Abstract
We use Quillen's theorem and algebraic geometry to investigate the modular transformation properties of some quantities of interest in string theory. In particular, we show that the spin structure dependence of the chiral Dirac determinant on a Riemann surface is given by Riemann's theta function. We use this result to investigate the modular invariance of multiloop heterotic string amplitudes.
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Actually, this particular example is well-known to physicists. See, for example, Jackiw, R.: Topological methods in field theory. Les Houches lectures 1983. What is new here is the better geometrical understanding in terms of holomorphic line bundles, and the idea that holomorphy is a powerful tool for understanding determinants
The generalization of Quillen's theorem has been independently derived by Belavin and Knizhnik. We thank Stephen Della Pietra for pointing out an error in an earlier version of Eq. (4.15). We also thank Phil Nelson and Joe Polchinski for discussions on the application of Eq. (4.15) to holomorphic factorization on moduli space, and on the important difference between Eq. (4.15) and Eq. (4.16)
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In preparation
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After this work was completed we received a preprint in which this question is answered in the affirmative. See Manin, Yu.I.: The partition function of the Polyakov string can be expressed in terms of theta functions. Phys. Lett. (submitted)
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Alvarez-Gaumé, L., Moore, G. & Vafa, C. Theta functions, modular invariance, and strings. Commun.Math. Phys. 106, 1–40 (1986). https://doi.org/10.1007/BF01210925
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DOI: https://doi.org/10.1007/BF01210925