In Szabo's book on string theory he calculates the vacuum to vacuum one loop (genus one)(adsbygoogle = window.adsbygoogle || []).push({});

diagram.

The contributions can be organized according to different spin structures (periodic or antiperiodic along

the two cycles of the torus). The spin structures are (+,+). (-,-) , (-,+) and (+,-). It is not important

what the exact definition is for the rest of the question. The (+,+) structure happens to vanish

identically so it won't play a role.

Now, under the modular transformation [itex] \tau \rightarrow -1/\tau[/itex], the spin structures transform as

(-,-) -> (-,-)

(-,+) -> (+,-)

(+,-) -> (-,+)

Basically, the transformation switches the two indices.

Under the transformation [itex] \tau \rightarrow \tau+ 1 [/itex], they transform as

(-,-) -> (+,-)

(-,+) -> (-,+)

(+,-) -> (-,-)

The rule is that the first index changes if the second index is a minus.

So far so good.

Now, inhis equation 4.53, he writes that, up to an overall constant, the only modular invariant combination is

(-,-) - (+,-) - (-,+)

This is clearly invariant under the first modular transformation but not under the second one! The first two terms

would need to have the same sign.

Now, I thought at first that this was simply a typo (I found several within a few pages). But the

rest of the discussion, in particular the recovery of the GSO projection, relies heavily

on the first two terms having opposite signs.

So I am probably misunderstanding something obvious. Can anyone clarify the situation?

Thanks!

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# Simple question about a calculation in superstring theory

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