ADHM linear data, stability, costability and FI levels

Fix integers $N\ge 1$ and $k\ge 0$ and equip $V={\mathbb{C}}^k$ and $W={\mathbb{C}}^N$ with the standard Hermitian inner products $⟨u,v⟩={\sum }_j\overline{u_j}{v}_j$, which are conjugate-linear in the first variable. For a linear map $T:E\to F$ between these spaces, its Hermitian adjoint $T^{†}:F\to E$ is characterized by $⟨Tu,v{⟩}_F=⟨u,T^{†}v{⟩}_E$; in the standard bases its matrix is the conjugate transpose. For $A,B\in {\mathrm{End}}_{\mathbb{C}}(V)$, define their commutator by $[A,B]:=AB-BA$. Let

${\mathcal{D}}_{k,N}={\mathrm{End}}_{\mathbb{C}}(V{)}^2\times {\mathrm{Hom}}_{\mathbb{C}}(W,V)\times {\mathrm{Hom}}_{\mathbb{C}}(V,W),\mathrm{Herm}(V)=\{H\in {\mathrm{End}}_{\mathbb{C}}(V):H^{†}=H\}\mathrm{.}$

Here $\mathrm{Herm}(V)$ is regarded as a real vector space. A framed ADHM linear datum is $x=(B_1,B_2,I,J)\in {\mathcal{D}}_{k,N}$. The complex and real moment maps are the maps ${\mu }_{\mathbb{C}}:{\mathcal{D}}_{k,N}\to {\mathrm{End}}_{\mathbb{C}}(V)$ and ${\mu }_{\mathbb{R}}:{\mathcal{D}}_{k,N}\to \mathrm{Herm}(V)$ defined by

${\mu }_{\mathbb{C}}(x)=[B_1,B_2]+IJ,{\mu }_{\mathbb{R}}(x)=[B_1,B_1^{†}]+[B_2,B_2^{†}]+I{I}^{†}-J^{†}J\mathrm{.}$

The datum is stable if the vectors $B_{i_1}\cdots B_{i_m}I(w)$, with $m\ge 0$, $i_j\in \{1,2\}$ and $w\in W$, span $V$. It is costable if there is no nonzero subspace of $V$ invariant under both $B_i$ and contained in $\ker J$. It is regular if it is stable and costable. The group $GL(V)$ acts by $g\cdot x=(g{B}_1{g}^{-1},g{B}_2{g}^{-1},gI,J{g}^{-1})$. Its unitary subgroup is $U(V)$.

Write $X^s=\{x:{\mu }_{\mathbb{C}}(x)=0,x\text{ stable}\}$, $X^{\mathrm{r}\mathrm{e}\mathrm{g}}=\{x\in X^s:x\text{ costable}\}$ and, for real $\zeta \ge 0$, $X_{\zeta }^U=\{x\in X^s:{\mu }_{\mathbb{R}}(x)=\zeta I_V\}$. Denote the corresponding orbit sets by $X^s\mathrm{/}GL(V)$, $X^{\mathrm{r}\mathrm{e}\mathrm{g}}\mathrm{/}GL(V)$ and $X_{\zeta }^U\mathrm{/}U(V)$, with quotient topologies when discussing their analytic spaces. A word frame consists of $k$ columns obtained by fixed words in $B_1,B_2$ applied to fixed standard basis vectors of $W$.

For $T_{\mathbb{C}}=({\mathbb{C}}^{\ast }{)}^2\times ({\mathbb{C}}^{\ast }{)}^N$, write $t_1,t_2,e_1,\dots ,e_N$ for the coordinate characters and put $h=\mathrm{diag}(e_1^{-1},\dots ,e_N^{-1})$. Its action is $(t_1,t_2,h)\cdot x=(t_1{B}_1,t_2{B}_2,I{h}^{-1},t_1{t}_{2}hJ)$. The compact torus is its subgroup on which all characters have modulus one. The framing representation thus has character ${\sum }_{\alpha }{e}_{\alpha }^{-1}$ and its dual has character ${\sum }_{\alpha }{e}_{\alpha }$.

These are finite-dimensional matrix definitions. They do not assert a real/complex quotient correspondence, a sheaf or ASD identification, or a noncompact pushforward. In particular, positive-FI stability and zero-FI regularity are distinct conditions; neither an endpoint identification nor an ordinary ASD interpretation follows from this definition.

Nekrasov, §§3.1–3.3: ADHM data and FI moment-map conventions

Nakajima–Yoshioka, §3.1: framed-sheaf and regular-locus boundary