Virtual Localisation Formula for $ SL_{\eta} $-Oriented Theories

In this paper, we extend the Virtual Localization Formula of Levine to a wide class of motivic ring spectra, obtaining in particular a localization formula for virtual fundamental classes in Witt theory $\mathrm{KW}$. Applying standard tools of $\mathbb A^1$-intersection theory to any $SL_{\eta}$-oriented spectra $\mathrm A$, we obtain an additive presentation of $\mathrm A(BN)$, for $N$ the normaliser of the torus in $SL_2$. Then we establish an equivariant Atiyah-Bott localization theorem for $\mathrm A_N^{\mathrm{BM}}(X)$ and we conclude with the $N$-equivariant virtual localisation formula. Of independent interest, we also describe the ring structure of $\mathrm{KW}(BN)$.