Abstract
Let G be a simply connected compact Lie group. Let L e ( G ) be the based loop group with the base point e which is the identity element. Let ν e be the pinned Brownian motion measure on L e ( G ) and let α ∈ L 2 ( ⋀ 1 T ⁎ L e ( G ) , ν e ) ∩ D ∞ , p ( ⋀ 1 T ⁎ L e ( G ) , ν e ) ( 1 < p < 2 ) be a closed 1-form on L e ( G ) . Using results in rough path analysis, we prove that there exists a measurable function f on L e ( G ) such that d f = α . Moreover we prove that dim ker □ = 0 for the Hodge–Kodaira type operator □ acting on 1-forms on L e ( G ) .
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