From Chern–Simons to Tomonaga–Luttinger

Maggiore, Nicola

doi:10.1142/s0217751x18500136

Cited by 11 publications

(12 citation statements)

References 24 publications

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“…We recover here the physical interpretation of the parameter v appearing in the boundary conditions (2.13) as the chiral velocity of the edge modes living on the boundary of CS theory, which are known, in the framework of FQHE, to be measurable quantities. In most Hall systems, the observed chiral velocity is constant, and this is achieved by a CS theory built on a flat 3D spacetime with planar boundary [52]. The novelty we are finding here, is that v is now a local quantity, in particular depending on time, which is a consequence of considering the CS theory on a curved, instead of flat, spacetime.…”

Section: Holographic Contactmentioning

confidence: 82%

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

Bertolini,

Gambuti,

Maggiore

2021

Preprint

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The abelian Chern-Simons theory is considered on a cylindrical spacetime R×D, 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.

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Section: Holographic Contactmentioning

confidence: 82%

“…As we shall see, it will be crucial for the determination of the boundary algebra and of the 2D theory holographically induced on the boundary. Notice that it holds for any bulk metric, and it is simply the curved extension of its flat counterpart [52]. From (2.23), at vanishing external sources J k (x) = 0 (i.e.…”

Section: Ward Identitymentioning

confidence: 98%

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

Bertolini,

Gambuti,

Maggiore

2021

Preprint

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“…In addition, we chose to keep 3D covariance on the boundary. A more general, non covariant boundary term could have been written [26,27,28].…”

Section: The Model: Action Boundary Conditions and Ward Identitiesmentioning

confidence: 99%

Topologically Protected Duality on The Boundary of Maxwell-BF Theory

Blasi

Maggiore

2019

Symmetry

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The Maxwell-BF theory with a single-sided planar boundary is considered in Euclidean four dimensional spacetime. The presence of a boundary breaks the Ward identities which describe the gauge symmetries of the theory, and, using standard methods of quantum field theory, the most general boundary conditions and a nontrivial current algebra on the boundary are derived. The electromagnetic structure which characterizes the boundary is used to identify the three dimensional degrees of freedom, which turn out to be formed by a scalar field and a vector field, related by a duality relation.The induced three dimensional theory shows a strong-weak coupling duality which separates different regimes described by different covariant actions. The role of the Maxwell term in the bulk action is discussed, together with the relevance of the topological nature of the bulk action for the boundary physics.

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“…The study of TQFT with boundary led to remarkable results, mainly in condensed matter physics. The Hall systems have been understood in terms of a 3D CS theory with planar boundary [9,10,11,12], and the Topological Insulators (TI), which represent the other important topological phase of matter [13,14,15,16], are described by the topological BF theory with planar boundary, in 3D and 4D [17,18,19,20]. In both cases it has ben possible to show the existence, on the boundary, of conserved currents forming an algebra of the Kaç-Moody (KM) type [21,22], with central charge proportional to the inverse of the action coupling constant.…”

Section: Introductionmentioning

confidence: 99%

Topological BF description of 2D accelerated chiral edge modes

Bertolini,

Fecit,

Maggiore

2022

Preprint

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In this paper we consider the topological abelian BF theory with radial boundary on a generic 3D manifold. Our aim is to study if, where and how the boundary keeps memory of the details of the background metric. We find that some features are topologically protected and do not depend on the bulk metric. These are the Kaç-Moody algebra formed by the conserved currents and its central charge, which is proportional to the inverse of the bulk action coupling constant. We then derive the 2D action holographically induced on the boundary, which depends on two scalar fields, and which can be decoupled in two Luttinger actions describing two chiral bosons moving on the edge of the 3D bulk. The outcome is that these edge excitations are accelerated, as a direct consequence of the non-flat nature of the bulk spacetime. The chiral velocities of the edge modes, indeed, acquire a local dependence through the determinant of the induced metric on the boundary. We find three possibilities for the motion of the edge quasiparticles: same directions, opposite directions and a single-moving mode. But, requiring that the Hamiltonian of the 2D theory is bounded by below, the case of edge modes moving in the same direction is ruled out: systems involving parallel Hall currents (for instance Fractional Quantum Hall Effect with ν = 2/5) cannot be described by a BF theory with boundary, independently from the geometry of the bulk spacetime, because of positive energy considerations. We are therefore left with physical situations characterized by edge excitations moving with opposite velocities (examples are FQHE with ν = 1 − 1/n, with n positive integer, and Helical Luttinger Liquids phenomena) or a single-moving mode (Quantum Anomalous Hall). A strong restriction is obtained by requiring Time Reversal symmetry, which uniquely identifies modes with equal and opposite velocities, and we know that this is the case of Topological Insulators. The novelty, with respect to the flat bulk background, is that the modes have local velocities, which corresponds to Topological Insulators with accelerated edge modes.

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From Chern–Simons to Tomonaga–Luttinger

Cited by 11 publications

References 24 publications

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

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

Topologically Protected Duality on The Boundary of Maxwell-BF Theory

Topological BF description of 2D accelerated chiral edge modes

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