Superconformal Invariance and the Geography¶of Four-Manifolds

The correlation functions of supersymmetric gauge theories on a four-manifold X can sometimes be expressed in terms of topological invariants of X. We show how the existence of superconformal fixed points in the gauge theory can provide nontrivial information about four-manifold topology. In particular, in the example of gauge group SU(2) with one doublet hypermultiplet, we derive a theorem relating classical topological invariants such as the Euler character and signature to sum rules for Seiberg–Witten invariants. A short account of this paper can be found in [1]. Received: 19 December 1998 / Accepted: 7 March 1999     

Torsion Structure in Riemann-Cartan Manifold and Dislocation

The U(1) gauge structure of torsion and dislocation in three dimensional Riemann-Cartan manifold have been studied. The local topological structure of dislocation have been presented by so-called topological method in which the quantum number is by Hopf indices and Brouwer degree. Furthermore, the relationship between the dislocation lines and Wilson lines of the U(1) gauge theory is discussed by using the Chern-Simons theory.     

Quantum Deformation of Lattice Gauge Theory

A quantum deformation of 3-dimensional lattice gauge theory is defined by applying the Reshetikhin-Turaev functor to a Heegaard diagram associated to a given cell complex. In the root-of-unity case, the construction is carried out with a modular Hopf algebra. In the topological (weak-coupling) limit, the gauge theory partition function gives a 3-fold invariant, coinciding in the simplicial case with the Turaev-Viro one. We discuss bounded manifolds as well as links in manifolds. By a dimensional reduction, we obtain a q-deformed gauge theory on Riemann surfaces and find a connection with the algebraic Alekseev-Grosse-Schomerus approach. Received: 29 April 1996 / Accepted: 24 September 1996     

Reducibility and Gribov Problem in Topological Quantum Field Theory

In spite of its simplicity and beauty, the Mathai–Quillen formulation of cohomological topological quantum field theory with gauge symmetry suffers two basic problems: i) the existence of reducible field configurations on which the action of the gauge group is not free and ii) the Gribov ambiguity associated with gauge fixing, i. e. the lack of global definition on the space of gauge orbits of gauge fixed functional integrals. In this paper, we show that such problems are in fact related and we propose a general completely geometrical recipe for their treatment. The space of field configurations is augmented in such a way to render the action of the gauge group free and localization is suitably modified. In this way, the standard Mathai–Quillen formalism can be rigorously applied. The resulting topological action contains the ordinary action as a subsector and can be shown to yield a local quantum field theory, which is argued to be renormalizable as well. The salient feature of our method is that the Gribov problem is inherent in localization, and thus can be dealt within a completely equivariant setting, whereas gauge fixing is free of Gribov ambiguities. For the stratum of irreducible gauge orbits, the case of main interest in applications, the Gribov problem is solvable. Conversely, for the strata of reducible gauge orbits, the Gribov problem cannot be solved in general and the obstruction may be described in the language of sheaf theory. The formalism is applied to the Donaldson–Witten model. Received: 22 July 1996 / Accepted: 21 October 1996     

Mutual Chern-Simons theory and its applications in condensed matter physics

In this paper, the mutual Chern-Simons (MCS) theory is introduced as a new kind of topological gauge theory in 2+1 dimensions. We use the MCS theory in gapped phase as an effective low energy theory to describe the Z ₂ topological order of the Kitaev-Wen model. Our results show that the MCS theory can catch the key properties for the Z ₂ topological order. On the other hand, we use the MCS theory as an effective model to deal with the doped Mott insulator. Based on the phase string theory, the t-J model reduces to a MCS theory for spinons and holons. The related physics in high T _c cuprates is discussed.      

Topologically massive gauge theories

Gauge vector and gravity models are studied in three-dimensional space-time, where novel, gauge invariant, P and T odd terms of topological origin give rise to masses for the gauge fields. In the vector case, the massless Maxwell excitation, which is spinless, becomes massive with spin 1. When interacting with fermions, the quantum theory is infrared and ultraviolet finite in perturbation theory. For non-Abelian models, topological considerations lead to a quantization condition on the dimensionless coupling constant-mass ratio. Ordinary Einstein gravity is trivial, but when augmented by our mass term, it acquires a propagating, massive, spin 2 mode. This theory is ghost-free and causal, although of third-derivative order. Quantum calculations are presented in both the Abelian and non-Abelian vector models, to exhibit some of the delicate aspects of infrared behavior, and regularization dependence.     

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 using the properties of the Atiyah-Singer index theorem, the nomtrivial 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.     

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.     

Polynomial Invariants for Torus Knots¶and Topological Strings

We make a precision test of a recently proposed conjecture relating Chern–Simons gauge theory to topological string theory on the resolution of the conifold. First, we develop a systematic procedure to extract string amplitudes from vacuum expectation values (vevs) of Wilson loops in Chern–Simons gauge theory, and then we evaluate these vevs in arbitrary irreducible representations of SU(N) for torus knots. We find complete agreement with the predictions derived from the target space interpretation of the string amplitudes. We also show that the structure of the free energy of topological open string theory gives further constraints on the Chern–Simons vevs. Our work provides strong evidence towards an interpretation of knot polynomial invariants as generating functions associated to enumerative problems. Received: 1 May 2000 / Accepted: 6 November 2000     

Melting Crystal, Quantum Torus and Toda Hierarchy

Searching for the integrable structures of supersymmetric gauge theories and topological strings, we study melting crystal, which is known as random plane partition, from the viewpoint of integrable systems. We show that a series of partition functions of melting crystals gives rise to a tau function of the one-dimensional Toda hierarchy, where the models are defined by adding suitable potentials, endowed with a series of coupling constants, to the standard statistical weight. These potentials can be converted to a commutative sub-algebra of quantum torus Lie algebra. This perspective reveals a remarkable connection between random plane partition and quantum torus Lie algebra, and substantially enables to prove the statement. Based on the result, we briefly argue the integrable structures of five-dimensional supersymmetric gauge theories and A-model topological strings. The aforementioned potentials correspond to gauge theory observables analogous to the Wilson loops, and thereby the partition functions are translated in the gauge theory to generating functions of their correlators. In topological strings, we particularly comment on a possibility of topology change caused by condensation of these observables, giving a simple example.     

