scholarly journals Time-reversal symmetry, anomalies, and dualities in (2+1)$d$

2018 ◽  
Vol 5 (1) ◽  
Author(s):  
Clay Cordova ◽  
Po-Shen Hsin ◽  
Nathan Seiberg
Keyword(s):  
Time Reversal ◽  
Gauge Theories ◽  
Dirac Fermion ◽  
Global Symmetry ◽  
Time Reversal Symmetry ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Long Distance ◽  
Free Theory ◽  
Topological Quantum

We study continuum quantum field theories in 2+1 dimensions with time-reversal symmetry \cal T. The standard relation {\cal T}^2=(-1)^F is satisfied on all the “perturbative operators” i.e. polynomials in the fundamental fields and their derivatives. However, we find that it is often the case that acting on more complicated operators {\cal T}^2=(-1)^F {\cal M} with \cal M a non-trivial global symmetry. For example, acting on monopole operators, \cal M could be \pm1±1 depending on the magnetic charge. We study in detail U(1)U(1) gauge theories with fermions of various charges. Such a modification of the time-reversal algebra happens when the number of odd charge fermions is 2 ~{\rm mod }~4, e.g. in QED with two fermions. Our work also clarifies the dynamics of QED with fermions of higher charges. In particular, we argue that the long-distance behavior of QED with a single fermion of charge 22 is a free theory consisting of a Dirac fermion and a decoupled topological quantum field theory. The extension to an arbitrary even charge is straightforward. The generalization of these abelian theories to SO(N)SO(N) gauge theories with fermions in the vector or in two-index tensor representations leads to new results and new consistency conditions on previously suggested scenarios for the dynamics of these theories. Among these new results is a surprising non-abelian symmetry involving time-reversal.

scholarly journals THE RELATIONS BETWEEN GENERALIZED FIELDS AND SUPERFIELDS FORMALISMS OF THE BATALIN–VILKOVISKY METHOD OF QUANTIZATION

2004 ◽  
Vol 19 (14) ◽  
pp. 2339-2353 ◽  
Author(s):  
ÖMER F. DAYI
Keyword(s):  
General Solution ◽  
Master Equation ◽  
Gauge Theories ◽  
Relativistic Particle ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Yang Mills ◽  
Topological Quantum ◽  
Topological Quantum Field Theories

A general solution of the Batalin–Vilkovisky master equation was formulated in terms of generalized fields. Recently, a superfields approach of obtaining solutions of the Batalin–Vilkovisky master equation is also established. Superfields formalism is usually applied to topological quantum field theories. However, generalized fields method is suitable to find solutions of the Batalin–Vilkovisky master equation either for topological quantum field theories or the usual gauge theories like Yang–Mills theory. We show that by truncating some components of superfields with appropriate actions, generalized fields formalism of the usual gauge theories result. We demonstrate that for some topological quantum field theories and the relativistic particle both of the methods possess the same field contents and yield similar results. Inspired by the observed relations, we give the solution of the BV master equation for on-shell N=1 supersymmetric Yang–Mills theory utilizing superfields.

scholarly journals DIFF(Σ) AND METRICS FROM HAMILTONIAN-TQFT'S IN 2+1 DIMENSIONS

1993 ◽  
Vol 08 (24) ◽  
pp. 2277-2283 ◽  
Author(s):  
ROGER BROOKS
Keyword(s):  
Topological Field Theories ◽  
Gauge Theories ◽  
Dimensional Space ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Two Dimensional ◽  
Canonical Gravity ◽  
Topological Quantum ◽  
Topological Field

The constraints of BF topological gauge theories are used to construct Hamiltonians which are anti-commutators of the BRST and anti-BRST operators. Such Hamiltonians are a signature of topological quantum field theories (TQFTs). By construction, both classes of topological field theories share the same phase spaces and constraints. We find that, for (2+1)- and (1+1)-dimensional space-times foliated as M=Σ × ℝ, a homomorphism exists between the constraint algebras of our TQFT and those of canonical gravity. The metrics on the two-dimensional hypersurfaces are also obtained.

scholarly journals Discrete theta angles, symmetries and anomalies

2021 ◽  
Vol 10 (2) ◽  
Author(s):  
Po-Shen Hsin ◽  
Ho Tat Lam
Keyword(s):  
Gauge Theory ◽  
Gauge Theories ◽  
Topological Quantum Field Theory ◽  
Group Symmetry ◽  
Global Symmetry ◽  
Quantum Field ◽  
Additional Symmetry ◽  
Gauge Groups ◽  
Topological Quantum ◽  
Symmetry Protected

