nLab
1-dimensional Chern-Simons theory

Context

\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

Ingredients

Definition

Examples

Quantum field theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

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

tr:𝔲(n). 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 Σ\Sigma is the groupoid of Lie algebra valued 1-forms Ω 1(Σ,𝔲)\Omega^1(\Sigma, \mathfrak{u}). After identifying Σ\Sigma \subset \mathbb{R} this may be identified with the space of 𝔲(n)\mathfrak{u}(n)-valued functions.

The action functional is simply the trace operation

S CS(ϕ)= Σtr(ϕ). 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.

For a group character, on a coadjoint orbit

For GG a suitable Lie group (compact, semi-simple and simply connected) the Wilson loops of GG-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=0n = 0 induces a 1d Chern-Simons theory whose Chern-Simons form is a Liouville form of the symplectic form.

This case is discussed in …

References

For the first Chern class

A discussion of 1d CS theory in the context of large NN-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

Further discussion is in section 5.7 of

For a symplectic Lie 0-algebroid

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

Revised on January 4, 2013 04:32:50 by Urs Schreiber (89.204.135.106)