$a\times b=c$ in $2+1$D TQFT
- URL: http://arxiv.org/abs/2012.14689v4
- Date: Wed, 2 Jun 2021 15:38:06 GMT
- Title: $a\times b=c$ in $2+1$D TQFT
- Authors: Matthew Buican, Linfeng Li, and Rajath Radhakrishnan
- Abstract summary: We study the implications of the anyon fusion equation $atimes b=c$ on global properties of $2+1$D topological quantum field theories (TQFTs)
We argue that the appearance of such fusions for non-abelian $a$ and $b$ can also be an indication of zero-form symmetries in a TQFT.
- Score: 2.982218441172364
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: We study the implications of the anyon fusion equation $a\times b=c$ on
global properties of $2+1$D topological quantum field theories (TQFTs). Here
$a$ and $b$ are anyons that fuse together to give a unique anyon, $c$. As is
well known, when at least one of $a$ and $b$ is abelian, such equations
describe aspects of the one-form symmetry of the theory. When $a$ and $b$ are
non-abelian, the most obvious way such fusions arise is when a TQFT can be
resolved into a product of TQFTs with trivial mutual braiding, and $a$ and $b$
lie in separate factors. More generally, we argue that the appearance of such
fusions for non-abelian $a$ and $b$ can also be an indication of zero-form
symmetries in a TQFT, of what we term "quasi-zero-form symmetries" (as in the
case of discrete gauge theories based on the largest Mathieu group, $M_{24}$),
or of the existence of non-modular fusion subcategories. We study these ideas
in a variety of TQFT settings from (twisted and untwisted) discrete gauge
theories to Chern-Simons theories based on continuous gauge groups and related
cosets. Along the way, we prove various useful theorems.
Related papers
- A Unified Framework for Uniform Signal Recovery in Nonlinear Generative
Compressed Sensing [68.80803866919123]
Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $mathbfx*$ rather than for all $mathbfx*$ simultaneously.
Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples.
We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy.
arXiv Detail & Related papers (2023-09-25T17:54:19Z) - Algebraic Aspects of Boundaries in the Kitaev Quantum Double Model [77.34726150561087]
We provide a systematic treatment of boundaries based on subgroups $Ksubseteq G$ with the Kitaev quantum double $D(G)$ model in the bulk.
The boundary sites are representations of a $*$-subalgebra $Xisubseteq D(G)$ and we explicate its structure as a strong $*$-quasi-Hopf algebra.
As an application of our treatment, we study patches with boundaries based on $K=G$ horizontally and $K=e$ vertically and show how these could be used in a quantum computer
arXiv Detail & Related papers (2022-08-12T15:05:07Z) - Monogamy of entanglement between cones [68.8204255655161]
We show that monogamy is not only a feature of quantum theory, but that it characterizes the minimal tensor product of general pairs of convex cones.
Our proof makes use of a new characterization of products of simplices up to affine equivalence.
arXiv Detail & Related papers (2022-06-23T16:23:59Z) - Symmetry-resolved entanglement entropy in critical free-fermion chains [0.0]
symmetry-resolved R'enyi entanglement entropy is known to have rich theoretical connections to conformal field theory.
We consider a class of critical quantum chains with a microscopic U(1) symmetry.
For the density matrix, $rho_A$, of subsystems of $L$ neighbouring sites we calculate the leading terms in the large $L$ expansion of the symmetry-resolved R'enyi entanglement entropies.
arXiv Detail & Related papers (2022-02-23T19:00:03Z) - Uncertainties in Quantum Measurements: A Quantum Tomography [52.77024349608834]
The observables associated with a quantum system $S$ form a non-commutative algebra $mathcal A_S$.
It is assumed that a density matrix $rho$ can be determined from the expectation values of observables.
Abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables.
arXiv Detail & Related papers (2021-12-14T16:29:53Z) - Quantum double aspects of surface code models [77.34726150561087]
We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry.
We show how our constructions generalise to $D(H)$ models based on a finite-dimensional Hopf algebra $H$.
arXiv Detail & Related papers (2021-06-25T17:03:38Z) - $T\bar T$-deformed Fermionic Theories Revisited [0.0]
We revisit $Tbar T$ deformations of $d=2$ theories with fermions with a view toward the quantization.
We ask about different $Tbar T$ deformations, such as manifestly supersymmetric $Tbar T$ and also more generally via the symmetric energy-momentum.
arXiv Detail & Related papers (2021-04-19T18:00:07Z) - The Geometry of Time in Topological Quantum Gravity of the Ricci Flow [62.997667081978825]
We continue the study of nonrelativistic quantum gravity associated with a family of Ricci flow equations.
This topological gravity is of the cohomological type, and it exhibits an $cal N=2$ extended BRST symmetry.
We demonstrate a standard one-step BRST gauge-fixing of a theory whose fields are $g_ij$, $ni$ and $n$, and whose gauge symmetries consist of (i) the topological deformations of $g_ij$, and (ii) the ultralocal nonrelativistic limit of space
arXiv Detail & Related papers (2020-11-12T06:57:10Z) - Skein-Theoretic Methods for Unitary Fusion Categories [0.0]
Unitary fusion categories (UFCs) have gained increased attention due to emerging connections with quantum physics.
We consider a fusion rule of the form $qotimes cong mathbf1oplusbigoplusk_i=1x_i$ in a UFC $mathcalC$, and extract information using the graphical calculus.
arXiv Detail & Related papers (2020-08-17T07:35:56Z) - Quantum computation and measurements from an exotic space-time R4 [0.0]
A valid subgroup $H$ of index $d$ in $G$ leads to a'magic' state $left|psirightrangle$ in $d$-dimensional Hilbert space.
A new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of 'quantum gravity'
arXiv Detail & Related papers (2020-01-22T15:16:03Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.