Knowlpedia

Transgression form

A differential form whose exterior derivative is the difference of two characteristic forms coming from different connections

Let π:P→M\pi:P\to M be a with Lie algebra g\mathfrak g, and let ω0,ω1\omega_0,\omega_1 be two on PP with corresponding Ω0,Ω1∈Ω2(P;g)\Omega_0,\Omega_1\in\Omega^2(P;\mathfrak g).

Fix an integer n≥1n\ge1 and an Ad⁡\operatorname{Ad}-invariant symmetric multilinear map

p:g×⋯×g⏟n factors⟶K,K∈{R,C},p:\underbrace{\mathfrak g\times\cdots\times\mathfrak g}_{n\ \text{factors}}\longrightarrow \mathbb K,\qquad \mathbb K\in\{\mathbb R,\mathbb C\},

i.e. an invariant polynomial datum as used to build a . Define the affine path of connections

ωt:=ω0+t(ω1−ω0),t∈[0,1],\omega_t := \omega_0 + t(\omega_1-\omega_0),\qquad t\in[0,1],

and set η:=ω1−ω0∈Ω1(P;g)\eta:=\omega_1-\omega_0\in\Omega^1(P;\mathfrak g). (By , η\eta is horizontal and Ad⁡\operatorname{Ad}-equivariant.) Let Ωt\Omega_t be the curvature of ωt\omega_t.

The transgression form associated to pp and the pair (ω0,ω1)(\omega_0,\omega_1) is the (2n−1)(2n-1)-form Tp(ω0,ω1)T_p(\omega_0,\omega_1) on MM uniquely characterized by the requirement that its pullback to PP is

π∗Tp(ω0,ω1)  =  n∫01p ⁣(η∧Ωt n−1) dt,\pi^*T_p(\omega_0,\omega_1) \;=\; n\int_0^1 p\!\big(\eta\wedge \Omega_t^{\,n-1}\big)\,dt,

where Ωt n−1\Omega_t^{\,n-1} denotes the wedge product of (n−1)(n-1) copies of Ωt\Omega_t and p(η∧Ωtn−1)p(\eta\wedge \Omega_t^{n-1}) means the K\mathbb K-valued form obtained by feeding the g\mathfrak g-valued factors into pp.

Because the integrand is basic (horizontal and GG-invariant), Tp(ω0,ω1)T_p(\omega_0,\omega_1) is well-defined on the base.

Transgression identity

The exterior derivative satisfies the difference of the corresponding Chern--Weil forms is exact:

d Tp(ω0,ω1)=cw⁡p(ω1)−cw⁡p(ω0)on M,d\,T_p(\omega_0,\omega_1)=\operatorname{cw}_p(\omega_1)-\operatorname{cw}_p(\omega_0) \quad\text{on }M,

which is the content of the . Specializing ω0\omega_0 to a reference connection produces the usual .

Examples
  1. Degree 1 (linear invariant polynomial). For n=1n=1 and an Ad⁡\operatorname{Ad}-invariant linear functional p:g→Rp:\mathfrak g\to\mathbb R, the formula reduces to
    π∗Tp(ω0,ω1)=p(ω1−ω0),dTp=cw⁡p(ω1)−cw⁡p(ω0).\pi^*T_p(\omega_0,\omega_1)=p(\omega_1-\omega_0), \qquad dT_p = \operatorname{cw}_p(\omega_1)-\operatorname{cw}_p(\omega_0).
  1. Degree 2 on a trivial bundle (the usual Chern--Simons 3-form). On a trivial bundle and in a global gauge, a connection is represented by a g\mathfrak g-valued 1-form AA (see ). For p(X,Y)=tr⁡(XY)p(X,Y)=\operatorname{tr}(XY) (degree 22), using the product flat connection as reference (its local potential is A0=0A_0=0, while its principal connection form is not zero) gives the standard 3-form
    CS⁡(A)=tr⁡ ⁣(A∧dA+23A∧A∧A),\operatorname{CS}(A)=\operatorname{tr}\!\Big(A\wedge dA+\frac{2}{3}A\wedge A\wedge A\Big),
    with d CS⁡(A)=tr⁡(F∧F)d\,\operatorname{CS}(A)=\operatorname{tr}(F\wedge F), where F=dA+A∧AF=dA+A\wedge A is the .
  1. Abelian case. If GG is abelian (e.g. U(1)U(1)), then Ad⁡\operatorname{Ad} is trivial and A∧A=0A\wedge A=0. For the complex-valued polynomial p:iR↪Cp:i\mathbb R\hookrightarrow\mathbb C, the local expression between two U(1)U(1)-connections is T(A0,A1)=A1−A0T(A_0,A_1)=A_1-A_0; these differences glue to a global form, and dT=dA1−dA0dT=dA_1-dA_0.
References
  1. Hessel Posthuma, Notes on Chern–Simons Theory, §1.2, Theorem 1.10 and equation (1.3), p. 4. Author-hosted notes.