Space Of Flat Connections Research Articles

Dirac structures on the space of connections

We shall give a twisted Dirac structure on the space of irreducible connections on a SU(n)-bundle over a three-manifold, and give a family of twisted Dirac structures on the space of irreducible connections on the trivial SU(n)-bundle over a four-manifold. The twist is described by the Cartan 3-form on the space of connections. It vanishes over the subspace of flat connections. So the spaces of flat connections are endowed with ( non-twisted ) Dirac structures. The Dirac structure on the space of flat connections over the three-manifold is obtained as the boundary restriction of a corresponding Dirac structure over the four-manifold. We discuss also the action of the group of gauge transformations over these Dirac structures.

Twisted character varieties, covering spaces, and gerbes

For a finite-index subgroup \(\tilde \prod \) of a finitely generated group ∏, we establish various relations between (twisted) character varieties of ∏ and \(\tilde \prod \). We interpret our results geometrically in terms of the moduli space of flat connections on principal bundles twisted by a flat bundle gerbe over a manifold with the fundamental group ∏.

Aharonov–Bohm quantum systems on a punctured 2-torus

This paper deals with Aharonov–Bohm (A-B) quantum systems on a punctured two-dimensional torus from the geometric and the operator theoretic point of view. First, flat connections on the U(1)-bundles over the punctured 2-torus are studied, which serve as vector potentials for A-B effect magnetic fields. It is proved that the moduli space of flat connections is identified with the (N + 1)-dimensional torus TN+1, if the punctured torus has N > 0 pinholes at which solenoids are assumed to penetrate the 2-torus. For a given point of TN+1, an associated flat connection is constructed in terms of the Weierstrass zeta function on C together with an inhomogeneous linear function on R2. A-B quantum systems are defined in terms of position operators and momentum operators coupled with the A-B potentials. Necessary and sufficient conditions are given for two A-B quantum systems to be unitarily equivalent. Further, the A-B Hamiltonian is defined and analysed from the viewpoint of operator theory. The deficiency indices of the A-B Hamiltonian are determined to be (N + M, N + M), where M is the number of solenoids whose fluxes are not quantized. Finally, the eigenvalue problem is studied for the A-B Hamiltonian with all fluxes quantized to obtain eigenvalues together with eigenfunctions which are described in terms of the Weierstrass sigma functions.

Geometry of C-flat connections, coarse graining and the continuum limit

A notion of effective gauge fields which does not involve a background metric is introduced. The role of scale is played by cellular decompositions of the base manifold. Once a cellular decomposition is chosen, the corresponding space of effective gauge fields is the space of flat connections with singularities on its codimension two skeleton, AC-flat∕G¯M,⋆⊂A¯M∕G¯M,⋆. If cellular decomposition C2 is finer than cellular decomposition C1, there is a coarse graining map πC2→C1:AC2-flat∕G¯M,⋆→AC1-flat∕G¯M,⋆. We prove that the triple (AC2-flat∕G¯M,⋆,πC2→C1,AC1-flat∕G¯M,⋆) is a principal fiber bundle with a preferred global section given by the natural inclusion map iC1→C2:AC1-flat∕G¯M,⋆→AC2-flat∕G¯M,⋆. Since the spaces AC-flat∕G¯M,⋆ are partially ordered (by inclusion) and this order is directed in the direction of refinement, we can define a continuum limit, C→M. We prove that, in an appropriate sense, limC→MAC-flat∕G¯M,⋆=A¯M∕G¯M,⋆. We also define a construction of measures in A¯M∕G¯M,⋆ as the continuum limit (not a projective limit) of effective measures.

Moduli space of potentials in the Aharonov-Bohm effect

We present a simplified derivation of the fact that the set of gauge equivalence classes or moduli space of flat connections (potentials) in the abelian Aharonov-Bohm effect, is isomorphic to the circle. The length of this circle is the absolute value of the electric charge.

The real locus of an involution map on the moduli space of flat connections on a Riemann surface

It is known that every nonorientable surface Σ has an orientable double cover Σ~. The covering map induces an involution on the moduli space ℳ~ of gauge equivalence classes of flat G-connections on Σ~. We identify the relation between the moduli space ℳ and the fixed point set of the moduli space ℳ~. In particular, ℳ is isomorphic to the fixed point set of ℳ~ if and only if the order of the center of G is odd. One important application is that we give a way to construct a minimal Lagrangian submanifold of the moduli space ℳ~.

Around Polygons in ℝ3 and S3

We survey certain moduli spaces in low dimensions and some of the geometric structures that they carry, and then construct identifications among all of these spaces. In particular, we identify the moduli spaces of polygons in ℝ3 and S 3, the moduli space of restricted representations of the fundamental group of a punctured 2-sphere, the moduli space of flat connections on a punctured sphere, the moduli space of parabolic bundles on a sphere, the moduli space of weighted points on ℂℙ1 and the symplectic quotient of SO(3) acting diagonally on (S 2) n . All of these spaces depend on parameters and some the above identifications require the parameters to be small. One consequence of this work is that these spaces are all biholomorphic with respect to the most natural complex structures they can each be given.

Symplectic Geometry of the Moduli Space of Flat Connections on a Riemann Surface: Inductive Decompositions and Vanishing Theorems

AbstractThis paper treats the moduli spaceg,1(Λ) of representations of the fundamental group of a Riemann surface of genusgwith one boundary component which send the loop around the boundary to an element conjugate to exp Λ, where Λ is in the fundamental alcove of a Lie algebra. We construct natural line bundles overg,1(Λ) and exhibit natural homology cycles representing the Poincaré dual of the first Chern class. We use these cycles to prove differential equations satisfied by the symplectic volumes of these spaces. Finally we give a bound on the degree of a nonvanishing element of a particular subring of the cohomology of the moduli space of stable bundles of coprime rankkand degreed.

