I'm interested in learning about topological QFT including Chern Simons theory, Jones polynomial, Donaldson theory and Floer homology – basically the kind of things Witten worked on in the 80s. I'm looking for pedagogical reviews rather than original articles. Though these things sit at the interface of mathematics and physics I'm interested in them more as a physics student. I remember someone asking for a suggested reading list for topological QFT in mathoverflow. The suggested papers were almost uniformly rigorous mathematics written by mathematicians. I am not looking for something like that.
[Physics] Reading list in topological QFT
chern-simons-theorymathematical physicsquantum-field-theoryresource-recommendationstopological-field-theory
Related Solutions
The work being done on TQFT by physicists and mathematicians is wonderful, but in no way should you think it somehow captures what is important in QFT to physicists trying to explain the real world. QFT as applied to the real world has particle-like excitations, non-trivial correlation functions depending on the spacetime distance between operators, spontaneous symmetry breaking, non-trivial renormalization group flows and so on and calculations are done in 4d flat Minkoswki space because this is an excellent approximation to the local spacetime metric. TQFT, as the name implies, captures information about the topology of manifolds and would be completely boring on $R^{3,1}$. You can't use it to study pion scattering, or compute the short distance interaction between quarks, or the production cross-section for the Higgs boson or really anything that particle theorists do with QFT. So the answer to your first question is no. The axioms of TQFT do not capture what physicists feel is important in QFT. It isn't really a question of adding more parameters, they are just very different beasts. To your question of whether mathematicians have gotten rid of too much for physicists to care, the answer is yes, for most physicists. As I said earlier, for all practical purposes on can do particle theory on flat Minkowski space and the only structure that really matters is the causal structure. None of this is meant to denigrate work on TQFT of course.
There are several different relations between Chern-Simons/WZW models, and there are several way to show these. A nice paper doing this in a concrete way is Elitzur et al Nucl.Phys. B326 (1989) 108.
The Chern-Simons theory on a compact spatial manifold give rise to a finite dimensional Hilbert space (only global degrees of freedom) which turns out to be isomorphic to the space of conformal blocks of a WZW model (which is also finite dimensional, since there are finite number of WZW primaries under the associated affine Lie algebra).
If you however put the theory on a manifold with boundary, there will be local degrees of freedom near the boundary and the Hilbert space infinite dimensional (the dynamics of the boundary degrees of freedom are controlled by a WZW model). Let me go through a simple example of the latter type, you can fill out the detailed calculations.
The action is given by
$$ S[a] = \frac k{4\pi}\int_\mathcal M\text{tr}\left(a\wedge\text d a + \frac 23 a\wedge a\wedge a\right).$$ One can show that for $k\in\mathbb Z$ and the boundary condition $a_0\big|_{\partial\mathcal M} = 0$, $e^{iS[a]}$ is gauge invariant and the equations for motions well-defined. Next we need to fix the gauge appropriately, lets now assume our three-manifold has the following simple form $\mathcal M = \mathbb R\times\Sigma$. Make a temporal decomposition $\text d = \partial_0\text dx^0 + \tilde{\text d}$, where $\tilde{\text d} = \partial_i\text dx^i$, and $a = \tilde a_0 + \tilde a$, where $\tilde a_0 = a_0\text dx^0$ and $\tilde a = a_i\text dx^i$ $(i=1,2)$. With this decomposition we get the following action
$$S[a] = -\frac k{4\pi}\int_\mathcal M\text{tr}\left(\tilde a\wedge\partial_0\tilde a\right)\wedge\text dx^0 + \frac k{2\pi}\int_\mathcal M\text{tr}\left(\tilde a_0\wedge \tilde f\right),$$ where $\tilde f = \tilde{\text d}\tilde a +\tilde a\wedge\tilde a$. It is clear that $\tilde a_0$ is just a Lagrange multiplier and we fix the gauge as $a_0 = 0$ (everywhere, not just on the boundary). Alternatively, integrate out $a_0$ and we get $\delta(\tilde f)$ in the path-integral. We therefore have the following action and constraint
$$S[\tilde a, \tilde a_0=0] = -\frac k{4\pi}\int_\mathcal M\text{tr}\left(\tilde a\wedge\partial_0\tilde a\right)\wedge\text dx^0, \qquad \tilde f = \tilde{\text d}\tilde a +\tilde a\wedge\tilde a=0.$$ Thus the phase-space of the theory is the moduli space of flat connections on $\Sigma$. Whether the phase-space has finite or infinite volume, depends of whether $\Sigma$ has a boundary or not.
