Gauge Invariant Frame-like Formalism in Infinite Spin Theory

• Gauge Invariant Frame-like Formalism is a unified framework that systematically describes infinite spin fields using an infinite chain of coupled forms.
• It employs specific cross-terms and modified gauge transformations to maintain full invariance for both completely symmetric and mixed-symmetry tensor fields.
• Its extension to curved (A)dS spaces reveals that unitary infinite spin theories are viable only in Minkowski and Anti-de Sitter backgrounds.

The gauge invariant frame-like formalism provides a unified and manifestly gauge-invariant Lagrangian framework for describing infinite (continuous) spin representations of the Poincaré group, with natural extensions to mixed-symmetry tensor fields and curved (Anti-)de Sitter (A)dS spaces. This approach systematically constructs an infinite chain of coupled fields and associated auxiliary variables, achieving full gauge invariance via cross-terms and specific modifications to the gauge symmetries. The formalism delivers an explicit Lagrangian realization of continuous spin states and clarifies which classes of curved backgrounds admit unitary infinite spin field theories.

1. Frame-like Lagrangian Construction for Infinite Spin Fields

The frame-like description employs a hierarchy of one-forms and higher-degree forms endowed with Lorentz indices, corresponding to the stepwise Young tableau structure of the underlying representation. For the bosonic completely symmetric case, the Lagrangian involves one-forms $h_\mu^{a(k)}$ and auxiliary one-forms $\omega_\mu^{a(k),b}$, each subject to trace and symmetry constraints dictated by the Young pattern. For mixed-symmetry types, additional forms are included as required by the specific tableau $Y(k, l)$ or its fermionic counterpart $Y(k+\frac{1}{2}, l+\frac{1}{2})$.

The frame-like Lagrangian is organized as a chain (or "tower") over $k$: $$\mathcal{L}_0 = \sum_{k=1}^{\infty} \left[ L_0(h_\mu^{a(k)}) + e_0^2 L_2(h_\mu^{a(k)}) \right],$$ where $L_0$ and $L_2$ are the kinetic and mass-like terms, respectively (see eq. (18) of the source). Cross-terms "glue" fields of adjacent $k$ and provide compensators: $$\mathcal{L}_\text{cross} \sim \sum_{k=1}^{\infty} e_k \{\ldots [\omega(a(k-1)), h(a(k-1)) ] \ldots \},$$ ensuring full gauge invariance (eq. (19)). The gauge transformations themselves include $k$-dependent couplings: $$\delta h^{a(k)} = (\ldots) x^{a(k)} + e_k \{ \ldots \},$$ with $e_k$ fixed recursively by consistency (eq. (20)).

To generalize to mixed-symmetry fields, forms of appropriate rank and symmetry are introduced (e.g., a two-form $P$ for the physical field, one-form $Q$ for the connection), with their own cross-couplings and gauge structure, as detailed for $Y(k+1,l+1)$ types.

2. Infinite Spin Representations and the Limit Process

Continuous spin (infinite spin) representations of the Poincaré group are characterized by a continuous parameter and encode an unbounded spectrum of "spin-like" states. In the frame-like formalism, these arise as limits where the number of symmetric indices $k \to \infty$ while the "depth" $l$ of the second Young tableau row is held fixed, with $l$ labelling the class of the representation.

The infinite spin limit is implemented by taking $k \to \infty$, and the parameters entering the tower (e.g., $e_k$) are promoted to sequences or functions of $k$: $$e_k^2 \sim -2K x_k^2 + [C_0 - C_0^2 + \ldots] x_k + \ldots ,$$ where $x_k = k(d + k - 3)$ (see eq. (23)). This process constructs an infinite, gauge-invariant chain of fields, and in the limit, the aggregated system realizes a continuous spin representation (Khabarov et al., 2017).

3. Generalization to Curved Spacetimes

The formalism naturally extends to (A)dS backgrounds by replacing ordinary derivatives with (A)dS-covariant ones, $D_\mu$, and compensating for curvature-induced gauge-variance by introducing mass-like (but not genuine mass) terms: $$\mathcal{L}_\text{mass-like} \sim b_k \{ \ldots \} h^{a(k-1)} h^{a(k-1)},$$ which are required to cancel curvature-induced variations (see eq. (3) in Section 1.1).