Moduli Space of Flat Connections as a Poisson Manifold

In this talk we describe the Poisson structure of the moduli space of flat connections on a two dimensional Riemann surface in terms of lattice gauge fields and Poisson–Lie groups.

Toric structures on the moduli space of flat connections on a Riemann surface II: Inductive decomposition of the moduli space

Toric structures on the moduli space of flat connections on a Riemann surface II: Inductive decomposition of the moduli space

Poisson reduction of twisted Heisenberg Double and finite-dimensional integrable systems

We clarify the special Poisson reduction procedure for the twisted Heisenberg Double and show that the Poisson structure on the reduced space coincides with the one of the moduli space of flat connections on a Riemann surface of genus one.

Degeneration of Riemann surfaces and intermediate polarization of the moduli space of flat connections

We construct a family of polarizations of the moduli space of flatSU(n)-connections on a closed 2-manifold of genusg(≧ 2). These are generalizations of various polarizations known until now. That is, our family of polarizations includes Weitsman's real polarizations in the case ofn=2 [17], as well as the Kahler polarizations which are well known since [2] and [18]. Our construction is based on an original formulation of degeneration of Riemann surfaces. The relation between our polarizations and the complex structures of the moduli spaces of parabolic bundles are also studied.

Degeneration of Riemann surfaces and intermediate polarization of the moduli space of flat connections

Degeneration of Riemann surfaces and intermediate polarization of the moduli space of flat connections

On Some Geometric Invariants Associated to the Space of Flat Connections on an Open Space

AbstractA geometric invariant is associated to the parabolic moduli space on a marked surface and is related to the symplectic structure of the moduli space.

A symplectic structure for connections on surfaces with boundary

For compact surfaces with one boundary component, and semisimple gauge groups, we construct a closed gauge invariant 2-form on the space of flat connections whose boundary holonomy lies in a fixed conjugacy class. This form descends to the moduli space under the action of the full gauge group, and provides an explicit description of a symplectic structure for this moduli space.

GAUGE-INVARIANT HAMILTONIAN FORMULATION OF LATTICE YANG-MILLS THEORY AND THE HEISENBERG DOUBLE

It is known that to get the usual Hamiltonian formulation of lattice Yang-Mills theory in the temporal gauge A0=0 one should place on each link a cotangent bundle of a Lie group. The cotangent bundle may be considered as a limiting case of a so-called Heisenberg double of a Lie group which is one of the basic objects in the theory of Lie-Poisson and quantum groups. It is shown in the paper that there is a generalization of the usual Hamiltonian formulation to the case of the Heisenberg double. The physical phase space of the (1+1)-dimensional γ-deformed Yang-Mills model is proved to be equivalent to the moduli space of flat connections on a two-dimensional torus.

A new 2-form for connections on surfaces with boundary

A natural gauge-invariant 2-form is introduced on the space of connections over a compact oriented surface with boundary. It is shown that this 2-form descends to the moduli space of flat connections, under a group of based gauge transformations. An explicit expression (in terms of holonomy variations) is given for the resulting 2-form, and it is related to the symplectic structure of the extended moduli space introduced by L. C. Jeffrey.

Half density quantization of the moduli space of flat connections and witten's semiclassical manifold invariants

LET PpB denote the moduli space of flat connections on (the trivial SU(2) bundle on) a compact, oriented two-manifold P. The space s”, is a stratified space containing an open dense set Yg which is a symplectic manifold of (real) dimension 6g - 6. There exists a natural line bundle 9 -+ ,Fb, with hermitian metric ( , ), whose restriction to Y9 has a connection V with curvature given by the symplectic form o on Y9. If we equip the two-manifold Xg with a metric, the resulting Hodge star operator turns ,Yg into a Klhler manifold, with w as the Kahler form. Setting aside for the moment concern with the singularities of Pg,, we arrive at the usual setting for geometric quantization: we are given a symplectic manifold (qg, o), a line bundle 3’ + 9, with connection of curvature CO, and a polarization of the space of sections of 3. The quantization CYY~(C~) = H”(Pg, Yk) of this system has been of importance in attempts to construct a topological field theory. A topological field theory would assign to every two-manifold Cg the quantization XDk(Cg) of the system (Pg, CO, 9, V), which must be proved independent of the choice of Kahler structure on -Fg, coming from a choice of metric on X9. The space Xk(Xg) can be given the structure of a Hilbert space with the metric given by ((s1,s~)) = j(si>s~)~~~-~. A topological field theory would also assign an element sN of the quantization Xk(Xg) to every three-manifold N3 whose boundary is given by 8N3 = Cg. If this assignment satisfied the axioms of topological field theory, the space Zk(Eg) would be equipped with a representation of the group Diff(Xg) of diffeomorphisms of Eg. Furthermore, we would obtain invariants of closed three-manifolds as follows. Let N be a closed three-manifold, and let N = H be a

Flat connections and polyubles

The Poisson structure of the moduli space of flat connections on a two dimensional Riemann surface is described in terms of lattice gauge fields and Poisson-Lie groups.

Hyper-K�hler induction and self-duality on Riemann surfaces

Universal hyper-Kahler spaces are constructed from Lie groups acting on flat Kahler manifolds. These spaces are used to describe the moduli space of solutions of Hitchin's equation — self-duality equations on a Riemann surface — as the contangent bundle of the moduli space of flat connections on a Riemann surface.