Gauge theories in various dimensions often admit discrete theta angles, that arise from gauging a global symmetry with an additional symmetry protected topological (SPT) phase. We discuss how the global symmetry and ’t Hooft anomaly depends on the discrete theta angles by coupling the gauge theory to a topological quantum field theory (TQFT). We observe that gauging an Abelian subgroup symmetry, that participates in symmetry extension, with an additional SPT phase leads to a new theory with an emergent Abelian symmetry that also participates in a symmetry extension. The symmetry extension of the gauge theory is controlled by the discrete theta angle which comes from the SPT phase. We find that discrete theta angles can lead to two-group symmetry in 4d4d QCD with SU(N),SU(N)/\mathbb{Z}_kSU(N),SU(N)/ℤk or SO(N)SO(N) gauge groups as well as various 3d3d and 2d2d gauge theories.

TOPOLOGICAL QUANTUM FIELD THEORIES ASSOCIATED TO FINITE GROUPS AND CROSSED G-SETS

1992 ◽  
Vol 01 (01) ◽  
pp. 1-20 ◽  
Author(s):  
DAVID N. YETTER
Keyword(s):  
Gauge Group ◽  
Quantum Groups ◽  
Finite Groups ◽  
Gauge Theories ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Link Invariants ◽  
Topological Quantum ◽  
Topological Quantum Field Theories

Using methods suggested by the work of Turaev and Viro [11, 12], we provide a detailed construction of topological quantum field theories associated to finite crossed G-sets. Our construction of theories associated to finite groups fills in some details implicit in Dijkgraaf and Witten's [3] discussion of topological gauge theories with finite gauge group, while the theories associated to finite crossed G-sets simultaneously extend Dijkgraaf and Witten's [3] results to 3-manifolds equipped with links and Freyd and Yetter's [5] construction of link invariants from crossed G-sets from links in the 3-sphere to links in arbitrary 3-manifolds. Topological interpretations of the manifold and link invariants associated to these TQFT's are provided. We conclude discussion of our results as a toy model for QFT and of their relation to quantum groups.

TOPOLOGICAL QUANTUM FIELD THEORIES AND GAUGE INVARIANCE IN STOCHASTIC QUANTIZATION

1991 ◽  
Vol 06 (16) ◽  
pp. 2793-2803 ◽  
Author(s):  
Laurent Baulieu
Keyword(s):  
Gauge Theories ◽  
Three Dimensional ◽  
Langevin Equations ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Topological Quantum ◽  
Brst Charge ◽  
Deep Relationship

The Langevin equations describing the quantization of gauge theories have a geometrical structure. We show that stochastically quantized gauge theories are governed by a single differential operator. The latter combines supersymmetry and ordinary gauge transformations. Quantum field theory can be defined on the basis of a Hamiltonian of the type [Formula: see text], where Q has has deep relationship with the conserved BRST charge of a topological gauge theory, and [Formula: see text] is its adjoint. We display the examples of Yang-Mills theory and of 2D gravity. Interesting applications are for first order actions, in particular for the theories defined by the three dimensional Chern Simons action as well as the “two dimensional” ∫M2TrϕF.

scholarly journals a×b=c in 2+1D TQFT

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 468
Author(s):  
Matthew Buican ◽  
Linfeng Li ◽  
Rajath Radhakrishnan
Keyword(s):  
Gauge Theories ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Mathieu Group ◽  
Gauge Groups ◽  
Global Properties ◽  
Topological Quantum ◽  
The One ◽  
Chern Simons

We study the implications of the anyon fusion equation a×b=c on global properties of 2+1D 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, M24), 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.

scholarly journals The Power of Duality — Exact Results in 4D SUSY Field Theory

1997 ◽  
Vol 12 (29) ◽  
pp. 5171-5182 ◽  
Author(s):  
N. Seiberg
Keyword(s):  
Gauge Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Long Distance ◽  
Strongly Coupled ◽  
Weakly Coupled ◽  
Wide Range ◽  
Distance Behavior ◽  
Physical Phenomena

