Zero-FI regular ADHM and framed twistor data

Fix integers $N\ge 2$ and $k\ge 0$. Let $V={\mathbb{C}}^k$ and $W={\mathbb{C}}^N$ with their standard Hermitian inner products. A raw framed ADHM datum is $x=(B_1,B_2,I,J)$ with $B_i\in {\mathrm{End}}_{\mathbb{C}}(V)$, $I:W\to V$, and $J:V\to W$. Its complex and real moment expressions are

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

The datum is stable when the vectors $B_{i_1}\cdots B_{i_m}I(w)$, for $m\ge 0$, $i_j\in \{1,2\}$, and $w\in W$, span $V$. It is costable when no nonzero subspace of $V$ is invariant under both $B_1,B_2$ and contained in $\ker J$. It is regular when it is both stable and costable. The group $GL(V)$ acts by $g\cdot x=(g{B}_1{g}^{-1},g{B}_2{g}^{-1},gI,J{g}^{-1})$ and its unitary subgroup is $U(V)$. The regular zero-FI real quotient is the orbit set

${\mathcal{A}}_{k,N}^{\mathrm{r}\mathrm{e}\mathrm{g}}=\{x:{\mu }_{\mathbb{C}}(x)=0,{\mu }_{\mathbb{R}}(x)=0,x\text{ stable and costable}\}\mathrm{/}U(V)\mathrm{.}$

Let $S^4=\mathbb{H}{\mathbb{P}}^1$ have the unit round metric and the orientation whose restriction to the stereographic coordinates $z_1=x^0+i{x}^1$, $z_2=x^2+i{x}^3$ on ${\mathbb{R}}^4=S^4\setminus \{\mathrm{\infty }\}$ is $d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$. For a smooth right principal $U(N)$-bundle $P\to S^4$, let $E=P{\times }_{U(N)}{\mathbb{C}}^N$ be its Hermitian associated vector bundle, let $A$ be a smooth unitary connection, and let $\mathrm{\Phi }:E{\mathrm{\mid }}_{\mathrm{\infty }}\to {\mathbb{C}}^N$ be a unitary framing. Write $F_A$ for the anti-Hermitian curvature matrix and define

$q_N(P,A)=-\frac{1}{8{\pi }^2}{\int }_{S^4}{\mathrm{tr}}_N(F_A\wedge F_A)\mathrm{.}$

The anti-self-dual equation is $F_A^{+}=(F_A+{\ast }_g{F}_A)\mathrm{/}2=0$. For integer $k\ge 0$, the framed ASD sector ${\mathcal{M}}_{k,N}^{\mathrm{A}\mathrm{S}\mathrm{D}}$ is the set of such framed connections satisfying $F_A^{+}=0$ and $q_N(P,A)=-k$, modulo smooth unitary gauge transformations whose value at $\mathrm{\infty }$ is the identity in the framing. This definition alone does not assert that every sector exists or that $q_N$ is integral; the exact $U(N)$ charge normalization is a separate lemma below.

Identify ${\mathbb{H}}^2$ with ${\mathbb{C}}^4$ using a fixed right quaternionic structure $j$ with $j^2=-1$. The twistor space is $Z=\mathbb{C}{\mathbb{P}}^3={\mathbb{P}}_{\mathbb{C}}({\mathbb{H}}^2)$, the projection $\pi :Z\to \mathbb{H}{\mathbb{P}}^1=S^4$ sends a complex line $\mathrm{\ell }$ to the quaternionic line $\mathrm{\ell }+j\mathrm{\ell }$, and the antiholomorphic involution is $\sigma ([v])=[jv]$. A real twistor line is a fibre $L_x={\pi }^{-1}(x)\cong \mathbb{C}{\mathbb{P}}^1$; $\sigma$ restricts to the antipodal map on each such line. Fix homogeneous coordinates $[{\lambda }_1:{\lambda }_2:{\mu }_1:{\mu }_2]$ so $\sigma$ is induced by $({\lambda }_1,{\lambda }_2,{\mu }_1,{\mu }_2)\mapsto (-{\bar{\lambda }}_2,{\bar{\lambda }}_1,-{\bar{\mu }}_2,{\bar{\mu }}_1)$. The finite real line over $z$ has equations ${\mu }_1=z_1{\lambda }_1-{\bar{z}}_2{\lambda }_2$, ${\mu }_2=z_2{\lambda }_1+{\bar{z}}_1{\lambda }_2$; the line at infinity is ${\lambda }_1={\lambda }_2=0$. These equations fix the quaternionic identification and its orientation. In the chart ${\lambda }_1=1,{\lambda }_2=0$, vertical complex orientation followed by the positive orientation of $(z_1,z_2)$ is the complex orientation of $Z$.

