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Conference Paper

The holomorphic bosonic string


Gwilliam,  Owen
Max Planck Institute for Mathematics, Max Planck Society;

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Gwilliam, O., & Williams, B. (2018). The holomorphic bosonic string. In Topology and quantum theory in interaction (pp. 213-258). Providence: American Mathematical Society.

Cite as: https://hdl.handle.net/21.11116/0000-0003-AD51-F
We describe and analyze a holomorphic version of the bosonic string in the formalism of quantum field theory developed by Costello and collaborators, which provides a powerful combination of renormalization theory and the Batalin-Vilkovisky formalism. Our focus here is on the case in which the target space-time is a vector space. We identify the critical dimension as an obstruction to satisfying the quantum master equation, and when the
obstruction vanishes, we construct a one-loop exact quantization. Moreover, we show how the factorization algebra of observables recovers the BRST cohomology of the string and use this perspective to give a new construction of its Gerstenhaber structure. Finally, we show how the factorization homology along closed manifolds encodes the determinant line bundle over the moduli
space of Riemann surfaces. An auxiliary goal of this paper is to give an exposition of this QFT formalism with the holomorphic bosonic string theory as the running example.