A key outcome is that infinite spin gauge-invariant systems, as constructed, admit physically acceptable (i.e., unitary) solutions only in Minkowski and Anti de Sitter space. In de Sitter space (with positive curvature $K>0$), the unitarity constraint forces coefficients such as $e_k^2$ in the infinite chain to become negative for some $k$, precluding a well-defined, unitary infinite spin theory. In contrast, for Anti de Sitter ($K<0$), a wide range of unitary continuous spin spectra remains possible, subject to positivity of the field norms across the entire tower (see the analysis around eqs. (22)-(23)).

However, the precise correspondence between the constructed infinite-spin AdS field theories and specific representation theory of the AdS group remains unresolved. Determining which irreducible modules these Lagrangian systems realize is an open question.

4. Mixed Symmetry Fields and Generalizations

Mixed symmetry fields, such as those associated with Young tableaux $Y(k+1,l+1)$, represent an essential extension beyond the completely symmetric or antisymmetric cases. Such fields are modeled by introducing components for each relevant symmetry type, along with a set of tailored gauge symmetries and compensator fields.

The frame-like Lagrangian for mixed-symmetry fields follows the same organizing principle: an infinite chain of forms, each carrying the appropriate symmetry, cross-coupled to ensure overall gauge invariance. Notably, in the infinite spin limit ($k \to \infty$ at fixed $l$), these constructions yield a more general class of continuous spin field theories that are not restricted to symmetric representations. The parameter $l$ functions as a label for the representation class.

Curiously, these generalized mixed symmetry systems often admit partially massless subsectors (upon further tuning of parameters), or can split into decoupled sectors, enriching the classification of continuous spin theories in both flat and (A)dS backgrounds.

5. Unitarity, Constraints, and Open Problems

A central aspect of the infinite spin frame-like constructions is the need for unitarity—positivity of kinetic terms and consistency of the functional inner product across the full chain. The detailed unitarity analysis hinges on the sign-definiteness of certain quadratic forms, typically in the $e_k^2$ or related parameters, as $k \to \infty$: $$P(x) = -K x^2 + (B K + C) x + D \qquad\text{(see eq. (80))}$$ where $P(x)$ arises from the mass-like sector or the kinetic normalization of fields (see Appendix of (Khabarov et al., 2017)). The region of parameter space admitting unitarity in (A)dS is thus delimited, and in de Sitter is shown to be empty for infinite spin.

Open questions remain regarding:

• The explicit AdS group representation content of these unitary infinite spin models.
• The consistent formulation of interacting continuous spin (infinite spin) theories, either self-coupled or coupled to matter or gravity.
• The broader classification of partially massless limits within the infinite spin tower, and the characterization of the parameter space for which these partially massless subsectors decouple or become interacting.

Potential extensions include the systematic construction of infinite spin mixed-symmetry fermionic fields, possibly within a supersymmetric or super-Poincaré generalization.

6. Summary Table: Infinite Spin Frame-like Formalism Features

Feature	Description	Curved Space Comments (A)dS
Field content	Infinite chain of forms (one-/two-forms, etc.)	Covariant forms, mixed symmetry as required
Gauge invariance	Achieved via hierarchy of cross-terms and compensators	Mass-like terms restore invariance in (A)dS
Unitarity	Positive-definiteness of kinetic norms in the chain	Unit. infinite spin models: AdS only
Mixed-symmetry fields	Arbitrary $Y(k,l)$ types, parametrized by fixed $l$	Parametric dependence on representation labels
Continuous spin limit	$k \to \infty$ at fixed $l$ in the field tower	Ensures unbounded gauge chain, physical series
Open problems	AdS rep. theory, interaction construction, classification	Representation identification, higher interactions

7. Outlook and Future Research

The gauge invariant frame-like formalism provides a robust foundation for the systematic paper of infinite (continuous) spin representations within field theory, with clear prescriptions for completely symmetric and mixed symmetry types, and with explicit Lagrangian realizations in flat and (A)dS spaces. While unitary infinite spin field theories are realized in Minkowski and Anti de Sitter settings, the precise representation-theoretic status in AdS remains unidentified. Open directions include the construction of consistent interactions, the classification of partially massless subsystems, systematic exploration of the unitarity parameter regions, and the extension to supersymmetric or higher-structure frameworks.

These advances deepen the understanding of the landscape of relativistic field theories and their spectrum—providing concrete models for exotic representations, as well as clarifying the interplay between gauge invariance, locality, and the infinite-dimensional representations of spacetime symmetry groups (Khabarov et al., 2017).