- #1

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## Main Question or Discussion Point

The electromagnetic action in the language of differential geometry is given by

##\displaystyle{S \sim \int F \wedge \star F},##

where ##A## is the one-form potential and ##F={\rm d}A## is the two-form field strength.

At the extremum of the action ##S##, ##F## is constrained by ##{\rm d}F=0## and ##{\rm d}\star F=0##.

Now, generalise the above action to

##\displaystyle{S \sim \int H \wedge \star H}##

where ##B## is the two-form potential and ##H={\rm d}B## is the three-form field strength.

At the extremum of the action ##S##, ##H## is constrained by ##{\rm d}H=0## and ##{\rm d}\star H=0##.

Are there any qualitative differences between the two sets of equations in ##d+1##-dimensions?

##\displaystyle{S \sim \int F \wedge \star F},##

where ##A## is the one-form potential and ##F={\rm d}A## is the two-form field strength.

At the extremum of the action ##S##, ##F## is constrained by ##{\rm d}F=0## and ##{\rm d}\star F=0##.

Now, generalise the above action to

##\displaystyle{S \sim \int H \wedge \star H}##

where ##B## is the two-form potential and ##H={\rm d}B## is the three-form field strength.

At the extremum of the action ##S##, ##H## is constrained by ##{\rm d}H=0## and ##{\rm d}\star H=0##.

Are there any qualitative differences between the two sets of equations in ##d+1##-dimensions?