Berry Phases on Virasoro Orbits

TL;DR

This work shows that unitary Virasoro representations carry exact Berry phases arising from tracing closed conformal-transform paths on Virasoro coadjoint orbits, computable via the Maurer–Cartan form. When left- and right-moving sectors are combined, these phases describe boundary gravitons in AdS$_3$ and amount to a gravitational generalization of Thomas precession, recoverable as a sum of SL$(2,\mathbb{R})$ phases in suitable limits. The main result is an explicit expression for the Virasoro Berry phase in terms of the path $f(t)$ and the weights $(h,c)$, with particular emphasis on circular paths and superboosts, yielding interpretable quantities such as geometric actions and memory. These phases probe the geometry of the infinite-dimensional Virasoro coadjoint orbits and connect to gravitational memory through Brown–Henneaux diffeomorphisms and the space of boundary vacua, suggesting avenues for realization in driven CFTs and extensions to broader symmetry algebras.

Abstract

We point out that unitary representations of the Virasoro algebra contain Berry phases obtained by acting on a primary state with conformal transformations that trace a closed path on a Virasoro coadjoint orbit. These phases can be computed exactly thanks to the Maurer-Cartan form on the Virasoro group, and they persist after combining left- and right-moving sectors. Thinking of Virasoro representations as particles in AdS_3 dressed with boundary gravitons, the Berry phases associated with Brown-Henneaux diffeomorphisms provide a gravitational extension of Thomas precession.

Figures (3)

• Figure 1: A path $f(t)$ in $G$ such that $f(T)=f(0)\cdot h$ with $h\in G_{\phi}$ is mapped on a closed curve in $G/G_{\phi}$ by the projection $\pi:G\rightarrow G/G_{\phi}:f\mapsto f\cdot G_{\phi}$. When $G/G_{\phi}$ is a coadjoint orbit, the Berry phase along $f(t)$ coincides with the flux of the Kirillov-Kostant symplectic form through any surface enclosed by the projected path on $G/G_{\phi}$. For $f(t)=f(0)\cdot h(t)$ with $h(t)\in G_{\phi}$, the projected path is a point and the Berry phase vanishes.
• Figure 2: A circle with uniformly spaced dots is acted upon by a superboost (\ref{['s75']}) of order $n=3$ with positive $\lambda$. The dots converge towards the points $\varphi=0,2\pi/3,4\pi/3$ and move away from $\varphi=\pi/3,\pi,5\pi/3$. Analogous observations apply to all superboosts.
• Figure 3: Boundary gravitons in AdS$_3$ span an infinite-dimensional manifold ${\cal M}\cong\text{Diff}\,S^1/\text{PSL}(2,\mathbb{R})\times\text{Diff}\,S^1/\text{PSL}(2,\mathbb{R})$. Any loop $\gamma$ in ${\cal M}$ encloses a two-dimensional surface embedded in ${\cal M}$. The symplectic flux through that surface coincides with the Berry phase picked up by boundary gravitons as they undergo a family of conformal transformations whose projection on ${\cal M}$ is the path $\gamma$.