Chern-Simons Gauge Theory and Ginzburg-Landau Type Theories: (Ⅰ) Abelian Cases

We show that after the spontaneous symmetry breaking has taken place the Abelian Higgs models with or without the Chern-Simons term would satisfy the same field equation with the massive Chern-Simons gauge theory and have the same anyon solutions. So we regard the massive Chern-Simons gauge theory as an effective gauge theory of the Abelian Higgs models.     

A Toy Model of the M5-Brane: Anomalies of Monopole Strings in Five Dimensions

We study a five-dimensional field theory which contains a monopole (string) solution with chiral fermion zero modes. This monostring solution is a close analog of the fivebrane solution of M-theory. The cancellation of normal bundle anomalies parallels that for the M-theory fivebrane; in particular, the presence of a Chern-Simons term in the low-energy effective U(1) gauge theory plays a central role. We comment on the relationship between the microscopic analysis of the world-volume theory and the low-energy analysis and draw some cautionary lessons for M-theory.     

Quantum field theory and link invariants

A skein relation for the expectation values of Wilson line operators in three-dimensional SU(N) Chern-Simons gauge theory is derived at first order in the coupling constant. We use a variational method based on the properties of the three-dimensional field theory. The relationship between the above expectation values and the known link invariants is established.     

Some comments on Chern-Simons gauge theory

Following M. F. Atiyah and R. Bott [AB] and E. Witten [W], we consider the space of flat connections on the trivialSU(2) bundle over a surfaceM, modulo the space of gauge transformations. We describe on this quotient space a natural hermitian line-bundle with connection and prove that if the surfaceM is now endowed with a complex structure, this line bundle is isomorphic to the determinant bundle. We show heuristically how path-integral quantisation of the Chern-Simons action yields holomorphic sections of this bundle.I.M.S. and T.R.R. supported by DOE grant DE-FG02-88ER 25066. J.W. supported by NSF Mathematical Sciences post-doctoral research scholarship 8807291     

SU(2) Lattice Gauge Theory Formulation of the (2+1)-Dimensional Antiferromagnetic Heisenberg Model

The equivalence of 2+1 antiferromagnetic Heisenberg model and the SU(2) Kogut Susskind lattice gauge theory is recapitulated and the naive Euclidean lattice action of the threedimensional an tiferromagnetic Heisen berg model is derived. The three-dimensional lattice gauge fermion theory is formulated to give the consistent lattice gauge theory of antiferromagnetic Heisenberg model. In continu um limit the two copies of two flavor fermions are resulted, which give the negative results of the microscopic derivation of the Chern-Simons terms. The Chern-Simons terms, the gauge invariant problem of effective action and the '%hiralityn are discussed.     

Real polarization of the moduli space of flat connections on a Riemann surface

We prove that the moduli space of flatSU(2) connections on a Riemann surface has a real polarization, that is, a foliation by lagrangian subvarieties. This polarization may provide an alternative quantization of the Chern-Simons gauge theory in higher genus, in line with the results of [11] for genus one.Supported by NSF Mathematical Sciences Postdoctoral Research Fellowship DMS 88-07291     

Quantization of the Chern invariant polynomial and its topological quantum ground state

The symplectic quantization of the Chern invariant polynomial on a 2n-dimensional manifold is considered. The ground state wave functional as a representation of the vacuum state of the theory is constructed in terms of the Chern-Simons form. An important consequence of these results is that the existence of the Chern-Simons wave functional is not exclusive of topological four-dimensional gauge theory.     

Chern-Simons gauge theory on orbifolds: Open strings from three dimensions

Chern-Simons gauge theory is formulated on three-dimensional $\text{Z}$₂ orbifolds. The locus of singular points on a given orbifold is equivalent to a link of Wilson lines. This allows one to reduce any correlation function on orbifolds to a sum of more complicated correlation functions in the simpler theory on manifolds. Chern-Simons theory on manifolds is known to be related to two-dimensional (2D) conformal field theory (CFT) on closed-string surfaces; here it is shown that the theory on orbifolds is related to 2D CFT of unoriented closed- and open-string models, i.e. to worldsheet orbifold models. In particular, the boundary components of the worldsheet correspond to the components of the singular locus in the 3D orbifold. This correspondence leads to a simple identification of the open-string spectra, including their Chan-Paton degeneration, in terms of fusing Wilson lines in the corresponding Chern-Simons theory. The correspondence is studied in detail, and some exactly solvable examples are presented. Some of these examples indicate that it is natural to think of the orbifold group $\text{Z}$₂ as a part of the gauge group of the Chern-Simons theory, thus generalizing the standard definition of gauge theories.     

