SU (3) Chern-Simons field theory in three-manifolds

We present the solution of the non-Abelian SU (3) Chern-Simons field theory defined in a generic three-manifold which is closed, connected and orientable. The surgery rules, which permit us to solve the theory, are derived and several examples of vacuum expectation values of Wilson line operators are computed. The three-manifold invariant associated with the non-Abelian SU (3) Chern-Simons model is defined and its values are computed for various three-manifolds.     

Spinor Decomposition of SU(2) Gauge Potential and Spinor Structures of Chern-Simons and Chern Density

In this paper, the decomposition of SU(2) gauge potential in terms of Pauli spinor is studied. Using thisdecomposition, the spinor structures of Chern-Simons form and the Chern density are obtained. Furthermore, the knotquantum number of non-Abelian gauge theory can be expressed by the Chern-Simons spinor structure, and the secondChern number is characterized by the Hopf indices and the Brouwer degrees of φ-mapping.     

Fermionic Magnons, non-Abelian spinons, and the spin quantum Hall effect from an exactly solvable spin-1/2 Kitaev model with SU(2) symmetry

We introduce an exactly solvable SU(2)-invariant spin-1/2 model with exotic spin excitations. With time reversal symmetry (TRS), the ground state is a spin liquid with gapless or gapped spin-1 but fermionic excitations. When TRS is broken, the resulting spin liquid exhibits deconfined vortex excitations which carry spin-1/2 and obey non-Abelian statistics. We show that this SU(2) invariant non-Abelian spin liquid exhibits the spin quantum Hall effect with quantized spin Hall conductivity σ(xy)(s)=?/2π, and that the spin response is effectively described by the SO(3) level-1 Chern-Simons theory at low energy. We further propose that a SU(2) level-2 Chern-Simons theory is the effective field theory describing the topological structure of the non-Abelian SU(2) invariant spin liquid.     

Spinor Decomposition of SU（2） Gauge Potential and Spinor Structures of Chern-Simons and Chern Density

In this paper, the decomposition of SU(2) gauge potential in terms of Pauli spinor is studied. Using this decomposition, the spinor structures of Chern Simons form and the Chern density are obtained. Furthermore, the knot quantum number of non-Abelian gauge theory can be expressed by the Chern-Simons spinor structure, and the second Chern number is characterized by the Hopf indices and the Brouwer degrees of Φ-mapping.     

0-extended supergravity and Chern-Simons theories

We give generalizations of extended Poincaré supergravity with arbitrarily many supersymmetries in the absence of central charges in three dimensions by gauging its intrinsic global SO(N) symmetry. We call these ₀ (Aleph-null) supergravity theories. We further couple a non-Abelian supersymmetric Chern-Simons theory and an Abelian topological BF theory to ₀ supergravity. Our result overcomes the previous difficulty for supersymmetrization of Chern-Simons theories beyond N = 4. This feature is peculiar to the Chern-Simons and BF theories including supergravity in three dimensions. We also show that dimensional reduction schemes for four-dimensional theories such as N = 1 self-dual supersymmetric Yang-Mills theory or N = 1 supergravity theory that can generate ₀ globally and locally supersymmetric theories in three dimensions. As an interesting application, we present ₀ supergravity Liouville theory in two dimensions after appropriate dimensional reduction from three dimensions.     

On non-Abelian Thomas-Fermi screening

The Thomas-Fermi screening of non-Abelian gauge fields by fermions or screening of gluon fields in quark matter is discussed. It is described by an effective mass term which is, as with hard thermal loops, related to the eikonal for a Chern-Simons theory and the Wess-Zumino-Witten action.     

Quantal Conserved Charges in Gauge Theory

Starting from the configuration-space generating functional for gauge theory obtained by using the Faddeev-Popov method,the conservation laws at the quantum level for the gauge-invariant system are derived.Appling to non-Abel Chern-Simons(CS)theory,the quantum BRS conserved charge and quantuml conserved angular momentum for the non-Abelian CS fields coupled to Fermion field are deduced.The property of fractional spin in CS theory is discussed.     

0-extended supergravity and Chern-Simons theories

We give generalizations of extended Poincaré supergravity with arbitrarily many supersymmetries in the absence of central charges in three dimensions by gauging its intrinsic global SO(N) symmetry. We call these ₀ (Aleph-null) supergravity theories. We further couple a non-Abelian supersymmetric Chern-Simons theory and an Abelian topological BF theory to ₀ supergravity. Our result overcomes the previous difficulty for supersymmetrization of Chern-Simons theories beyond N = 4. This feature is peculiar to the Chern-Simons and BF theories including supergravity in three dimensions. We also show that dimensional reduction schemes for four-dimensional theories such as N = 1 self-dual supersymmetric Yang-Mills theory or N = 1 supergravity theory that can generate ₀ globally and locally supersymmetric theories in three dimensions. As an interesting application, we present ₀ supergravity Liouville theory in two dimensions after appropriate dimensional reduction from three dimensions.     

Quantal symmetries in the non-linear σ model with Maxwell and non-Abelian Chern-Simons terms

The quantal symmetry property of the CP1 nonlinear （y model with Maxwell non-Abelian Chern- Simons terms in （2＋1） dimension is studied. In the Coulomb gauge, the system is quantized by using the Faddeev-Senjanovic （FS） path-integral formalism. Based on the quantaum Noether theorem, the quantal conserved angular momentum is derived and the fractional spin at the quantum level in this system is presented.     

