Worldline Formulations of Covariant Fracton Theories

We develop worldline formulations of covariant fracton gauge theories. These are a one-parameter family of gauge theories of a rank-two symmetric tensor field, invariant under a scalar gauge transformation involving a double derivative. These theories, which can be interpreted as linearized gravity theories invariant under longitudinal diffeomorphisms, provide a covariant framework for studying Lorentz-breaking fracton quasiparticles, which are excitations with restricted mobility due to dipole-moment conservation. We construct three worldline models. The first two are obtained by deducing their constraint structure directly from the spacetime gauge transformations. By applying BRST quantization, we show that these models reproduce the BV spectrum and the associated BRST transformations of two specific fracton theories. The third model is defined as a deformation of the second one: although free, it is analyzed by drawing inspiration from the standard treatment of interacting worldline systems, and is shown to capture almost the entire family of covariant fracton theories. Finally, we discuss the gauge-fixing, comparing the BV-BRST spacetime perspective with the worldline analogue of the ``Siegel gauge" employed in string field theory.

💡 Research Summary

The paper develops a first‑quantized, worldline description of covariant fracton gauge theories, which are rank‑two symmetric tensor gauge theories invariant under the scalar gauge transformation δΛ hμν = ∂μ∂νΛ. These theories, originally introduced as Lorentz‑breaking models describing fracton quasiparticles with conserved charge and dipole moment, can be embedded in a covariant framework that resembles linearized gravity but with only longitudinal diffeomorphisms. The authors aim to reproduce the full Batalin‑Vilkovisky (BV) formulation of these spacetime theories from a worldline perspective, using BRST quantization to generate the complete field‑antifield spectrum.

The work is organized around three worldline models:

1. Tensor Model – The authors introduce a pair of complex symmetric tensor oscillators (αμν, \barαμν) in phase space together with the usual coordinates (xμ, pμ). The primary constraint is L = αμν pμ pν (and its conjugate \bar L). Their Poisson brackets close with the free Hamiltonian H = p², forming a first‑class algebra. After constructing the nilpotent BRST charge and imposing the physical state condition, the resulting cohomology reproduces the BV spectrum {λ, hμν, hμν, λ} and the BRST transformations of the covariant fracton theory. This model corresponds to the parameter relation 2α − β = 0, i.e. a specific line in the (α, β) parameter space.

2. Vector Model – Here a pair of complex vector oscillators (αμ, \barαμ) is used. The constraint becomes quadratic both in momenta and oscillators: L = (α·p) (α·p). To achieve closure, a third constraint ℓ is introduced, leading to a first‑class algebra with structure functions (rather than constant structure constants). Remarkably, the usual Hamiltonian constraint H = p² cannot be retained without breaking consistency, so the model is defined without it. Nevertheless, the BRST analysis yields a consistent worldline description of a different fracton theory, now satisfying 2α + 3β = 0 (valid for D ≥ 4).

3. Deformed Vector Model – Building on the vector model, the authors deform the constraint algebra by adding non‑linear functions of L, \bar L, and ℓ, inspired by techniques used for interacting worldline systems. Although the model remains free (no background interactions are turned on), the deformation allows it to reproduce essentially the entire two‑parameter family of covariant fracton theories. The only exceptions are three special loci: (i) 2α − β = 0 (already covered by the tensor model), (ii) β = 0 (a scalar‑only limit), and (iii) 2α + (D − 1)β = 0 (the traceless limit where an extra Weyl‑type scaling symmetry appears). Thus the deformed model provides a near‑complete worldline realization of the covariant fracton landscape.

After constructing the models, the paper turns to gauge fixing. From the spacetime side, the BV‑BRST formalism introduces antighosts, Nakanishi–Lautrup fields, and a gauge‑fixing fermion to solve the master equation. On the worldline side, the authors adopt an analogue of the Siegel gauge familiar from string field theory: they impose L = \bar L = 0 directly on the worldline Hilbert space via Lagrange multipliers. Both procedures lead to the same physical spectrum, but the worldline Siegel gauge is computationally simpler and makes the reduction of unphysical degrees of freedom more transparent. The comparison highlights that while BV‑BRST is more flexible (e.g., for interacting extensions), the worldline approach offers an intuitive and economical gauge‑fixing scheme.

Overall, the paper achieves several notable results: (i) it provides explicit worldline actions whose BRST cohomology reproduces the BV formulation of covariant fracton gauge theories; (ii) it identifies three distinct worldline realizations (tensor, vector, deformed vector) that together cover almost the entire parameter space of these theories; (iii) it clarifies the role of the Hamiltonian constraint, showing that a consistent fracton worldline model can exist without it, contrary to conventional particle models; and (iv) it establishes a clear correspondence between spacetime BV‑BRST gauge fixing and the worldline Siegel gauge. These developments open a new avenue for studying fracton physics, especially for perturbative calculations (heat‑kernel coefficients, scattering amplitudes) and non‑perturbative phenomena (pair production, strong‑field effects) using worldline techniques. Future work may extend the deformed model to include genuine interactions, explore supersymmetric extensions, or apply the formalism to holographic duals of fracton phases.