Complex cobordism ring and conformal field theory over \({\mathbb{Z}}\)
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.
scientific article; zbMATH DE number 4199060