Topology of Quantum Modified Moduli Spaces

TL;DR

This paper proves that certain supersymmetric Yang-Mills theories with quantum modified moduli spaces have trivial homotopy groups up to dimension 4, implying they do not support skyrmions or vortices, and discusses implications for their effective actions.

Contribution

It establishes the triviality of homotopy groups for all such theories with a single-constraint quantum modified moduli space, extending previous specific cases.

Findings

Homotopy groups π_j are trivial for j=0,1,2,3,4 in these theories.

Theories with trivial H^5() can still admit Wess-Zumino-Witten terms.

Examples with trivial H^5() are identified in existing literature.

Abstract

We prove that all SYM theories that have a quantum modified moduli space $\m$ defined by a single constraint equation have trivial homotopy groups $\pi_j(\m)$ 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.