equations. The reason behind these structures is that moduli of stable maps ($\ke=\infty$) admit forgetful lci morphisms $$\mathrm{Q}^\infty_{g,k+1}({W/\!\!/{{\bf G}}},\beta){\longrightarrow}\mathrm{Q}^\infty_{g,k}({W/\!\!/{{\bf G}}},\beta)$$ for which the virtual classes behave functorially. These morphisms may fail to exist for stability parameters $\ke\leq 1$. Even when they do exist, the virtual classes often are not compatible with the pull-back. In particular, for a general target ${W/\!\!/{{\bf G}}}$, the unit cohomology class $\one$ will be the unit for the CohFT only in the Gromov-Witten range of the stability parameter $\ke$. The appropriate versions of string and dilaton equations for all $\ke$ in the case of semi-positive triples $(W,{{\bf G}},\theta)$ are discussed later in the paper.

Equivariant theory and non-compact targets {#equivariant}
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Suppose that $W$ admits an additional action by an algebraic torus ${{{\bf {T}}}}\cong(\CC^*)^n$, which commutes with action of ${{\bf G}}$. There are induced actions on $[W/{{\bf G}}]$, on ${W/\!\!/{{\bf G}}}$, and on $W/_{\mathrm {aff}}{{\bf G}}$. The projective morphism ${W/\!\!/{{\bf G}}}{\longrightarrow}W/_{\mathrm {aff}}{{\bf G}}$ is ${{\bf {T}}}$-equivariant. Further, there are also induced ${{\bf {T}}}$-actions on the moduli spaces ${\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)}$ (and on the graph spaces recalled in §3 below).

We will always assume in this paper that the locus of ${{\bf {T}}}$-fixed points in the affine quotient 