but much less tractable, "heavy-light" blocks. The results exhibit an initial decay, followed by erratic behavior and recurrences. We also connect this result to gravitational contributions to anomalous dimensions of double trace operators by using the Lorentzian inversion formula to extract the latter. Inverting the stress tensor block provides a pedagogical example of inversion formula machinery.

Figures (3)

• Figure 1: Plot of Wilson line correlator at early time. We plot the real part of $\ln \left( V_{hh'}(t) \over V_{hh'(0)}\right)$ versus ${6t\over c}$, with the parameter choices $h=3.23$, $h'=4.91$, $r=0.9$.
• Figure 2: Same parameters as in Fig. \ref{['fig:fa']} but now over larger time range
• Figure 3: Same parameters as in Fig. \ref{['fig:fa']} but now at late time