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Betti numbers apear in the cohomology theory in algebraic topology. How exactly do they arise in a) string theory and b) in supersymmetry? A more general question would be what are the cohomological constructions that appear in string theory and supersymmetry (other than BRST which is mathematically well defined) ?
I do not remember somewhere in specific the Betti numbers but rather their Dolbeault cohomology counterparts, the Hodge numbers . The idea is that if we are given some topological space (satisfying the Calabi-Yai conditions etc) we need a way to count the number of free parameters in the representative of its $SU(N)$ holonomy metric. This is not a new idea, we do the same when we choose the compactification radius when we perform compactifications on $S^1$. The number of parameters (for the topological space I started talking about) is given in terms of the Hodge numbers which are a topological invariant of the corresponding complex manifold.
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