M. Martellini

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.

Perturbative aspects of the Chern-Simons field theory

Abstract The quantization of the non-abelian Chern-Simons theory in three dimensions is performed in the framework of the BRS formalism. General covariance is preserved on the physical subspace. The perturbative analysis at two loops confirms that the model is finite. To this order, the radiative corrections to the two- and three-point proper vertices vanish.

Chern-Simons holonomies and the appearance of quantum groups

Abstract The monodromy representation defined by the quantum holonomies and realized on the physical state space of the Chern-Simons model is investigated. It is shown that this representation naturally extends to a representation of the braid group related to a particular solution of the quantum Yang-Baxter equation. The presence of a hidden quantum group symmetry is established.

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.

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.

Chern-Simons model and new relations between the HOMFLY coefficients

Abstract Using the connection between the vacuum expectation value of the Wilson line operator W T (C) in the Chern-Simons field theory and the HOMFLY polynomial associated with the knot C, we derive new relations between the HOMFLY coefficients. An explicit analytic expression for the second coefficient of the Alexander-Conway polynomial is presented.

Deformations of complex structures and representations of Krichever-Novikov algebras

Abstract By applying the conformal Ward identities we study the representations of the Krichever-Novikov algebras associated to conformal field theories on compact Riemann surfaces. We compute the matrix elements between primary states of the KN generators corresponding to deformations of the complex structure. We show that these matrix elements depend on the derivatives of the partition function with respect to the moduli. The effects of this dependence on the highest weight representations is discussed.

Conformal invariance on a generic Riemann surface in the presence of a non-flat background metric

Abstract By using bases of meromorphic differentials, recently proposed by Krichever and Novikov, we calculate the brackets of the moments of the energy-momentum tensor of a bosonic string theory in the presence of a generic target metric. To do this we expand the target metric in normal coordinates. To the lowest perturbative order we find that the BRST operator is nilpotent if we impose the condition that the Ricci tensor of the target metric vanish.

Non-perturbative two-dimensional quantum gravity, KdV Hierarchy and solutions of the string equations

Abstract We investigate the relation between the multicritical one-matrix random model and the moduli space parametrizing the solutions of the KdV hierarchy and give a geometrical interpretation of the non-perturbative free parameters of 2D-QG. From our viewpoint the string equations, in the asymptotic limit, give the boundary conditions of the stationary generalized KdV equation which contains all the information about the theory. We thus study those stationary solutions, matching the asymptotic behaviour of the solutions of the string equations, which are real and free of singularities in the scaling variable. We also discuss the consequences of these requirements on the geometry of such solutions.