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We define an extended field theory in dimensions $1+1+1$, that takes the formof a `quasi 2-functor' with values in a strict 2-category$\widehat{\mathcal{H}am}$, defined as the `completion of a partial 2-category'$\mathcal{H}am$, notions which we define. Our construction extends Wehrheim andWoodward's Floer Field theory, and is inspired by Manolescu and Woodward'sconstruction of symplectic instanton homology. It can be seen, in dimensions$1+1$, as a real analog of a construction by Moore and Tachikawa.  Our construction is motivated by instanton gauge theory in dimensions 3 and4: we expect to promote $\widehat{\mathcal{H}am}$ to a (sort of) 3-category viaequivariant Lagrangian Floer homology, and extend our quasi 2-functor todimension 4, via equivariant analogues of Donaldson polynomials.

A two-category of Hamiltonian manifolds, and a (1+1+1) field theory