# Simple question about a calculation in superstring theory

1. Jan 25, 2009

### 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!

2. Jan 28, 2009

### Haelfix

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

3. Jan 28, 2009