I'll talk about cylindrical contact homology, which is a relatively well-established part of SFT. It's a variant of Hamiltonian Floer homology for contact manifolds. Let $\alpha$ be a contact 1-form on a closed manifold $V$, and let $A\colon LV\to \mathbb{R}$, $\gamma\mapsto \int_{S^1}{\gamma^*\alpha}$ be the action functional on its loopspace. It's invariant under rotation of loops, hence not a Morse function. The critical points are the 1-periodic orbits of the Reeb vector field.

Here are three things we could try to construct:

(i) The Floer cohomology of $A$. Each geometric Reeb orbit with multiplicity contributes $H^*(S^1)$ to the cochain complex.

(ii) The $S^1$-equivariant Floer cohomology of $A$. Each geometric Reeb orbit with multiplicity contributes $H^\ast_{S^1}(S^1)$ to the complex (the $S^1$ action on itself depends on the multiplicity).

(iii) The Floer homology of $A$ on $LV/S^1$ (over $\mathbb{Q}$). Each geometric Reeb orbit with multiplicity contributes $\mathbb{Q}$ to the complex.

Which of these things work? None of them, without substantial modification. All of them, with modifications. They are called (i) symplectic cohomology; (ii) circle-equivariant symplectic cohomology; (iii) cylindrical contact homology. In each case, the differential essentially counts pseudo-holomorphic maps $S^1\times \mathbb{R}\to V\times \mathbb{R}$, asymptotic to periodic Reeb orbits, but with subtle differences.

Since (iii) is a quotient construction we have to allow loop-rotation; so we mark a standard point on $S^1\times -\infty$, an arbitrary point on $S^1\times +\infty$, and insist that these markers map to chosen points on the Reeb orbits. In (i), we would use the standard marker also on $S^1\times +\infty$; allowing it to vary defines a loop-rotation (BV) operator on symplectic cohomology.

See Bourgeois-Oancea's recent Inventiones paper for info on the relationship between these constructions.