Revealing the conformal symmetry of the discrete series scalars in dS${}_2$

Technical Summary: Revealing the conformal symmetry of the discrete series scalars in dS$_2$

Problem Statement
The paper addresses the realization of global conformal symmetry for a specific class of free massive scalar fields $\phi$ in two-dimensional constant curvature spacetimes, specifically de Sitter space (dS$_2$) and Anti-de Sitter space (AdS$_2$). These fields possess a mass squared parameter $m_k^2 = -k(k+1)R/2$ (where $R$ is the Ricci scalar and $k \in \mathbb{Z}_{\geq 0}$), corresponding to the discrete series representations of the de Sitter group.

While it is established that these theories admit conserved rank-$(k+1)$ symmetric traceless tensors $F_{\mu_1\dots\mu_{k+1}}$ which decompose into (anti)-holomorphic currents satisfying global conformal Ward identities, a critical gap remained: the explicit action of the global conformal symmetry group on the scalar field $\phi$ itself was undetermined. Standard conformal transformations do not act on massive scalars in the usual way, and previous literature suggested the symmetry might be non-local. Furthermore, it was unclear whether these theories admit a traceless stress tensor capable of generating these transformations, particularly in dS$_2$ where the mass parameter is tachyonic and unitarity requires careful definition.

Methodology
The author employs a geometric reformulation of the scalar field theory to resolve these issues. The core methodology involves the following steps:

1. Equivalence to Conformal Killing Tensors: The general solution to the Klein–Gordon equation for the discrete series scalars is re-expressed in terms of a symmetric, traceless conformal Killing tensor $W_{\mu_1\dots\mu_k}$. Specifically, the scalar field is written as the $k$-fold divergence of this tensor: $\phi = \nabla_{\mu_1}\dots\nabla_{\mu_k} W^{\mu_1\dots\mu_k}$. This equivalence relies on an intertwining relation (Eq. 2.6) between the Klein–Gordon operator and the conformal Killing tensor equation, valid under appropriate boundary conditions (periodic in dS$_2$, vanishing Dirichlet in AdS$_2$).
2. Defining the Symmetry Transformation: By treating $W_{\mu_1\dots\mu_k}$ as the fundamental variable, the author defines the conformal symmetry transformation. Since $W_{\mu_1\dots\mu_k}$ is Weyl invariant (in its contravariant form), its transformation under a global conformal isometry (a combination of a coordinate transformation and a Weyl rescaling) is well-defined. The transformation of the scalar field $\phi$ is then derived by applying this transformation to the divergence relation.
3. Constructing the Stress Tensor: The author constructs a stress tensor $T_{\mu\nu}$ by varying the action with respect to the metric, treating $W_{\mu_1\dots\mu_k}$ as the independent field. This stress tensor is shown to be traceless and conserved only after imposing the equations of motion and boundary conditions.
4. Quantization and Charge Analysis: The paper analyzes the mode expansions of the fields in both AdS$_2$ (upper half-plane) and dS$_2$ (Riemann sphere). Special attention is paid to the zero-norm modes in dS$_2$ which obstruct standard quantization. A physical Hilbert space is defined by restricting the operator algebra to shift-invariant operators and a vacuum state annihilated by specific shift generators.

Key Contributions and Results

• Non-local Action on Scalars, Local Action on Tensors: The paper derives an explicit formula (Eq. 3.7) for the global conformal symmetry transformation acting on the scalar field $\phi$. The result demonstrates that while the transformation acts locally on the conformal Killing tensor $W_{\mu_1\dots\mu_k}$, it acts non-locally on the scalar field $\phi$ in dS$_2$ (where the Weyl parameter $\sigma \neq 0$). This confirms the symmetry is an on-shell symmetry acting on the solution space rather than an off-shell symmetry of the action.
• Traceless Stress Tensor: A traceless stress tensor $T_{\mu\nu}$ is explicitly constructed in terms of $W_{\mu_1\dots\mu_k}$ and the conserved current $F_{\mu_1\dots\mu_{k+1}}$ (Eq. 4.4). While $T_{\mu\nu}$ is a local functional of $W$, it is non-local in $\phi$. Crucially, this tensor is shown to be traceless and conserved on-shell.
• Global Conformal Generators: The integrated charges of the stress tensor components are computed. In AdS$_2$, the generators for $m = -1, 0, 1$ reproduce the $SL(2, \mathbb{R})$ isometry algebra. In dS$_2$, the analysis reveals that the stress tensor components $T(z)$ and $\bar{T}(\bar{z})$ are not single-valued local operators due to monodromy relations involving zero-mode shift operators ($p_n$). However, upon restricting to the physical Hilbert space (where shift operators vanish), the integrated charges for $m = -1, 0, 1$ become well-defined and generate the global conformal algebra $SL(2, \mathbb{R}) \times SL(2, \mathbb{R})$.
• Primary Fields: The paper establishes that the currents $F(z)$ and $\bar{F}(\bar{z})$ behave as global conformal primaries with weights $(k+1, 0)$ and $(0, k+1)$ in the physical Hilbert space. The tensor components $W(z)$ and $\bar{W}(\bar{z})$ behave as primaries with weights $(-k, 0)$ and $(0, -k)$, up to polynomial shift ambiguities inherent to the zero modes.

Significance
The paper provides a geometric realization of the global conformal symmetry for discrete series scalars in dS$_2$, resolving the ambiguity regarding how this symmetry acts on the fundamental scalar field. By leveraging the equivalence to conformal Killing tensors, the author demonstrates that the symmetry is manifest locally in the tensor description but inherently non-local in the scalar description.

The work clarifies the structure of the stress tensor in these theories, showing that while a traceless stress tensor exists, it is only well-defined as a generator of global symmetries after imposing the equations of motion and appropriate boundary conditions. In dS$_2$, the analysis highlights the subtleties of defining a unitary theory with tachyonic masses, specifically the necessity of removing zero-mode shift operators to define a physical vacuum and single-valued stress tensor charges.

The author notes that this equivalence between a scalar field and a conformal Killing tensor is a new example of field theory equivalences in two dimensions. The paper concludes by suggesting that a formulation treating $W$ and $F$ as independent variables could lead to a theory with a full Virasoro algebra, reminiscent of $\beta\gamma$ systems, though this remains a direction for future investigation rather than a result of the current work.