For semisimple Lie algebra targets
For discrete group targets
For discrete 2-group targets
For Lie 2-algebra targets
For targets extending the super Poincare Lie algebra
(such as the supergravity Lie 3-algebra, the supergravity Lie 6-algebra)
Chern-Simons-supergravity
for higher abelian targets
for symplectic Lie n-algebroid targets
for the $L_\infty$-structure on the BRST complex of the closed string:
higher dimensional Chern-Simons theory
topological AdS7/CFT6-sector
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By the general mechanism of ∞-Chern-Simons theory, every invariant polynomial of total degree 2 induces a 1-dimensional Chern-Simons-like theory.
By the general mechanism of ∞-Chern-Simons theory there is a Chern-Simons action functional associated to the first Chern class, or rather to the corresponding invariant polynomial, which is simply the trace map on the unitary Lie algebra
This yields an action functional for a 1-dimensional QFT as follows:
The configuration space over a 1-dimensional $\Sigma$ is the groupoid of Lie algebra valued 1-forms $\Omega^1(\Sigma, \mathfrak{u})$. After identifying $\Sigma \subset \mathbb{R}$ this may be identified with the space of $\mathfrak{u}(n)$-valued functions.
The action functional is simply the trace operation
Degenerate as this situation is, it can be useful to regard the trace as a Chern-Simons action functional.
Arguments for a role in large $N$ gauge theory are in (Nair 06).
The spectral action is of this form.
For $G$ a suitable Lie group (compact, semi-simple and simply connected) the Wilson loops of $G$-principal connections are equivalently the partition functions of a 1-dimensional Chern-Simons theory.
This appears famously in the formulation of Chern-Simons theory with Wilson lines. More detailes are at orbit method.
A symplectic manifold regarded as a symplectic Lie n-algebroid with $n = 0$ induces a 1d Chern-Simons theory whose Chern-Simons form is a Liouville form of the symplectic form.
This case is discussed in …
A discussion of 1d CS theory in the context of large $N$-gauge theory is in
An exposition of this theory formulated via an extended Lagrangian in higher geometric quantization is in section 1 of
Further discussion is in section 5.7 of
A 1d Chern-Simons theory with target a cotangent bundle is discussed in
Last revised on July 17, 2024 at 12:58:48. See the history of this page for a list of all contributions to it.