Enore Guadagnini

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.

Quantum Field Theory and Link Invariants

Abstract 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.

Braids and Quantum Group Symmetry in {Chern-Simons} Theory

Abstract The monodromy matrices defined by the quantum holonomies acting on the physical state space of the Chern-Simons theory are derived. Up to equivalence, these matrices are reconstructed by means of a matrix-valued gauge connection satisfying the Gauss law. In terms of this connection, the relation of the Chern-Simons model with conformal field theory and quantum group is established. The braid group representation realized on the physical states is obtained. The quantum group symmetry appears as a hidden symmetry of the quantized theory.

Scale invariant sigma models on homogeneous spaces

Abstract A class of two-dimensional scale invariant σ -models on homogeneous spaces is presented. We discuss the symmetry properties of the models and show that they are finite. We compute at two loops the central charge of the Virasoro algebra associated with the energy-momentum tensor.

Chern-Simons Field Theory and Quantum Groups

We study the Gauss constraint of the Chern-Simons theory in presence of sources. We solve this constraint in terms of a matrix-valued gauge connection. The associated holonomies define a representation of the braid group, which commutes with the action of a quantum group.

Beta functions and central charge of supersymmetric sigma models with torsion

Abstract We present a method for the computation of the renormalization group β-functions and the central charge in two-dimensional supersymmetric sigma models in a gravitational background. The two-loops results are exhibited. We use the Pauli-Villars regularization which preserves supersymmetry and permits an unambiguous treatment of the model with torsion. The central charge we derive for a general manifold is in agreement with the expression found on group manifolds.

Current algebra in sigma models on homogeneous spaces

Abstract The commutator algebra of the currents for two-dimensional scale invariant sigma models on homogeneous spaces is constructed. The central extensions of the Kac-Moody algebras and the central charge of the Virasoro algebra associated with the energy-momentum tensor are computed.

Chiral anomalies and supersymmetry

Abstract The structure of the chiral anomalies in supersymmetric gauge theories is studied. We discuss in detail the relationship between the two pictures: the manifestly SUSY-covariant approach and the Wess-Zumino gauge. A prescription for the computation of the anomalies in higher dimensions is presented and the resulting expression in six dimensions is reported.

Topological field theory and surgery on three-manifolds

Abstract The solution of the SU(2) quantum Chern-Simons field theory defined on a closed, connected and orientable three-manifold is presented. The vacuum expectation values of Wilson line operators, associated with framed links in a generic manifold, are computed in terms of the expectation values of the three-sphere. The method consists of using an operator realization of Dehn surgery. The rules, corresponding to the surgery instructions in the three-sphere, are derived and the three-manifold invariant defined by the Chern-Simons theory is constructed. Several examples are considered and explicit results are reported.

Large-N Chern-Simons field theory

Abstract We consider the SU ( N ) Chern-Simons field theory and study the behaviour of the observables, which are defined by the expectation values of Wilson line operators, in the large- N limit. We determine the structure of the knot invariants which are obtained in this limit and some of their properties are derived; it is proved, for example, that for the product of gauge-invariant operators the factorization property is satisfied. A new relation connecting each knot with its double is derived.