nLab D=1 Chern-Simons theory

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

$$tr : \mathfrak{u}(n) \to \mathbb{R} \,.$$

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

$$S_{CS}(\phi) = \int_\Sigma tr(\phi) \,.$$

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

  • D=1 Chern-Simons theory

  • D=2 Chern-Simons theory

  • D=3 Chern-Simons theory

  • D=4 Chern-Simons theory

  • D=5 Chern-Simons theory

  • D=6 Chern-Simons theory

  • D=7 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)