Introduction of a boundary in topological field theories

Amoretti, Andrea; Braggio, Alessandro; Caruso, Giacomo; Maggiore, Nicola; Magnoli, Nicodemo

doi:10.1103/physrevd.90.125006

Cited by 21 publications

(18 citation statements)

References 28 publications

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“…It turns out that it is necessary to break Lorentz invariance even on the boundary plane, and this distinguishes our approach from the usual one [10]. In fact, if Lorentz invariance on the 2D boundary is imposed, no compatible boundary conditions for the fields can be obtained.…”

Section: Discussionmentioning

confidence: 99%

From Chern–Simons to Tomonaga–Luttinger

Maggiore

2018

Int. J. Mod. Phys. A

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: A single-sided boundary is introduced in the three-dimensional Chern-Simons model. It is shown that only one boundary condition for the gauge fields is possible, which plays the twofold role of chirality condition and bosonization rule for the two-dimensional Weyl fermion describing the degrees of freedom of the edge states of the Fractional Quantum Hall Effect. It is derived the symmetry on the boundary which determines the effective two dimensional action, whose equation of motion coincides with the continuity equation of the Tomonaga-Luttinger theory. The role of Lorentz symmetry and of discrete symmetries on the boundary is also discussed.

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Section: Discussionmentioning

confidence: 99%

From Chern–Simons to Tomonaga–Luttinger

Maggiore

2018

Int. J. Mod. Phys. A

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“…( 2), a tensorial Chern-Simons theory for a single two-form field B in 4+1-D dimensions is a total derivative and does not provide any (fractional) Hall current in the bulk [34]. In 4+1-D, only time-reversal-invariant BF-like theories are still well defined in the bulk and give rise to 3+1-D (non-chiral) dynamical gauge theories on the boundary [34,41]. Thus, the 4D FQHE can have a physical realization only in terms of point-like quasiparticles and the corresponding effective topological action is given in terms of Chern-Simons terms built from one-form gauge fields [20][21][22].…”

Section: D Fqhe For Membranesmentioning

confidence: 99%

Fractional Quantum Hall Effect for Extended Objects: from Skyrmionic Membranes to Dyonic Strings

Palumbo

2021

Preprint

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It is well known that in two spatial dimensions the fractional quantum Hall effect (FQHE) deals with point-like anyons that carry fractional electric charge and statistics. Moreover, in presence of a SO(3) order parameter, point-like skyrmions emerge and play a central role in the corresponding quantum Hall ferromagnetic phase. In this work, we show that in six spatial dimensions, the FQHE for extended objects shares very similar features with its two-dimensional counterpart. In the higher-dimensional case, the electromagnetic and hydrodynamical one-form gauge fields are replaced by three-form gauge fields and the usual point-like anyons are replaced by membranes, namely two-dimensional extended objects that can carry fractional charge and statistics. We focus on skyrmionic membranes, which are associated to a SO(5) order parameter and give rise to an higher-dimensional generalizaton of the quantum Hall ferromagnetism. We show that skyrmionic membranes naturally couple to the curved background through a generalized Wen-Zee term and can give us some insights about the chiral conformal field theory on the boundary. We then present a generalization of the Witten effect in six spatial dimensions by showing that one-dimensional extended monopoles (magnetic strings) in the bulk of the FQH states can acquire electric charge through an axion field by becoming dyonic strings.

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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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Introduction of a boundary in topological field theories

Cited by 21 publications

References 28 publications

From Chern–Simons to Tomonaga–Luttinger

From Chern–Simons to Tomonaga–Luttinger

Fractional Quantum Hall Effect for Extended Objects: from Skyrmionic Membranes to Dyonic Strings

Topological BF description of 2D accelerated chiral edge modes

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