Abstract
We point out that different choices of Gliozzi-Scherk-Olive (GSO) projections in superstring theory can be conveniently understood by the inclusion of fermionic invertible phases, or equivalently topological superconductors, on the worldsheet. This allows us to find that the unoriented Type 00 string theory with \Omega^2=(-1)^{f}Ω2=(−1)f admits different GSO projections parameterized by nn mod 8, depending on the number of Kitaev chains on the worldsheet. The presence of nn boundary Majorana fermions then leads to the classification of D-branes by KO^n(X)\oplus KO^{-n}(X)KOn(X)⊕KO−n(X) in these theories, which we also confirm by the study of the D-brane boundary states. Finally, we show that there is no essentially new GSO projection for the Type II worldsheet theory by studying the relevant bordism group, which classifies corresponding invertible phases. In two appendixes the relevant bordism group is computed in two ways.
Highlights
The two Type II superstring theories, Type IIA and Type IIB, arise due to a difference in the specifics of this projection. It was pointed out in [6, 7] that the GSO projection can be interpreted as a sum over the possible spin structures on the worldsheet, with different consistent GSO projections corresponding to different ways of assigning complex phases to spin structures, in a manner consistent with cutting and gluing of the worldsheet
We will find by a standard algebraic topology computation that any invertible phase one can add on the worldsheet is either the Arf invariant associated to the orientation double cover, or a continuum version of the Haldane chain
In the two sections, we give a more detailed analysis of the classification of D-branes. This is done from two different perspectives: in Sec. 4 we study D-branes via boundary fermions and tachyon condensation, while in Sec. 5 we utilize the boundary state formalism
Summary
In perturbative formulations of superstring theories, one treats the 2d worldsheets of strings as 2d quantum field theories with fermions. The two Type II superstring theories, Type IIA and Type IIB, arise due to a difference in the specifics of this projection It was pointed out in [6, 7] that the GSO projection can be interpreted as a sum over the possible spin structures on the worldsheet, with different consistent GSO projections corresponding to different ways of assigning complex phases to spin structures, in a manner consistent with cutting and gluing of the worldsheet. This point of view makes manifest the all-genus consistency of known GSO projections, which is not evident in the one-loop analysis often presented to beginners of string theory e.g. in [8]. The discussions up to this point can be summarized schematically as follows: boundary anomaly : bulk invertible phase ∼ properties of D-branes : choice of GSO projection
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