Holographic Symmetry Algebras

• Holographic Symmetry Algebras are algebraic structures that encode higher-dimensional quantum symmetries and connect them to bulk topological orders.
• They use categorical data to describe complex symmetry phenomena such as higher-form, non-invertible symmetries and their anomalies.
• These algebras clarify operator product expansions, phase classifications, and anomaly inflows across diverse frameworks like celestial holography and AdS/CFT.

A holographic symmetry algebra is an algebraic structure that captures the symmetry content of a quantum theory in a manner reflective of, and determined by, the properties of a higher-dimensional (typically topological or gravitational) bulk theory. The unifying principle is the notion that nontrivial symmetries, including higher-form and non-invertible symmetries, as well as their anomalies, are most naturally described by categorical data associated to topological quantum field theory in one higher spacetime dimension. Across conformal field theories, higher-spin gravities, gauge theories at null infinity, and condensed-matter systems, holographic symmetry algebras organize selection rules, operator product expansions (OPEs), and phase classifications via their bulk-topological avatars.

1. Algebraic Higher Symmetry and Fusion n-Categories

In an $n$-dimensional quantum system, the algebraic higher symmetry is encoded in the set of local symmetric operators—those that commute with all symmetry generators—which form a closed operator algebra. The fundamental objects are "charge objects" or topologically stable symmetry defects (e.g., point, line, membrane excitations), whose fusion and condensation obey the axioms of a fusion $n$-category $\mathcal{R}$: $$\mathcal{R} = \{\text{charge objects of the symmetry}\}.$$ The fusion of simple objects $X$ and $Y$ is governed by

$$X\otimes Y \simeq \bigoplus_{Z\in\operatorname{Irr}(\mathcal{R})} N_{X,Y}^Z Z,\quad N_{X,Y}^Z\in\mathbb{N}.$$

If $W_X(S^\alpha)$ creates a defect of type $X$ on an $\alpha$-sphere, then the composition

$$W_X(S^\alpha)W_Y(S^\alpha) = \sum_Z N_{X,Y}^Z W_Z(S^\alpha)$$

exactly matches the categorical fusion data. This structure generalizes the notion of higher-group symmetries and encapsulates non-invertible symmetries beyond group-theoretic frameworks (Kong et al., 2020).

2. Categorical/Holographic Symmetry and Anomaly Inflow

Restricting to the "symmetric subspace" (the sector invariant under all defect operators) is equivalent to imposing a gravitational anomaly or, categorically, to encountering a topological order in $n+1$ dimensions. For a given fusion $n$-category $\mathcal{R}$, there exists a unique anomaly-free bulk topological order $\mathcal{M}^{n+1}$ satisfying: $$\mathcal{M}^{n+1} = (\mathcal{R}), \quad \Omega\mathcal{M}^{n+1} = \mathcal{R}, \quad \Omega^2\mathcal{M}^{n+1} = Z_1(\mathcal{R}),$$ where $\Omega$ is the boundary map and $Z_1$ is the categorical center. The partition function on a closed $(n+1)$-manifold with background $\mathcal{R}$-connections $A$ exhibits an anomaly inflow: $$Z_n[A] = \left\langle \exp\Bigl(i\int_{M^{n+1}} \omega_{n+1}(A)\Bigr)\right\rangle_\text{bulk}.$$ This holographically realizes the notion that symmetries are shadows ("categorical symmetries") of bulk topological order, with anomalies precisely captured by the bulk (Kong et al., 2020, Chatterjee et al., 2022).

3. Construction and Algebraic Structure in Celestial Holography

In four-dimensional gauge and gravitational theories with asymptotic boundaries, the relevant symmetries are encoded by infinite-dimensional current algebras acting as 2D chiral (vertex) algebras on the celestial sphere. For instance, in gauge theory and gravity, one constructs towers of soft currents $R^{k,a}(z,\bar z)$ and $H^k(z,\bar z)$ whose modes generate a hierarchical algebra: $$[R^{k,a}_n, R^{\ell,b}_{n'}] = -if^{ab}{}_c \cdot (\cdots) R^{k+\ell-1,c}_{n+n'}.$$ The subalgebra of holomorphic highest-weight modes closes as a loop algebra (for gauge theory) or lies within the $w_{1+\infty}$ algebra in gravity: $$[\widehat{R}^{k,a}_m, \widehat{R}^{\ell,b}_n] = -if^{ab}{}_c \, \widehat{R}^{k+\ell-1,c}_{m+n}.$$ This structure is an explicit realization of a holographic symmetry algebra, pairing the higher-spin or soft structure of scattering amplitudes to chiral symmetry on the boundary (Guevara et al., 2021, Jiang, 2021). In the MHV sector of gravity and Yang-Mills, the algebra is a semidirect product of $w_{1+\infty}$ (or $S$-algebra) with an abelian algebra of negative-helicity soft generators, fully determining amplitude recursion relations (Banerjee et al., 4 Aug 2025).

