Nonlinear W(infinity) Algebra as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity

Abstract: We investigate the asymptotic symmetry algebra of (2+1)-dimensional higher spin, anti-de Sitter gravity. We use the formulation of the theory as a Chern-Simons gauge theory based on the higher spin algebra hs(1,1). Expanding the gauge connection around asymptotically anti-de Sitter spacetime, we specify consistent boundary conditions on the higher spin gauge fields. We then study residual gauge transformation, the corresponding surface terms and their Poisson bracket algebra. We find that the asymptotic symmetry algebra is a nonlinearly deformed W(infinity) algebra with classical central charges. We discuss implications of our results to quantum gravity and to various situations in string theory.

• The paper introduces a nonlinear W∞ algebra that extends the Virasoro framework through the inclusion of an infinite tower of higher spin fields.
• It employs a Chern-Simons gauge theory formulation to define boundary conditions that preserve classical central charges in 3D AdS gravity.
• The findings provide significant insights for the AdS/CFT correspondence and offer a consistent approach to finite spin truncation in higher spin theories.

Asymptotic Symmetry and Higher Spin Gravity in Three Dimensions

This paper presents an in-depth analysis of the asymptotic symmetry algebra of (2+1)-dimensional higher spin anti-de Sitter (AdS) gravity, employing a Chern-Simons gauge theory framework based on the hs(1,1) algebra. Unlike traditional studies focusing on Einstein-Hilbert gravity, this work addresses a more intricate landscape by considering the implications of incorporating an infinite tower of higher spin (HS) fields, thereby significantly expanding the potential applications and theoretical insights into quantum gravity and string theory.

The authors systematically define boundary conditions that restrict the behavior of the gauge fields at asymptotic infinity, ensuring conformity under an infinite-dimensional group of transformations. These boundary conditions are crucial for maintaining the consistency and integrity of the gauge fields in the AdS space, ultimately leading to the derivation of a nonlinearly realized W∞ algebra, endowed with classical central charges.

The foundational basis of this paper lies in extending the Virasoro algebra, which arises as the asymptotic symmetry group of pure (2+1) AdS gravity. Here, the study reveals that while the central charge in the Virasoro algebra substructure persists as in pure gravity, the incorporation of a richer algebraic structure enables the addition of classical central charges, thus enriching the symmetry algebra beyond the classical understanding.

Key Contributions

1. Extension to Nonlinear W∞ Algebra: The central innovation demonstrated is the transformation of the classical W∞ algebra into a nonlinear form with preserved classical central charges, distinctively set by the AdS radius. This mathematical structure provides a broader framework for examining asymptotic symmetries in three-dimensional spacetimes and is an inherent characteristic of the higher spin extension of AdS gravity.
2. Implications for AdS/CFT Correspondence: The findings in this paper have pronounced implications in the context of the AdS/CFT correspondence, where understanding the holographic duality at weak coupling regimes often remains challenging. The introduction of HS gauge fields in the holographic framework offers potential pathways to studying CFTs with infinitely many towers of HS currents.
3. Truncation Consistency: A significant portion of the research ensures that truncation of the higher spin algebra remains consistent up to a specific finite spin, particularly within the sl(3,R) algebra, providing a manageable scope for practical computation and analysis without infringing the mathematical coherence required by the Jacobi identities.

Implications and Future Directions

The established nonlinear W∞ algebra not only expands the scope for theoretical exploration within the domains of quantum gravity and string theory but also posits intriguing possibilities for the classification and understanding of classical solutions in HS AdS gravity. These solutions might encompass an infinite set of conserved charges akin to more traditional physical quantities like mass and angular momentum.

The speculative connections drawn between HS AdS gravity and other dimensions, such as potential linkages to four-dimensional self-dual gravity, may unveil unexplored gravitational correspondences, enriching the contemporary understanding of gravitational symmetries and holographic principles.

The work also invites further examination of potential supersymmetric extensions and the inclusion of spin-1 currents, indicating a broader spectrum of applications within theoretical physics. This research sets a foundational stage for investigating new facets of quantum gravity and establishes a compelling argument for the inclusion of higher spin theories in understanding fundamental interactions.

In conclusion, this paper contributes a robust mathematical framework with potential theoretical advancements in understanding the asymptotic symmetry properties of HS AdS gravity. Its findings could significantly influence both the theoretical modeling and practical computation of complex gravitational systems across multiple dimensions.