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Question: What is the challenge and the current status to define the 3d Chern-Simons(-Witten) (CSW) theory on a simplicial complex or on a discrete lattice? (Or is there a no-go or an obstruction behind this attempt?)


See also an open problem in the post of Daniel Moskovich:

Open problem: Construct a discrete $3$-dimensional Chern-Simons theory, compatible with gauge symmetry, replacing the path integrals of the smooth picture (which are not mathematically well-defined) with finite dimensional integrals.


Background info:

Here the CSW theory is a three-dimensional (3d) topological quantum field theory (TQFT) whose configuration space is the space of G-principal bundles with connection on a bundle and whose Lagrangian $\mathcal{L}_{\text{CS}}(A)$ is given by the Chern-Simons form of such a connection (for a simply connected compact Lie group G; though there may be a more general version of it).

Since we are talking about defining the quantum theory, in physics, it means that we like to define the following continuum path integral/partition function $Z$ on a simplicial complex or on a discrete lattice on a 3-manifold $M^3$, summing over all the gauge-inequivalent configurations: $$ Z=\int [DA] \exp[i k S_{\text{CS}}(A)] =\int [DA] \exp[i k \int_{M^3} \mathcal{L}_{\text{CS}}(A)]$$ $$=\int [DA] \exp[i \frac{k}{4 \pi} \int_{M^3} \text{Tr}(A \wedge dA+ \frac{2}{3} A^3 )]. $$ Here Tr is an invariant quadratic form on the Lie algebra of G, and $A$ a connection on a G bundle E. If E is trivial, the connection $A$ can be regarded as a Lie algebra valued one form, and we can define the Chern-Simons functional by this familiar formula. The $k$ is called the level.

If G is a connected, simply connected compact Lie group, then a G bundle on a 3-manifold is necessarily trivial.

If there are non-trivial bundles, it means that we consider the gauge group G; but not a connected, simply connected group whose Lie algebra equals Lie(G).

If the bundle E is not trivial, we can define the path integral in a 4-manifold $N^4$: Since any 3-manifold $M^3$ can be realized as the boundary of a 4-manifold $N^4$, as follows (see this Ref). This is a more general definition of the Chern-Simons functional: $$ Z=\int [DA] \exp[i \frac{k}{4 \pi} \int_{N^4} \text{Tr}(F \wedge F)], $$


More details: To define the continuum CSW theory on the discrete simplicial complex/lattice, in principle, we should be able to reproduce the following properties (seen in the continuum CSW theory), from the discrete simplicial complex/lattice calculation, exactly:

  1. Hilbert space. Define on a genus-$g$ Riemann surface ($\Sigma_g^2$) with a time circle $S^1$, so $\Sigma_g^2 \times S^1$ we get the dimensions of Hilbert space.

  2. The 2d open boundary has the Wess–Zumino–Witten (WZW) model.

  3. Certain theories (certain level $k$) can be defined only on the spin manifold.

  4. Knot invariants of CSW theory. (Can be computed with the Wilson line insertions into the path integral $Z$.)

  5. The framing of Wilson lines.

...

etc.

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    $\begingroup$ I'm not an expert on this at all (so I don't know if this is at all close to what you want) but I heard about this work arxiv.org/abs/1502.00641 (a particular discretization of abelian CS theory that applies to surface graphs with a 1:1 correspondence between faces and vertices) in some talks a few years ago and found it quite nice. It features an appearance of Hall's marriage theorem (unexpected, to me at least). $\endgroup$ – j.c. Jul 19 '18 at 4:08

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