Hilbert Lie Group Research Articles

Regularized mean curvature flow for invariant hypersurfaces in a Hilbert space and its application to gauge theory

In this paper, we investigate a regularized mean curvature flow starting from an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group whose orbits are minimal regularizable submanifolds. We prove that, if the initial invariant hypersurface satisfies a certain kind of horizontally convexity condition and some additional conditions, then it collapses to an orbit of the Hilbert Lie group action along the regularized mean curvature flow. In the final section, we state a vision for applying the study of the regularized mean curvature flow to the gauge theory.

The Yang-Mills heat equation with finite action in three dimensions

The existence and uniqueness of solutions to the Yang-Mills heat equation is proven over R 3 \mathbb {R}^3 and over a bounded open convex set in R 3 \mathbb {R}^3 . The initial data is taken to lie in the Sobolev space of order one half, which is the critical Sobolev index for this equation over a three dimensional manifold. The existence is proven by solving first an augmented, strictly parabolic equation and then gauge transforming the solution to a solution of the Yang-Mills heat equation itself. The gauge functions needed to carry out this procedure lie in the critical gauge group of Sobolev regularity three halves, which is a complete topological group in a natural metric but is not a Hilbert Lie group. The nature of this group must be understood in order to carry out the reconstruction procedure. Solutions to the Yang-Mills heat equation are shown to be strong solutions modulo these gauge functions. Energy inequalities and Neumann domination inequalities are used to establish needed initial behavior properties of solutions to the augmented equation.

On the classification of infinite-dimensional irreducible Hermitian-symmetric affine coadjoint orbits

In the finite-dimensional setting, every Hermitian-symmetric space of compact type is a coadjoint orbit of a finite-dimensional Lie group. It is natural to ask whether every infinite-dimensional Hermitian-symmetric space of compact type, which is a particular example of an Hilbert manifold, is transitively acted upon by a Hilbert Lie group of isometries. In this paper we give the classification of infinite-dimensional irreducible Hermitian-symmetric affine coadjoint orbits of connected simple L *-groups of compact type using the notion of simple roots of non-compact type. The key step is, given an infinite-dimensional symmetric pair (, ), where is a simple L *-algebra of compact type and a subalgebra of , to construct an increasing sequence of finite-dimensional subalgebras n of together with an increasing sequence of finite-dimensional subalgebras n of such that , , and such that the pairs ( n , n ) are symmetric. Comparing with the classification of Hermitian-symmetric spaces given by W. Kaup, it follows that any Hermitian-symmetric space of compact or non-compact type is an affine-coadjoint orbit of an Hilbert Lie group.

Connectivity properties of moment maps on based loop groups

For a compact, connected, simply-connected Lie group G, the loop group LG is the infinite-dimensional Hilbert Lie group consisting of H^1-Sobolev maps S^1-->G. The geometry of LG and its homogeneous spaces is related to representation theory and has been extensively studied. The space of based loops Omega(G) is an example of a homogeneous space of $LG$ and has a natural Hamiltonian T x S^1 action, where T is the maximal torus of G. We study the moment map mu for this action, and in particular prove that its regular level sets are connected. This result is as an infinite-dimensional analogue of a theorem of Atiyah that states that the preimage of a moment map for a Hamiltonian torus action on a compact symplectic manifold is connected. In the finite-dimensional case, this connectivity result is used to prove that the image of the moment map for a compact Hamiltonian T-space is convex. Thus our theorem can also be viewed as a companion result to a theorem of Atiyah and Pressley, which states that the image mu(Omega(G)) is convex. We also show that for the energy functional E, which is the moment map for the S^1 rotation action, each non-empty preimage is connected.

On harmonic maps into gauge groups

We continue the work of the paper [J. Math. Phys. 37, 4076 (1996)] on the existence of globally defined harmonic maps from the Minkowski plane R1,1 into an infinite-dimensional Hilbert Lie group. We prove that the Cauchy problem for harmonic maps from R1,1 into Hilbert loop groups Hs(LG) can be solved globally for all remaining cases 12<s⩽34 and we obtain similar results for harmonic maps from R1,1 into certain Hilbert Lie gauge groups Hs(Sn,G) (n⩾2). These answer the two questions remaining from the above paper.

Constrained quantisation and θ-angles II

Certain aspects of the quantisation of 2D Yang-Mills theory are analysed with a new constrained quantisation technique, which quantises classical symplectic reduction. To apply this new method, it is necessary to take the gauge group as a Hilbert Lie group, and use the Wiener measure on a certain completion of the gauge group. The physical Hilbert space, the Hamiltonian and the Wilson loop observables for 2D Yang-Mills with a semisimple compact structure group are explicitly constructed using this technique. A new theory of θ-angle, not particular to 2D, emerges naturally. An interesting relation between Fock space coherent states and heat kernels on compact Lie groups is also found. All computations are explicit and simple in the sense that all results originate from the fundamental properties of Wiener measure. An example of SO(3) Yang-Mills is given. The possibility and difficulties of generalising our computations to higher dimensions by utilising the Levy measure (which generalises the Wiener measure) is briefly discussed.

