Duality in twisted N=4 supersymmetric gauge theories in four dimensions

We consider a twisted version of the four-dimensional N=4 supersymmetric Yang-Mills theory with gauge groups SU(2) and SO(3), and bare masses for two of its chiral multiplets, thereby breaking N=4 down to N=2. Using the wall-crossing technique introduced by Moore and Witten within the u-plane approach to twisted topological field theories, we compute the partition function and all the topological correlation functions for the case of simply-connected spin four-manifolds of simple type. By including 't Hooft fluxes, we analyse the properties of the resulting formulae under duality transformations. The partition function transforms in the same way as the one first presented by Vafa and Witten for another twist of the N=4 supersymmetric theory in their strong coupling test of S-duality. Both partition functions coincide on K3. The topological correlation functions turn out to transform covariantly under duality, following a simple pattern which seems to be inherent in a general type of topological quantum field theories.Comment: 60 pages, phyzz

Mass Perturbations in Twisted N=4 Supersymmetric Gauge Theories

Mass perturbations of the twisted N=4 supersymmetric gauge theory considered by Vafa and Witten to test S-duality are studied for the case of Kahler four-manifolds. It is shown that the resulting mass-perturbed theory can be regarded as an equivariant extension associated to a U(1) symmetry of the twisted theory, which is only present for Kahler manifolds. In addition, it is shown that the partition function, the only topological invariant of the theory, remains invariant under the perturbation.Comment: 36 pages, phyzzx, a footnote and two references adde

Chern-Simons Invariants of Torus Links

We compute the vacuum expectation values of torus knot operators in Chern-Simons theory, and we obtain explicit formulae for all classical gauge groups and for arbitrary representations. We reproduce a known formula for the HOMFLY invariants of torus links and we obtain an analogous formula for Kauffman invariants. We also derive a formula for cable knots. We use our results to test a recently proposed conjecture that relates HOMFLY and Kauffman invariants.Comment: 20 pages, 5 figures; v2: minor changes, version submitted to AHP. The final publication is available at http://www.springerlink.com/content/a2614232873l76h6

Coset Constructions in Chern-Simons Gauge Theory

Coset constructions in the framework of Chern-Simons topological gauge theories are studied. Two examples are considered: models of the types ${U(1)_p\times U(1)_q\over U(1)_{p+q}}\cong U(1)_{pq(p+q)}$ with $p$ and $q$ coprime integers, and ${SU(2)_m\times SU(2)_1\over SU(2)_{m+1}}$. In the latter case it is shown that the Chern-Simons wave functionals can be identified with t he characters of the minimal unitary models, and an explicit representation of the knot (Verlinde) operators acting on the space of $c<1$ characters is obtained.Comment: 15 page

Topological Massive Sigma Models

In this paper we construct topological sigma models which include a potential and are related to twisted massive supersymmetric sigma models. Contrary to a previous construction these models have no central charge and do not require the manifold to admit a Killing vector. We use the topological massive sigma model constructed here to simplify the calculation of the observables. Lastly it is noted that this model can be viewed as interpolating between topological massless sigma models and topological Landau-Ginzburg models.Comment: 20 pages, Phyzzx. Revised version to appear in Nucl. Phys. B. The construction of the model is clarified and there are a few minor change

Mathai-Quillen Formulation of Twisted N=4 Supersymmetric Gauge Theories in Four Dimensions

We present a detailed description of the three inequivalent twists of N=4 supersymmetric gauge theories. The resulting topological quantum field theories are reobtained in the framework of the Mathai-Quillen formalism and the corresponding moduli spaces are analyzed. We study their geometric features in each case. In one of the twists we make contact with the theory of non-abelian monopoles in the adjoint representation of the gauge group. In another twist we obtain a topological quantum field theory which is orientation reversal invariant. For this theory we show how the functional integral contributions to the vacuum expectation values leading to topological invariants notably simplify.Comment: 70 pages, macropackage phyzz

Chern-Simons Gauge Theory: Ten Years After

A brief review on the progress made in the study of Chern-Simons gauge theory since its relation to knot theory was discovered ten years ago is presented. Emphasis is made on the analysis of the perturbative study of the theory and its connection to the theory of Vassiliev invariants. It is described how the study of the quantum field theory for three different gauge fixings leads to three different representations for Vassiliev invariants. Two of these gauge fixings lead to well known representations: the covariant Landau gauge corresponds to the configuration space integrals while the non-covariant light-cone gauge to the Kontsevich integral. The progress made in the analysis of the third gauge fixing, the non-covariant temporal gauge, is described in detail. In this case one obtains combinatorial expressions, instead of integral ones, for Vassiliev invariants. The approach based on this last gauge fixing seems very promising to obtain a full combinatorial formula. We collect the combinatorial expressions for all the Vassiliev invariants up to order four which have been obtained in this approach.Comment: 62 pages, 21 figures, lecture delivered at the workshop "Trends in Theoretical Physics II", Buenos Aires, November 199

Twisted Baryon Number in N=2 Supersymmetric QCD

We show that the baryon number of N=2 supersymmetric QCD can be twisted in order to couple the topological field theory of non-abelian monopoles to $Spin^c$-structures. To motivate the construction, we also consider some aspects of the twisting procedure as a gauging of global currents in two and four dimensions, in particular the role played by anomalies.Comment: 14 pages, late