  We study quantum field theories with boundary by utilizing non-invertible
symmetries. We consider three kinds of boundary conditions of the four
dimensional $\mathbb{Z}_2$ lattice gauge theory at the critical point as
examples. The weights of the elements on the boundary is determined so that
these boundary conditions are related by the Kramers-Wannier-Wegner (KWW)
duality. In other words, it is required that the KWW duality defects ending on
the boundary is topological. Moreover, we obtain the ratios of the hemisphere
partition functions with these boundary conditions; this result constrains the
boundary renormalization group flows under the assumption of the conjectured
g-theorem in four dimensions.

High Energy Physics - Theory;Statistical Mechanics;High Energy Physics - Lattice