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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).

https://arxiv.org/abs/1502.00641 defines $\mathbb{R}^k$-group Chern-Simons theory on spatial discrete simplicial complex, with continuous time.

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).

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.

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).

https://arxiv.org/abs/1502.00641 defines $\mathbb{R}^k$-group Chern-Simons theory on spatial discrete simplicial complex, with continuous time.

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).

https://arxiv.org/abs/1502.00641 defines $\mathbb{R}^k$-group Chern-Simons theory on spatial discrete simplicial complex, with continuous time.

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. Our discrete path integral also has exact 1-symmetry of the Chern-Simons theory.

https://arxiv.org/abs/1502.00641 defines $\mathbb{R}^k$-group Chern-Simons theory on spatial discrete simplicial complex, with continuous time.

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.

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).

https://arxiv.org/abs/1502.00641 defines $\mathbb{R}^k$-group Chern-Simons theory on spatial discrete simplicial complex, with continuous time.