# Introduction to instanton counting Note prepared for [Geometry of instantons and quivers](https://www.mathconf.org/giq2026), 22-26 June, 2026, Lille ## Abstract The aim of these lectures is to explain basic notions of instantons in relation to various mathematical concepts in algebraic geometry, representation theory, and enumerative geometry. Here is the plan of four lectures. 1. Yang-Mills theory in four dimensions and instantons We start with a quick review of Yang-Mills (YM) theory and then introduce the instanton with emphasis on its motivation and also the role of Anti-Self-Dual (ASD) connections. 2. ADHM construction of instantons on $\mathbb{S^4}$ We discuss the Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction of instantons on $\mathbb{S^4}$. We see the construction of ASD connections and computation of the 2nd Chern class integral. We will briefly mention a relation to the Atiyah-Singer index theorem and the twistor. 3. Instanton moduli space We introduce the instanton moduli space based on the ADHM construction, discussing the equivariant action, GIT quotient, and its relation to the Hilbert scheme of points on $\mathbb{C}^2$ (and also its higher-rank version). We will mention a general framework of Nakajima's quiver variety. 4. Equivariant integral and localization formula We discuss general ideas to compute the (equivariant) integral over the instanton moduli space, based on the equivariant localization formula, analysis of fixed points under the torus action, etc. We derive Nekrasov's formula of the instanton partition function (equivariant volume of the instanton moduli space) and discuss its higher-dimensional analogues yielding enumerative invariants of algebraic varieties in three and four dimensions, e.g., Donaldson-Thomas invariants and Pandharipande-Thomas invariants. #### Main references - Donaldson, Kronheimer, *[The Geometry of Four-manifolds](https://books.google.fr/books/about/The_Geometry_of_Four_manifolds.html?id=LbHmMtrebi4C)* - Freed, Uhlenbeck, *[Instantons and Four-Manifolds](https://doi.org/10.1007/978-1-4613-9703-8)* - Blaine Lawson, Michelsohn, *[Spin geometry](https://doi.org/10.1515/9781400883912)* - Nakajima, *[Lectures on Hilbert Schemes of Points on Surfaces](https://doi.org/10.1090/ulect/018)* ## Lecture 1 ### Setup Let $X$ be a closed oriented Riemannian four-manifold, $P \to X$ be a principal $G$-bundle with the structure group $G$. We denote the corresponding Lie algebra by $\mathfrak{g} = \operatorname{Lie} G$. Let $\operatorname{Ad}: G \to \operatorname{Aut}(\mathfrak{g}) \subset \mathrm{GL}(\mathfrak{g})$. We define an adjoint bundle $\operatorname{ad}(P) = P \times_{\operatorname{Ad}} \mathfrak{g}$. We denote a connection on $P$ by $A$. Then, with an exterior covariant derivative on adjoint-valued forms, $d_A = d + A$, we define a curvature $$F_A = d_A^2 \in \Omega^2(X,\operatorname{ad}(P))$$ which is obstruction to flatness. More precisely, it is understood as $d_A^2 = [F_A,\cdot]$. Locally, on a trivilizing open set $U \subset X$, we may regard the connection as a $\mathfrak{g}$-valued one-form, $A \in \Omega^1(X,\mathfrak{g})$. Therefore, we have $$F_A = dA + A \wedge A.