# Lagrangian invariant but Action is gauge invariant

## 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 = $\int$d$^{3}$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 $\int$d$^{3}$x F$_{\lambda}$$\epsilon$$^{\lambda\mu\nu}$$\partial$$_{\mu}$$\partial$$_{\nu}$$\Lambda$