An Analytic LT-equivariant Index and Noncommutative Geometry

2017/01/21 by Takata, Doman
#Differential Geometry (math.DG) #FOS: Mathematics #Operator Algebras (math.OA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1701.06055

Abstract

Let T be a circle and LT be its loop group. Let M be an infinite dimensional manifold equipped with a nice LT-action. We construct an analytic LT-equivariant index for M, and justify it in terms of noncommutative geometry. More precisely, we construct a Hilbert space H consisting of "L²-sections of a Clifford module bundle" and a "Dirac operator" D which acts on H. Then, we define an analytic index of D valued in the representation group of LT, so called Verlinde ring. We also define a "twisted crossed product LT\ltimes_τC₀(M)," although we cannot define each concept "function algebra for M vanishing at infinity," "function from LT to a C^*-algebra vanishing at infinity," and a Haar measure on LT. Moreover we combine all of them in terms of spectral triples and verify that the triple has an infinite spectral dimension. Lastly, we add some applications including Borel-Weil theory for LT.

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