On the Wess-Zumino-Witten models on Riemann surfaces

We give a formulation of the Wess-Zumino-Witten models on Riemann surfaces of arbitrary genus in which the Ward identities for the current algebras become complete. It requires twisting the models in a non-abelian way by Lie group elements. The Ward identities are written in terms of twisted Poincaré series for Schottky groups and the zero modes of the currents are defined by a Lie derivation acting on the twists. Furthermore, we identify the denominator of the chiral partition function and we argue that both the numerator and the denominator of the partition function satisfy a heat equation on the moduli space.     

A criterion for global hyperbolicity of left-invariant Lorentz metrics on Lie groups

We show that every left-invariant Lorentz metric on a non-abelian simply connected Lie group is globally hyperbolic whenever its restriction to the commutator ideal of the Lie algebra is positive definite. We also show that a left-invariant Lorentz metric on the three-dimensional Heisenberg group is globally hyperbolic if and only if its restriction to the center of the Lie algebra is positive definite or degenerate.     

Some non-abelian phase spaces in low dimensions

A non-abelian phase space, or a phase space of a Lie algebra, is a generalization of the usual (abelian) phase space of a vector space. It corresponds to a para-Kähler structure in geometry. Its structure can be interpreted in terms of left-symmetric algebras. In particular, a solution of an algebraic equation in a left-symmetric algebra which is an analogue of classical Yang–Baxter equation in a Lie algebra can induce a phase space. In this paper, we find that such phase spaces have a symplectically isomorphic property. We also give all such phase spaces in dimension 4 and some examples in dimension 6. These examples can be a guide for a further development.     

Higher-dimensional Chern-Simons theory and link invariants

A higher-dimensional generalization of (topological) Chern-Simons theory based on the concept of non-homogeneous differential forms is analyzed. A topological matter multiplet in an arbitrary irrep of the Lie group playing the role of an observable measuring “non-abelian linking numbers” of higher-dimensional links is introduced. The theory as a multiple highly on-shell reducible gauge system is quantized in the framework of the Batalin-Vilkovisky antifield-antibracket formalism. The corresponding monodromy matrix is derived in an explicitly covariant path-integral manner     

Semi-classical spectrum of the homogeneous sine-Gordon theories

The semi-classical spectrum of the homogeneous sine-Gordon theories associated with an arbitrary compact simple Lie group G is obtained and shown to be entirely given by solitons. These theories describe quantum integrable massive perturbations of Gepner's G-parafermions whose classical equations-of-motion are non-abelian affine Toda equations. One-soliton solutions are constructed by embeddings of the SU(2) complex sine-Gordon soliton in the regular SU(2) subgroups of G. The resulting spectrum exhibits both stable and unstable particles, which is a peculiar feature shared with the spectrum of monopoles and dyons in N = 2 and N = 4 supersymmetric gauge theories.     

Thermoalgebras and path integral

Using a representation for Lie groups closely associated with thermal problems, we derive the algebraic rules of the real-time formalism for thermal quantum field theories, the so-called thermo-field dynamics (TFD), including the tilde conjugation rules for interacting fields. These thermo-group representations provide a unified view of different approaches for finite-temperature quantum fields in terms of a symmetry group. On these grounds, a path integral formalism is constructed, using Bogoliubov transformations, for bosons, fermions and non-abelian gauge fields. The generalization of the results for quantum fields in topology is addressed.     

General covariance of the non-abelian DBI-action: checks and balances

We perform three tests on our proposal to implement diffeomorphism invariance in the non-abelian D0-brane DBI action as a base-point independence constraint between matrix Riemann normal coordinate systems. First, we show that T-duality along an isometry correctly interchanges the potential and kinetic terms in the action. Second, we show that the method to impose base-point independence using an auxiliary dN²-dimensional nonlinear sigma model also works for metrics which are curved along the brane, provided a physical gauge choice is made at the end. Third, we show that without alteration this method is applicable to higher order in velocities. Testing specifically to order 4, we elucidate the range of validity of the symmetrized trace approximation to the non-abelian DBI action.     

Gauge invariance structure of quantum chromodynamics

Perturbative QCD may be subdivided into separately gauge-invariant sectors according to the projection of non-abelian color weights onto linearly independent basis elements. We exploit the general Lie group structure of the theory to give an algorithm for finding these gauge-invariant sets and present several examples of its use. The planar sector and the systematics of the non-planar corrections are defined for any gauge theory. Our gauge set classification has implications for QCD bound states, finite order perturbative QCD calculations, the study of QCD infrared singularities and for the question of convergence of the perturbation series.     

Non-abelian exclusion statistics

We introduce the notion of ‘order-k non-abelian exclusion statistics’. We derive the associated thermodynamic equations by employing the thermodynamic Bethe ansatz for specific non-diagonal scattering matrices. We make contact with results obtained by different methods and we point out connections with ‘fermionic sum formulas’ for characters in a conformal field theory. As an application, we derive thermodynamic distribution functions for quasi-holes over a class of non-abelian quantum Hall states recently proposed by Read and Rezayi.     

