# 3. Topological field theories in two-dimensions$.$

As in my previous two posts (1 & 2), a unitary two-dimensional TQFT is specified by a set of real numbers $$\{\lambda_i\}$$ such that the partition function is $$Z(\lambda)=\sum_{i=1}^n\lambda_i^{g-1}\tag1$$

My main question is whether there is a natural, "first principles" construction of such a TQFT other than by specifying these real numbers. This question is inspired by the similar situation in $$d=3$$, where the "instrinsic data" is a modular tensor category, but there is also the Chern-Simons construction which supposedly describes an arbitrary three-dimensional TQFT. In particular, Chern-Simons actions are specified by some Lie Group. So the intrinsic data is the MTC, but it can be conveniently packaged into a choice of Lie Group. I wonder whether a similar situation holds in two-dimensional TQFTs:

1. Can every TQFT$$_2$$ be obtained in the path-integral formalism? (By this I mean continuous field theory rather than a lattice prescription). What is the corresponding Lagrangian?

2. Is there a set of basic "building blocks" such that any other TQFT can be obtained from them via a set of simple operations? These would be akin to the simple groups in three-dimensions, such that any other Lie Group can be obtained through direct product, extensions, and quotients.

Well I guess in $$2d$$ we can write a continuum topological term, that is a metric independent term which also has a zero Hamiltonian this is the definition of the topological term after all, as
$$S=\int_M dA$$ where $$A$$ is 1-form $$U(1)$$ gauge field. You can extend this to non-abelian case as well I think that would answer your question on packing MTC to Lie Group(for the non-abelian case we would have $$S=\text{tr}(\int_M dA+A\wedge A$$ i am not extremely sure about the non-abelian case though ).
We also can extend this abelian case to all even dimensions. In $$4d$$ this would be just theta term. In odd dimensions, the topological term is just d dimensional chern simons term.
For $$d=n+1>2$$ we can have a general topological term now in the lattice as
$$\begin{equation} S_{\text{top}} = i \pi \int_Y (A_{n-1} \cup_{n-3} A_{n-1} + A_{n-1} \cup_{n-2} \delta A_{n-1}), \end{equation}$$ where $$A_{n-1}$$ is $$(n-1)$$-form gauge fields, a $$(n-1)$$-cochain $$A_{n-1} \in C^{n-1}(Y,\mathbb{Z}_2)$$, and $$Y$$ is $$(n+1)$$-dimensional spacetime manifold. The continuum limit of this is quite trivial for d=3 for higher dimensional case I have no idea how to take continuum limit.