$$ Two connections differing by gauge transformation should be regarded as equivalent. We define the gauge group, $$\mathscr{G} = \operatorname{Aut}(P)$$ The gauge group consists of bundle automorphisms covering the identity. After choosing a trivialization, write $$g:X\to G$$ and then, $$A \longmapsto g^{-1}Ag+g^{-1}dg$$ Globally, gauge transformations are automorphisms of the principal bundle; locally they appear as $G$-valued functions. ### Variational problem We define the **Yang--Mills (YM) functional** of the connection, $$\operatorname{YM}(A) = ||F_A||^2 = \int_X |F_A|^2 \operatorname{dvol} = \int_X \left< F_A,F_A \right> = \int_X \operatorname{tr} (F_A \wedge *F_A)$$ which is often called the YM action denoted by $S_\text{YM}(A)$ in the physics literature. Here we denote by $*$ the Hodge star oprator, and the norm uses the Riemannian metric on $X$ and the invariant inner product on $\mathfrak{g}$ (Killing form). We look for critical points of this functional. Write $A_t = A + t a$ with $a \in \Omega^1(X,\operatorname{ad}(P))$. We vary the connection in the direction of an ajoint-valued one-form. Using the local formula, we have $$ F_{A_t} = d(A + t a) + (A + ta) \wedge (A + ta) = F_A + t d_A a + O(t^2)$$ from which we conclude the linearization of the curvature, $$\left.\frac{d}{dt} F_{A_t}\right|_{t = 0} = d_A a$$ Then, the derivative of the YM functional is given by $$\frac{d}{dt} \operatorname{YM}(A) = 2 \int_X \left< d_A a, F_A \right> = 2 \int_X \left< a, d_A^* F_A \right>$$ where we applied the integral by parts for the second equality. Therefore, the critical point equation reads $$ d_A^*F_A = 0$$ This is the **Yang--Mills equation**, a non-linear second order partial differential equation, which is elliptic modulo gauge transformation. ### Self-duality in four dimensions In four dimensions, we have $$ * : \Omega^2(X) \to \Omega^2(X)$$ and hence $*^2 = 1$ with eigenvalues, $\pm 1$. Every two-form decomposes uniquely into self-dual (SD) and anti-self-dual (ASD) parts: $$\Omega^2(X) = \Omega_+^2(X) \oplus \Omega_-^2(X)$$ $$ \omega = \omega^+ + \omega^-$$ such that $* \omega^\pm = \pm \omega^\pm$. Curveture therefore splits into two geometrically distinguished pieces, $$ F_A = F_A^+ + F_A^-$$ Then, since the Hodge star is self-adjoint involution, $F_A^\pm$ are orthogonal with respecto the inner product, $$||F_A||^2 = ||F_A^+||^2 + ||F_A^-||^2$$ Meanwhile, we have \begin{align} \left< F_A, *F_A \right> & = \operatorname{tr} F_A \wedge F_A \\ & = \operatorname{tr} F_A^+ \wedge F_A^+ + \operatorname{tr} F_A^- \wedge F_A^- \\ & = \operatorname{tr} F_A^+ \wedge * F_A^+ - \operatorname{tr} F_A^- \wedge * F_A^- \\ & = |F_A^+|^2 - |F_A^-|^2 \end{align} Combining these two expressions, we have \begin{align} \operatorname{YM}(A) & = - \int_X \operatorname{tr} F_A \wedge F_A + 2 ||F_A^+||^2 \\ & = 8 \pi^2 k + 2 ||F_A^+||^2 \end{align} where $$k = \frac{-1}{8 \pi^2} \int_X \operatorname{tr} F_A \wedge F_A = \left< c_2(P), [X] \right> \in \mathbb{Z}$$ is called the *second Chern number* (second Chern class evaluated with the fundamental class), also called the *instaton charge*, or *topological charge*. By Chern--Weil theory, this four-form represents the second Chern class, so although it is written in terms of curvature, its integral depends only on the bundle topology. Since $||F_A^+||^2 \ge 0$, we have $$\operatorname{YM}(A) \ge 8 \pi^2 k$$ The equality holds for the **anti-self-dual (ASD) connection**, a connection obeying the **anti-self-dual Yang--Mills (ASDYM) equation**, $F_A^+ = 0$. The ASD connection is also called the **instanton**, while the SD connection is instead called the anti-instanton. Thus, the instanton minimizes the YM functional in a fixed topological sector. We remark that the YM equation is a second order PDE, while the ASDYM equation is a first order PDE, both elliptic modulo gauge