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.

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 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.

QFT and unification of knot theories

2014 ◽  
Vol 29 (24) ◽  
pp. 1430025
Author(s):  
Alexey Sleptsov
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Knot Theory ◽  
Topological Quantum Field Theory ◽  
Quantum Field ◽  
Knot Invariants ◽  
Topological Quantum

We discuss relation between knot theory and topological quantum field theory. Also it is considered a theory of superpolynomial invariants of knots which generalizes all other known theories of knot invariants. We discuss a possible generalization of topological quantum field theory with the help of superpolynomial invariants.

SUPERSYMMETRIC TENSOR GAUGE THEORIES

1989 ◽  
Vol 04 (14) ◽  
pp. 1343-1353 ◽  
Author(s):  
T.E. CLARK ◽  
C.-H. LEE ◽  
S.T. LOVE
Keyword(s):  
Gauge Theory ◽  
Gauge Symmetry ◽  
Sigma Models ◽  
Gauge Field ◽  
Gauge Theories ◽  
Symmetric Tensor ◽  
Global Symmetry ◽  
Invariant Action ◽  
Symmetry Transformations ◽  
Linear Sigma Models

The supersymmetric extensions of anti-symmetric tensor gauge theories and their associated tensor gauge symmetry transformations are constructed. The classical equivalence between such supersymmetric tensor gauge theories and supersymmetric non-linear sigma models is established. The global symmetry of the supersymmetric tensor gauge theory is gauged and the locally invariant action is obtained. The supercurrent on the Kähler manifold is found in terms of the supersymmetric tensor gauge field.

scholarly journals Symmetry-enriched quantum spin liquids in (3 + 1)d

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Po-Shen Hsin ◽  
Alex Turzillo
Keyword(s):  
Gauge Theory ◽  
Gauge Field ◽  
Quantum Spin ◽  
Global Symmetry ◽  
Spin Liquids ◽  
Systematic Method ◽  
Bosonic Field ◽  
Quantum Phases ◽  
Symmetry Protected ◽  
Weyl Fermions

Abstract We use the intrinsic one-form and two-form global symmetries of (3+1)d bosonic field theories to classify quantum phases enriched by ordinary (0-form) global symmetry. Different symmetry-enriched phases correspond to different ways of coupling the theory to the background gauge field of the ordinary symmetry. The input of the classification is the higher-form symmetries and a permutation action of the 0-form symmetry on the lines and surfaces of the theory. From these data we classify the couplings to the background gauge field by the 0-form symmetry defects constructed from the higher-form symmetry defects. For trivial two-form symmetry the classification coincides with the classification for symmetry fractionalizations in (2 + 1)d. We also provide a systematic method to obtain the symmetry protected topological phases that can be absorbed by the coupling, and we give the relative ’t Hooft anomaly for different couplings. We discuss several examples including the gapless pure U(1) gauge theory and the gapped Abelian finite group gauge theory. As an application, we discover a tension with a conjectured duality in (3 + 1)d for SU(2) gauge theory with two adjoint Weyl fermions.

TOWARDS A QUANTUM ALGORITHM FOR THE PERMANENT

2007 ◽  
Vol 05 (01n02) ◽  
pp. 223-228 ◽  
Author(s):  
ANNALISA MARZUOLI ◽  
MARIO RASETTI
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Computation ◽  
Quantum Computer ◽  
Quantum Algorithm ◽  
Topological Quantum Field Theory ◽  
Quantum Field ◽  
Topological Quantum

We resort to considerations based on topological quantum field theory to outline the development of a possible quantum algorithm for the evaluation of the permanent of a 0 - 1 matrix. Such an algorithm might represent a breakthrough for quantum computation, since computing the permanent is considered a "universal problem", namely, one among the hardest problems that a quantum computer can efficiently handle.

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