We consider the theory of a generic rank-2 tensor field in three spacetime dimensions, which involves a symmetric tensor field transforming under infinitesimal diffeomorphisms, and a vector field, whose gauge transformation depends on a local vector parameter. The gauge fixing shows a non-trivial structure, and some non-intuitive possibilities are listed. Despite the fact that the theory is not topological, the energy-momentum tensor vanishes on-shell, which justifies the “quasi-topological” appellation we give to this theory. We show that the theory has three degrees of freedom. Moreover, we find an interesting physical interpretation, which consists in a generalized planar electromagnetism and in the emergence of two vector charges with restricted mobility. These are typical fractonic behaviours which can be related to the so-called traceless scalar and vector charge theories.

Ireland

Erica Bertolini

Italy

Alberto Blasi

Nicola Maggiore

Quasi-topological fractons: a 3D dipolar gauge theory

Fracton phases of matter constitute an interesting point of contact between condensed matter and high-energy physics. The limited mobility property of fracton quasi-particles finds applications in many different contexts, including quantum information, spin liquids, elasticity, hydrodynamics, gravity and holography. In this paper we adopt a field theoretical approach to investigate the three dimensional action of a rank-2 symmetric tensor field invariant under the covariant fracton symmetry. The theory appears as a non-topological higher rank generalization of the ordinary Chern-Simons model, depending only on the traceless part of the tensor gauge field. After defining a field strength, a rank-2 traceless “electric” field and a “magnetic” vector field are identified, in analogy with the standard Chern-Simons ones. Once matter is introduced, a Hall-like behaviour with fractonic features emerges. In particular, our model shows a Hall-like dipole current, together with a vectorial “flux-attachment” relation for dipoles. This gives a possible starting point for a fracton-vortex duality. A gauge-fixing term is then introduced, from which propagators are computed and the counting of the degrees of freedom is performed. Finally, the energy-momentum tensor is shown to be conserved and the integrated energy density is proved to be zero, which reminds the topological nature of the standard Chern-Simons model.

Daniel Shaikh

Hall-like behaviour of higher rank Chern-Simons theory of fractons

In this paper we consider four-dimensional (4D) linearized gravity (LG) with a planar boundary, where the most general boundary conditions are derived following Symanzik’s approach. The boundary breaks diffeomorphism invariance, and this results in a breaking of the corresponding Ward identity. From this, on the boundary we find two conserved currents that form an algebraic structure of the Kac-Moody type, with a central charge proportional to the action “coupling.” Moreover, we identify the boundary degrees of freedom, which are two symmetric rank-2 tensor fields, and derive the symmetry transformations, which are diffeomorphisms. The corresponding most general 3D action is obtained and a contact with the higher dimensional theory is established by requiring that the 3D equations of motion coincide with the 4D boundary conditions. Through this kind of holographic procedure, we find two solutions: LG for a single tensor field and LG for two tensor fields with a mixing term. Curiously, we find that the Symanzik’s 4D boundary term that governs the whole procedure contains a mass term of the Fierz-Pauli type for the bulk graviton.

Theory of a symmetric tensor field with boundary: Kac-Moody algebras in linearized gravity

In this paper we study the consequences of the introduction of a flat boundary on a four-dimensional (4D) covariant rank-2 gauge theory described by a linear combination of linearized gravity and covariant fracton theory. We show that this theory gives rise to a Maxwell-Chern-Simons-like theory of two rank-2 traceless symmetric tensor fields. This induced three-dimensional (3D) theory can be physically traced back to the traceless scalar charge theory of fractons, where the Chern-Simons-like term plays the role of a matter contribution. By further imposing time reversal invariance on the boundary, the Chern-Simons-like term disappears. Importantly, on the boundary of our 4D gauge theory we find a generalized U(1) Kaç-Moody algebra and the induced 3D theory is characterized by the conservation of the dipole moment.

Giandomenico Palumbo

Covariant fracton gauge theory with boundary

We show that the main properties of the fracton quasiparticles can be derived from a generalized covariant Maxwell-like action. Starting from a rank-2 symmetric tensor field <math><msub><mi>A</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math>, we build a partially symmetric rank-3 tensor field strength <math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>μ</mi><mi>ν</mi><mi>ρ</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math> which obeys a kind of Bianchi identity. The most general action invariant under the covariant “fracton” transformation <math><msub><mi>δ</mi><mrow><mtext>fract</mtext></mrow></msub><msub><mi>A</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mo>∂</mo><mi>μ</mi></msub><msub><mo>∂</mo><mi>ν</mi></msub><mi>Λ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> consists of two independent terms: one describing linearized gravity (LG) and the other referable to fractons. The whole action can be written in terms of <math><msub><mi>F</mi><mrow><mi>μ</mi><mi>ν</mi><mi>ρ</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math>, and the fracton part of the invariant Lagrangian writes as <math><msup><mi>F</mi><mn>2</mn></msup><mo>(</mo><mi>x</mi><mo>)</mo></math>, in analogy with Maxwell theory. The canonical momentum derived from the fracton Lagrangian coincides with the tensor electric field appearing in the fracton literature, and the field equations of motion, which have the same form as the covariant Maxwell equations (<math><msup><mo>∂</mo><mi>μ</mi></msup><msub><mi>F</mi><mrow><mi>α</mi><mi>β</mi><mi>μ</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></math>), can be written in terms of the generalized electric and magnetic fields and yield two of the four Maxwell equations (generalized electric Gauss and Ampère laws), while the other two (generalized magnetic Gauss and Faraday laws) are consequences of the “Bianchi identity” for the tensor <math><msub><mi>F</mi><mrow><mi>μ</mi><mi>ν</mi><mi>ρ</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math>, as in Maxwell theory. In the covariant generalization of the fracton theory, the equations describing the fracton limited mobility, i.e., the charge and dipole conservation, are not external constraints, but rather consequences of the field equations of motion, hence of the invariant action and, ultimately, of the fracton covariant symmetry. Finally, we increase the known analogies between LG and fracton theory by noting that both satisfy the generalized Gauss constraint which underlies the limited mobility property, which one would not expect in LG.

