# Geometric (fibre bundle) picture of heterotic string theory?

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Does there exist a fibre bundle structure for heterotic string theory that allows to include both, the left moving 26 D bosonic sector and the 10 D right moving supersymmetric sector, in a consistent way?

edited Dec 26, 2016

What do you mean by "fibre bundle structure"? To define a background for heterotic string theory, one needs to choose a background gauge connection on a given principal bundle (of group $E_8 \times E_8$ or $Spin(32)/(\mathbb{Z}/2)$ depending on the version of the heterotic string) but I don't know if it is what you mean.

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The answer is yes according with the following papers;

1. Keke Li, Character-valued index theorems in supersymmetric string theories;  Class. Quantum Grav. 5 (1988) 95-109.

2.

In the paper 1, for the heterotic string with SO(32), the following action is used

where x are bosons ,  psi_ are  right moving fermions; and psi+ are left moving fermions.

According with paper 2 we have the following theorem

Then, combining the papers 1 and 2, it is possible to consider the heterotic string S0(32) with d = 10;  as a principal fiber bundle with structure group  SO(32)xSO(10).   In this case the bosons and the right-moving fermions are living in the associated fiber bundle corresponding to the principal fiber bundle  with structure group SO(10).  The left-moving fermions are living in the associated fiber bundle corresponding to the principal fiber bundle with structure group SO(32).   The bosons and the right-moving fermions are coupled to the curvature tensor for SO(10).  The left.moving fermions are coupled to the Yang-Mills field for SO(32).

Do you agree?

answered Dec 26, 2016 by (1,105 points)

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