formula, $$\label{equiv invariants}
\langle \delta_1\psi_1^{a_1},\dots,\delta_k\psi_k^{a_k}
\rangle^\ke_{g,k,\beta}:=\int_{[({\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)})^{{{\bf {T}}}}]^{\mathrm{vir}}}\frac{i^*(\prod_j  ev_j^*(\delta_j)\psi_j^{a_j})}{\mathrm{e}(N^{\mathrm{vir}})},$$ with $i:({\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)})^{{{\bf {T}}}} \hookrightarrow{\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)}$. Both the pairing and the invariants take values in the field $K$ (or in $\Lambda$, if we take the insertions from $H^*_{{{\bf {T}}},\mathrm{loc}}({W/\!\!/{{\bf G}}},\Lambda)$).

If ${W/\!\!/{{\bf G}}}$ is [*projective*]{} and we take all insertions in the non-localized equivariant cohomology $H^*_{{{\bf {T}}}}({W/\!\!/{{\bf G}}},\QQ)$, then the ${{\bf {T}}}$-equivariant invariants may be defined without localization and take values in the ring $\QQ[\lambda_1,\dots,\lambda_n]$. Upon specializing $\lambda_1=\dots=\lambda_n=0$ we recover the non-equivariant theory from the previous subsection.

The properness of the moduli spaces over the affine quotient implies immediately that the evaluation maps are proper, so the push-forward $(ev_i)_*$ is well-defined for all targets. The invariant may then be defined as the pairing $$\label{push-forward by ev}
\left{\langle}\delta_1, (ev_1)_*\left ([{\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)}]^{\mathrm{vir}}\cap \prod_{j=2}^k  ev_j^*(\delta_j)\prod_{j=1}^k\psi_j^{a_j}\right)\right {\rangle}.$$

Twisted theories
----------------

A ${{\bf G}}$-representation $E$ is called [*convex*]{} (respectively, [*concave*]{}) if the equivariant vector bundle $W\times E$ on $W$ is generated by ${{\bf G}}$-equivariant global sections (respectively, it has no non-zero ${{\bf G}}$-equivariant global sections). Vector bundles $\underline{E}:= W^s\times_{{\bf G}}E$ on ${W/\!\!/{{\bf G}}}$ induced by representations will be called [*homogeneous*]{}. We fix a $1$-dimensional torus ${\bf U}\cong \CC^*$, acting trivially on $W$ and by multiplication on 