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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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For compact $U^k(1)$ Abelian group, we recently have a paper https://arxiv.org/abs/1906.08270 to define its Chern-Simons theory on spacetime discrete simplicial complex. The quantized coefficient of the $U^k(1)$ Chern-Simons term is given by the $K$-matrix (symmetric integer matrix with even diagonals). Our theory is a bosonic theory and the spacetime does not needs to be spin. So we solved the Open problem for compact Abelian group, with an additional bonus: our discrete path integral also has exact 1-symmetry of the Chern-Simons theory.

The gauge symmetry is not important in discrete simplicial complex, and requiring it may be misleading. So we did not require it. But we do require our discrete Chern-Simons theory to reproduce the correspond topological order (or TQFT).

Our approach does not work for fermionic Chern-Simons theory where the $K$-matrix is a symmetric integer matrix with some odd diagonals. Our approach also depends a choice of branch structure of spacetime simplicial complex (which may correspond to the framing structure of continuous spacetime manifold).

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