Meromorphic Open-String Vertex Algebras and Riemannian Manifolds

Abstract

Let M be a Riemannian manifold. For p ∈ M, the tensor algebra

T(\widehat{T_{p}M}\mathrm{\_})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$T({\widehat {T_{p}M}}\_)$$\end{document}

of the negative part of the affinization

\widehat{T_{p}M}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\widehat {T_{p}M}}$$\end{document}

of the tangent space T_pM of M at p has a natural structure of a meromorphic open-string vertex algebra. These meromorphic open-string vertex algebras form a vector bundle over M with a connection. The author constructs a sheaf

\mathcal{V}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal V}$$\end{document}

of meromorphic open-string vertex algebras on the sheaf of parallel sections of this vector bundle. Using covariant derivatives, he constructs a representation on the space of smooth functions of the algebra of parallel tensor fields. These representations are used to construct a sheaf

\mathcal{W}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal W}$$\end{document}

of left

\mathcal{V}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal V}$$\end{document}

-modules generated by the sheaf of smooth functions. In particular, the author obtains a meromorphic open-string vertex algebra V_M as the global sections on M of the sheaf

\mathcal{V}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal V}$$\end{document}

and a left V_M-module W_M as the global sections on M of the sheaf

\mathcal{W}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal W}$$\end{document}

. He shows that the Laplacian on M is in fact a component of a vertex operator for the left V_M-module W_M restricted to the space of smooth functions.

Keywords

Meromorphic open-string vertex algebra / Riemannian manifold / Sheaf of left modules / 53C20 / 81T40 / 17B69

References

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