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Duality refers to two equivalent descriptions of the same theory fromdifferent points of view. Recently there has been tremendous progress informulating and understanding possible dualities of quantum many body theoriesin $2+1$-spacetime dimensions. Of particular interest are dualities thatdescribe conformally invariant quantum field theories in $(2+1)d$. These ariseas descriptions of quantum critical points in condensed matter physics. Theappreciation of the possible dual descriptions of such theories has greatlyenhanced our understanding of some challenging questions about such quantumcritical points. Perhaps surprisingly the same dualities also underlie recentprogress in our understanding of other problems such as the half-filled Landaulevel and correlated surface states of topological insulators. Here we providea pedagogical review of these recent developments from a point of view gearedtoward condensed matter physics.

Duality between $(2+1)d$ Quantum Critical Points