Maxwell theory of fractons

We consider the theory of a symmetric tensor field in 4D, invariant under a subclass of infinitesimal diffeomorphism transformations, where the vector diff parameter is the 4-divergence of a scalar parameter. The resulting gauge symmetry characterizes the “fracton” quasiparticles and identifies a theory which depends on a dimensionless parameter, which cannot be reabsorbed by a redefinition of the tensor field, despite the fact that the theory is free of interactions. This kind of “electromagnetic gauge symmetry” is weaker that the original diffeomorphism invariance, in the sense that the most general action contains, but is not limited to, linearized gravity, and we show how it is possible to switch continuously from linearized gravity to a mixed phase where both gravitons and fractons are present, without changing the degrees of freedom of the theory. The gauge fixing procedure is particularly rich and rather peculiar, and leads to the computation of propagators which in the massive case we ask to be tachyonic-free, thus constraining the domain of the parameter of the theory. Finally, a closer contact to fractons is made by the introduction of a parameter related to the “rate of propagation”. For a particular value of this parameter the theory does not propagate at all, and we guess that, for this reason, the resulting theory should be tightly related to the fracton excitations.

The theory of symmetric tensor field: From fractons to gravitons and back

The Abelian Chern-Simons theory is considered on a cylindrical spacetime <math><mrow><mi>R</mi><mo>×</mo><mi>D</mi></mrow></math>, in a not necessarily flat Lorentzian background. As in the flat bulk case with planar boundary, we find that also on the radial boundary of a curved background a Kaç-Moody algebra exists, with the same central charge as in the flat case, which henceforth depends neither on the bulk metric nor on the geometry of the boundary. The holographically induced theory on the 2D boundary is topologically protected, in the sense that it describes a Luttinger liquid, no matter which the bulk metric is. The main result of this paper is that a remnant of the 3D bulk theory resides in the chiral velocity of the edge modes, which is not a constant like in the flat bulk case, but it is local, depending on the determinant of the induced metric on the boundary. This result may provide a theoretical framework for the recently observed accelerated chiral bosons on the edge of some Hall systems.

United Kingdom

Giulio Gambuti

Notes from the bulk: Metric dependence of the edge states of Chern-Simons theory

Modifying gravity at large distances by means of a massive graviton may explain the observed acceleration of the Universe without Dark Energy. The standard paradigm for Massive Gravity is the Fierz–Pauli theory, which, nonetheless, displays well known flaws in its massless limit. The most serious one is represented by the vDVZ discontinuity, which consists in a disagreement between the massless limit of the Fierz–Pauli theory and General Relativity. Our approach is based on a field-theoretical treatment of Massive Gravity: General Relativity, in the weak field approximation, is treated as a gauge theory of a symmetric rank-2 tensor field. This leads us to propose an alternative theory of linearized Massive Gravity, describing five degrees of freedom of the graviton, with a good massless limit, without vDVZ discontinuity, and depending on one mass parameter only, in agreement with the Fierz–Pauli theory.

Fierz–Pauli theory reloaded: from a theory of a symmetric tensor field to linearized massive gravity

We consider the harmonic gauge condition in linearized gravity, seen as a gauge theory for a symmetric tensor field. Once the harmonic gauge condition is implemented, as customary, according to the Faddeev-Popov procedure, the gauge fixed action still depends on one gauge parameter. Consequently, the harmonic gauge appears to be a class of conditions, rather than a particular one. This allows to give a physical motivation for the covariant harmonic gauge(s), which emerges when the gravitational perturbation is given a mass term. In fact, for a particular choice of harmonic gauge, we find a theory of linearized massive gravity displaying five degrees of freedom, as it should, and which is not affected by the vDVZ discontinuity, differently from what happens in the standard Fierz-Pauli theory.

A note on harmonic gauge(s) in massive gravity

We consider a generic description of interacting dynamical Dark Energy, characterized by an equation of state with a time dependent coefficient w(t), and which may interact with both radiation and matter. Without referring to any particular cosmological model, we find a differential equation which must be satisfied by w(t) and involving the function Q(t) which describes the interaction between Dark Energy and the other cosmological fluids. The relation we find represents a constraint for various models of interacting dynamical Dark Energy. In addition, an observable is proposed, depending on kinematic variables and on density parameters, which may serve as a new test for $$\Lambda $$ <math><mi>Λ</mi></math> CDM.