Complexity of warped conformal field theory

TL;DR

This paper explores the complexity of warped conformal field theories (WCFTs) using holographic volume complexity and circuit complexity based on Virasoro-Kac-Moody symmetries, revealing linear divergence structures and scaling behaviors in the semiclassical limit.

Contribution

It introduces a detailed analysis of WCFT complexity through holographic and group-theoretic methods, highlighting their divergence structures and scaling properties.

Findings

Holographic volume complexity exhibits a linear UV divergence similar to local CFT2.

Circuit complexity based on symmetry gates scales linearly with time in special solutions.

Both complexities scale linearly with the Kac-Moody level parameter k in the semiclassical limit.

Abstract

Warped conformal field theories in two dimensions are exotic nonlocal, Lorentz violating field theories characterized by Virasoro-Kac-Moody symmetries and have attracted a lot of attention as candidate boundary duals to warped AdS$_3$ spacetimes, thereby expanding the scope of holography beyond asymptotically AdS spacetimes. Here we investigate WCFT$_2$\,s using \emph{circuit complexity} as a tool. First we compute the holographic volume complexity (CV) which displays a linear UV divergence structure, more akin to that of a local CFT$_2$ and has a very complicated dependence on the Virasoro central charge $c$ and the $U(1)$ Kac-Moody level parameter $k$. Next we consider circuit complexity based on Virasoro-Kac-Moody symmetry gates where the complexity functional is the geometric (group) action on coadjoint orbits of the Virasoro-Kac-Moody group. We consider a special solution to extremization equations for which complexity scales linearly with ``time''. In the semiclassical limit (large $c,k$, while $c/k$ remains finite and small) both the holographic volume complexity and circuit complexity scales linearly with $k$.