Higher-Spin Gravity in Two Dimensions with Vanishing Cosmological Constant

Published 9 Apr 2026 in hep-th | (2604.08093v1)

Abstract: In this paper, we use a version of the BF formulation of two-dimensional dilaton gravity that allows to define a gauge theory of the two-dimensional Poincaré or Maxwell algebras and several of their higher-spin generalisations, both of finite and infinite dimension. The spectrum of the two-dimensional higher-spin gravity with vanishing cosmological constant based on the extended, infinite-dimensional higher-spin algebra is shown to contain an infinite collection of scalar degrees of freedom with a continuum of ever increasing mass, corresponding to the twisted-(co)adjoint representation. We comment on an approach to include backreaction of the scalar fields on the gravity sector at the level of formal equations of motion, thereby providing a first example of a fully interacting higher-spin gravity theory with vanishing cosmological constant in two dimensions.

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Summary

The paper establishes a novel BF-type formulation that extends 2D dilaton gravity to include higher-spin fields and interacting matter sectors.
It constructs finite and infinite-dimensional higher-spin algebras by overcoming algebraic obstructions inherent to non-semisimple Poincaré structures.
A continuous spectrum of massive scalar fields emerges via dual-space decomposition, offering new insights into holography and integrability in flat backgrounds.

Higher-Spin Gravity in Two Dimensions with Vanishing Cosmological Constant

Introduction

This work develops a framework for two-dimensional higher-spin gravity in the zero cosmological constant regime, circumventing the standard obstructions arising from the structure of the Poincaré (and Maxwell) algebras. The approach systematically extends the BF-type formulation of dilaton gravity—including the Jackiw-Teitelboim (JT) model—to accommodate higher-spin fields and propagating matter sectors, generalizing both finite and infinite-dimensional higher-spin algebras. The analysis is rooted in the algebraic properties of the two-dimensional spacetime symmetry groups and their extensions, with a dual-space (adjoint/coadjoint) formalism employed to overcome the degeneracy of bilinear forms in these algebras. The resulting theories are the first examples of fully interacting, higher-spin-dilaton-matter gravities in two dimensions with $\Lambda=0$ and are significant for their analytic tractability and potential holographic implications.

Algebraic Framework and BF Formulation

Standard BF formulations in two-dimensional gravity rely on the existence of a non-degenerate, invariant bilinear form on the gauge algebra. For semisimple cases (e.g., AdS $_2$ scenarios, $\mathfrak{so}(2,1)$ ), the Killing form suffices. However, the Poincaré algebra ( $\mathfrak{iso}(1,1)$ ), central for Minkowski backgrounds, is non-semisimple; no such bilinear form exists. The paper addresses this by utilizing a BF action defined via the canonical pairing between the algebra and its dual rather than a standard trace:

$S[A, B^*] = \int (B^*, F),$

with $A$ in the adjoint and $B^*$ in the coadjoint representation. This enables well-defined dynamics even when the gauge algebra possesses Abelian ideals and lacks invariant non-degenerate metrics.

Higher-Spin Extensions

Finite and Infinite Towers

Finite-dimensional higher-spin extensions are constructed as contractions of $\mathfrak{sl}(N, \mathbb{R})$ leading to finite towers of gauge fields. Infinite-dimensional generalizations use novel quotients of universal enveloping algebras, e.g., $hs[\lambda]$ in AdS $_2$ , yielding infinite towers of higher-spin fields. The flat-space analog $_2$ 0 is introduced for Minkowski backgrounds.

The analysis distinguishes two major classes of higher-spin modules:

Adjoint modules: Infinite-dimensional for $_2$ 1 due to the absence of highest- or lowest-weight representations in $_2$ 2. Contrasts with the $_2$ 3 case, where $_2$ 4 decomposes into finite-dimensional modules.
Twisted-adjoint modules: Induced by an involutive automorphism $_2$ 5; these represent matter (scalar) sectors transforming in infinite-dimensional representations with shifting mass parameters.

