Regulated ADHM $N=2^{\ast }$ Instanton Series

Fix integers $N\ge 2$ and $k\ge 0$ and a real stability parameter $\zeta >0$. Let $V={\mathbb{C}}^k$ and $W={\mathbb{C}}^N$ with Hermitian structures, and let ${\mathcal{S}}_{k,N}^{\zeta }$ be the locus of quadruples $(B_1,B_2,I,J)$ with $B_i\in {\mathrm{End}}_{\mathbb{C}}(V)$, $I:W\to V$, and $J:V\to W$, satisfying $[B_1,B_2]+IJ=0$ and the condition that the vectors $B_{i_1}\cdots B_{i_m}I(w)$ span $V$ for all words of length $m\ge 0$ and $w\in W$. The group $GL(V)$ acts algebraically by $g\cdot (B_1,B_2,I,J)=(g{B}_1{g}^{-1},g{B}_2{g}^{-1},gI,J{g}^{-1})$. These are algebraic coordinates on the finite-dimensional complex data space; this formal quotient assigns no spacetime units to $B_i,I,J$. The parameter $\zeta$ labels the positive stability chamber and is not assigned a physical dimension here. Let ${\pi }_{k,N}:{\mathcal{S}}_{k,N}^{\zeta }\to X_{k,N}^{\zeta }$ be the positive-FI geometric quotient in the classical variety sense: ${\pi }_{k,N}$ is a surjective invariant morphism whose fibres are precisely the $GL(V)$-orbits; $U\subseteq X_{k,N}^{\zeta }$ is open exactly when ${\pi }_{k,N}^{-1}(U)$ is open; and for every such $U$ pullback identifies ${\mathcal{O}}_{X_{k,N}^{\zeta }}(U)$ with ${\mathcal{O}}_{{\mathcal{S}}_{k,N}^{\zeta }}({\pi }_{k,N}^{-1}(U){)}^{GL(V)}$. Under the stable positive-FI hypotheses, the supplied stable ADHM quotient result gives this quotient a separated smooth finite-type complex structure of dimension $2kN$ and a finite invariant affine word-slice atlas. The parameter $\zeta$ labels the positive stability chamber; this is not, by definition, the zero-FI regular quotient or the moduli space of anti-self-dual connections.

Let $T_{\mathbb{C}}=({\mathbb{C}}^{\times }{)}^2\times ({\mathbb{C}}^{\times }{)}^N$ act algebraically, and let $K=(S^1{)}^2\times (S^1{)}^N$ be its maximal compact subgroup. The action is $(t_1,t_2,h)\cdot (B_1,B_2,I,J)=(t_1{B}_1,t_2{B}_2,I{h}^{-1},t_1{t}_{2}hJ)$, where $h$ acts on $W$ with weights $e_{\alpha }^{-1}$. Thus $\mathbf{W}={\sum }_{\alpha =1}^N{e}_{\alpha }^{-1}$ and ${\mathbf{W}}^{\vee }={\sum }_{\alpha =1}^N{e}_{\alpha }$. Write ${ϵ}_i=c_1^K(t_i)$ and $a_{\alpha }=c_1^K(e_{\alpha })$; each has mass dimension one in units $c=\mathrm{\hslash }=1$. Let $q$ be a dimensionless formal charge-counting variable.

Assume the complexified torus action has a finite proper fixed locus $F_{k,N}$ on each quotient. Extend the torus to $T_m=T_{\mathbb{C}}\times {\mathbb{C}}_m^{\times }$ with maximal compact subgroup $K_m=K\times S_m^1$; the flavor factor acts trivially on $X_{k,N}^{\zeta }$. Use the product right Milnor model $E{K}_m$, and write $B{K}_m=E{K}_m\mathrm{/}{K}_m$ and $X_{k,N,K_m}^{\zeta }=E{K}_m{\times }_{K_m}{X}_{k,N}^{\zeta }$, with diagonal quotient relation $(e,x)\cdot t=(e\cdot t,t^{-1}\cdot x)$. Equivariant cohomology means $H_{T_m}^{\ast }(X_{k,N}^{\zeta };\mathbb{Q})=H^{\ast }(X_{k,N,K_m}^{\zeta };\mathbb{Q})$. The derivative action makes $T{X}_{k,N}^{\zeta }$ a $K_m$-equivariant complex vector bundle, and its Borel descent is $E{K}_m{\times }_{K_m}T{X}_{k,N}^{\zeta }\to X_{k,N,K_m}^{\zeta }$. Assume each component of this Borel base has CW type and that the descended bundle, including its tensor products below, is numerable; Chern classes are formed componentwise on a CW model and transported by homotopy invariance. For a numerable complex rank-$n$ bundle $E$ on a connected CW model, its Chern classes are the unique coefficients in the projective-bundle relation $x_E^n-c_1(E)x_E^{n-1}+\cdots +(-1{)}^n{c}_n(E)=0$, where $P(E)$ is the bundle of complex lines in the fibers and $x_E=c_1({\gamma }_E)$ for the tautological line ${\gamma }_E\to P(E)$. We use their rational images. For a character $\chi :T_m\to {\mathbb{C}}^{\times }$, let $L_{\chi }=E{K}_m{\times }_{K_m}{\mathbb{C}}_{\chi {\mathrm{\mid }}_{K_m}}\to B{K}_m$ be its associated line, with right equivalence $(e,z)\cdot t=(e\cdot t,\chi (t{)}^{-1}z)$, and write $u_{\chi }=c_1(L_{\chi })$. Put $\tilde{m}=u_{{\chi }_m}$ for the defining flavor character, of mass dimension one like ${ϵ}_i$ and $a_{\alpha }$, and let $L_{-1}=L_{{\chi }_m^{-1}}$; thus its additive weight is $-\tilde{m}$. At a fixed point with tangent characters ${\chi }_1,\dots ,{\chi }_{2kN}$ and additive weights $w_j=u_{{\chi }_j}$, pullback to $B{K}_m$ gives character lines $L_{{\chi }_j{\chi }_m^{-1}}$. The top Chern class of a rank-$n$ complex bundle equals the Euler class of its complex-oriented real bundle, and the Euler class is natural and multiplicative for ordered direct sums; for a character line its Euler class is its first Chern class. Therefore the top Chern restriction is ${\prod }_{j=1}^{2kN}(w_j-\tilde{m})$.

