Explicit Rank-One One-Instanton ADHM Quotient

Let $V=W=\mathbb{C}$, let $k=N=1$, and let $\mathcal{S}$ be the scalar ADHM data $(b_1,b_2,i,j)\in {\mathbb{C}}^4$ satisfying $ij=0$ and stability $i\mathrm{\ne }0$. Let $GL(V)={\mathbb{C}}^{\times }$ act by $(b_1,b_2,i,j)\mapsto (b_1,b_2,\lambda i,{\lambda }^{-1}j)$. The geometric quotient here means a surjective invariant morphism $\pi :\mathcal{S}\to X$ whose fibres are the $GL(V)$-orbits, for which $U\subseteq X$ is open exactly when ${\pi }^{-1}(U)$ is open, and for which ${\mathcal{O}}_X(U)\cong {\mathcal{O}}_{\mathcal{S}}({\pi }^{-1}(U){)}^{GL(V)}$ for every open $U$. Let $T=({\mathbb{C}}^{\times }{)}^2\times {\mathbb{C}}_h^{\times }$ act by $(t_1,t_2,h)\cdot (b_1,b_2,i,j)=(t_1{b}_1,t_2{b}_2,h^{-1}i,t_1{t}_{2}hj)$, and extend it by a flavor torus ${\mathbb{C}}_m^{\times }$ acting trivially on $X$. The quotient is isomorphic to ${\mathbb{C}}^2$ with coordinates $b_1,b_2$; its fixed locus under $T_m$ is the origin and its tangent characters there are $t_1,t_2$, with additive weights ${ϵ}_1=c_1(t_1)$ and ${ϵ}_2=c_1(t_2)$.

Let $K_m=(S^1{)}^2\times S_h^1\times S_m^1$ and use the product right Milnor model $E{K}_m$. Write $X_{K_m}=E{K}_m{\times }_{K_m}X$ with the diagonal quotient relation, and define $H_{T_m}^{\ast }(X;\mathbb{Q})=H^{\ast }(X_{K_m};\mathbb{Q})$. The derivative action makes $TX$ a rank-two equivariant complex vector bundle; its Borel descent, tensored with the inverse-flavor line $L_{-1}=E{K}_m{\times }_{K_m}{\mathbb{C}}_{{\chi }_m^{-1}}$, is a rank-two complex bundle over $X_{K_m}$. Let $\tilde{m}=c_1(E{K}_m{\times }_{K_m}{\mathbb{C}}_{{\chi }_m})$. Here $c_i^{T_m}$ means the ordinary Chern class of this descended bundle, rationalized; for a numerable rank-$n$ bundle on a connected CW model, $c_i$ is defined by the unique projective-bundle relation $x_E^n-c_1(E)x_E^{n-1}+\cdots +(-1{)}^n{c}_n(E)=0$, where $x_E$ is the first Chern class of the tautological line on the projective bundle of fiber lines. The Borel base in this example is the total space of the rank-two bundle associated to the linear $T_m$-action on ${\mathbb{C}}^2$ over $B{K}_m=({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}{)}^4$; it deformation retracts to its zero section, and the descended bundle is numerable. At the origin the top Chern class restricts to $({ϵ}_1-\tilde{m})({ϵ}_2-\tilde{m})$, since top Chern equals the Euler class of the complex-oriented real bundle and the Euler class multiplies over the two character lines. Let $S_{T_m}$ be generated by first Chern classes of nontrivial integral characters, and use the character-localized proper-fixed-locus trace $p_{\ast }^{\mathrm{l}\mathrm{o}\mathrm{c}}=p_{F\ast }\circ (i_{\ast }{)}^{-1}$ when the localized fixed-inclusion map $i_{\ast }$ is invertible; here $F=\{0\}$ is proper. The complex-oriented normalized Borel–Moore fundamental class $[X{]}_{K_m}$ has degree $4$. If the localized pushforward of $c_2^{T_m}(TX\otimes L_{-1})\cap [X{]}_{K_m}$ is defined, then the charge-one coefficient is $I_{1,1}^{\mathrm{a}\mathrm{d}\mathrm{j}}(\tilde{m})=\frac{({ϵ}_1-\tilde{m})({ϵ}_2-\tilde{m})}{{ϵ}_1{ϵ}_2}\mathrm{.}$ At charge zero the quotient is a point with rank-zero insertion and coefficient $1$.

Classical geometric quotient conditions

Normalized equivariant Borel–Moore conventions

Proper-fixed-locus localization formula

Equivariant tangent bundle

Topological Chern class construction

Euler class by the Thom construction

Top Chern class equals the real Euler class

Euler class naturality and direct-sum product

Rational torus classifying ring and character weights