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Schur \(Q\)-functions and a Kontsevich-Witten genus. (English) Zbl 0937.55003
Mahowald, Mark (ed.) et al., Homotopy theory via algebraic geometry and group representations. Proceedings of a conference on homotopy theory, Evanston, IL, USA, March 23-27, 1997. Providence, RI: American Mathematical Society. Contemp. Math. 220, 255-266 (1998).
The author gives a homotopy-theoretical interpretation of Virasoro operations in Witten’s theory of two-dimensional topological gravity as endomorphisms of an ordinary cohomology theory with coefficients in the localization \(\Delta[q_1^{-1}]\) of I. Schur’s ring \(\Delta\) of \(Q\)-functions. The central construction of topological gravity is a partition function which is defined by a family of maps from compactifications of Riemann moduli spaces (for each genus) to the complex cobordism spectrum tensored with the rational numbers. The main result of the paper is the construction of a morphism from the complex cobordism spectrum to the spectrum of cohomology theory with coefficients in the localization \(\Delta[q_1^{-1}]\). This morphism (called the Kontsevich-Witten genus) when composed with the above family sends the fundamental class of the moduli space to a highest-weight vector for a naturally defined Virasoro action on \(\Delta\) tensored with the rational numbers. The resulting theory has many of the features of a vertex operator algebra.
For the entire collection see [Zbl 0901.00044].
Reviewer: T.E.Panov (Moskva)

55N22 Bordism and cobordism theories and formal group laws in algebraic topology
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
55N35 Other homology theories in algebraic topology
55P42 Stable homotopy theory, spectra
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