Topological Cosets via Anyon Condensation and Applications to Gapped $\mathrm{\bf{QCD_{2}}}$

The coset construction of two-dimensional conformal field theory (2D CFT) defines a 2D CFT by taking the quotient of two previously known chiral algebras. In this work, we use the methods of non-abelian (non-invertible) anyon condensation to describe 2D topological cosets, defined by the special case where the quotient of chiral algebras is a conformal embedding. In this case, the coset has zero central charge, and the coset theory is thus purely topological. Using non-abelian anyon condensation we describe in general the spectrum of line and local operators as well as their fusion, operator product expansion, and the action of the lines on local operators. An important application of our results is to QCD$_{2}$ with massless fermions in any representation that leads to a gapped phase, where topological cosets (conjecturally) describe the infrared fixed point. We discuss several such examples in detail. For instance, we find that the $Spin(8)_{1}/SU(3)_{3}$ and $Spin(16)_{1}/Spin(9)_{2}$ topological cosets appearing at the infrared fixed point of appropriate QCD$_{2}$ theories are described by $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ triality and $\mathbb{Z}_{2} \times \mathrm{Rep(S_{3})}$ fusion categories respectively. Additionally, using this setup, we argue that chiral $Spin(8)$ QCD$_{2}$ with massless chiral fermions in the vectorial and spinorial representations is not only gapped, but moreover trivially gapped, with a unique ground state.