Given a simple self-dual quantum Hamiltonian $H=KB+\Gamma \tilde{B}$, where $K$ and $\Gamma$ are coupling constants, and the condition that $[B,[B,[B,\tilde{B}]\left]\right]=16[B,\tilde{B}]$, then we construct an infinite set of conserved charges $Q_{2n}$; $[H,Q_{2n}]=0\mathrm{}$. In simple models, like the two-dimensional Ising or Baxter eight-vertex, these charges appear in the associated quantum theories and are equivalent to those which result from the transfer-matrix formulation and exact quantum integrability of the system. The power of our result is that it is an operator statement and does not refer to the number of dimensions or the nature of the space-time manifold: lattice, continuum, or loop space. It is suggested how the establishment of this link between duality and integrability could be used to exploit the Kramers-Wannier-type self-duality of the four-dimensional $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ gauge theory to find hidden symmetry.

• Received 16 November 1981

DOI:https://doi.org/10.1103/PhysRevD.25.1587