Set ${\mathcal{E}}_{\tilde{m}}=c_{2kN}^{T_m}(T{X}_{k,N}^{\zeta }\otimes L_{-1})\in H_{T_m}^{4kN}(X_{k,N}^{\zeta };\mathbb{Q})$, where $L_{-1}$ in the tensor product denotes pullback from $B{K}_m$. At a fixed point with tangent weights $w_1,\dots ,w_{2kN}$, its restriction is ${\prod }_{j=1}^{2kN}(w_j-\tilde{m})$ by the preceding construction. Let $S_{T_m}$ be generated by the first Chern classes of nontrivial integral characters of $T_m$, put $R_{T_m}=H^{\ast }(B{K}_m;\mathbb{Q})=\mathbb{Q}[{ϵ}_1,{ϵ}_2,a_1,\dots ,a_N,\tilde{m}]$, and let $K_{N,m}=\mathrm{Frac}(R_{T_m})$. When the character-localized Borel–Moore concentration map and proper fixed-locus trace define $p_{k,\ast }^{\mathrm{l}\mathrm{o}\mathrm{c}}$, set $I_{k,N}^{\mathrm{a}\mathrm{d}\mathrm{j}}(\tilde{m})=p_{k,\ast }^{\mathrm{l}\mathrm{o}\mathrm{c}}({\mathcal{E}}_{\tilde{m}}\cap [X_{k,N}^{\zeta }{]}_{K_m})\in S_{T_m}^{-1}{R}_{T_m}\subset K_{N,m}\mathrm{.}$ This is a localized pushforward, not an ordinary pushforward along a nonproper map. The concentration isomorphism and trace are required inputs and are not asserted to exist by this definition.

When these data are supplied for every $k\ge 0$, define the coefficientwise formal series $Z_{N,\mathrm{a}\mathrm{d}\mathrm{j}}^{\mathrm{l}\mathrm{o}\mathrm{c}}(q;\tilde{m})={\sum }_{k\ge 0}{q}^k{I}_{k,N}^{\mathrm{a}\mathrm{d}\mathrm{j}}(\tilde{m})\in K_{N,m}⟨\text{\hspace{0.17em}⁣}⟨q⟩\text{\hspace{0.17em}⁣}⟩\mathrm{.}$ It has no asserted analytic convergence. In the source's Euclidean convention the real mass $m_E$ has mass dimension one, $m=i{m}_E$, and the equivariant mass is $\tilde{m}=m+({ϵ}_1+{ϵ}_2)\mathrm{/}2$. For the adopted round $S_r^4$ model, $N=2$, ${ϵ}_1={ϵ}_2=r^{-1}$, and hence $\tilde{m}=i{m}_E+r^{-1}$. This mass shift is a source-qualified convention. The S4 postulate assumes a supersymmetry-compatible noncommutative regulator agreeing near each pole with the flat-space regulator; it does not construct that regulator or a continuum path-integral measure.

Positive-FI framed ADHM quotient and torus action

Classical geometric quotient conditions

Existence and geometry of the stable ADHM quotient

Normalized equivariant Borel–Moore groups

Localized proper-fixed-locus pushforward

Equivariant tangent bundle

Topological Chern class construction

Chern class naturality and Whitney sum

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

Source-qualified $N=2^{\ast }$ U(2) S4 model

Nekrasov, §1 equation (1.3), §§3.3–3.6

Nakajima and Yoshioka, abstract, §3.1 and §4

Nekrasov and Okounkov, §§2.2, 2.4, 3.1 and 6.1

Pestun, §1, §4.3, and §5 equations (5.11)–(5.12)