For the spinor convention, use the standard spin structure on $S^4$ and $\mathrm{Spin}(4)\cong SU(2{)}_{+}\times SU(2{)}_{-}$. Let $S^{+},S^{-}$ be the associated rank-two complex spinor bundles, so $T_{\mathbb{C}}{S}^4\cong S^{+}\otimes S^{-}$. The twistor fibre at $x$ is $\mathbb{P}(S_x^{+})$, and the complex two-plane associated with $\mathrm{\ell }\in \mathbb{P}(S_x^{+})$ is $\mathrm{\ell }\otimes S_x^{-}$. Use the Levi-Civita connection of the round metric for the horizontal distribution of $\pi$; the horizontal $(0,1)$ subspace at $[\mathrm{\ell }]\in L_x$ projects to this complex alpha-plane. The displayed orientation is the twistor orientation: the vertical complex orientation followed by the quotient orientation on $S^4$ agrees with the complex orientation on $Z$. It is chosen so the curvature component that vanishes on every such plane is precisely $F^{+}=(F+{\ast }_{g}F)\mathrm{/}2$. In spinor components, $F_{AB}=F_{(AB)}$ denotes the symmetric spinor corresponding to $F^{+}$ under ${\mathrm{\Lambda }}_{+}^2{T}_x^{\ast }{S}^4\otimes \mathbb{C}\cong {\mathrm{Sym}}^2(S_x^{+}{)}^{\ast }$; restriction to the plane is $F_{AB}{\pi }^A{\pi }^B$, which vanishes for every $[\pi ]\in \mathbb{P}(S_x^{+})$ exactly when $F^{+}=0$.

A linear monad on $Z$ is a complex of holomorphic bundles

${\mathcal{O}}_Z(-1)\otimes U\stackrel{a}{\to }{\mathcal{O}}_Z\otimes K\stackrel{b}{\to }{\mathcal{O}}_Z(1)\otimes U^{\mathrm{\prime }},$

where $U,U^{\mathrm{\prime }},K$ are finite-dimensional complex vector spaces, $b\circ a=0$, $a$ is injective as a sheaf map, $b$ is surjective, and the cohomology sheaf $\ker b\mathrm{/}\mathrm{im}a$ is locally free. It is framed when that cohomology has a specified unitary trivialization on $L_{\mathrm{\infty }}$. For a holomorphic bundle $\mathcal{E}$ on $Z$ that is holomorphically trivial on every real line, a positive Hermitian Ward reality datum is a holomorphic nondegenerate pairing $b:\mathcal{E}\otimes {\sigma }^{\ast }\bar{\mathcal{E}}\to {\mathcal{O}}_Z$ satisfying $b_{\sigma z}(v,u)=\overline{b_z(u,v)}$. Here $b_z$ is complex linear in $u\in {\mathcal{E}}_z$ and conjugate linear in $v\in {\mathcal{E}}_{\sigma z}$; the conjugate pullback is holomorphic because $\sigma$ is antiholomorphic. For holomorphic sections $s,t$ on $L_x$, the function $b_z(s(z),t(\sigma z))$ is holomorphic on the compact line, hence constant. It defines $h_x(s,t)$; require this Hermitian form to be positive definite for every $x$. The metric on each fibre of $\mathcal{E}{\mathrm{\mid }}_{L_x}$ is transported from this form by holomorphic evaluation. Thus positive forms are induced by the pairing, not independent arbitrary choices. A framed Ward bundle includes a unitary framing on $L_{\mathrm{\infty }}$ for this induced form. Smoothness and parallelness of the induced section-bundle metric are conclusions of the Ward construction.

The compact torus $T=(S^1{)}^2\times (S^1{)}^N$ acts on ${\mathbb{R}}^4\cong {\mathbb{C}}^2$ by $(z_1,z_2)\mapsto (t_1{z}_1,t_2{z}_2)$ and on $W$ by $h=\mathrm{diag}(e_1^{-1},\dots ,e_N^{-1})$. Its raw ADHM action is

$(t_1,t_2,h)\cdot (B_1,B_2,I,J)=(t_1{B}_1,t_2{B}_2,I{h}^{-1},t_1{t}_{2}hJ)\mathrm{.}$

On the framed ASD sector, the same torus acts by inverse pullback under the indicated rotation of $S^4$ and by postcomposition of the framing with $h$. On $Z$, the rotations act by $[{\lambda }_1:{\lambda }_2:{\mu }_1:{\mu }_2]\mapsto [{\lambda }_1:t_1{t}_2{\lambda }_2:t_1{\mu }_1:t_2{\mu }_2]$, equivalently the projective spin lift to $\mathbb{P}(S^{+})$; the central sign in the spin double cover acts trivially after projectivization, and the induced action commutes with $\sigma$. The framing torus acts on the framed monad cohomology by the same diagonal representation.

ADHM linear data, stability, costability and FI levels

Trace-normalized instanton number and self-dual sectors

The Axiom of Choice

For each $x\in S^4$, the fibre line $L_x$ is included in $Z$ and projects to $x$. The square commutes because the projection is constant on a fibre; the diagram records the chosen twistor-fibre convention.