Chern-Simons expectation values and quantum horizons from loop quantum gravity and the Duflo map

We report on a new approach to the calculation of Chern-Simons theory expectation values, using the mathematical underpinnings of loop quantum gravity, as well as the Duflo map, a quantization map for functions on Lie algebras. These new developments can be used in the quantum theory for certain types of black hole horizons, and they may offer new insights for loop quantum gravity, Chern-Simons theory and the theory of quantum groups.     

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.     

Quantum field theory and link invariants

A skein relation for the expectation values of Wilson line operators in three-dimensional SU(N) Chern-Simons gauge theory is derived at first order in the coupling constant. We use a variational method based on the properties of the three-dimensional field theory. The relationship between the above expectation values and the known link invariants is established.     

New hierarchies of knot polynomials from topological Chern-Simons gauge theory

In this Letter, we report on a study of the expectation values of Wilson loops in D=3 topological Chern-Simons theory associated with the fundamental representation of the simple Lie algebras SO(n) and Sp(n). The skein relations satisfied by these expectation values are derived by conformal field-theory techniques. New hierarchies of invariant polynomials for knots in S ³ can be derived from these relations (at least) up to ten crossings. The N=3 Akutsu-Wadati polynomials are a special case with G=SO(3). The expectation value of the Wilson loops for a couple of simple unknotted circles is identified to the Weyl character.Work supported in part by U.S. National Science Foundation Grant PHY8706501.Work supported in party by Chinese National Science Foundation through Nankai University.     

Chern–Simons theory,Stokes’ theorem,and the Duflo map

We consider a novel derivation of the expectation values of holonomies in Chern–Simons theory, based on Stokes’ Theorem and the functional properties of the Chern–Simons action. It involves replacing the connection by certain functional derivatives under the path integral. It turns out that ordering choices have to be made in the process, and we demonstrate that, quite surprisingly, the Duflo isomorphism gives the right ordering, at least in the simple cases that we consider. In this way, we determine the expectation values of unknotted, but possibly linked, holonomy loops for SU(2) and SU(3), and sketch how the method may be applied to more complicated cases. Our manipulations of the path integral are formal but well motivated by a rigorous calculus of integration on spaces of generalized connections which has been developed in the context of loop quantum gravity.     

Membranes,higher Hopf maps,and phase interactions

By coupling an antisymmetric three-indexed gauge potential to a thick or solitonic membrane appearing in an SO(5) nonlinear σ model, we find that the physics describes the second Hopf map S⁷→S⁴. The resulting phase interaction between membranes provides an integral representation of the topological invariant. Using quarternionic fields, we can write the Hopf term in a local form. Remarkably, the three-indexed abelian gauge potential may be identified with the Chern-Simons form of an SU(2) gauge potential constructed out of the quanternion fields.     

Ghosts,knots and Kontsevich integrals

It is shown how ghost propagation in the Hamiltonian formulation of Chern-Simons Field Theory is the physics underlying the Kontsevich integrals: the expectation values of Wilson loops computed to the appropriate order in Perturbation Theory to describe the topology of a knot.     

Two-dimensional topological gravity and intersection theory on the moduli space of holomorphic bundles

We define a two-dimensional topological Yang-Mills theory for an arbitrary compact simple Lie group. This theory is defined in terms of intersection theory on the moduli space of flat connections on a two-dimensional surface and corresponds physically to a two-dimensional reduction and truncation of four-dimensional topological Yang-Mills theory. Two-dimensional topological Yang-Mills theory defines a topological matter system and may be naturally coupled to two-dimensional topological gravity. This topological Yang-Mills theory is also closely related to Chern-Simons gauge theory in 2 + 1 dimensions. We also discuss a relation between SL (2, ) Chern-Simons theory and two-dimensional topological gravity.     

Three-dimensional fermionic determinants,Chern-Simons and nonlinear field redefinitions

The three-dimensional abelian fermionic determinant of a two component massive spinor in flat euclidean space-time is reset to a pure Chern-Simons action through a nonlinear redefinition of the gauge field.     

Vassiliev Knot Invariants and Chern-Simons Perturbation Theory to All Orders

At any order, the perturbative expansion of the expectation values of Wilson lines in Chern-Simons theory gives certain integral expressions. We show that they all lead to knot invariants. Moreover these are finite type invariants whose order coincides with the order in the perturbative expansion. Together they combine to give a universal Vassiliev invariant. Received: 26 March 1996 / Accepted: 7 November 1996     

BRST Formalism in Self-Dual Chern-Simons Theory with Matter Fields

We apply BRST method to the self-dual Chern-Simons gauge theory with matter fields and the generators of symmetries of the system from an elegant Lie algebra structure under the operation of Poisson bracket. We discuss four different cases: abelian, nonabelian, relativistic, and nonrelativistic situations and extend the system to the whole phase space including ghost fields. In addition, we obtain the BRST charge of the field system and check its nilpotence of the BRST transformation which plays an important role such as in topological quantum field theory and string theory.     