Stochastic Tools on Hilbert Manifolds: Interplay with Geometry and Physics

Projections via the action of a Hilbert Lie group of a class of semi-martingales (given by Ito fields) defined on Hilbert manifolds are investigated. Using Ito calculus, we show that the drift term arising in the projected process can be interpreted in terms of a regularised trace of the second fundamental form of the orbits. For group actions with finite dimensional orbit space, we introduce a notion of strongly harmonic functions resp. regularised Brownian motion, which project onto harmonic functions resp. onto Brownian motion, whenever the orbits are minimal (in a regularised sense). We relate this projection procedure of semi-martingales to the Faddeev-Popov procedure in gauge field theory.

On the generating functional of a convolution semigroup on a hilbert-lie group

On the generating functional of a convolution semigroup on a hilbert-lie group

MASSLESS PARTICLES, ELECTROMAGNETISM, AND RIEFFEL INDUCTION

The connection between space-time covariant representations (obtained by inducing from the Lorentz group) and irreducible unitary representations (induced from Wigner’s little group) of the Poincaré group is re-examined in the massless case. In the situation relevant to physics, it is found that these are related by Marsden-Weinstein reduction with respect to a gauge group. An analogous phenomenon is observed for classical massless relativistic particles. This symplectic reduction procedure can be (‘second’) quantized using a generalization of the Rieffel induction technique in operator algebra theory, which is carried through in detail for electromagnetism. Starting from the so-called Fermi representation of the field algebra generated by the free abelian gauge field, we construct a new (‘rigged’) sesquilinear form on the representation space, which is positive semi-definite, and given in terms of a Gaussian weak distribution (promeasure) on the gauge group (taken to be a Hilbert Lie group). This eventually constructs the algebra of observables of quantum electromagnetism (directly in its vacuum representation) as a representation of the so-called algebra of weak observables induced by the trivial representation of the gauge group.

Uniqueness of Ground States for Schrödinger Operators over Loop Groups

Over the space L of based loops in a compact connected Lie group G there is a natural gradient operator ∇ induced by the left action on L of the Hilbert Lie group K0 which consists of finite energy elements of L. Denote by μ pinned Brownian motion measure on L. The Schrödinger operator ∇ + ∇ + V acting in L2(L, μ) is shown to have a unique ground state over each homotopy class in L. The proof of uniqueness is reduced to the proof of ergodicity of the left action of K0 on (L, μ) for a simply connected group G of compact type. Ergodicity, in turn, is proved by first characterizing the Itô expansion coefficients of K0 invariant functions and then regularizing such functions via a smooth approximation of multiple Stratonovich integrals. As a byproduct of this method one obtains an isometric isomorphism of L2(G, heat kernel measure at time one) with a natural completion of the universal enveloping algebra of G, thus extending to the noncommutative case the well known isometry between L2(Rd, Gauss measure) and the space of symmetric tensors over Rd.

GAUGE THEORY ON OPEN MANIFOLDS OF BOUNDED GEOMETRY

On compact manifolds (Mn, g) and for r>n/2+1 the configuration space [Formula: see text] is a well-defined object. [Formula: see text] is an affine space with a Sobolev space as vector space, and [Formula: see text] a Hilbert Lie group which acts smoothly and properly on [Formula: see text]. [Formula: see text] is a stratified space with Hilbert manifolds as strata. The existence problem has been solved for many interesting cases by Cliff Taubes and the description of the moduli space of instantons has been given by Donaldson. On noncompact manifolds none of the approaches of the compact case is further valid. We present here an intrinsic, self-consistent approach for gauge theory on open manifolds of bounded geometry up to order n/2+2. The main idea is to endow the space CP of gauge potentials and the gauge group with an intrinsic Sobolev topology. Bounded geometry of the underlying manifold and the considered connections provides all the Sobolev theorems which are needed to prove the existence of instantons if G=SU(2). We prove the existence of instantons if (M4, g) satisfies a certain spectral condition and has a positive definite L2 intersection form.

A Garding domain for representations of some Hilbert Lie groups

Given a Banach representation of a Hilbert Lie group, the Lie algebra \(\mathfrak{G}\) of which is the closure of the union of an increasing sequence of finite dimensional subalgebras, we construct a Garding domain on which we differentiate the group representation to a representation of a dense subalgebra of \(\mathfrak{G}\).

The mean curvature flow for invariant hypersurfaces in a Hilbert space with an almost free group action

In this paper, we study the regularized mean curvature flow starting from invariant hypersurfaces in a Hilbert space equipped with an isometric almost free Hilbert Lie group action whose orbits are minimal regularizable submanifolds, where "almost free" means that the stabilizers of the group action are finite. First we obtain the evolution equations for some geometric quantities along the regularized mean curvature flow. Next, by using the evolution equations, we prove a horizontally strongly convexity preservability theorem for the regularized mean curvature flow. From this theorem, we derive the strongly convexity preservability theorem for the mean curvature flow starting from compact Riemannian suborbifolds in the orbit space (which is a Riemannian orbifold) of the Hilbert Lie group action.