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Topology of Quantum Modified Moduli Spaces

We prove that all SYM theories that have a quantum modified moduli space $\m$ defined by a single constraint equation have trivial homotopy groups $π_j(\m)$ for $j=0,1,2,3$ and 4. This implies that none of these theories admit skyrmions or vortexes, a fact that had only been proved for supersymmetric QCD with $N_f=N_c$ and $Sp(2n)$ with $2n+2$ fundamentals, whereas those of them with a nontrivial $H^5 (\m)$ admit Wess-Zumino-Witten terms in their effective actions. Contrary to expectations, examples of quantum modified moduli spaces with a trivial $H^5 (\m)$ are found in the literature.