Area-dependent quantum field theory with defects

Area-dependent quantum field theory is a modification of two-dimensional topological quantum field theory, where one equips each connected component of a bordism with a positive real number - interpreted as area - which behaves additively under glueing. As opposed to topological theories, in area-dependent theories the state spaces can be infinite-dimensional. We introduce the notion of regularised Frobenius algebras and show that area-dependent theories are in one-to-one correspondence to commutative regularised Frobenius algebras. We provide a state-sum construction for area-dependent theories, which includes theories with defects. Defect lines are labeled by dualisable bimodules over regularised algebras. We show that the tensor product of such bimodules agrees with the fusion of defect lines, which is defined as the limit where the area separating two defect lines is taken to zero. All these constructions are exemplified by two-dimensional Yang-Mills theory with compact gauge group and with Wilson lines as defects, which we treat in detail.