arXiv subjects

J. M. F. Labastida

Publications and source records attributed to J. M. F. Labastida.

35 records · Page 2Linked to original sources

arXiv1996

Primitive Vassiliev Invariants and Factorization in Chern-Simons Perturbation Theory

The general structure of the perturbative expansion of the vacuum expectation value of a Wilson line operator in Chern-Simons gauge field theory is analyzed. The expansion is organized according to the independent group structures that appear at each order. It is shown that the analysis is greatly simplified if the group factors are chosen in a certain way that we call canonical. This enables us to show that the logarithm of a polinomial knot invariant can be written in terms of primitive Vassiliev invariants only.

q-alg↗

arXiv1996

Twisted N=2 Supersymmetry with Central Charge and Equivariant Cohomology

We present an equivariant extension of the Thom form with respect to a vector field action, in the framework of the Mathai-Quillen formalism. The associated Topological Quantum Field Theories correspond to twisted $N=2$ supersymmetric theories with a central charge. We analyze in detail two different cases: topological sigma models and non-abelian monopoles on four-manifolds.

hep-th↗

arXiv1995

A Relation Between the Kauffman and the HOMFLY Polynomials for Torus Knots

Polynomial invariants corresponding to the fundamental representation of the gauge group $SO(N)$ are computed for arbitrary torus knots in the framework of Chern-Simons gauge theory making use of knot operators. As a result, a formula which relates the Kauffman and the HOMFLY polynomials for torus knots is presented.

q-alg↗

arXiv1995

Polynomial Invariants for SU(2) Monopoles

We present an explicit expression for the topological invariants associated to $SU(2)$ monopoles in the fundamental representation on spin four-manifolds. The computation of these invariants is based on the analysis of their corresponding topological quantum field theory, and it turns out that they can be expressed in terms of Seiberg-Witten invariants. In this analysis we use recent exact results on the moduli space of vacua of the untwisted $N=1$ and $N=2$ supersymmetric counterparts of the topological quantum field theory under consideration, as well as on electric-magnetic duality for $N=2$ supersymmetric gauge theories.

hep-th↗

arXiv1995

Topological Quantum Field Theory: A Progress Report

A brief introduction to Topological Quantum Field Theory as well as a description of recent progress made in the field is presented. I concentrate mainly on the connection between Chern-Simons gauge theory and Vassiliev invariants, and Donaldson theory and its generalizations and Seiberg-Witten invariants. Emphasis is made on the usefulness of these relations to obtain explicit expressions for topological invariants, and on the universal structure underlying both systems.

hep-th↗

arXiv1995

Vassiliev Invariants for Torus Knots

Vassiliev invariants up to order six for arbitrary torus knots $\{ n , m \}$, with $n$ and $m$ coprime integers, are computed. These invariants are polynomials in $n$ and $m$ whose degree coincide with their order. Furthermore, they turn out to be integer-valued in a normalization previously proposed by the authors.

q-alg↗

arXiv1995

Non-Abelian Monopoles on Four-Manifolds

We present a non-abelian generalization of Witten monopole equations and we analyze the associated moduli problem, which can be regarded as a generalization of Donaldson theory. The moduli space of solutions for SU(2) monopoles on Kähler manifolds is discussed. We also construct, using the Mathai-Quillen formalism, the topological quantum field theory corresponding to the new moduli problem. This theory involves the coupling of topological Yang-Mills theory to topological matter in four dimensions

hep-th↗

arXiv1995

A TOPOLOGICAL LAGRANGIAN FOR MONOPOLES ON FOUR-MANIFOLDS

We present a topological quantum field theory which corresponds to the moduli problem associated to Witten's monopole equations for four-manifolds. The construction of the theory is carried out in purely geometrical terms using the Mathai-Quillen formalism, and the corresponding observables are described. These provide a rich set of new topological quantites.

