Generalized Geometric Loop Groups of Complex Manifolds, Gaussian Quasi-Invariant Measures on Them and Their Representations

Abstract

Loop groups are very important in differential geometry, algebraic topology, and theoretical physics [7, 9, 16, 37, 47], but nothing was known about Gaussian quasi-invariant differentiable measures on them. Only the simplest possible representations associated with path integrals were constructed for loop groups of the circle, i.e., for the manifold M = S1 and (real) Riemannian manifolds N [37]. On the other hand, quasi-invariant measures can be used for construction of regular unitary representations [30–32,35]. Moreover, quasi-invariant measures are helpful for investigation of a group itself. In the previous papers of the author [33, 34], loop groups of Riemannian manifolds M and N were investigated, where either M = S is an n-dimensional real sphere, n = 1, 2, . . . , or M = S∞ is the unit sphere in a real separable Hilbert space l2(R). This was progress in comparison with previous works of others authors, which considered only loop groups for the simplest case M = S1. This paper treats arbitrary complex separable connected metrizable manifolds M and N . For example, products of odd-dimensional real spheres S2n−1×S2m−1 can be endonved with structures of complex manifolds in different ways [29]. Other numerous examples of complex manifolds can be found in [25] and references therein; they are domains in Cn, the complex torus Cn/D, where D is a discrete additive subgroup of Cn generated by a basis τ1, . . . , τ2n of C n over R; the quotient space G/D of a complex Lie group G by a discrete subgroup D, submanifolds of the complex Grassmann manifold Gp,q(C), and also their different products and submanifolds. In general, there are complex compact manifolds which are not Kahler manifolds [28,36]. For construction of loop groups here, we use manifolds M with some mild additional conditions. When M is finite-dimensional over C, we assume that it is compact. This condition is not very restrictive, since each locally compact space admits the Alexandrov (one-point) compactification (see [13, Theorem 3.5.11]). When M is infinite-dimensional over C, it is assumed that M is embedded as a closed bounded subset in the corresponding Banach space XM over C. This is necessary for defining a group structure on the quotient space of a free loop space. The free loop space is considered as consisting of continuous functions f : M → N which are holomorphic on M \ M ′ and preserving distinguished points f(s0) = y0, where M ′ is a closed real submanifold depending on f of codimension codimRM ′ = 1, s0 ∈ M , and y0 ∈ N are distinguished points. There are two reasons to consider such a class of mappings. The first is the need to correctly define compositions of elements in the loop group (see below). The second is the isoperimetric inequality for holomorphic loops, which can impose the condition that the loop be constant on a sufficiently small neighborhood of s0 in M if this loop is in some small neighborhood of w0, where w0(M) := {y0} is a constant loop (see [23, Remark 3.2]). In this paper, loop groups of different classes are considered. Classes analogous to Gevrey classes of f : M \M ′ → N are considered for construction of dense loop subgroups and quasi-invariant measures. Henceforth, we consider only orientable manifolds M and N , since for a nonorientable manifold, there always exists its orientable double covering manifold (see [1, 6.5]). Loop commutative monoids with cancellation property are quotients of families of mappings f from M into a manifold N with f(s0) = y0