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 [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!