Spectrum of the non-abelian phase in Kitaev’s honeycomb lattice model

The spectral properties of Kitaev’s honeycomb lattice model are investigated both analytically and numerically with the focus on the non-abelian phase of the model. After summarizing the fermionization technique which maps spins into free Majorana fermions, we evaluate the spectrum of sparse vortex configurations and derive the interaction between two vortices as a function of their separation. We consider the effect vortices can have on the fermionic spectrum as well as on the phase transition between the abelian and non-abelian phases. We explicitly demonstrate the 2ⁿ-fold ground state degeneracy in the presence of 2n well separated vortices and the lifting of the degeneracy due to their short-range interactions. The calculations are performed on an infinite lattice. In addition to the analytic treatment, a numerical study of finite size systems is performed which is in exact agreement with the theoretical considerations. The general spectral properties of the non-abelian phase are considered for various finite toroidal systems.     

Non-abelian Gauge Symmetry for Fields in Phase Space: a Realization of the Seiberg-Witten Non-abelian Gauge Theory

The Seiberg-Witten formalism has been realized as an electrodynamics in phase space (associated to the Dirac equation written in phase space) and this fact is explored here with non-abelian gauge group. First, a physically heuristic presentation of the Seiberg-Witten approach is carried out for non-abelian gauge in order to guide the calculation procedures. These results are realized by starting with the Lagrangian density for the free Dirac field in phase space. Then a field strength is derived, where the non-abelian gauge group is the SU(2), corresponding to an isospin (non-abelian) field theory in phase space. An application to nucleon is then discussed.

     

Improved methods for supergraphs

We introduce some new techniques into superfield perturbation theory which allow considerable simplifications in calculations. As a result, we show that all contributions to the effective action can be written as integrals over a single d⁴θ. We also give the background group field formalism for supersymmetric non-abelian gauge theories. To illustrate our methods, we give examples of loop calculations: in particular, we show that in O(4) extended supersymmetric non-abelian gauge theories all one-loop propagator corrections cancel identically (both infinite and finite parts) and that these theories, at one loop, are finite and have no renormalizations (in the Fermi-Feynman gauge).     

Characterizing neutral modes of fractional states in the second Landau level

Fractionally charged quasiparticles, which obey non-abelian statistics, were predicted to exist in the ν=8/3, ν=5/2, and ν=7/3 fractional quantum Hall states (in the second Landau level). Here we present measurements of charge and neutral modes in these states. For both ν=7/3 and ν=8/3 states, we found a quasiparticle charge e=1/3 and an upstream neutral mode only in ν=8/3-excluding the possibility of non-abelian Read-Rezayi states and supporting Laughlin-like states. The absence of an upstream neutral mode in the ν=7/3 state also proves that edge reconstruction was not present in the ν=7/3 state, suggesting its absence also in ν=5/2 state, and thus may provide further support for the non-abelian anti-pfaffian nature of the ν=5/2 state.     

On the equivalence between the Wess-Zumino action and the free Fermi theory in two dimensions

Using the functional technique we prove the bosonization rules of Witten for the currents in a non-abelian two-dimensional theory with a particular regularization of the Fermi theory.     

Approaches to a non-abelian antisymmetric tensor gauge field theory

Two distinct attempts at constructing a theory of non-abelian antisymmetric tensor gauge fields (ATGF's) are considered. First, a recently proposed geometry of abelian ATGF's is reviewed and then generalized to the non-abelian case. The resulting geometric action is non-local and is invariant under non-local gauge transformations; in the local limit the action describes free fields. Lattice actions for both the abelian and non-abelian ATGF theories are also presented. In the second approach, a lattice action for non-abelian ATGF's is constructed using a plaquette variables that carry four internal indices. The continuum limit is also a non-interacting theory.     

Non-Abelian Quantum Hall States—Exclusion Statistics, K-Matrices, and Duality

We study excitations in edge theories for non-abelian quantum Hall states, focussing on the spin polarized states proposed by Read and Rezayi and on the spin singlet states proposed by two of the authors. By studying the exclusion statistics properties of edge-electrons and edge-quasiholes, we arrive at a novel K-matrix structure. Interestingly, the duality between the electron and quasihole sectors links the pseudoparticles that are characteristic for non-abelian statistics with composite particles that are associated to the pairing physics of the non-abelian quantum Hall states.     

The Nambu string versus the Dirac string in non-trivial QCD

The global topology of a non-trivial non-abelian gauge theory which displays a remarkable interplay between the Nambu string and the Dirac string is analysed, it is suggested that the non-trivial character of QCD, rather than the non-abelian nature of it, could be directly responsible for the confinement of color.     

S-duality of color-ordered amplitudes

Recently, it has been proposed that the S-matrix elements on the world volume of an abelian D₃-brane are consistent with the Ward identity associated with the S-duality. In this paper we extend this study to the case of multiple D₃-branes. We speculate that the S-matrix elements are consistent with the S-dual Ward identity irrespective of the ordering of the external states. Imposing this symmetry on the particular case of the S-matrix element of one Kalb–Ramond, one transverse scalar and two non-abelian gauge bosons, we will find the linear S-duality transformation of the commutator of two non-abelian gauge field strengths. Using this transformation and the standard S-duality transformations of the supergravity fields, all other non-abelian S-matrix elements of one closed and three open string states can be found by the S-duality proposal. We will show that the predicted S-matrix elements are reproduced exactly by explicit calculations.