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It's continuation of question: Abelian theory with confiment in 4d (Polyakov book)

  1. It's quite simple to construct theory of abelian 2-form with gauge transformation: $$ A_{[\mu\nu]} \to A_{[\mu\nu]} + \partial_{[\mu}\alpha_{\nu]} $$ Or in form language ($A=A_{[\mu\nu]}dx^\mu dx^\nu $ is two form): $$ A\to A +d\alpha $$ Field strength: $$ F_{[\mu\nu\rho]} = \partial_{[\mu}A_{\nu\rho]} $$ $$ F=dA $$ $$ S = \int d^4x \;F\wedge\star F = \int d^4x \;F^{[\mu\nu\rho]}F_{[\mu\nu\rho]} $$

  2. How to generalize this to non-abelian 2-form?

It is not clear even how define non-abelian transformation.

Qmechanic
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Nikita
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1 Answers1

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This is largely open, in the sense that there are multiple proposals for non-abelian 2-forms (and also, for higher forms).

One I like (that is not mine...) is the relatively early proposal of C. Hofman Nonabelian 2-forms. That paper is fairly self-contained too.