Question about SO(32) heterotic - Type I duality, M-theory perspective

[PelleJW]

I'm having trouble understanding the arguments in http://arxiv.org/abs/hep-th/9510209

They propose that M-theory on the orbifold R^9 \times S^1 \times S^1 / Z_2(1) yields Type I' on R^9 \times S^1, wheras M-theory on R^9 \times S^1 / Z_2 \times S^1(2) is E_8 \times E_8 on R^9 \times S^1. What exactly is the difference between (1) and (2)? Does it imply a different ordering of applying the orbifold and the circular compactification? In other words: what happens when the two last factors are exchanged?

 

[fzero]

I'm having trouble understanding the arguments in http://arxiv.org/abs/hep-th/9510209

They propose that M-theory on the orbifold R^9 \times S^1 \times S^1 / Z_2(1) yields Type I' on R^9 \times S^1, wheras M-theory on R^9 \times S^1 / Z_2 \times S^1(2) is E_8 \times E_8 on R^9 \times S^1. What exactly is the difference between (1) and (2)? Does it imply a different ordering of applying the orbifold and the circular compactification? In other words: what happens when the two last factors are exchanged?

Check the bottom of page 12. In the Type I' theory (1), the gauge group is $SO(16)\times SO(16)$, while the heterotic theory has $E_8\times E_8$ gauge group. So Horava and Witten argue that we get from (2) to (1) by turning on Wilson lines to break $E_8\times E_8 \rightarrow SO(16)\times SO(16)$.

This illustrates an important concept about these kinds of dualities. In general, we have a theory characterized by some moduli space of coupling constants, geometric parameters (like radii) and Wilson lines. At some particular point or neighborhood in the moduli space, the theory can be clearly expressed in terms of the degrees of freedom of some perturbatively defined string (or 11d) theory, call it X. At a different point, the obvious description is in terms of some other theory Y. We conclude that the theories X and Y, after including the appropriate compactification data, share the same moduli space. This is what we mean when we say that X and Y are dual to one another.

When the modulus that we have to vary to get from the perturbative description X to that of Y is a coupling constant, we might call this a strong-weak duality or S-duality. When the modulus is a radius of a circle, it is a T-duality. But a specific type of duality might also include the other types of moduli, like Wilson lines, and we don't have a special name for this.