# Does a Chern-Simons term break the $F \rightarrow \star F$ symmetry?

When is the electro-magnetic duality $F \rightarrow \star F$ a symmetry of a theory? I know it holds for free Yang-Mills, but would for instance a Chern-Simons term break it or a coupling to matter? And is the duality a part of the Lorentz symmetry as they both mix electric and magnetic fields? So if a theory is Lorentz symmetric, is it also symmetric under the electro-magnetic duality?

• What dimensions are you working in? How do you produce your C-S form? Have you written down the Duality transformation explicitly and Lorentz-transformed both sides? As it is, you are tossing words around expecting a revealing essay in return. – Cosmas Zachos Jun 27 '18 at 14:33
• I am working in three dimensions with a $U(1)$ gauge group, therefore my Yang-Mills term looks like $F \wedge \star F$ and my chern-simons term is $F \wedge A$. To make my question as specific as possible; is the lagrangian $\frac{1}{e^2} F \wedge \star F + k \ F \wedge A$ still invariant under the electro-magnetic duality $F \rightarrow \star F$ – Graphite Jun 27 '18 at 15:10
• How many indices does your $\star F$ have in 3 dimensions? – Cosmas Zachos Jun 27 '18 at 15:16
• one, since $F$ is a two form and the hodge star maps it to 3-2 = 1 form. – Graphite Jun 27 '18 at 15:38
• Right ...so can you write that duality map explicitly in components in your question? – Cosmas Zachos Jun 27 '18 at 15:43