Worldline Formulations of Fracton Gauge Theories

• The paper presents a worldline formulation that recasts covariant fracton gauge theories into BRST-quantized particle models, elucidating key gauge constraints.
• It demonstrates how double-derivative gauge symmetry, linked to charge and dipole conservation, underpins the restricted mobility of fracton excitations.
• The review compares tensor and vector oscillator models and shows how deformation parameters interpolate between fracton actions and linearized gravity.

Covariant fracton gauge theories are a class of field theories in which the fundamental gauge symmetry is enacted by double derivatives of a scalar gauge parameter acting on a symmetric rank-two tensor field. These theories, structurally related to linearized gravity but with reduced (longitudinal) diffeomorphism invariance, underpin the restricted mobility of fracton excitations resulting from conservation laws for both charge and dipole moment. Recent work has developed worldline (first-quantized) formulations of these theories, recasting their field-theoretic content and gauge constraints into particle-like, BRST-quantized models. This article reviews the key mathematical structures, physical content, and technical implementations of worldline formulations of covariant fracton gauge theories, with attention to their unique symmetries, constraint algebra, models, and relation to gauge fixing and quantization.

1. Fracton Gauge Symmetry and Covariant Field Theory

The defining attribute of covariant fracton gauge theories is the gauge transformation

$$\delta_\Lambda h_{\mu\nu} = \partial_\mu \partial_\nu \Lambda$$

where $h_{\mu\nu}$ is a symmetric rank-two field and $\Lambda$ is a scalar gauge parameter. This structure differs from ordinary (linearized gravity) diffeomorphisms, $$\delta_\xi h_{\mu\nu} = \partial_\mu \xi_\nu + \partial_\nu \xi_\mu$$, amounting to the restriction to longitudinal transformations $\xi_\mu = \partial_\mu \Lambda$.

The covariant action for fracton gauge theories typically takes the form

$$S^{(\text{fr})} = \int d^D x\, [\alpha\, f^{\mu\nu} f_{\mu\nu} + \beta\, f^{\mu\nu}{}_{\nu} f_{\mu}{}^{\nu}],$$

where $f_{\mu\nu}$ is a derived field strength tensor (with possible variations in explicit definition), constructed to be invariant under the above gauge transformation. The quadratic form of this action allows a covariant, Maxwell-analogue kinetic term for the tensor field while its gauge structure ensures the conservation of both total charge and dipole moment. This guarantees the limited mobility—”fractonic"—property of corresponding excitations.

2. Worldline Models: Constraint Structure and BRST Quantization

Worldline formulations encode the field-theoretic gauge symmetries as first-class constraints on the classical phase space of a particle model, expanded with bosonic oscillator variables to handle the tensor indices. For $h_{\mu\nu}$, two essential approaches are identified (Fecit et al., 20 Aug 2025):

• Tensor Model: Introduces symmetric tensor oscillators $(\alpha^{\mu\nu}, \bar{\alpha}_{\mu\nu})$, with constraints including $H = p^2$ (Hamiltonian), $L = \alpha^{\mu\nu} p_\mu p_\nu$, $\bar{L} = \bar{\alpha}^{\mu\nu} p_\mu p_\nu$, and the number operator $J = \alpha^{\mu\nu} \bar{\alpha}_{\mu\nu}$.
• Vector Model: Employs vector oscillators $(\alpha^\mu, \bar{\alpha}_\mu)$, defining $L = (\alpha^\mu p_\mu)^2$, $\bar{L} = (\bar{\alpha}^\mu p_\mu)^2$, a mixed constraint $\ell = (\alpha^\mu p_\mu)(\bar{\alpha}^\nu p_\nu)$, and $J = \frac{1}{2}\alpha^\mu\bar{\alpha}_\mu$.

The constraints encode the required invariance under double-momentum saturation (mirroring $\partial_\mu \partial_\nu$). Quantization proceeds via the BRST procedure: each first-class constraint is paired with a fermionic ghost field, and the nilpotent BRST charge $Q$ generates the physical cohomology. For example, in the tensor model, the key BRST transformation for the string field $\Psi$ reads

$$Q\Psi = [\text{BRST operator constructed from constraints}]\,\Psi,$$

and, when analyzed on component fields,

$$s h_{\mu\nu} = \partial_\mu \partial_\nu \lambda, \quad s\lambda = 0,$$

realizing the covariant fracton gauge symmetry with ghost field $\lambda$. The full Batalin–Vilkovisky (BV) spectrum of fields, ghosts, and antifields is thereby recovered from the worldline perspective.

