Topological Yang-Mills Theory Research Articles

Stochastic quantization of the Chern-Simons theory

We discuss stochastic quantization of d=3 dimensional non-Abelian Chern-Simons theory. We demonstrate that the introduction of an appropriate regulator in the Langevin equation yields a well-defined equilibrium limit, thus leading to the correct propagator. We also analyze the connection between d=3 Chern-Simons and d=4 topological Yang-Mills theories, showing the equivalence between the corresponding regularized partition functions. Finally, we discuss the introduction of a non-trivial kernel as an alternative regularization.

Currents and anomalies in topological Yang-Mills theory

The quantum properties of topological Yang-Mills theory are investigated in the light of the N = 2 supersymmetry observed in flat space. We construct a unique system of covariantly (partially) conserved currents which develop anomalies while preserving BRS invariance of the theory. In particular, the one-loop renormalized energy-momentum tensor is free of purely gravitational contributions and can be written as a BRS variation. We study the consequences of changing the renormalization prescriptions inherited from the N = 2 supersymmetry to those consistent with BRS. Most of our conclusions are verified by explicit calculations. As a byproduct we derive the formula of Atiyah, Hitchin and Singer for the dimension of the moduli space of self-dual Yang-Mills fields. Finally strong arguments are given that the full system of Donaldson polynomials and the quantum BRS current are not renormalized beyond one-loop.

Infinite tower extension of topological Yang-Mills theory

Introducing an infinite number of fields which form a half-infinite chain representation of the BRST algebra, we construct an extended topological Yang-Mills theory. The properties of the theory are discussed from the standpoint of the covariant canonical quantization. It is shown that the extended topological Yang-Mills theory becomes a cohomological topological field theory or a theory equivalent to a quantum Yang-Mills theory with an anti-self-dual constraint according to subsidiary conditions imposed on the state-vector space.

Complex geometrical interpretation of BRST algebra in topological Yang-Mills theory

Abstract We show that the BRST and anti-BRST algebra can be obtained by constructing a complex vector bundle over (M × X), where M is the real four-manifold and X an additional complex manifold. Witten's BRST algebra is derived from a hermitian holomorphic vector bundle, and the complex manifold X can be identified with the instanton moduli space.

An Extended Topological Yang-Mills Theory

Introducing infinite number of fields, we construct an extended version of the topological Yang-Mills theory. The properties of the extended topological Yang-Mills theory (ETYMT) are discussed from standpoint of the covariant canonical quantization. It is shown that the ETYMT becomes a cohomological topological field theory or a theory equivalent to a quantum Yang-Mills theory with anti-self-dual constraint according to subsidiary conditions imposed on state-vector space. On the basis of the ETYMT, we may understand a transition from an unbroken phase to a physical phase (broken phase)

Local BRST symmetry and superfield formulation of the Donaldson-Witten theory

We apply a symmetry principle for defining topological quantum field theories, based on local BRST invariance, in the case of topological Yang-Mills theory. This principle selects the self-duality and Lorentz gauge functions of the Donaldson-Witten theory. We also present a superfield formalism, which may unveil the geometrical meaning of this principle as reparametrization invariance in an appropriate superspace, with no dependence on the supermetric.

Local topological gauge symmetry

We present a symmetry principle, based on a local version of topological BRST invariance, that selects gauge conditions appropriate for topological quantum field theories. We illustrate how it works in supersymmetric quantum mechanics, a toy model for topological Yang-Mills theory in four dimensions. The latter will appear in a separate publication.

Skein relations for any representation of any Lie group

>A general approach to monodromy skein relations for any pair of irreducible representations of any semi-simple Lie group in the framework of topological Yang-Mills theory is proposed. Two key examples yielding, among other things, the most general (Yang-Mills theory based) form of the HOMFLY and Kauffman/Dubrovnik skein relations are given.

Topological sigma models with loop groups as the target manifolds.

We construct two-dimensional topological sigma-models with loop groups as the target manifolds and study their properties. These models are related to four-dimensional topological Yang-Mills theory as a consequence of a theorem of Atiyah, which identifies the instanton moduli spaces of these two types of theories in the simplest cases. The field-theory implications of this identification of the moduli spaces are also discussed.

Superfield description of N=2 topological supergravity

A free-field formulation of N=2 topological supergravity is obtained with a suitable gauge fixing of two-dimensional topological Yang-Mills theory, taking as the gauge group a contraction of Osp(2|2). We also give a superfield description of the action and its symmetry currents.

