Abstract
We point out that the choice of phases in Gliozzi-Scherk-Olive projections can be accounted for by a choice of fermionic symmetry-protected topological phases on the world sheet of the string. This point of view not only easily explains why there are essentially two type II theories, but also predicts that there are unoriented type 0 theories labeled by n mod 8 and that there is an essentially unique choice of the type I world sheet theory. We also discuss the relationship between this point of view and the K-theoretic classification of D-branes.
Highlights
Introduction.—The most traditional method of studying superstring theory is via superstring perturbation theory
There, it was pointed out that the GSO projection is a summation over the spin structure of the world sheet and that different GSO projections correspond to different possible phases assigned to spin structures in a way compatible with the cutting and the gluing of the world sheet
It was found there that the different signs appearing in type IIA and type IIB GSO projections are given by an invariant of the spin structure known as the Arf invariant
Summary
We point out that the choice of phases in Gliozzi-Scherk-Olive projections can be accounted for by a choice of fermionic symmetry-protected topological phases on the world sheet of the string. There is a general classification of possible invertible phases, or equivalently SPT phases, in terms of bordism groups [7,8,9] This means that, with the technology currently available to us, we can understand the consistency of a given GSO projection, and enumerate all possible GSO projections. It is known that eight copies of the Kitaev chain protected by this symmetry are continuously connected to a completely trivial theory [10] In this case, the partition function of the low-energy limit of the Kitaev chain is known as the Arf-Brown-Kervaire (ABK) invariant, and is of order 8. When we perform the GSO projection, or equivalently, when we sum over the pin− structures, we can include n copies of the ABK invariant This leads to a series of unoriented type 0 string theories, labeled by n mod 8.
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