Recently the vacuum structure of a large class of four-dimensional (supersymmetric) quantum field theories was determined exactly. These theories exhibit a wide range of interesting new physical phenomena. One of the main new insights is the role of "electric–magnetic duality." In its simplest form it describes the long distance behavior of some strongly coupled, and hence complicated, "electric theories" in terms of weakly coupled "magnetic theories." This understanding sheds new light on confinement and the Higgs mechanism and uncovers new phases of four-dimensional gauge theories. We review these developments and speculate on the outlook.

scholarly journals THE POWER OF DUALITY — EXACT RESULTS IN 4D SUSY FIELD THEORY

2001 ◽  
Vol 16 (27) ◽  
pp. 4365-4376 ◽  
Author(s):  
N. SEIBERG
Keyword(s):  
Gauge Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Long Distance ◽  
Strongly Coupled ◽  
Weakly Coupled ◽  
Wide Range ◽  
Distance Behavior ◽  
Physical Phenomena

Recently the vacuum structure of a large class of four-dimensional (supersymmetric) quantum field theories was determined exactly. These theories exhibit a wide range of interesting new physical phenomena. One of the main new insights is the role of "electric–magnetic duality." In its simplest form it describes the long distance behavior of some strongly coupled, and hence complicated, "electric theories" in terms of weakly coupled "magnetic theories." This understanding sheds new light on confinement and the Higgs mechanism and uncovers new phases of four-dimensional gauge theories. We review these developments and speculate on the outlook.

GENERAL COVARIANCE, TOPOLOGICAL QUANTUM FIELD THEORIES AND FRACTIONAL STATISTICS

1992 ◽  
Vol 07 (02) ◽  
pp. 209-234 ◽  
Author(s):  
J. GAMBOA
Keyword(s):  
Hamiltonian Formalism ◽  
Field Theories ◽  
General Covariance ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Fractional Statistics ◽  
Multiply Connected ◽  
Topological Quantum ◽  
Gauge Fixing ◽  
Topological Quantum Field Theories

Topological quantum field theories and fractional statistics are both defined in multiply connected manifolds. We study the relationship between both theories in 2 + 1 dimensions and we show that, due to the multiply-connected character of the manifold, the propagator for any quantum (field) theory always contains a first order pole that can be identified with a physical excitation with fractional spin. The article starts by reviewing the definition of general covariance in the Hamiltonian formalism, the gauge-fixing problem and the quantization following the lines of Batalin, Fradkin and Vilkovisky. The BRST–BFV quantization is reviewed in order to understand the topological approach proposed here.

scholarly journals DMRG study of strongly interacting $\mathbb{Z}_2$ flatbands: a toy model inspired by twisted bilayer graphene

2020 ◽  
Vol 3 (2) ◽  
Author(s):  
Paul Eugenio ◽  
Ceren Dag
Keyword(s):  
Magnetic Field ◽  
Time Reversal ◽  
Ground States ◽  
Bilayer Graphene ◽  
Chern Number ◽  
Strong Interactions ◽  
Time Reversal Symmetry ◽  
Density Matrix Renormalization ◽  
Topological Quantum ◽  
Twisted Bilayer Graphene

Strong interactions between electrons occupying bands of opposite (or like) topological quantum numbers (Chern=\pm1=±1), and with flat dispersion, are studied by using lowest Landau level (LLL) wavefunctions. More precisely, we determine the ground states for two scenarios at half-filling: (i) LLL’s with opposite sign of magnetic field, and therefore opposite Chern number; and (ii) LLL’s with the same magnetic field. In the first scenario – which we argue to be a toy model inspired by the chirally symmetric continuum model for twisted bilayer graphene – the opposite Chern LLL’s are Kramer pairs, and thus there exists time-reversal symmetry (\mathbb{Z}_2ℤ2). Turning on repulsive interactions drives the system to spontaneously break time-reversal symmetry – a quantum anomalous Hall state described by one particle per LLL orbital, either all positive Chern |{++\cdots+}\rangle|++⋯+⟩ or all negative |{--\cdots-}\rangle|−−⋯−⟩. If instead, interactions are taken between electrons of like-Chern number, the ground state is an SU(2)SU(2) ferromagnet, with total spin pointing along an arbitrary direction, as with the \nu=1ν=1 spin-\frac{1}{2}12 quantum Hall ferromagnet. The ground states and some of their excitations for both of these scenarios are argued analytically, and further complimented by density matrix renormalization group (DMRG) and exact diagonalization.

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