# Simple question about a calculation in superstring theory

• nrqed
In summary, Szabo's book on string theory discusses the calculation of the vacuum to vacuum one loop (genus one) diagram. The contributions are organized according to different spin structures and can be transformed under modular transformations. However, there seems to be a discrepancy in the modular invariance of the combination of spin structures used to reproduce the GSO projection. This leads to confusion and possible mistakes in Szabo's book.
nrqed
Homework Helper
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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!

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

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

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.

## 1. What is superstring theory and how does it relate to other theories of physics?

Superstring theory is a theoretical framework that attempts to explain the fundamental nature of the universe by describing all matter and energy as vibrations of tiny strings. It is a candidate for a theory of everything, as it aims to merge the theories of general relativity and quantum mechanics. It has also been shown to be consistent with other theories, such as the Standard Model of particle physics.

## 2. How is a calculation performed in superstring theory?

In superstring theory, calculations involve complex mathematical equations and techniques such as perturbation theory and conformal field theory. These calculations are used to predict the behavior of strings and their interactions in different scenarios, such as in the early universe or in black holes.

## 3. What is the significance of superstring theory in the scientific community?

Superstring theory is a major area of research in theoretical physics and has generated a lot of interest and debate in the scientific community. Its potential to provide a unified explanation for all physical phenomena has made it a popular topic among physicists, and its predictions have been tested in a variety of experiments.

## 4. Can superstring theory be proven or disproven?

Currently, there is no conclusive evidence to prove or disprove superstring theory. It is a highly complex and abstract theory, making it difficult to test directly. However, some of its predictions have been supported by experimental evidence, giving some credibility to the theory.

## 5. How does superstring theory address the concept of extra dimensions?

Superstring theory proposes that there are more than the three dimensions of space and one dimension of time that we experience in our everyday lives. Instead, it suggests that there are 10 or 11 dimensions, with the extra dimensions being compactified or curled up at the subatomic level. This helps to explain the behavior of particles at a quantum level and provides a possible explanation for the force of gravity.

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