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We show that in (3+1)-D space-time dimensions, the O(6) non-linear sigmamodel, with a level-1 Wess-Zumino-Witten term, exhibits the electromagneticduality. If we name the six components of the sigma field as the Neel andvalence-bond-solid (VBS) order parameters, the hedgehogs of the Neel and VBSorder play the role of monopole and charge. The duality corresponds to theexchange of monopole and charge hence the exchange of the Neel and VBS order.The condensation of monopole can trigger a direct Neel $\leftrightarrow$ VBSphase transition. We conjecture that the critical point is self-dual, which isa generalization of the deconfined quantum critical point in (2+1)-D. In thistheory, there exist two deconfined phases where the Neel/VBS hedgehogs aremassive but cost finite energy. This leads to two fractionalized phases withparticles carrying fractional spin or VBS quantum numbers and gapless gaugebosons.

The electromagnetic duality and the 3+1D O(6) non-linear sigma model with a level-1 Wess-Zumino-Witten term