transformation. We also remark that, by the **Bianchi identity**, $d_A F_A = 0$, we immediately see that the instanton solves the YM equation, $$d_A^* F_A = - d_A F_A = 0$$ The (A)SDYM equation, $F_A^\pm = 0$, is interpreted as *a half* of the flatness equation, $F_A = 0$, which is a reasonable equation in four dimensions: $A$ has three degrees of freedom modulo gauge transformation, and $F_A^\pm$ also has three, while $F_A$ has six. ## Lecture 2 ### Finiteness condition and topology Set $X = \mathbb{R}^4$ and $G=$ SU(2). We are in particular looking for finite-action connections on $\mathbb R^4$, $$ \operatorname{YM}(A) = \int_{\mathbb{R}^4} |F_A|^2 \operatorname{dvol} < \infty $$ From this finiteness condition, curvature decays sufficiently fast at infinity: We have asymptotically flat connection, $F_A \to 0$, implying that the connection is asymptotically pure gauge (Mauler--Cartan form), $A \to g^{-1} dg$. This gives a map, $$ g : \mathbb{S}^3 \longrightarrow \mathrm{SU}(2) $$ which determines the topology. Indeed, the degree of this map gives the instanton charge, $$\operatorname{deg}(g) = k$$ where $$ \operatorname{deg}(g) = \frac{1}{24 \pi^2} \int_{\mathbb{S}^3} \operatorname{tr} \left( g^{-1} dg \wedge g^{-1} dg \wedge g^{-1} dg \right) $$ In this way, we may compactify $\mathbb{R}^4$ by one point, $\mathbb{S}^4 = \mathbb{R}^4 \cup \{\infty\}$, turning infinity into an equatorial $\mathbb{S}^3$. This is consistent with the stereographic projection, which identifies $\mathbb R^4$ with $\mathbb S^4$ minus one point, $\mathbb S^4\setminus\{\infty\} \cong \mathbb R^4$, and finite action guarantees that the missing point can be filled back in topologically. ### Instanton solutions on $\mathbb{S}^4$ We then show that the ASDYM equation has nontrivial solutions for $X = \mathbb{S}^4$. Our idea is a quaternion analogue of the Hopf fibration, \begin{align} \mathrm{U}(1) & = \mathbb{S}^1 \longrightarrow \mathbb{S}^3 \longrightarrow \mathbb{S}^2 = \mathbb{CP}^1 \\ \mathrm{Sp}(1) = \mathrm{SU}(2) & = \mathbb{S}^3 \longrightarrow \mathbb{S}^7 \longrightarrow \mathbb{S}^4 = \mathbb{HP}^1 \end{align} #### Two dimensions $\mathbb{S}^2 = \mathbb{CP}^1$ From the homogeneous coordinate, $(z_0;z_1) \in \mathbb{CP}^1$, we define the affine corrdinate of $\mathbb{CP}^1$, $z = z_1/z_0 \in \mathbb{C}$, which is interpreted as the complex coordinate of the stereographic projection. In this setup, curvature of the principal $\mathrm{U}(1)$-bundle on $\mathbb{CP}^1$ is identified with the Kähler form of the Fubini-Study metric, $$F = 2\omega = i \partial \bar{\partial} \log (1 + |z|^2) = i \frac{d \bar{z} \wedge d{z}}{(1+|z|^2)^2}$$ where $\partial$, $\bar{\partial}$ are the Dolbeault operators. In our convention, we have $$\int_{\mathbb{CP}^1} \omega = \operatorname{vol}(\mathbb{CP}^1) = \pi$$ from which we obtain $c_1 = \frac{i}{2\pi} F$, such that $\left< c_1, [\mathbb{CP}^1] \right> = 1$. #### Four dimensions $\mathbb{S}^4 = \mathbb{HP}^1$ In order to study the instanton on $\mathbb{S}^4$, the quaternion notation is natural because $\mathbb R^4 \cong \mathbb H$, and SU(2) is the group of unit quaternions. We write $$x = x_1 + x_2 i + x_3 j + x_4 k \in \mathbb{H}$$ where $(i,j,k)$ are the standard imaginary quaternion units, $i^2 = j^2 = k^2 = ijk = -1$, and its conjugate denoted by $$ \bar{x} = x_1 - x_2 i - x_3 j - x_4 k $$ The norm is given by $$|x|^2 = x \bar{x} = \sum_{a=1}^4 x_a^2$$ We remark that imaginary quaternions are identitifed with $\mathfrak{su}(2)$, $\operatorname{Im} \mathbb{H} = \mathfrak{sp}(1) = \mathfrak{su}(2)$. As in the case of $\mathbb{CP}^1$, we obtain the curvature of the $\mathrm{SU}(2)$-bundle from the corresponding Kähler form, $$ F = \frac{d\bar{x} \wedge dx}{(1 + |x|^2)^2}$$ which is seen to be $\mathfrak{su}(2)$-valued under the above-mentioned identification. It is clearly ASD from the quaternionic structure, \begin{align} \frac{1}{2} d\bar{x} \wedge dx & = i (dx_1 \wedge dx_2 - dx_3 \wedge dx_4) \\ & \ + j (dx_1 \wedge dx_3 - dx_2 \wedge dx_4) \\ & \ + k (dx_1 \wedge dx_4 - dx_2 \wedge dx_3) \end{align} and hence $*F = - F$. We remark that there is no contribution proportional to the unit "1", corresponding to the $\mathfrak{u}(1)$-factor. Changing the orientation, $F \propto dx \wedge d\bar{x}$, it becomes SD, $*F = F$. Moreover, recalling, $$\frac{1}{2} \int_{\mathbb{HP}^1} \omega \wedge \omega = \operatorname{vol}(\mathbb{HP}^1) = \frac{\pi^2}{6} $$ we identiy $c_2 = \frac{-1}{8\pi^2} F \wedge F$. This curvature agrees with the so-called **BPST (Belavin--Polyakov--Schwartz--Tyupkin) instanton** solution of charge $k = 1$. #### Twistor description We have seen that the quaternionic Hopf fibration, $\mathbb{S}^3 \to \mathbb{S}^7 \to \mathbb{S}^4 = \mathbb{HP}^1$, plays a crucial role in the construction of the instanton solution. There is a complex projective analogue of this construction $$ \mathbb{CP}^1 \to \mathbb{CP}^3 \to \mathbb{S}^4 $$ where $\mathbb{CP}^3 = S^7 / \mathrm{U}(1)$ and $\mathbb{CP}^1 = S^3 / \mathrm{U}(1)$. In this context, $\mathbb{CP}^3$ is called the **twistor space** of $\mathbb{S}^4$. There is a one-to-one correspondence between the ASD connection on $\mathbb{S}^4$ and the holomorphic vector bundle on $\mathbb{CP}^3$. ### Moduli problem The expression allows us to consider a modified solution, $$ F = \frac{d\bar{x} \wedge dx}{(\lambda + |x - x_0|^2)^2}$$ where $\lambda \in \mathbb{R}_{>0}$ and $x_0 \in \mathbb{H}$, corresponding to rescale and translation of the configuration. Hence, we have a 5-parameter family of the instanton solution. We remark that, under the gauge transformation, the curvature behaves $F \mapsto g^{-1} F g$, preserving $\lambda$ and $x_0$ in the above expression. We define the moduli space as the space of ASD connections modulo gauge equivalence, $${\mathscr{M}}_k=\{A \mid F_A^+=0, \left< c_2,[X] \right> = k \}/\mathscr{G}$$ The BPST instanton shows that this moduli space is nonempty for $X = \mathbb{R}^4$ ($\mathbb{S}^4$) with $k=1$. On $\mathbb R^4$, one often fixes the gauge transformation at infinity (the global part of the gauge choice), $$\mathscr G_0=\{g:X\to G\mid g(\infty)=1\}$$ leading to the framed moduli space, $${\mathscr M}_k^\text{fr}=\{A\mid F_A^+=0\}/\mathscr G_0$$ Framing remembers the asymptotic trivialization. In general, we have $\operatorname{dim} {\mathscr{M}}_k = 8k-3$, while $\operatorname{dim} {\mathscr{M}}_k^\text{fr} = 8k$ for $G = \mathrm{SU}(2)$. The deformation equation is given by the linearization, $d_A^+ a = 0$, together with the gauge fixing, $d_A^* a = 0$, and hence $d_A^* \oplus d_A^+$. This is the elliptic operator governing infinitesimal moduli. In this way, we may know the moduli space dimension via the index theorem. ## Lecture 3 Let us discuss the **Atiyah--Drinfeld--Hitchin--Manin (ADHM) construction** of instantons on $\mathbb{S}^4$. For this purpose, we recall how to obtain the non-trivial connection for the Hopf fibration, $\mathrm{U}(1) \hookrightarrow \mathbb{S}^3 \to \mathbb{S}^2 = \mathbb{CP}^1$. Let $$\mathbb{S}^3 = \{ \xi = (\xi_1,\xi_2) \in \mathbb{C}^2 \mid \left< \xi, \xi \right> = 1 \}$$ and $x \in \operatorname{Im}\mathbb{H} \cong \mathfrak{su}(2)$. In a matrix notation, we may write $$x = i \begin{pmatrix} x_3 & x_1 + i x_2 \\ x_1 - i x_2 & - x_3 \end{pmatrix}$$ We define $D = x - \lambda 1$ with $\lambda \in \mathbb{C}$ and consider the kernel, $\ker D$. In order to have a non-null kernel, we need to put $\lambda = \pm i |x|$. Then, we may write $$\xi = \frac{1}{\sqrt{1 + |z|^2}} \begin{pmatrix} z \\ 1 \end{pmatrix} \in \ker D$$ where $z = (x_1 + i x_2)/(x_3 \pm |x|) \in \mathbb{C}$ is the corresponding affine coordinate of $\mathbb{CP}^1$. Then, the connection is given by $$A = \left< \xi, d \xi \right> = \frac{1}{2} \frac{\bar{z} dz - z d\bar{z}}{1 + |z|^2} = i \Im \frac{\bar{z}dz}{1+|z|^2}$$ which leads to the curvature mentioned before. ### ADHM construction Let us discuss the ADHM construction of $k$-instanton solution on $\mathbb{R}^4 \cong \mathbb{C}^2$ for $G = \mathrm{SU}(n)$. We write the coordinate $(z_1,z_2) \in \mathbb{C}^2$ where $z_1 = x_1 + i x_2$ and $z_2 = x_3 + i x_4$. Let $N = \mathbb{C}^n$, $K = \mathbb{C}^k$. We define linear maps, $B_i = \operatorname{Hom}(K,K)$ $(i=1,2)$, $I = \operatorname{Hom}(N,K)$, $J = \operatorname{Hom}(K,N)$. We define the so-called (zero-dimensional) Dirac operator, $$ D = \begin{pmatrix} B_1 - z_1 1_K & B_2 - z_2 1_K & I \\ - B_2^\dagger + \bar{z}_2 1_K & B_1^\dagger - \bar{z}_1 1_K & - J^\dagger \end{pmatrix} : K \otimes \mathbb{C}^2 \oplus N \to K \otimes \mathbb{C}^2$$ We may also write it in a quaternion notation, $$ \begin{pmatrix} B_1 - z_1 1_K & B_2 - z_2 1_K \\ - B_2^\dagger + \bar{z}_2 1_K & B_1^\dagger - \bar{z}_1 1_K \end{pmatrix} \cong B - x 1_K , \quad B \in \operatorname{End}(K) \otimes \mathbb{H}, \ x \in \mathbb{H} $$ We impose the condition $D D^\dagger = \Delta \otimes 1_{\mathbb{C}^2}$ (quaternion scalar condition) with $\Delta : K \to K$ (asymptotically $\Delta \to |x| 1_K$ at infinity), leading to the ADHM equations (also called the moment map equations), \begin{align} \mu_{\mathbb{R}} & := [B_1,B_1^\dagger] + [B_2,B_2^\dagger] + II^\dagger - J^\dagger J = 0 \\ \mu_{\mathbb{C}} & := [B_1, B_2] + IJ = 0 \end{align} Moreover, we assume that $\Delta$ is invertible ($D$ : full-rank), which will be related to the stability condition of the moduli space. From the full-rank condition, we have $\dim \ker D = n$. We write the orthonormal basis of $\ker D$ as $\{\psi_a\}_{a=1,\ldots,n}$, such that $\left< \psi_a, \psi_b \right> = \delta_{a,b}$, and the corresponding basis matrix, $$ \Psi = (\psi_1,\ldots,\psi_n) : N \to K \otimes \mathbb{C}^2 \oplus N $$ We also define a projector from $K \otimes \mathbb{C}^2 \oplus N$ onto $N$, $$P := \Psi \Psi^\dagger = 1_{K \otimes \mathbb{C}^2 \oplus N} - D^\dagger(\Delta^{-1} \otimes 1_{\mathbb{C}^2}) D $$ **Proposition** : $A = \left<\Psi,d \Psi\right>$ is ASD. *Proof* : The corresponding curvature is given by $$F = dA + A \wedge A = d \Psi^\dagger (1 - \Psi \Psi^\dagger) d\Psi = \Psi^\dagger (d D^\dagger) (\Delta^{-1} \otimes 1_{\mathbb{C}^2}) (dD) \Psi $$ Recalling that $dD = - 1_K dx$, $dD^\dagger = - 1_K d\bar{x}$ in the quaternion basis, the curvature is proportional to $d\bar{x} \wedge dx$, which is ASD. $\square$ **Proposition** : The instanton charge is given by $k = \dim K$. *Proof* : From the curvature formula above, we may have (Osborn's formula), $$ \operatorname{tr} F \wedge F = - \partial^2 \partial^2 \log \det \Delta^{-1}$$ From the asymptotic behavior