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.     

Instantons, quivers and noncommutative Donaldson-Thomas theory

We construct noncommutative Donaldson-Thomas invariants associated with abelian orbifold singularities by analyzing the instanton contributions to a six-dimensional topological gauge theory. The noncommutative deformation of this gauge theory localizes on noncommutative instantons which can be classified in terms of three-dimensional Young diagrams with a colouring of boxes according to the orbifold group. We construct a moduli space for these gauge field configurations which allows us to compute its virtual numbers via the counting of representations of a quiver with relations. The quiver encodes the instanton dynamics of the noncommutative gauge theory, and is associated to the geometry of the singularity via the generalized McKay correspondence. The index of BPS states which compute the noncommutative Donaldson-Thomas invariants is realized via topological quantum mechanics based on the quiver data. We illustrate these constructions with several explicit examples, involving also higher rank Coulomb branch invariants and geometries with compact divisors, and connect our approach with other ones in the literature.     

Monopoles and topological field theory

We present a topological quantum field theory for magnetic monopoles in an SU(N) Yang-Mills-Higgs model. This field theory is obtained by gauge fixing the topological action defining the monopole charge. This work extends to the three-dimensional case the quantization of invariant polynomials in four dimensions. We choose the Bogomolny self-duality equations as gauge conditions for the magnetic monopole topological field theory. In this way the geometrical equation discussed e.g. in Atiyah and Hitchin's work are recovered as ghost equations of motion. We give the cocycles of the corresponding topological symmetry. In the N→∞ limit interesting phenomena occur. The functional integration is forced to cover only the moduli space and the role of the ghosts stemming from the gauge fixing process is to provide a smooth semiclassical approximation.     

具有全局对称性的强关联拓扑物态的规范场论

在有对称性保护的条件下,拓扑能带绝缘体等自由费米子体系的拓扑不变量可以在能带结构计算中得到.但是,为了得到强关联拓扑物质态的拓扑不变量,我们需要全新的理论思路.最典型的例子就是分数量子霍尔效应:其低能有效物理一般可以用Chern-Simons拓扑规范场论来计算得到;霍尔电导的量子化平台蕴含着十分丰富的强关联物理.本文将讨论存在于玻色和自旋模型中的三大类强关联拓扑物质态:本征拓扑序、对称保护拓扑态和对称富化拓扑态.第一类无需考虑对称性,后两者需要考虑对称性.理论上,规范场论是一种非常有效的研究方法.本文将简要回顾用规范场论来研究强关联拓扑物质态的一些研究进展.具体内容集中在"投影构造理论"、"低能有效理论"、"拓扑响应理论"三个方面.     

State sum invariants of three-manifolds: A combinatorial approach to topological quantum field theories

The construction of Turaev and Viro involving quantum 6j-symbols and giving rise to invariants of closed, compact three-manifolds is extended. It leads to invariants of coloured graphs on the boundary of compact three-manifolds. This allows one to derive surgery formulas when cutting along an arbitrary two-manifold. In particular all axioms of a topological quantum field theory may be verified and the dimensions of the associated Hilbert spaces are given by the square of the Verlinde formula.     

Supersymmetric instantons and topological quantum field theory

We present an action which generates the supersymmetric self-dual equations corresponding to euclidean super Yang-Mills theory in four dimensions. By adding additional constraint fields with new local symmetries, the classical equations of this system are the usual super self-dual equations when a gauge is chosen for the constraint fields. This construction is a supersymmetric generalization of the Labastida-Pernici action which corresponds to a gauge unfixed version of Witten's topological quantum field theory. We discuss some topological prospects for this model, and the role of supersymmetric instantons in Donaldson theory.     

Weil algebra structure and geometrical meaning of BRST transformation in topological quantum field theory

The Weil algebra structure of the BRST transformation of topological quantum field theory is investigated. This structure appears in the gauge and ghost fields sector and is common to both topological quantum field theory and BRS gauge fixed non-abelian gauge theory. By the Weil algebra structure, we can derive the descent equations of topological quantum field theory which generate the Donaldson polynomials. The algebraic structure also reveals the geometrical meaning of the ghost fields ψ and ? in topological quantum field theory as the components of the total curvature.     

Embedded graph invariants in Chern-Simons theory

Chern-Simons gauge theory, since its inception as a topological quantum field theory, has proved to be a rich source of understanding for knot invariants. In this work the theory is used to explore the definition of the expectation value of a network of Wilson lines — an embedded graph invariant. Using a generalization of the variational method, lowest-order results for invariants for graphs of arbitrary valence and general vertex tangent space structure are derived. Gauge invariant operators are introduced. Higher order results are found. The method used here provides a Vassiliev-type definition of graph invariants which depend on both the embedding of the graph and the group structure of the gauge theory. It is found that one need not frame individual vertices. However, without a global projection of the graph there is an ambiguity in the relation of the decomposition of distinct vertices. It is suggested that framing may be seen as arising from this ambiguity — as a way of relating frames at distinct vertices.     

