Lagrangian invariant but Action is gauge invariant

[creepypasta13]

Homework Statement

So I'm having some difficulty with my QFT assignment. I have to solve the following problem.

In three spacetime dimensions (two space plus one time) an antisymmetric Lorentz tensor
F^{\mu\nu} = -F^{\nu\mu} is equivalent to an axial Lorentz vector, F^{\mu\nu} = e^{\mu\nu\lambda}F_{\lambda}. Consequently, in 3D
one can have a massive photon despite unbroken gauge invariance of the electromagnetic
field A_{\mu}. Indeed, consider the following Lagrangian:

L = -(1/2)*F_{\lambda}F^{\lambda} + (m/2)*F_{\lambda}A^{\lambda} (6)

where

F_{\lambda}(x) = (1/2)*\epsilon_{\lambda\mu\nu}F^{\mu\nu} = \epsilon_{\lambda\mu\nu}\partial^{\mu}A^{\nu},

or in components, F_{0} = -B, F1 = +E^{2}, F_{2} = -E^{1}.

(a) Show that the action S = \intd^{3}x*L is gauge invariant (although the Lagrangian (6) is not invariant).
So I tried substituting A^{\lambda} -> A^{\lambda'} = A^{\lambda} + \partial^{\lambda}\Lambda
and F^{\lambda} -> F^{\lambda'} = \epsilon^{\lambda\mu\nu}\partial_{\mu}A_{\nu}'

then I obtained L' = L + (1/2)*[ F_{\lambda} \epsilon^{\lambda\mu\nu}\partial_{\mu} \partial_{\nu} \Lambda + some other terms]

What I don't understand is how these leftover terms would vanish after being integrated (to obtain S'), but they don't all vanish if they are not integrated (since L is not invariant). Is there some kind of special mathematical trick I have to use? I just don't see how I can integrate terms like \intd^{3}x F_{\lambda}\epsilon^{\lambda\mu\nu}\partial_{\mu}\partial_{\nu}\Lambda

 

[aready]

Well, I can see how Fλ ϵλμν∂μ ∂ν Λ would vanish. The \mu and \nu are symmetric wrt to exchange for the partial derivatives, but the indices of the \epsilon symbol are totally antisymmetric. Multiplying a symmetric tensor and an antisymmetry one gives zero always.