Complexity of warped conformal field theory

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 $k$.

34 pages, New references added; revised discussions of Virasoro-Kac-Moody circuits in section 3 although the final conclusions remain unchanged; updated discussion of complexity for warped conformal field theory. Version accepted for publication in EPJC

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