For simplicity, let us restrict to the simple manifold $\mathcal M = \mathbb R\times D^2$, where $\Sigma=D^2$ is the $2$-disc. Since $\pi_1(\mathcal M)=0$, there no non-trivial Wilson loops/holonomies (since the Wilson loop only depend on the homotopy class of a curve, for flat-connections) and there are thereby no topological degrees of freedom. In this case we can solve the flat connection constraint $\tilde f = 0$ by letting the gauge field be a pure-gauge
$$\tilde a = -\tilde dUU^{-1},$$ where $U:\mathcal M\rightarrow G$ is a single-valued, group-valued function. Here, $U$, parametrize the local degrees of freedom (of the Chern-Simons theory) modulo gauge redundancies. The action that determines the dynamics of $U$ is found by substituting $\tilde a$ in the above action. Using the coordinates $(t,r,\theta)$, we find
\begin{align*} S_{CWZW}&[U] = S\left[\tilde a=-\tilde{\text d} UU^{-1},\tilde a_0=0\right],\\ &= \frac k{4\pi}\int_{\partial\mathcal M}\text{tr}\left(\partial_\theta U^{-1}\partial_tU\right)\text d^2x + \frac k{12\pi}\int_{\mathcal M}\text{tr}\left([\text d UU^{-1}]^3\right),\\ &= \frac k{4\pi}\int_{\partial\mathcal M}\text{tr}\left(\partial_\theta U^{-1}\partial_tU\right)\text d^2x + \frac k{12\pi}\int_{\mathcal M}\text{tr}\left(\epsilon^{\mu\nu\rho}\partial_\mu UU^{-1}\partial_\nu UU^{-1}\partial_\rho UU^{-1}\right)\text d^3 x. \end{align*} Formally, one also has to check that the path-integral does not come with any Jacobian
$$\int\mathcal D\tilde a\delta(\tilde f) = \int\mathcal DU,$$ where $\mathcal DU$ comes from the Haar measure of $G$. This shows what you were looking for, that the partition function of the Chern-Simons theory is determined by a (chiral) WZW model on the boundary. For more general $\mathcal M$, one can do a similar calculation with a few extra elements. See the above reference for details.
There are of course other ways to show this relation. One can for example show that the Dirac bracket on the phase-space (moduli space of flat-connections) reduce to the affine Lie algebra $\hat{\mathfrak g}_k$ (which is the Chiral algebra of the WZW model). There are also approaches where functional equations for the wave functional are derived. One can also use canonical quantization as Witten does, exploding the fact that the moduli space of flat connections (modulo gauge transformation) is a Kähler manifold and the symplectic form represents the first Chern class of a holomorphic line bundle. This last approach is more abstract and less direct, than the one taken above.
Best Answer
The relation is very deep and has a rich mathematical structure, so (unfortunately) most stuff will be written in a more formal, mathematical way. I can't say anything about Donaldson theory or Floer homology, but I'll mention some resources for Chern-Simons theory and its relation to the Jones Polynomial.
There is first of all the original article by Witten - Quantum field theory and the Jones polynomial. A related article is this one (paywall) by Elitzur, Moore, Schwimmer and Seiberg.
A very nice book is from Kauffman called Knots and Physics. Also the book by Baez and Munaiin has two introductory chapters on Chern-Simons theory and its relation to link invariants.
There are also some physical applications of Chern-Simons Theory. For instance, it appears as an effective (longe wavelength) theory of the fractional quantum Hall effect. Link invariants, such as the Jones polynomial, can be related to a generalized form of exchange statistics. See this review article: abs/0707.1889. See also this book by Lerda for more on this idea of generalized statistics.