Simple question about a calculation in superstring theory

[nrqed]

In Szabo's book on string theory he calculates the vacuum to vacuum one loop (genus one)
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 \tau \rightarrow -1/\tau, the spin structures transform as

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

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

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

Basically, the transformation switches the two indices.

Under the transformation \tau \rightarrow \tau+ 1, 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!

 

[Haelfix]

Hi Nrqed, eyeballing the problem as stated I agree that there must be a typo somewhere. Did you find it?

 

[nrqed]

Hi Nrqed, eyeballing the problem as stated I agree that there must be a typo somewhere. Did you find it?

Hi, thanks for replying.

No, I still don't get it. It does not seem to be a typo because his choice of sign is necessary in order to reproduce the GSO projection. But clearly, the linear combination he picked is not modular invariant.

However, I looked at the first volume of GSW and they use a different expression for the GSO projection! Szabo uses 1-(-1)^F while GSW use 1+(-1)^F. This does not seem to be a typo in Szabo either because it is written in several places.

So if I would use the expression of GSW *and* change the combination of spin structures to pick the one modular invariant then things would work out ok. But that would mean many many non-trivial mistakes in Szabo. So it feels more like I am missing something.