Holonomy groups, complex structures andD=4 topological Yang-Mills theory

We study conditions for the existence of extended supersymmetry in topological Yang-Mills theory. These conditions are most conveniently formulated in terms of the holonomy group of the underlying manifold, on which the topological Yang-Mills theory is defined. For irreducible manifolds we find that extended supersymmetries are in 1–1 correspondence with covariantly constant complex structures. Therefore, the topological Yang-Mills theory on any Kähler manifold possesses one additional supersymmetry and on any hyper Kähler manifold there are three additional supersymmetries. The Donaldson map, which plays a crucial role in the construction of the topological invariants, is generalized for Kähler manifolds, thus providing candidates for new invariants of complex manifolds.

A four-dimensional Yang-Mills approach to polynomial invariants of links

A four-dimensional description of polynomial invariants of links in the framework of quantum topological Yang-Mills theory is proposed. Monodromy skein relations corresponding to intersections of deformed Seifert surfaces of knots as well as ordinary (HOMFLY) skein relations for the simplest case of the defining representation of the gauge group G=SU( N) are derived.

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, R ) Chern-Simons theory and two-dimensional topological gravity.

Two-dimensional topological gravity, topological matter and intersection theory on moduli space

We propose a two-dimensional topological matter system associated to an arbitrary compact Lie group G. This topological quantum field theory is defined in terms of intersection theory on the space of based holomorphic maps from a Riemann surface to the based loop group ωG of G, and corresponds physically to the topological sigma model with ωG as the target space. This topological matter system may be naturally coupled to two-dimensional topological gravity and is also related to topological Yang-Mills theory in four dimensions. There appears also to be a connection between this theory and chiral models involving infinite-dimensional twistor space.

Quantization of the topological Yang-Mills theory

The structure of the conserved or partially conserved currents of the topological Yang-Mills theory is discussed. These currents are then used to show that at the one-loop level the β-function reproduces the N=2 supersymmetry value. The result holds even if N=2 supersymmetry is broken down to a singlet supersymmetry.

Gauge dependence of the eta function in Chern-Simons field theory and the Vilkovisky-DeWitt correction.

We use two one-parameter families of gauges to demonstrate the gauge dependence of {eta}(0) for quantized pure Chern-Simons field theory on {ital R}{sup 3}. The first family consists of local covariant gauges and all reproduce the Landau-DeWitt gauge result. The second family consists of nonlocal and noncovariant gauges which produce gauge-dependent results not proportional to the original background Chern-Simons action. However, when the one-loop quantum operators for these gauges are corrected by adding the appropriate Vilkovisky-DeWitt term, the gauge dependence is removed. All functional traces are evaluated using two methods, a momentum expansion and a covariant perturbation similar to the Schwinger-DeWitt expansion. Gauge dependence of the {beta} function in topological Yang-Mills theory is also discussed.

Topological field theory: Zero-modes and renormalization

We address the issue of the non-triviality of the observables in various Topological Field Theories by means of the explicit introduction of the zero-modes into the BRST algebra. Supersymmetric quantum mechanics and Topological Yang-Mills theory are dealt with in detail. It is shown that due to the presence of fermionic zero-modes the BRST algebra may be dynamically broken, leading to nontrivial observables, albeit the local cohomology being trivial. However, the metric and coupling constant independence of the observables are still valid. A renormalization procedure is given that correctly incorporates the zero-modes. Particular attention is given to the conventional gauge fixing in Topological Yang-Mills theories, with emphasis on the geometrical character of the fields and their rôle in the non-triviality of the observables.

Construction of topological W 3 gravity

Topological W 3 gravity is constructed in detail from its definition as topological Yang-Mills theory in two dimensions with the gauge group being a contraction of SL(3, R ). The gauge transformation of the Yang-Mills gauge field can be rewritten to resemble that of the spin-2 and -3 gauge fields describing W 3 gravity. This fact provides a local parametrization of the moduli space of the flat connections in terms of spin-2 and -3 Beltrami differentials on Riemann surfaces. The variations of the action with respect to the moduli give the stress tensor and the spin-3 W current, and the BRST charge can be expressed in terms of these currents in a way similar to that of W 3 gravity. The physical content of the model is briefly discussed.

Two-dimensional topological Yang-Mills theory

Two-dimensional euclidean (topological) quantum Yang-Mills theory on the compact manifold in the Lorentz gauge is analysed in the framework of the covariant path-integral approach. The Nicolai map for the partition function and for the Wilson loop observables is explicitly given.

Central extension of the fermionic algebra in topological quantum field theory

The fermionic symmetry algebra of the topological Yang-Mills theory is shown to admit a central extension with some topological number as the central charge.