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I was reading this Physics.SE thread. The OP said, (changed the last word to type HO)

For D = 10, there are 5 vacua. Or maybe it's more correct to say 4, since type I is S-dual to type HO.

This lead me to the 2 closely related questions:

  1. When considering the landscape of string theories in some $D$ dimensions, are the S-dual theories considered as the same theory, as suggested by the quote? What about the T-dual theories? I don't think the T-dual theories should, because in that dimension itself, they are different until and unless one dimension is compactified, when they become approximately equal.

  2. When it is said that the string theory/M-theory landscape in $4$ dimensions in $10^{520}$, are the S-dual theories counted as the same? I don't think they are, because there is no classification of this landscape yet.

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The landscape isn't supposed to be just a set of isolated elements ("depressions" in the landscape, I mean minima) but also the "scenery" in between them. The number $10^{500}$ refers to the number of minima.

In 10 dimensions, supersymmetry can't be broken so there are vacua that have moduli – especially the dilaton, i.e. the string coupling (but also the RR axion in the type IIB case) – that are continuous and parameterize inequivalent vacua. For example, 10-dimensional Minkowski type IIB vacua form a 2-real-dimensional set while the remaining 3 inequivalent theories (type I = the $SO(32)$ heterotic, $E_8\times E_8$ heterotic, and type IIA) are 1-real-parameter families. One must distinguish the families of different dimensions – they should never be just "added" because it would be adding apples or oranges.

However, it'c clear that the S-dual (or otherwise dual) theories describe the same $n$-dimensional moduli set, an $n$-parameter "element of the landscape", if you wish. We may say that the 10-dimensional vacua come in 4 classes that are 1-, 1-, 1-, 2-dimensional, respectively. We may also say that there are 5 limiting weakly coupled string descriptions of various 10-dimensional vacua.

However, for all the vacua in the set of $10^{500}$ vacua, as they have been estimated, $n=0$. So they're zero-dimensional classes with no moduli left. Semirealistic vacua can't have continuous adjustable moduli because they would produce new long-range (scalar-field-induced) forces which would conflict with the experimentally verified equivalence principle.

So there are no moduli left. We say that they are stabilized vacua. If some of remaining stabilized vacua are equivalent, they are counted as one vacuum, if they're inequivalent, they are counted as whatever the higher number of inequivalent vacua is. Each vacuum may potentially have several descriptions that are related to each others by dualities but as long as the resulting dynamics is equivalent, it's just one vacuum.

On the other hand, the stable values of the moduli may be numerous – there may be several values on the previously existing moduli space where the potential is minimized (several solutions to the $V'(\mu)=0$ equation) – and if that's so, they must be counted as many vacua despite the fact that they result from the same (one) $n$-parameter family before the potential was generated.

The vacua are counted literally as the number of inequivalent worlds. This says everything. A duality is a useful tool or insight that allows us to describe the physics of a universe in many ways (and is particularly useful if other ways start to fail) but the knowledge of dualities is in no way needed to define the number of the vacua.

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