ℵ0-extended supergravity and Chern-Simons theories

We give generalizations of extended Poincaré supergravity with arbitrarily many supersymmetries in the absence of central charges in three dimensions by gauging its intrinsic global SO(N) symmetry. We call these ℵ₀ (Aleph-null) supergravity theories. We further couple a non-Abelian supersymmetric Chern-Simons theory and an Abelian topological BF theory to ℵ₀ supergravity. Our result overcomes the previous difficulty for supersymmetrization of Chern-Simons theories beyond N = 4. This feature is peculiar to the Chern-Simons and BF theories including supergravity in three dimensions. We also show that dimensional reduction schemes for four-dimensional theories such as N = 1 self-dual supersymmetric Yang-Mills theory or N = 1 supergravity theory that can generate ℵ₀ globally and locally supersymmetric theories in three dimensions. As an interesting application, we present ℵ₀ supergravity Liouville theory in two dimensions after appropriate dimensional reduction from three dimensions.     

Skein theory and topological quantum registers: Braiding matrices and topological entanglement entropy of non-Abelian quantum Hall states

We study topological properties of quasi-particle states in the non-Abelian quantum Hall states. We apply a skein-theoretic method to the Read-Rezayi state whose effective theory is the SU(2)_K Chern-Simons theory. As a generalization of the Pfaffian (K = 2) and the Fibonacci (K = 3) anyon states, we compute the braiding matrices of quasi-particle states with arbitrary spins. Furthermore we propose a method to compute the entanglement entropy skein-theoretically. We find that the entanglement entropy has a nontrivial contribution called the topological entanglement entropy which depends on the quantum dimension of non-Abelian quasi-particle intertwining two subsystems.     

Aspects of effective action for super Chern-Simons-Matter models

We develop the superfield background field method and study the effective action in the N = 2, d3 supersymmetric Chern-Simons-matter systems. The one-loop low-energy effective action for non-Abelian supersymmetric Chern-Simons theory is computed to order F ⁴ by use of N = 2 superfield heat kernel techniques.     

Quantal global symmetry for a gauge-invariant system

Based on the configuration-space generating functional obtained by using the Faddeev-Popov trick for a gauge-invariant system, the Ward identities for global transformation are derived. The conservation laws at the quantum level for global symmetry transformation are also deduced. A preliminary application of the present formulation to non-Abelian Chern-Simons (CS) theory is given. The Ward identity and quantal BRS charge under the BRS transformation are deduced. The quantal conserved angular momentum is obtained and the fractional spin for CS theories is discussed.     

THE U(3) ANOMALY

The U(3) anomaly will be given based upon the geometrical and topological method which was used in deterurinating the anomalies for SU(N) and SO(N) gauge fields. It is shown that both the non-Abelian and the Abelian anomaly can be derived in unified treatment from the third Chern class and corresponding Chern-Simons secondary class of the U(3) group.     

Extended hubbard model with ring exchange: a route to a non-Abelian topological phase

We propose an extended Hubbard model on a 2D kagome lattice with an additional ring exchange term. The particles can be either bosons or spinless fermions. We analyze the model at the special filling fraction 1/6, where it is closely related to the quantum dimer model. We show how to arrive at an exactly soluble point whose ground state is the "d-isotopy" transition point into a stable phase with a certain type of non-Abelian topological order. Near the "special" values, d=2cos(pi/(k+2), this topological phase has anyonic excitations closely related to SU(2) Chern-Simons theory at level k.     

ANOf}ALIES OF ARBITRARY GAUGE GROUP AND ITS REDUCTION GROUP,EINSTEIN AND LORENTZ ANOMALIES

Based upon the properties of the characteristic classes and their Chern-Simons secondary characteristic classes. the "Abelian" anomalies in M²ⁿ⁺²,the Euler-Heisenberq effective actions in M²ⁿ⁺¹, as well as the non-Abelian anomalies in M²ⁿ for arbitrary gauge gzoup and its reduction subgroup have been investigated thoroughly and the application to the gravitational anomalies is made. It is shown that the "Ahelian" anomalies of such groups are equal to each other, their Eu1er-Xeisenberg actions are also closely related to each, other,and their non-Abelian anomalies are also equivalent if their common generating functional can be taken as a counter-term.For the gravitational anomalies we present the common genera- ting functional for both non-Abelian Einstein and Lorentz anomalies M⁴ⁿ⁺² and show the relationship between them.     

ANOMALIES,COHOMOLOGY AND ChIERN-SIMONS COCHAINS

We present a concept of the gauge transformations, establish the homomorphic relation between this conomology group and the Chern-Simons type. charaateric class sequence,and analyse their applacatons to the non-Abelian anomaly, the gauge invariant Wess-Zumino effective action and the Schwinger term etc.we show that the present approaches.includes faddeev's song's and zumino'sapproaches.     

Induced Chern-Simons Term in Three-Dimensional Space at High Temperature

Perturbative calculations of the high temperature ground-state axial vector current of fermion fields coupled to gauge fields lead to an appearance of an additional anomalous Chern-Simons topological mass term in three-dimensional effective action. In Abelian case contributions come only from the vacuum polarization graph, and in non-Abelian case contributions come from the vacuum polarization graph and the two triangle graphs.     

Quantum Symmetry for a System with a Singular Lagrangian Containing Subsidiary Condition

The quantization for a system with a singular Lagrangian containing subsidiary constrained conditions in configuration space is studied. The system is called constrained singular system. In certain case, the constrained singular system can be brought into the theoretical framework of the constrained Hamilton system. A modified Dirac-Bergmann algorithm for the calculation of constraints in the system is deduced. The path integral quantization is formulated by using the Faddeev-Senjanovic scheme, and the classical/quantum Noether theorem in canonical formalism are also established for such a system. The application of the results to study the fractional spin in non-Abelian Chern-Simons theory is given. We make a precise investigation of the fractional spin for such a system at the quantal level. A simple example is presented to show that the connection between the symmetry and conservation law in classical theories in general is no longer preserved in quantum theories.