Contents

Idea

By the general mechanism of ∞-Chern-Simons theory, every invariant polynomial of total degree 2 induces a 1-dimensional Chern-Simons-like theory.

Examples

For the first Chern class

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 ${\Omega }^1(\Sigma ,𝔲)$. After identifying $\Sigma \subset ℝ$ this may be identified with the space of $𝔲(n)$-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 a group character, on a coadjoint orbit

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.

For a symplectic Lie 0-algebroid

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 …

Related concepts

• higher dimensional Chern-Simons theory

  • 1d Chern-Simons theory

  • 2d Chern-Simons theory

  • 3d Chern-Simons theory

  • 4d Chern-Simons theory

  • 5d Chern-Simons theory

  • 6d Chern-Simons theory

  • 7d Chern-Simons theory

  • AKSZ sigma-models

  • string field theory

  • infinite-dimensional Chern-Simons theory

References

For the first Chern class

A discussion of 1d CS theory in the context of large $N$-gauge theory is in

• V.P. Nair, The Matrix Chern-Simons One-form as a Universal Chern-Simons Theory Nucl.Phys.B750:289-320,2006 (arXiv:hep-th/0605007)

An exposition of this theory formulated via an extended Lagrangian in higher geometric quantization is in section 1 of

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, A higher stacky perspective on Chern-Simons theory

Further discussion is in section 5.7 of

• Urs Schreiber, differential cohomology in a cohesive topos

For a symplectic Lie 0-algebroid

A 1d Chern-Simons theory with target a cotangent bundle is discussed in

• Ryan Grady, Owen Gwilliam, One-dimensional Chern-Simons theory and the $\hat{A}$ genus (arXiv:1110.3533)