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

Spinor Decomposition of SU(2) Gauge Potential and Spinor Structures of Chern-Simons and Chern Density

In this paper, the decomposition of SU(2) gauge potential in terms of Pauli spinor is studied. Using thisdecomposition, the spinor structures of Chern-Simons form and the Chern density are obtained. Furthermore, the knotquantum number of non-Abelian gauge theory can be expressed by the Chern-Simons spinor structure, and the secondChern number is characterized by the Hopf indices and the Brouwer degrees of φ-mapping.     

Spinor Decomposition of SU（2） Gauge Potential and Spinor Structures of Chern-Simons and Chern Density

In this paper, the decomposition of SU(2) gauge potential in terms of Pauli spinor is studied. Using this decomposition, the spinor structures of Chern Simons form and the Chern density are obtained. Furthermore, the knot quantum number of non-Abelian gauge theory can be expressed by the Chern-Simons spinor structure, and the second Chern number is characterized by the Hopf indices and the Brouwer degrees of Φ-mapping.     

Chern-Simons Gauge Theory and projectively flat vector bundles on 421-1421-1421-1

We consider a vector bundle on Teichmüller space which arises naturally from Witten's analysis of Chern-Simons Gauge Theory, and define a natural connection on it. In the case when the gauge group isU(1) we compute the curvature, showing, in particular, that the connection is projectively flat.Work supported by DOE grant DE-FG-02-88ER25066     

Coalgebra Bundles

We develop a generalised theory of bundles and connections on them in which the role of gauge group is played by a coalgebra and the role of principal bundle by an algebra. The theory provides a unifying point of view which includes quantum group gauge theory, embeddable quantum homogeneous spaces and braided group gauge theory, the latter being introduced now by these means. Examples include ones in which the gauge groups are the braided line and the quantum plane. Received: 22 February 1996 / Accepted: 29 May 1997     

On Chern–Simons and WZW Partition Functions

Direct analysis of the path integral reduces partition functions in Chern-Simons theory on a three-manifold M with group G to partition functions in a WZW model of maps from a Riemann surface ‡ to G. In particular, Chern-Simons theory on S³, S¹ 2 ‡, B³ and the solid torus correspond, respectively, to the WZW model of maps from S² to G, the G/G model for ‡, and Witten's gauged WZW path integral Ansatz for Chern-Simons states using maps from S² and from the torus to G. The reduction hinges on the characterization of {\cal A / G}_{n}$, the space of connections modulo those gauge transformations which are the identity at a point n, as itself a principal fiber bundle with affine-linear fiber.     

The Seiberg-Witten Map and Topology

The mapping of topologically nontrivial gauge transformations in noncommutative gauge theory to corresponding commutative ones is investigated via the operator form of the Seiberg-Witten map. The role of the gauge transformation part of the map is analyzed. Chern-Simons actions are examined and the correspondence to their commutative counterparts is clarified.     

Chern-Simons theory and fusion algebras

The connection between the Chern-Simons theory and some features of the two-dimensional conformal models is considered. By using the properties of the expectation values of the Wilson line operators, it is shown how the fusion rules emerge in the three-dimensional context. The case G=SU(2) is considered in detail. The fusion algebra is obtained from the tensor algebra of the gauge group by factorizing an appropriate invariant subalgebra generated by a null vector.     

Topological particle field theory,general coordinate invariance and generalized Chern-Simons actions

We show that recently proposed generalized Chern-Simons action can be identified with the field theory action of a topological point particle. We find the crucial correspondence which makes it possible to derive the field theory actions from a special version of the generalized Chern-Simons actions. We provide arguments that the general coordinate invariance in the target space and the flat connection condition as a topological field theory can be accommodated in a very natural way. We propose series of new gauge invariant observables.     

One-Dimensional Chern-Simons Theory

We study a one-dimensional toy version of the Chern-Simons theory. We construct its simplicial version which comprises features of a low-energy effective gauge theory and of a topological quantum field theory in the sense of Atiyah.     

The asymptotical behaviour of the Abelian Chern-Simons gauge theory with matter in curved space-time

We study the ultraviolet and infrared behaviour of the Chern-Simons gauge theory with matter in curved space-time. It is shown that theory under investigation is asymptotically supersymmetric in infrared limit and asymptotically finite in ultraviolet limit. The investigation of scalar-graviton effective coupling constant behaviour shows that the theory can be asymptotically conformally invariant.