Let K denote a simply connected compact Lie group and let G denote its
complexification. It is known that there exists an LK-biinvariant probability
measure on a natural completion of the complex loop group LG. It is believed
that there exist deformations which are positive line bundle valued and
reproduce the unitary structure for (projective) positive energy
representations of LK. These are notes which supplement lectures I have given
on these measures, explaining a number of conjectures concerning how these
measures are characterized, how they are computed, and how they are potentially
useful for formulating quantum sigma models.

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