Defects in G/HG/H coset, G/GG/G topological field theory and discrete Fourier–Mukai transform

In this paper we construct defects in coset G/HG/H theory. Canonical quantization of the gauged WZW model G/HG/H with N   defects on a cylinder and a strip is performed and the symplectomorphisms between the corresponding phase spaces and those of double Chern–Simons theory on an annulus and a disc with Wilson lines are established. Special attention to topological coset G/GG/G has been paid. We prove that a G/GG/G theory on a cylinder with N defects coincides with Chern–Simons theory on a torus times the time-line R with 2N   Wilson lines. We have shown also that a G/GG/G theory on a strip with N defects coincides with Chern–Simons theory on a sphere times the time-line R   with 2N+42N+4 Wilson lines. This particular example of topological field theory enables us to penetrate into a general picture of defects in semisimple 2D topological field theory. We conjecture that defects in this case described by a 2-category of matrices of vector spaces and that the action of defects on boundary states is given by the discrete Fourier–Mukai transform.