Dual-Space Decomposition

The paper constructs explicit adjoint and twisted-adjoint bases for $_2$ 6 and its completions. The adjoint basis reveals an infinite set of modules labeled by a continuous parameter, in contrast to the discrete spectra in AdS $_2$ 7. The twisted-adjoint basis, crucial for matter coupling, comprises a continuum of fields, with mass squared $_2$ 8 parameterized continuously. These modules provide diagonalization analogues to those arising in AdS higher-spin gravities but with key spectral and representation-theoretic differences.

Unfolded Equations and Matter Coupling

The extended algebra accommodates propagating matter via the "unfolded" formulation, in which a matter zero-form field $_2$ 9 (or its dual $\mathfrak{so}(2,1)$ 0) satisfies a covariant constancy equation under the twisted-adjoint (or twisted-coadjoint) action:

$\mathfrak{so}(2,1)$ 1

Decomposition into the twisted-adjoint basis gives decoupled equations for each scalar mode:

$\mathfrak{so}(2,1)$ 2

where the mass spectrum is explicitly determined by the underlying algebraic construction. A key claim is that, in the vanishing $\mathfrak{so}(2,1)$ 3 regime, the matter sector consists of a continuous tower of massive scalar fields, a feature directly traceable to the structure of the twisted-adjoint modules for $\mathfrak{so}(2,1)$ 4.

Interactions and Algebraic Deformation

The framework is extended to interacting models via algebraic deformations analogous to those in the AdS context. An explicit one-parameter family of associative algebra deformations (controlled by $\mathfrak{so}(2,1)$ 5) is introduced, yielding consistent non-linear dynamics with backreaction between the higher-spin gauge fields and the scalar sector. The compatibility of the adjoint/coadjoint and twisted-adjoint/twisted-coadjoint representations under these deformations is preserved, ensuring the formal integrability of the resulting systems.

Higher-Spin Extension of Maxwell (Cangemi-Jackiw) Gravity

An alternative route employs the Maxwell algebra ( $\mathfrak{so}(2,1)$ 6) and its infinite-dimensional Weyl extension ( $\mathfrak{so}(2,1)$ 7). The Weyl algebra, with its unique supertrace and $\mathfrak{so}(2,1)$ 8 grading via automorphism $\mathfrak{so}(2,1)$ 9, supports BF-type actions and deformations. The extended algebra ( $\mathfrak{iso}(1,1)$ 0) allows for BF formulations paralleling those for $\mathfrak{iso}(1,1)$ 1, potentially yielding distinct spectra and interaction patterns depending on the chosen spin-2 subalgebra and background. This flexibility highlights the model-dependence of spectral and gravitational properties in 2D higher-spin BF gravities.

Physical and Theoretical Implications

The presented construction constitutes the first explicit, fully interacting two-dimensional higher-spin gravity theory with propagating matter and vanishing cosmological constant. The continuous spectrum of massive scalar fields is in stark contrast to the discrete towers in the $\mathfrak{iso}(1,1)$ 2 case, with implications for boundary dynamics, asymptotic symmetry analysis, and holography (including connections to SYK-like dualities). The formalism's ability to incorporate backreaction and admit algebraic deformations lays groundwork for a broader exploration of integrability and exact solvability in low-dimensional quantum gravity.

Several directions for future development are identified:

Supersymmetric and colored extensions (e.g., $\mathfrak{iso}(1,1)$ 3 analogues).
Connections to area-preserving diffeomorphism algebras in large $\mathfrak{iso}(1,1)$ 4 and/or classical limits.
Systematic asymptotic analysis and holographic dictionary for flat backgrounds.
Explicit extraction of interaction vertices and black hole solutions.
Investigation of the spectral/representation-theoretic structure for Weyl-based higher-spin gravities.

Conclusion

By generalizing the BF formulation of dilaton gravity through a dual-space approach, this work realizes both finite and infinite-dimensional higher-spin gravities in two dimensions with $\mathfrak{iso}(1,1)$ 5. Extended by continuous towers of massive scalars in the matter sector, these models evade the algebraic obstructions prevalent in standard formulations and allow fully interacting, formally integrable dynamics. The algebraic mechanisms presented here clarify the spectrum and representation structure of such theories, setting firm foundations for both practical computation and further investigation into the dynamics, holography, and classification of higher-spin gravitational models in two dimensions.