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.

We can also write a general topological term that is valid for the general dimension whether it is even or odd. For example;

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.


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