Chern-Simons, General relativity, and notations

Question

Consider the Einstein-Maxwell-dilaton theory with an additional Chern-Simons term as in this paper
begin{equation}
S = int d^4 sqrt{-g} left[ frac{1}{2} R - frac{1}{2} (partialvarphi)^2 - frac{tau(varphi)}{4} F^2 - V(varphi) right] -frac{1}{2} int theta(varphi)  Fwedge F.
end{equation}
with $F=d A$ being the field strength. Take for example the gauge equations:
begin{equation}
d(taustar F + theta F) =0.
end{equation}
I have two questions coming from my ignorance on the topic:

What does the operation $wedge$ mean and what is the explicit expression for $F wedge F$?
What does the operation $star$ mean and what is the explicit expression for $taustar F$?

mike stone · Answer

The $wedge$ in $Fwedge F$ is the wedge product of two copies of the curvature  2-form
$$
F= frac 12 F_{munu} dx^mu wedge dx^nu=dA.
$$
where
$$
A= A_mu dx^mu 
$$
and
$$
dA= d(A_nu dx^nu ) = (partial_mu A_nu) dx^muwedge dx^nu= frac 12 (partial_mu A_nu-partial_nu A_mu) dx^muwedge dx^nu.
$$
Thus
$$
Fwedge F= frac 14 F_{munu}F_{rhosigma}dx^mu wedge dx^nu wedge dx^rhowedge  dx^sigma = frac 14 epsilon^{munurhosigma} F_{munu}F_{rhosigma}d^4x 
$$
The $star$ is the Hodge star dual
and $tau$ is just multiplication by the function $tau(phi)$.
If you plan to work in an  area using these things, you need to learn the calculus of differential forms. Indeed, even if you do not plan such work, you should learn this calculus as it opens many doors.

Chern-Simons, General relativity, and notations

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