Framed U(N) Instanton Moduli and ADHM Data

Assume the Axiom of Choice. Let $N\ge 2$ and let $S^4={\mathbb{R}}^4\cup \{\mathrm{\infty }\}$ be the one-point compactification, oriented so the coordinates $x^1,x^2,x^3,x^4$ on ${\mathbb{R}}^4$ have $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge d{x}^4>0$. Give $S^4$ a smooth round metric in the conformal class of the Euclidean metric on ${\mathbb{R}}^4$. A framed $U(N)$ connection is a smooth unitary connection $A$ on a Hermitian rank-$N$ vector bundle $E\to S^4$, together with a chosen unitary frame $\mathrm{\Phi }:E_{\mathrm{\infty }}\to {\mathbb{C}}^N$ of its fiber at $\mathrm{\infty }$. Gauge transformations are smooth unitary bundle automorphisms that equal the identity in this frame at $\mathrm{\infty }$. The curvature is $F_A$, and the Hodge operator is defined by $\alpha \wedge {\ast }_g\beta =⟨\alpha ,\beta {⟩}_{g}d{\mathrm{v}\mathrm{o}\mathrm{l}}_g$ on two-forms. Set $F_A^{+}=(F_A+{\ast }_g{F}_A)\mathrm{/}2$ and $F_A^{-}=(F_A-{\ast }_g{F}_A)\mathrm{/}2$; the anti-self-dual condition is $F_A^{+}=0$, and the self-dual condition is $F_A^{-}=0$.

Let $\mathfrak{u}(N)$ be the anti-Hermitian matrices and put $B_N(X,Y)=-{\mathrm{tr}}_N(XY)$. In the trace convention of the library, the charge of $A$ is $q_N(A)=(8{\pi }^2{)}^{-1}{\int }_{S^4}{B}_N(F_A\wedge F_A)$, and the positive anti-self-dual instanton number is $k(A)=-q_N(A)$. For $k\in {\mathbb{Z}}_{\ge 0}$, the framed anti-self-dual charge-$k$ moduli set ${\mathcal{I}}_{k,N}^{\mathrm{A}\mathrm{S}\mathrm{D}}$ consists of framed pairs $(E,A)$ with $F_A^{+}=0$ and $k(A)=k$, modulo the stated based gauge transformations. The integer-charge condition is part of the selected bundle sector; no analytic existence or uniqueness assertion is included in this definition. The trace and Hodge conventions agree with the supplied compact-four-manifold definition of $q_B$; its special $SU(2)$ integrality clause is not used to assert $U(N)$ integrality.

For $k\ge 0$, take Hermitian spaces $V={\mathbb{C}}^k$ and $W={\mathbb{C}}^N$. The ADHM data are $B_1,B_2\in {\mathrm{End}}_{\mathbb{C}}(V),I\in {\mathrm{Hom}}_{\mathbb{C}}(W,V),J\in {\mathrm{Hom}}_{\mathbb{C}}(V,W)\mathrm{.}$ Their data space is ${\mathcal{R}}_{k,N}={\mathrm{End}}_{\mathbb{C}}(V{)}^2\oplus {\mathrm{Hom}}_{\mathbb{C}}(W,V)\oplus {\mathrm{Hom}}_{\mathbb{C}}(V,W)$, and their complex and real moment-map expressions are ${\mu }_{\mathbb{C}}=[B_1,B_2]+IJ\in {\mathrm{End}}_{\mathbb{C}}(V),{\mu }_{\mathbb{R}}=[B_1,B_1^{†}]+[B_2,B_2^{†}]+I{I}^{†}-J^{†}J\in {\mathrm{End}}_{\mathbb{C}}(V)\mathrm{.}$ Here $[X,Y]=XY-YX$, and $†$ is the Hermitian adjoint. The two displayed formulas define maps ${\mu }_{\mathbb{C}}:{\mathcal{R}}_{k,N}\to {\mathrm{End}}_{\mathbb{C}}(V)$ and ${\mu }_{\mathbb{R}}:{\mathcal{R}}_{k,N}\to \mathrm{Herm}(V)$, where ${\mathcal{R}}_{k,N}$ is the stated product of four Hom spaces and $\mathrm{Herm}(V)$ is the real vector space of Hermitian endomorphisms. They are the complex and real ADHM moment-map expressions, with these formulas fixing their normalization. A datum is stable when ${\mathrm{span}}_{\mathbb{C}}\{B_{i_1}\cdots B_{i_m}I(w):m\ge 0,i_j\in \{1,2\},w\in W\}=V,$ where the word of length zero is the identity. Equivalently, no proper $B_1,B_2$-invariant subspace of $V$ contains $\mathrm{im}I$. The group $GL(V)$ consists of invertible complex-linear endomorphisms, and $U(V)$ is its subgroup preserving the chosen Hermitian inner product. The group $GL(V)$ acts by $g\cdot (B_1,B_2,I,J)=(g{B}_1{g}^{-1},g{B}_2{g}^{-1},gI,J{g}^{-1})$. For $\zeta >0$, the stable complex ADHM quotient ${\mathcal{M}}_{k,N}^{\mathrm{G}\mathrm{I}\mathrm{T},\zeta }$ is the geometric quotient of $\{{\mu }_{\mathbb{C}}=0,\text{ stable}\}$ by $GL(V)$ in the positive stability chamber; $\zeta$ labels that chamber and is not an additional coordinate on the complex equation set.

