Complex cobordism ring and conformal field theory over \({\mathbb{Z}}\)
type: Item
instance of: scholarly article
title: Complex cobordism ring and conformal field theory over \({\mathbb{Z}}\)
zbMATH Open document ID: 0726.57025
DOI: 10.1007/BF01445226
published in: Mathematische Annalen
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164885
review text: The boson Fock space \({\mathcal H}_{T,0}(k)\) in conformal field theory is a polynomial ring in infinitely many variables (over a field k). We give an explicit isomorphism of \({\mathbb{Q}}\)-vector spaces between the boson Fock space \({\mathcal H}_{T,0}({\mathbb{Q}})\) and the complex cobordism ring \(MU^*\otimes {\mathbb{Q}}\). We also give an isomorphism between the boson Fock space \({\mathcal H}_{T,0}({\mathbb{Q}})\) and the ring of ``universal Chern classes'' in such a way that the natural pairing ``Chern number'' is identified with the natural pairing of \({\mathcal H}_{T,0}({\mathbb{Q}})\) with itself, which appears in the theory of the KP hierarchy. The identification allows us to show that an arbitrary multiplicative sequence has an interpretation as a tau function in the theory of the KP hierarchy, and that the totality of multiplicative sequences can be identified with a fiber of the projection from Mikio Sato's universal Grassmann manifold onto the space of Lax operators. We also give a comparison of our identification with the homotopy- theoretic results, due to J. Morava and V. M. Bukhshtaber-A. V. Shokurov. Namely, \(MU^*\otimes {\mathbb{Q}}\) is naturally identified with the coordinate ring of the affine group scheme of automorphisms of the formal line fixing the origin, which is also equal to \({\mathcal H}_{T,0}({\mathbb{Q}})\). We give an automorphism of \({\mathcal H}_{T,0}({\mathbb{Q}})\), which connects the above two identifications. As a corollary, we can determine the image of \(MU^*\) in \({\mathcal H}_{T,0}({\mathbb{Q}})\) by our isomorphism.
Mathematics Subject Classification ID: 57R77
Mathematics Subject Classification ID: 81T40
zbMATH DE Number: 4199060
zbMATH Keywords: boson Fock space
zbMATH Keywords: conformal field theory
zbMATH Keywords: complex cobordism ring
zbMATH Keywords: multiplicative sequence
zbMATH Keywords: KP hierarchy
author: Toshiyuki Katsura
author: Yuji Shimizu
author: Kenji Ueno
reviewed by: Kenji Ueno
MaRDI profile type: Publication
cites work: Q4067782
cites work: Determinant bundles and Virasoro algebras
cites work: The Landweber-Novikov algebra and formal vector fields on the line
cites work: Q3901604
cites work: Q5513537
cites work: New bosonization and conformal field theory over \({\mathbb{Z}}\)
cites work: Geometric realization of conformal field theory on Riemann surfaces
cites work: Associated prime ideals and Hopf algebras
cites work: Q5797085
cites work: Characteristic Classes. (AM-76)
cites work: Characterization of Jacobian varieties in terms of soliton equations
cites work: Fock space representations of the Virasoro algebra. Intertwining operators
description: scientific article; zbMATH DE number 4199060