3. Deformed Models and the Classification of Fracton Theories

A continuous one-parameter family of worldline models is realized by deforming the vector model constraints through parameters $(\hat{\alpha},\hat{\beta})$, yielding

$$\ell_{(\hat{\alpha},\hat{\beta})} = (1+\hat{\beta})\alpha^\mu p_\mu \bar{\alpha}^\nu p_\nu + (\hat{\alpha}-\frac{1}{2}\hat{\beta})H.$$

This deformation interpolates among the possible quadratic covariant fracton actions, reproducing the corresponding double-derivative gauge invariance and accommodating nearly the complete space of parameter choices for $(\alpha,\beta)$ in the action, subject to the non-exceptional cases $\beta=0$, $2\alpha-\beta=0$, or $2\alpha+(D-1)\beta=0$. This mapping demonstrates the flexibility and completeness of the vector oscillator approach in capturing the landscape of covariant fracton gauge theories.

4. Interpretation via Linearized Gravity and Comparison of Gauge Fixings

Covariant fracton theories are unified with linearized gravity by their shared field content and overlapping symmetry structure, differing essentially by the restriction of gauge symmetry to longitudinal diffeomorphisms. The most general action invariant under $$\delta h_{\mu\nu} = \partial_\mu\partial_\nu\Lambda$$ is a linear combination of a "fractonic" (double-derivative) term and the standard Fierz–Pauli linearized gravity action: $$S_{\text{inv}}(g_1,g_2) = g_1 S_{\text{fract}} + g_2 S_{\text{LG}}.$$ Gauge-fixing presents nontrivial subtleties: while scalar gauge conditions (such as $$\partial_\mu\partial_\nu h^{\mu\nu}+k\partial^2 h=0$$) provide convenient inversion for the kinetic operator, they may impede the smooth reduction to the gravity limit or force propagation to the Landau gauge. Instead, a tensorial (vector) Lorenz-type gauge condition, $\partial^\mu h_{\mu\nu}+k\partial_\nu h=0$, ensures invertibility for generic $k$ and supports a continuous interpolation between fracton and gravity theories without extraneous degree-of-freedom reductions.

In the worldline context, gauge-fixing is mirrored by the use of a “Siegel gauge" condition—i.e., the string field $\Psi$ is rendered independent of the reparameterization ghost. This aligns the kinetic operator (via $\{b,Q\}=H$) with the d'Alembertian, yielding well-controlled quadratic actions.

5. Physical Content, Mobility Constraints, and Covariant Amplitudes

The worldline models correctly reproduce the unique constraint algebra responsible for fracton mobility restrictions. The double-derivative gauge symmetry enforces Gauss-type laws on the canonical momentum (electric field), yielding conservation laws for charge and dipole moment. On the worldline, these become nontrivial constraints on allowed propagating degrees of freedom:

• Isolated fractons are strictly immobile; all motion requires formation of bound states (e.g., dipoles).
• Physical excitations propagate only within subspaces determined by the underlying conservation law structure; in particle models, this entails "frozen" worldline sectors or constrained oscillator dynamics.

Rigid BRST quantization ensures that the amplitude computations (for example, in perturbative or path-integral approaches) remain consistent with these unusual kinematic constraints. The physical state cohomology of the BRST charge $Q$ projects onto the appropriate fracton excitation subspace, and matches that of the covariant field theory.

6. Technical Summary Table: Worldline Model Features

Model Type	Oscillators	Constraint Structure	Main Correspondence
Tensor Model	$\alpha^{\mu\nu},\bar{\alpha}_{\mu\nu}$	$H$, $L = \alpha^{\mu\nu}p_\mu p_\nu$, $\bar{L}$, $J$	$2\alpha - \beta = 0$ theory
Vector Model	$\alpha^\mu,\bar{\alpha}_\mu$	$L$, $\bar{L}$, $\ell$, $J$	$2\alpha + 3\beta = 0$
Deformed Vector	$\alpha^\mu,\bar{\alpha}_\mu$	Deformed $\ell_{(\hat{\alpha},\hat{\beta})}$, $H$	Generic $(\alpha,\beta)$ cases

7. Significance and Future Directions

Worldline formulations for covariant fracton gauge theories grant a first-quantized, particle-based description for highly constrained field theories, enabling efficient computations and opening the path toward advanced perturbative and non-perturbative analysis. The interplay between gauge fixing, BRST/BV quantization, and mobility constraint makes these approaches robust, capable of mirroring the field-theoretic spectrum, and potentially generalizable to curved spaces, boundary phenomena, or interacting (non-Abelian) fracton systems. These results establish covariant fracton gauge theories as a bridge between generalized Maxwell/linearized gravity dynamics and the phenomenology of restricted-mobility quantum matter, with worldline methods providing a technically precise and conceptually transparent analytical toolkit (Fecit et al., 20 Aug 2025).