We consider the O(6) nonlinear sigma model, with the level-1 Wess-Zumino-Witten term in $3+1$ space-time dimensions. If we divide the O(6) order parameter into two sets of three components, the solitons (hedgehogs) in the first and second set play the role of charge and monopole in the electromagnetic duality. The condensation of these solitons can trigger a direct transition between orders characterized by the two sets of order parameters. We conjecture that the critical point occurs when the theory is self-dual. Moreover, the theory possesses two deconfined phases, where finite-energy hedgehogs coexist with gapless U(1) gauge bosons.

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• Duality
• Effective field theory
• Path-integral methods
• Quantum field theory (low energy)
• Sigma models
• Topology