Holomorphic structure of massive scalar fields in $\text{(A)dS}_2$

TL;DR

This work uncovers a holomorphic splitting for discrete-series scalar fields in $( ext{A)dS}_{2}$ with integer scaling dimensions $\Delta=k+1$, realized through holomorphic currents $F$ and $\overline{F}$ and a mode-expansion framework that mirrors 2d CFT structures. It demonstrates that the full symmetry content is captured by a chiral algebra, a subalgebra of the massless theory when $k>0$, and identifies Virasoro-like generators that act non-locally for $k>0$. The authors classify linear and nonlinear (chiral) symmetries, construct their algebraic structure, and show the existence of infinite families of integrable deformations for $k\in\{0,1,2\}$, while arguing that no such deformations exist for $k>2$ in line with a related integrable-hierarchy conjecture. These results illuminate how holomorphicity constrains the dynamics and symmetry of massive scalars in curved spacetimes and point to connections with higher-spin gravity and potential worldsheet or string-theoretic realizations. Overall, the paper advances the understanding of how holomorphic currents govern the quantisation, symmetry, and integrability of discrete-series scalars in $ ext{(A)dS}_{2}$.

Abstract

Scalar field theories in $\text{(A)dS}_{2}$ with integer scaling dimensions $Δ= k+1$ are characterised by the existence of a pair of (anti-)holomorphic higher-spin currents. We explore the consequences of this to describe their quantisation and subsets of their linear and non-linear symmetries, taking care to treat $\text{AdS}_{2}$ and $\text{dS}_{2}$ separately. In particular, we point out that the theories admit mode expansions reminiscent of standard two-dimensional conformal field theories in complex coordinates, with which we are able to construct operators implementing global conformal and Virasoro symmetry. We further leverage holomorphicity of the currents to show that the full set of symmetries of theories with $k>0$ is captured by a chiral algebra, which is a subalgebra of the one in the $k=0$ (massless) theory. This allows us to identify integrable deformations for $k \in \{0,1,2\}$. We finally observe that a lack of integrable deformations for $k>2$ is a consequence of a known conjecture.

Figures (4)

• Figure 1: $\text{(E)AdS}_{2}$ in different coordinate systems. Depicted on the left is the Penrose diagram of describing the conformal compactification of Lorentzian $\text{AdS}_{2}$ with metric \ref{['eq: conformal ads metric']}. After analytic continuation and under the transformation \ref{['eq: complex coordinates ads']}, this is mapped to the UHP, depicted on the right. The two disjoint conformal boundaries of $\text{AdS}_{2}$ at $\rho = \pm \pi/2$ in conformal coordinates are mapped to the negative and positive parts of the real axis, respectively in red and blue.
• Figure 2: The action of the elements in the $\text{dS}_{2}$ isometry group $\text{SO}(3)$ on the subspaces of the solutions to the $\text{dS}_{2}$ KG equation \ref{['eq: ds kg eq']}. Note that elements in $\mathcal{Z}$ are mixed into $\mathcal{V}_{+}$ and $\mathcal{V}_{-}$ under the isometries.
• Figure 3: $\text{(E)dS}_{2}$ in different coordinate systems. Depicted on the left is the Penrose diagram of (the conformal compactification of) Lorentzian $\text{dS}_{2}$ in conformal coordinates with metric \ref{['eq: conformal ds metric']}. Under the transformation \ref{['eq: complex coordinates ds']}, this is mapped to the extended complex plane, depicted on the right.
• Figure 4: Action of various generators on the modes in mode expansion \ref{['eq: UHP F mode expansion']} of the holomorphic current $F(z)$. Included in grey are the modes which are non-normalisable (in $\text{AdS}_{2}$) and are excluded from the mode expansion. Shown in blue, below the expression, is the action of the $\text{PSL}(2,\mathbb{R})$ generator $\ell_{-1}$---this does not involve the modes in the gap. An extension of the global to local conformal transformations would require the existence of an operator such as $\ell_{-2}$, whose action on the mode expansion is indicated by orange dotted arrows above the expression. This excites non-normalisable modes in the gap, e.g. $\delta_{z^{-1}}z=kz^{-1}$.

Theorems & Definitions (2)

• proof
• proof