Linear bosonic quantum field theories arising from causal variational principles

We describe discrete symmetries of two-dimensional Yang–Mills theory with gauge group G associated with outer automorphisms of G, and their corresponding defects. We show that the gauge theory partition function with defects can be computed as a path integral over the space of twisted G-bundles and calculate it exactly. We argue that its weak-coupling limit computes the symplectic volume of the moduli space of flat twisted G-bundles on a surface. Using the defect network approach to generalised orbifolds, we gauge the discrete symmetry and construct the corresponding orbifold theory, which is again two-dimensional Yang–Mills theory but with gauge group given by an extension of G by outer automorphisms. With the help of the orbifold completion of the topological defect bicategory of two-dimensional Yang–Mills theory, we describe the reverse orbifold using a Wilson line defect for the discrete gauge symmetry. We present our results using two complementary approaches: in the lattice regularisation of the path integral, and in the functorial approach to area-dependent quantum field theories with defects via regularised Frobenius algebras.

     

Topological field patterns in the Yang-Mills theory

We present a formalism where the topological configurations of pure Yang-Mills theory are characterised using gauge fields alone. Here, we obtain an expression for the charges of these topologicalSO(3) gauge field configurations in terms of the Abelian vector potentials. In this formalism we analyse the ’t Hooft-Polyakov monopole solution.     

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.     

Topological electric charge

By treating magnetic charge as a gauge symmetry through the introduction of a magnetic pseudo four-vector potential, we show that it is possible to construct a topological electric charge from a theory which originally contains gauge magnetic charge. This is an explicit realization of the Montonen-Olive conjecture that there should exist a dual theory to the usual 't Hooft-Polyakov monopole theory in which the roles of the gauge and topological charges are reversed. The physical distinction between 't Hooft-Polyakov monopoles and the dual theory with electric charge is that the strong and weak coupling regimes are reversed. Physically this leads to the mass of the electrically charged soliton being on the order of (1/137)M _W as opposed to the much larger mass (on the order of 137M _W) of the magnetically charged soliton. Thus even forM _W in the TeV range such an electrically charged particle could be observed at some future accelerator.     

Exactly soluble diffeomorphism invariant theories

A class of diffeomorphism invariant theories is described for which the Hilbert space of quantum states can be explicitly constructed. These theories can be formulated in any dimension and include Witten's solution to 2+1 dimensional gravity as a special case. Higher dimensional generalizations exist which start with an action similar to the Einstein action inn dimensions. Many of these theories do not involve a spacetime metric and provide examples of topological quantum field theories. One is a version of Yang-Mills theory in which the only quantum states onS ³×R are the vacua. Finally it is shown that the three dimensional Chern-Simons theory (which Witten has shown is intimately connected with knot theory) arises naturally from a four dimensional topological gauge theory.On leave from the Department of Physics, University of California, Santa Barbara, CA, USA     

Consistent interactions of dual linearized gravity in <Emphasis Type="Italic">D</Emphasis>=5: couplings with a topological BF model

Under some plausible assumptions, we find that the dual formulation of linearized gravity in D=5 can be nontrivially coupled to the topological BF model in such a way that the interacting theory exhibits a deformed gauge algebra and some deformed, on-shell reducibility relations. Moreover, the tensor field with the mixed symmetry (2,1) gains some shift gauge transformations with parameters from the BF sector.     

Aharonov-bohm effect as the basis of electromagnetic energy inherent in the vacuum

The Aharonov-Bohm effect shows that the vacuum is structured, and that there can exist a finite vector potentialA in the vacuum when the electric field strengthE and magnetic flux densityB are zero. It is shown on this basis that gauge theory produces energy inherent in the vacuum. The latter is considered as the internal space of the gauge theory, containing a field made up of components ofA, to which a local gauge transformation is applied to produce the electromagnetic field tensor, a vacuum charge/current density, and a topological charge g. Local gauge transformation is the result of special relativity and introduces spacetime curvature, which gives rise to an electromagnetic field whose source is a vacuum charge current density made up ofA and g. The field carries energy to a device which can in principle extract energy from the vacuum. The development is given forU(1) andO(3) invariant gauge theory applied to electrodynamics. Former Edward Davies Chemical Laboratories, University College of Wales, Aberystwyth SY32 1NE, Wales, United Kingdom.     

Inner Structure of Statistical Gauge Potential in Chern-Simons-Ginzburg-Landau Theory

Based on the decomposition theory of the U(1) gauge potential, the inner structure of the statistical gauge potential in the Chern-Simons-Ginzburg-Landau (CSGL) theory is studied. We give a new creation mechanism of the statistical gauge potential. Furthermore, making use of the φ-mapping topological current theory, we obtain the precise topological expression of the statistical magnetic field, which takes the topological information of the vortices.     

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.     

Abelian Duality and Abelian Wilson Loops

We consider a pure U(1) quantum gauge field theory on a general Riemannian compact four manifold. We compute the partition function with Abelian Wilson loop insertions. We find its duality covariance properties and derive topological selection rules. Finally, we show that, to have manifest duality, one must assume the existence of twisted topological sectors besides the standard untwisted one.