Kauffman knot polynomials in classical abelian Chern-Simons field theory

Kauffman knot polynomial invariants are discovered in classical abelian Chern-Simons field theory. A topological invariant t^I(L) is constructed for a link L, where I is the abelian Chern-Simons action and t a formal constant. For oriented knotted vortex lines, t^I satisfies the skein relations of the Kauffman R-polynomial; for un-oriented knotted lines, t^I satisfies the skein relations of the Kauffman bracket polynomial. As an example the bracket polynomials of trefoil knots are computed, and the Jones polynomial is constructed from the bracket polynomial.     

Classical gauge theories as dynamical systems—Regularity and chaos

In this review we present the salient features of dynamical chaos in classical gauge theories with spatially homogeneous fields. The chaotic behaviour displayed by both abelian and non-abelian gauge theories and the effect of the Higgs term in both cases are discussed. The role of the Chern-Simons term in these theories is examined in detail. Whereas, in the abelian case, the pure Chern-Simons-Higgs system is integrable, addition of the Maxwell term renders the system chaotic. In contrast, the non-abelian Chern-Simons-Higgs system is chaotic both in the presence and the absence of the Yang-Mills term. We support our conclusions with numerical studies on plots of phase trajectories and Lyapunov exponents. Analytical tests of integrability such as the Painlevé criterion are carried out for these theories. The role of the various terms in the Hamiltonians for the abelian Higgs, Yang-Mills-Higgs and Yang-Mills-Chern-Simons-Higgs systems with spatially homogeneous fields, in determining the nature of order-disorder transitions is highlighted, and the effects are shown to be counter-intuitive in the last-named system.     

Witten's identity for Chern-Simons theory

We present a simple heuristic calculational scheme to relate the expectation value of Wilson loops in Chern-Simons theory to the Jones polynomial. We consider the exponential of the generator of homotopy transformations which produces the finite loop deformations that define the crossing change formulas of knot polynomials. Applying this operator to the expectation value of Wilson loops for an unspecified measure, we find a set of conditions on the measure and the regularization such that the Jones polynomial is obtained.     

Wilson lines in Chern-Simons theory and link invariants

The vacuum expectation values of Wilson line operators W(L) in the Chern-Simons theory are computed to second order to perturbation theory. The meaning of the framing procedure for knots is analyzed in the context of the Chern-simons field theory. The relation between W(L) and the link invariant polynomials is discussed. We derive an explicit analytic expression for the second coefficient of the Alexander-Conway polynomial, which is related to the Arf- and Casson-invariant. We present also some new relations between the HOMFLY coefficients.     

The Nature of Electronic Charge

Advances in gauge theories and unified theories have not thrown light on the meaning of electron. The problem of the origin of electronic charge is made precise, new insights gained from Weyl space are summarized, and the origin of charge in terms of fractional spin is suggested. A new perspective on the abelian Chern-Simons theory is presented to explain charge.     

Chern-Simons theory and fusion algebras

The connection between the Chern-Simons theory and some features of the two-dimensional conformal models is considered. By using the properties of the expectation values of the Wilson line operators, it is shown how the fusion rules emerge in the three-dimensional context. The case G=SU(2) is considered in detail. The fusion algebra is obtained from the tensor algebra of the gauge group by factorizing an appropriate invariant subalgebra generated by a null vector.     

Spectral asymmetry and vacuum charge in three-dimensional closed and open space

The parity violating effective action for three-dimensional fermions coupled to an abelian gauge background potential is determined in a non-perturbative manner. The three-dimensional spectral asymmetry is calculated on compact manifolds with and without boundary as well as in open space in terms of lower-dimensional topological objects. It is shown that by quantizing the theory in different vacuum sectors the anomalous contribution to the effective action may be modified by non-local terms. The coefficient of the parity violating Chern-Simons term is found to vary accordingly, in agreement with previous hamiltonian calculations.     

Chern-Simons term by spontaneous symmetry breaking in an abelian Higgs model

We show that the Chern-Simons terms can be generated by spontaneous symmetry breaking in a generalized abelian Higgs model in 2+1 dimensions. We analyze this model in some detail and show the existence of two (neutral) vortices of finite energy in each topologically nontrivial sector.     

Odd Chern-Simons Theory, Lie Algebra Cohomology and Characteristic Classes

We investigate the generic 3D topological field theory within the AKSZ-BV framework. We use the Batalin-Vilkovisky (BV) formalism to construct explicitly cocycles of the Lie algebra of formal Hamiltonian vector fields and we argue that the perturbative partition function gives rise to secondary characteristic classes. We investigate a toy model which is an odd analogue of Chern-Simons theory, and we give some explicit computation of two point functions and show that its perturbation theory is identical to the Chern-Simons theory. We give a concrete example of the homomorphism taking Lie algebra cocycles to Q-characteristic classes, and we reinterpret the Rozansky-Witten model in this light.