This talk will be devoted to the study of spaces of states associated to 3-manifolds in a topologically twisted 4d N=4 super Yang-Mills theory. I will explain their different mathematical

formalizations: via skein modules, Floer theory and quantum groups as well as mention some relationship between the approaches. I will also explain a recent proof of finite-dimensionality of zeroth cohomology for generic coupling constants. At the end of the talk I will describe a similar picture for spaces of states in a partially topologically twisted 5d N=2 super Yang-Mills. (This is joint work with Sam Gunningham and David Jordan)