hep-th↗

arXiv1994

Numerical knot invariants of finite type from Chern-Simons perturbation theory

Chern-Simons gauge theory for compact semisimple groups is analyzed from a perturbation theory point of view. The general form of the perturbative series expansion of a Wilson line is presented in terms of the Casimir operators of the gauge group. From this expansion new numerical knot invariants are obtained. These knot invariants turn out to be of finite type (Vassiliev invariants), and to possess an integral representation. Using known results about Jones, HOMFLY, Kauffman and Akutsu-Wadati polynomial invariants these new knot invariants are computed up to type six for all prime knots up to six crossings. Our results suggest that these knot invariants can be normalized in such a way that they are integer-valued.

hep-th↗

arXiv1994

Type B Topological Matter, Kodaira-Spencer Theory, and Mirror Symmetry

Perturbing usual type B topological matter with vector $(0,1)$-forms we find a topological theory which contains explicitly Kodaira-Spencer deformation theory. It is shown that, in genus zero, three-point correlation functions give the Yukawa couplings for a generic point in the moduli space of complex structures. This generalization of type B topological matter seems to be the correct framework to understand mirror symmetry in terms of two-dimensional topological field theories.

hep-th↗

arXiv1994

Topological Matter in Four Dimensions

Topological models involving matter couplings to Donaldson-Witten theory are presented. The construction is carried using both, the topological algebra and its central extension, which arise from the twisting of $N=2$ supersymmetry in four dimensions. The framework in which the construction is based is constituted by the superspace associated to these algebras. The models show new features of topological quantum field theories which could provide either a mechanism for topological symmetry breaking, or the analog of two-dimensional mirror symmetry in four dimensions.

hep-th↗

arXiv1994

The HOMFLY polynomial for torus links from Chern-Simons gauge theory

Polynomial invariants corresponding to the fundamental representation of the gauge group $SU(N)$ are computed for arbitrary torus knots and links in the framework of Chern-Simons gauge theory making use of knot operators. As a result, a formula for the HOMFLY polynomial for arbitrary torus links is presented.

hep-th↗

arXiv1993

Breaking of Topological Symmetry

The coupling of topological matter to topological Yang-Mills theory in four dimensions is considered and a model is presented. It is shown that, contrary to the two-dimensional case, this coupling leads to a breaking of the topological symmetry. This means that the vacuum expectation values of the observables of the theory loose their invariance under small deformations of the metric while the action of the model possesses all the symmetries corresponding to the case with no coupling.

hep-th↗

arXiv1992

Polynomials for Torus Links from Chern-Simons Gauge Theories

Invariant polynomials for torus links are obtained in the framework of the Chern-Simons topological gauge theory. The polynomials are computed as vacuum expectation values on the three-sphere of Wilson line operators representing the Verlinde algebra of the corresponding rational conformal field theory. In the case of the $SU(2)$ gauge theory our results provide explicit expressions for the Jones polynomial as well as for the polynomials associated to the $N$-state ($N>2$) vertex models (Akutsu-Wadati polynomials). By means of the Chern-Simons coset construction, the minimal unitary models are analyzed, showing that the corresponding link invariants factorize into two $SU(2)$ polynomials. A method to obtain skein rules from the Chern-Simons knot operators is developed. This procedure yields the eigenvalues of the braiding matrix of the corresponding conformal field theory.

hep-th↗

arXiv1992

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.

hep-th↗

arXiv1991

Topological Matter in Two Dimensions

Topological quantum field theories containing matter fields are constructed by twisting $N=2$ supersymmetric quantum field theories. It is shown that $N=2$ chiral (antichiral) multiplets lead to topological sigma models while $N=2$ twisted chiral (twisted antichiral) multiplets lead to Landau-Ginzburg type topological quantum field theories. In addition, topological gravity in two dimensions is formulated using a gauge principle applied to the topological algebra which results after the twisting of $N=2$ supersymmetry.

hep-th↗

arXiv1991

Analysis of Observables in Chern-Simons Perturbation Theory

Chern-Simons Theory with gauge group $SU(N)$ is analyzed from a perturbation theory point of view. The vacuum expectation value of the unknot is computed up to order $g^6$ and it is shown that agreement with the exact result by Witten implies no quantum correction at two loops for the two-point function. In addition, it is shown from a perturbation theory point of view that the framing dependence of the vacuum expectation value of an arbitrary knot factorizes in the form predicted by Witten.

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