Chern-Simons topological Lagrangians in odd dimensions and their Kaluza-Klein reduction

By clarifying the behavior of generic Chern-Simons secondary invariants under infinitesimal variation and finite gauge transformation, it is proved that they are eligible to be a candidate term in the Lagrangian in odd dimensions (2k ? 1 for gauge theories and 4k ? 1 for gravity). The coefficients in front of these terms may be quantized because of topological reasons. As a possible application, the dimensional reduction of such actions in Kaluza-Klein theory is discussed. The difficulty in defining the Chern-Simons action for topologically nontrivial field configurations is pointed out and resolved.     

Gravity as a goldstone field in the Lorentz gauge theory

A bundle formalism is applied to interpret the Einstein gravitational field in gauge theory; its topological invariants are discussed.     

Quadratic Lagrangians and Topology in Gauge Theory Gravity

We consider topological contributions to the action integral in a gauge theory formulation of gravity. Two topological invariants are found and are shown to arise from the scalar and pseudoscalar parts of a single integral. Neither of these action integrals contribute to the classical field equations. An identity is found for the invariants that is valid for non-symmetric Riemann tensors, generalizing the usual GR expression for the topological invariants. The link with Yang–Mills instantons in Euclidean gravity is also explored. Ten independent quadratic terms are constructed from the Riemann tensor, and the topological invariants reduce these to eight possible independent terms for a quadratic Lagrangian. The resulting field equations for the parity non-violating terms are presented. Our derivations of these results are considerably simpler than those found in the literature.     

Topological phases of quantum theories. Chern-Simons theory

We analyze the vacuum structure (degeneracy, nodes and symmetries) of some quantum theories with special emphasis on the study of its dependence on the geometry and topology of the classical configuration space. The study of the topological limit shows that many low energy properties of those quantum theories can be inferred from the structure of their topological phases. After reviewing some simple pure quantum mechanical models (planar rotor, magnetic monopole and quantum Hall effect) we focus on the study of the rich relationship existing between topologically massive gauge theories and their topological phases, Chern-Simons theories. In particular we show that, although in a finite volume the degeneracy of the quantum vacuum of gauge theories depends on the topology of the underlying Riemann surface, in an infinite volume the vacuum is unique. Finally, the topological structure of Chern-Simons theory is analyzed in a covariant formalism within a geometric regularization scheme. We discuss in some detail the structure of the different metric dependent contributions to the Chern-Simons partition function and the associated topological invariants.     

Geometrical meaning of braid statistics in (1+1)- and (2+1)-dimensional quantum field theory

Braids naturally arise as topological objects in the discussion of statistics in quantum mechanics of indistinguishable pointlike particles moving in a (2+1)-dimensional space-time. Conversely, they also play a role as algebraic invariants in the discussion of superselection rules in (1+1)-dimensional algebraic quantum field theory. Here we show how Abelian braid statistics in (1+1) dimensions may be interpreted geometrically by introducing the concept of antiparticles, thus clarifying the connection between the two approaches.     

Gauge theories of Josephson junction arrays

We show that the zero-temperature physics of planar Josephson junction arrays in the self-dual approximation is governed by an Abelian gauge theory with a periodic mixed Chern-Simons term describing the charge-vortex coupling. The periodicity requires the existence of (Euclidean) topological excitations which determine the quantum phase structure of the model. The electric-magnetic duality leads to a quantum phase transition between a superconductor and a superinsulator at the self-dual point. We also discuss in this framework the recently proposed quantum Hall phases for charges and vortices in presence of external offset charges and magnetic fluxes: we show how the periodicity of the charge-vortex coupling can lead to transitions to anyon superconductivity phases. We finally generalize our results to three dimensions, where the relevant gauge theory is the so-called BF system with an antisymmetric Kalb-Ramond gauge field.     

2+1-D topological quantum field theory and 2-D conformal field theory

We describe the relation between three dimensional topological quantum field theory and two dimensional conformal field theory. Some Applications to quantum knot invariants leading to the equivalence of Chern-Simons-Witten and Kohno's approaches are outlined.     

Invariants of topological quantum mechanics

We investigate a new topological invariant of the punctured plane using a Hamiltonian approach. The Hamiltonian is built out of topological invariants available on the punctured plane. On the other hand it is shown that the model is a generalized version, using the appropriate language of homotopy, of the superconformal quantum mechanics (gauge approach) recently proposed by L. Baulieuet al. This relationship allows a better understanding of the structure and results of the gauge approach and makes possible a proper identification of the topological invariants which emerge from it.     

Topological approach to examine the singularity of the axial-vector current in an Abelian gauge field theory (QED)

A topological way to distinguish divergences of the Abelian axial-vector current in quantum field theory is proposed. By usirg the properties of the Atiyah-Singer index theorem, the non-trivial Jacobian factor of the integration measure in the path-integral formulation of the theory is connected with the topological properties of the gauge field. The singularity of the fermion current related to the topological character can be correctly examined in a gauge background.