4. Deformation, Twistor Realizations, and Higher-Spin Extensions

The introduction of a cosmological constant, as in AdS$_4$, causes a canonical deformation of the holographic (celestial) symmetry algebra, realized through the modified Poisson structure on twistor space: $\{f,g\}_\Lambda = \{f,g\}_0 + \Lambda \Delta(f,g).$ For Hamiltonians $w^p_{m,a}$, the deformed bracket reads

$$\{w^p_{m,a}, w^q_{n,b}\}_\Lambda = (m(q-1) - n(p-1)) w^{p+q-2}_{m+n,a+b} - \Lambda(a(q-2)-b(p-2)) w^{p+q-1}_{m+n,a+b}.$$

This matches direct field-theoretic computations, and the algebraic deformation is linked directly to the appearance of the cosmological constant. Twistor-space uplift yields explicit Noether charges as contour integrals, with the algebra realized in the symplectic structure of the self-dual (A)dS theory (Bittleston et al., 2024, Kmec et al., 2 Jun 2025).

Higher-spin holographic symmetry algebras also occur in the context of AdS$_3$/CFT$_2$ dualities, where asymptotic symmetry analysis yields infinite-dimensional $W$-algebras such as $W_\infty[\lambda]$ and their super-extensions ($\mathcal{N}=2$ super-$W_\infty[\lambda]$), which encode both the representation theory and correlator structure in dual minimal/super-minimal CFTs (Gaberdiel et al., 2011, Hanaki et al., 2012).

5. Symmetry/Topological Order Correspondence and the Holographic Principle

The categorical formulation shows that any finite symmetry algebra of local operators in $n$ dimensions corresponds to a non-degenerate braided fusion $n$-category $\mathcal{C}$—a topological order in $n+1$-dimensions. Transparent patch operators formalize this correspondence: their fusion, braiding, associativity, and modular data (e.g., $F$-symbols, $S$- and $T$-matrices) encode the entire symmetry content: $$a\otimes b \cong \bigoplus_{c} N_{ab}^{c} c, \quad F\text{-symbols}, \quad S_{ab}, \; T_{aa}.$$ The topological holographic principle asserts: the algebra of boundary operators uniquely determines the bulk topological order. Conversely, a bulk topological order constrains the possible boundary symmetries and phases (Chatterjee et al., 2022). This underlies practical classification schemes for gapped phases, SPT/SET phases, and anomalies via boundary–bulk duality (Kong et al., 2020).

6. Holographic View in Non-Relativistic and Emergent Symmetry Contexts

In non-relativistic holography, e.g., Lifshitz spacetimes, the symmetry algebra (Lifshitz algebra or a prototypical family thereof) encodes a bulk symmetric structure (anisotropic scaling and translations) with a clear boundary bulk correspondence:

• Bulk: $(d+2)$-dimensional Lifshitz geometries with an extra holographic coordinate, with Killing algebra $a_3(z)$,
• Boundary: $(d+1)$-dimensional Lifshitz–Weyl geometries realized as orbits of the dilation generator,
• Coadjoint orbit classification yields possible particle-like excitations and their symmetry constraints (Figueroa-O'Farrill et al., 2022).

7. Anomalies, Equivalence, and Phase Classification

Anomalous higher symmetries are classified by autoequivalences of the categorical center $Z_1(\mathcal{R})$ that preserve the boundary-defining algebra. Two systems are holo-equivalent (i.e., share all entanglement and excitations data modulo superficial choices) if their categorical centers, and thus their bulks, agree: $$Z_1(\mathcal{R}) \simeq Z_1(\mathcal{R}') \Longleftrightarrow \mathcal{M} \simeq \mathcal{M}'.$$ The classification of gapped liquid phases (SPT/SET order) with a fixed categorical symmetry is then mapped to the problem of classifying gapped boundaries and domain walls of the parent bulk topological order. The bulk-boundary correspondence yields a complete description up to stacking of invertible phases (living in the cobordism $\Omega^{n+1}$ groups) (Kong et al., 2020).

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References:

(Kong et al., 2020) Algebraic higher symmetry and categorical symmetry— a holographic and entanglement view of symmetry (Chatterjee et al., 2022) Symmetry as a shadow of topological order and a derivation of topological holographic principle (Guevara et al., 2021) Holographic Symmetry Algebras for Gauge Theory and Gravity (Bittleston et al., 2024) On AdS$_4$ deformations of celestial symmetries (Gaberdiel et al., 2011) Symmetries of Holographic Minimal Models (Hanaki et al., 2012) Symmetries of Holographic Super-Minimal Models (Figueroa-O'Farrill et al., 2022) Lifshitz symmetry: Lie algebras, spacetimes and particles (Jiang, 2021) Holographic Chiral Algebra: Supersymmetry, Infinite Ward Identities, and EFTs (Kmec et al., 2 Jun 2025) S-algebra in Gauge Theory: Twistor, Spacetime and Holographic Perspectives (Banerjee et al., 4 Aug 2025) Holographic symmetry algebra for the MHV sector revisited (Sleight et al., 2016) Higher-Spin Algebras, Holography and Flat Space