For a real parameter $\zeta \ge 0$, the real moment-map presentation is the quotient of $\{{\mu }_{\mathbb{C}}=0,{\mu }_{\mathbb{R}}=\zeta {\mathrm{i}\mathrm{d}}_V,\text{ stable}\}$ by $U(V)$. At $\zeta >0$ this is called the resolved or positive-FI ADHM quotient; at $\zeta =0$ it is the zero-FI real moment-map quotient. The equations and quotient are definitions; this item does not assert that the real moment-map quotient is isomorphic to the stable complex quotient or to an instanton moduli space.

Write ${\mathbb{C}\mathbb{P}}^2$ for the complex projective plane, ${\mathrm{\ell }}_{\mathrm{\infty }}=\{[0:z_1:z_2]\}$ for its line at infinity, and ${\mathcal{O}}_{{\mathrm{\ell }}_{\mathrm{\infty }}}$ for the sheaf of holomorphic functions on that line. The affine chart ${\mathbb{C}\mathbb{P}}^2\setminus {\mathrm{\ell }}_{\mathrm{\infty }}$ is ${\mathbb{C}}^2$; projective compactification adds the whole line ${\mathrm{\ell }}_{\mathrm{\infty }}$, unlike the one-point compactification $S^4$. Let $\mathcal{O}(1)$ be the hyperplane line bundle and $\mathcal{O}(m)=\mathcal{O}(1{)}^{\otimes m}$ for $m\ge 0$. A coherent sheaf is a sheaf of ${\mathcal{O}}_{{\mathbb{C}\mathbb{P}}^2}$-modules that is locally finitely presented; it is torsion-free when no nonzero local section is annihilated by a nonzero local function. Its rank is the dimension of its generic fiber over the function field. Define its Hilbert polynomial by $P_E(m)={\sum }_{i=0}^2(-1{)}^i{\dim }_{\mathbb{C}}{H}^i({\mathbb{C}\mathbb{P}}^2,E\otimes \mathcal{O}(m))$ for all sufficiently large integers $m$. The framed Gieseker moduli space ${\mathcal{G}}_{k,N}$ is the moduli of pairs $(E,\mathrm{\Phi })$ with $E$ a rank-$N$ torsion-free coherent sheaf, $P_E(m)=N\left(\genfrac{}{}{0px}{}{m+2}{2}\right)-k$, and $\mathrm{\Phi }:E{\mathrm{\mid }}_{{\mathrm{\ell }}_{\mathrm{\infty }}}\cong {\mathcal{O}}_{{\mathrm{\ell }}_{\mathrm{\infty }}}^{\oplus N}$. Its locally-free locus is distinguished from the boundary of non-locally-free sheaves. The Uhlenbeck set-theoretic compactification is the disjoint union of strata ${\mathcal{I}}_{k-\mathrm{\ell },N}^{\mathrm{A}\mathrm{S}\mathrm{D}}\times {\mathrm{Sym}}^{\mathrm{\ell }}({\mathbb{R}}^4)$ for $0\le \mathrm{\ell }\le k$, where ${\mathrm{Sym}}^{\mathrm{\ell }}({\mathbb{R}}^4)=({\mathbb{R}}^4{)}^{\mathrm{\ell }}\mathrm{/}{S}_{\mathrm{\ell }}$ records ideal point defects. The Gieseker space and the Uhlenbeck space are distinct: the former retains torsion-free sheaf data at the small-instanton boundary, while the latter records only ideal points. No compactness of either full space is asserted here.

Write $z_1=x^1+i{x}^2$ and $z_2=x^3+i{x}^4$. The compact torus $T=(S^1{)}^2\times T_G$, with $T_G\subset U(W)$ the diagonal framing torus, acts on ADHM data by $(t_1,t_2,h)\cdot (B_1,B_2,I,J)=(t_1{B}_1,t_2{B}_2,I{h}^{-1},t_1{t}_{2}hJ)\mathrm{.}$ This convention makes ${\mu }_{\mathbb{C}}$ transform with character $t_1{t}_2$ and leaves ${\mu }_{\mathbb{R}}$ invariant. Let $e_{\alpha }$ denote the character of the dual of the $\alpha$th framing line, so the framing representation has character $\mathbf{W}={\sum }_{\alpha =1}^N{e}_{\alpha }^{-1}$ and its dual has character ${\mathbf{W}}^{\vee }={\sum }_{\alpha =1}^N{e}_{\alpha }$. For the rotation characters write $t_1,t_2$, and write ${ϵ}_i=c_1^T(t_i)$ and $a_{\alpha }=c_1^T(e_{\alpha })$ for their additive equivariant weights.

Nekrasov, §§3.1 and 3.3, equations (3.1)–(3.9)

Nakajima and Yoshioka, §3.1, PDF pp. 17–19

Trace-normalized charge and the self-dual/anti-self-dual sign

Full Axiom of Choice