Abstract
We construct a cocycle on an infinite dimensional generalization of ap-summable Fredholm module. Our framework is related to Connes' cyclic cohomology and is motivated by our work on index theory on infinite dimensional manifolds. Thep-summability condition is characteristic of dimensionO(p). We replace this assumption by the requirement that there exists an underlying heat kernel which is trace class. Then we use the heat kernel to regularize states in dimension-independent fashion. Our cocycle may be interpreted as an infinite dimensional Chern character.
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Blackadar, B.:K. Berlin, Heidelberg, New York: Springer 1986
Connes, A.: Noncommutative differential geometry. Publ. Math. IHES62, 257–360 (1986)
Connes, A.: Compact metric spaces, Fredholm modules and hyperfiniteness (preprint)
Connes, A.: Entire cyclic cohomology of Banach algebras and characters of θ-summable Fredholm modules (preprint)
Getzler, E., Szenes, A.: Preprint
Helton, J.W., Howe, R.: Traces of commutators of integral operators. Acta Math.135, 271–305 (1975)
Hochschild, G., Kostant, B., Rosenberg, A.: Differential forms on regular affine algebras. Trans. Am. Math. Soc.102, 383–408 (1962)
Jaffe, A., Lesniewski, A.: A priori estimates forN=2 Wess-Zumino models on a cylinder. Commun. Math. Phys.114, 553–576 (1988)
Jaffe, A., Lesniewski, A., Lewenstein, M.: Ground state structure in supersymmetric quantum mechanics. Ann. Phys.178, 313–329 (1987)
Jaffe, A., Lesniewski, A., Weitsman, J.: Index of a family of Dirac operators on loop space. Commun. Math. Phys.112, 75–88 (1987)
Jaffe, A., Lesniewski, A., Weitsman, J.: The two-dimensional,N=2 Wess Zumino model on a cylinder. Commun. Math. Phys.114, 147–166 (1988)
Jaffe, A., Lesniewski, A., Weitsman, J.: TheS 1→ℝ loop space and supersymmetric quantum fields. Ann. Phys. (to appear)
Loday, J.L., Quillen, D.: Cyclic homology and the Lie algebra homology of matrices. Comment. Math. Helvetici59, 565–591 (1984)
Quillen, D.: Superconnections and the Chern character. Topology24, 89–95 (1985)
Taylor, J.: Homology and cohomology for topological algebras. Adv. Math.9, 137–182 (1972)
Witten, E.: Constraints on supersymmetry breaking. Nucl. Phys. B202, 253–316 (1982)
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Communicated by A. Jaffe
Supported in part by the National Foundation under Grant DMS/PHY 86-45122
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Jaffe, A., Lesniewski, A. & Osterwalder, K. QuantumK-theory. Commun.Math. Phys. 118, 1–14 (1988). https://doi.org/10.1007/BF01218474
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DOI: https://doi.org/10.1007/BF01218474