Chern–Simons form

Chern–Simons forms are transgression forms that measure how Chern–Weil forms change when the connection changes. They are a central tool in geometry and gauge theory, and they are instances of the general transgression form mechanism.

Let $\pi:P\to M$ be a principal G-bundle . Let $\omega_0$ and $\omega_1$ be two principal connections on $P$, with curvatures $\Omega_0$ and $\Omega_1$.

Fix an $\mathrm{Ad}$-invariant symmetric polynomial $P$ of degree $k$ on $\mathfrak{g}$ (as in the definition of a Chern–Weil form ).

Definition (relative Chern–Simons form)

Define a path of connections

and let $\Omega_t$ be its curvature. The Chern–Simons form associated to $P$ and the pair $(\omega_0,\omega_1)$ is the $(2k-1)$-form on $P$

Fundamental transgression identity

The Chern–Simons form satisfies

This is a concrete case of the transgression theorem .

Because $P(\Omega_1)$ and $P(\Omega_0)$ are basic (see Chern–Weil forms are basic ), their difference descends to the base $M$, and the identity implies that the difference of the corresponding base forms is exact.

A common special case

If one takes $\omega_0$ to be a fixed reference connection (often chosen locally or globally if available), then $\mathrm{CS}_P(\omega_0,\omega_1)$ is a primitive for the difference of Chern–Weil forms. In gauge theory language, it is a secondary characteristic form whose behavior under gauge transformations is subtle; see Chern–Simons gauge transformation behavior .

Examples

1. Degree 1 (abelian case) Take $G=U(1)$ and the degree-1 invariant polynomial $P(X)=\frac{i}{2\pi}X$. Then $P(\Omega)$ is (a multiple of) the curvature 2-form, and the Chern–Simons form for $(\omega_0,\omega_1)$ simplifies to a multiple of the difference of connection 1-forms:

   $$\mathrm{CS}_P(\omega_0,\omega_1)=\frac{i}{2\pi}\,(\omega_1-\omega_0), \qquad d\,\mathrm{CS}_P(\omega_0,\omega_1)=\frac{i}{2\pi}(\Omega_1-\Omega_0).$$

2. The classical 3-dimensional Chern–Simons form For a matrix Lie group (such as $SU(n)$), take $k=2$ and $P(X,Y)=\mathrm{tr}(XY)$. On a local trivialization, write the local connection 1-form as $A$ and local curvature as $F$ (compare local connection 1-form and local curvature 2-form ). Then a standard representative of the Chern–Simons 3-form is

   $$\mathrm{CS}_3(A)=\mathrm{tr}\!\Bigl(A\wedge dA+\frac{2}{3}A\wedge A\wedge A\Bigr),$$

   and it satisfies

   $$d\,\mathrm{CS}_3(A)=\mathrm{tr}(F\wedge F).$$

   This uses the local curvature convention fixed in the convention F = dA + A wedge A .

3. Comparing two connections on the same bundle If $\omega_0$ and $\omega_1$ are two different principal connections on the same principal bundle, then the transgression identity shows that the two Chern–Weil forms $P(\Omega_0)$ and $P(\Omega_1)$ represent the same de Rham cohomology class on $M$, because their difference is exact. The explicit primitive is the Chern–Simons form $\mathrm{CS}_P(\omega_0,\omega_1)$.