of $\Delta$, $\Delta \to |x| 1_K$ at infinity, we obtain $c_2 = \operatorname{tr} 1_K = k$. $\square$ Here is another derivation. We have the following complex (ADHM complex), on the twistor description on $\mathbb{CP}^3$, $$ 0 \to K \otimes O(-1) \to (K \oplus K \oplus N) \otimes O \to K \otimes O(1) \to 0$$ Denoting the Chern character of $O(\pm 1)$ by $e^{\pm H}$, we have the Chern character of the instanton bundle denoted by $E$, $\operatorname{ch}(E) = n - k H^2$, from which, together with $c_1(E) = 0$, we identify $c_2(E) = k H^2$. ### ADHM moduli space In the ADHM construction, there is an analogue of the gauge transformation, $$\mathrm{U}(k) : (B_{1,2},I,J) \mapsto (g B_{1,2} g^{-1}, g I, J g^{-1})$$ which preserves the ADHM equation, $\mu = (\mu_{\mathbb{R}},\Re \mu_{\mathbb{C}}, \Im \mu_{\mathbb{C}}) = 0$, since $\mu \mapsto g \mu g^{-1}$. There is another group action, $$\mathrm{SU}(n) : (B_{1,2},I,J) \mapsto (B_{1,2}, I h^{-1}, h J)$$ which corresponds to rotation of the framing at infinity. We then define the **ADHM moduli space** as follows, $$\mathscr{M}_{n,k}^\text{ADHM} = \{(B_{1,2},I,J) \mid \mu = 0 \}/\mathrm{U}(k)$$ This moduli space is known to be hyper-Kähler due to the triple of the moment maps. **Proposition** : All the ASD connections on $\mathbb{S}^4$ can be constructed by the ADHM construction (surjective), and the corresponding ADHM data, obtained from the inverse ADHM construction, are unique (injective). From the existence and the uniqueness of the construction, we may describe the framed instanton moduli space on $\mathbb{S}^4$ based on the ADHM construction, $$ {\mathscr{M}}_{n,k}^\text{fr} = \mathscr{M}_{n,k}^\text{ADHM}$$ We count degrees of freedom of the moduli space, which will be the corresponding (virtual) dimension, $$ \dim_\mathbb{R} {\mathscr{M}}_{n,k}^\text{fr} = 2 \cdot 2 k^2 + 2 \cdot 2 nk - 3 k^2 - k^2 = 4 nk $$ We also write $\dim_\mathbb{C} {\mathscr{M}}_{n,k}^\text{fr} = 2 nk$. The $\mathrm{SU}(n)$-quotient gives rise to the *unframed* moduli space of the dimention, $\dim {\mathscr{M}}_{n,k} = 4nk - \dim \mathrm{SU}(n)$. Recall that, for $G = \mathrm{SU}(2)$, $k = 1$, the dimension of the unframed moduli space was given by $4 \cdot 2 \cdot 1 - 3 = 5$, as seen before. ### Singularity and compactification In the previous example, $$ F = \frac{d\bar{x} \wedge dx}{(\lambda + |x - x_0|^2)^2}$$ the parameter $\lambda > 0$ gives the size of the instanton, and the limit $\lambda \to 0$ causes the so-called small instanton singularity. In general, we have such a singularity at the boundary of the moduli space, $$ \mathscr{M}_{n,k}^\text{fr} \leadsto \mathscr{M}_{n,k-1}^\text{fr} \times \mathbb{R}^4 $$ This leads to the **Uhlenbeck compactification** of the instanton moduli space, $$\overline{\mathscr{M}}_{n,k} = \sqcup_{m=0}^k \mathscr{M}_{n,k-m}^\text{fr} \times \operatorname{Sym}^m (\mathbb{R}^4)$$ which is analogous to compactification of $\mathbb{R}^4$ with one point. ### Stability condition The Uhlenbeck compactification does not resolve the small instanton singularities. In order to resolve them, we consider the following deformation of the moduli space introduced by Nakajima, $$\mathscr{M}_{n,k}^\zeta = \{ (B_{1,2},I,J) \mid \mu_\mathbb{R} = \zeta 1_K, \mu_\mathbb{C} = 0 \}/U(k), \quad \zeta \neq 0$$ which also appeared as the moduli space of instantons on the non-commutative space, in the work of Nekrasov and Schwarz. **Definition** : Stability condition on the linear maps (quiver representation) is: No proper $S \subset K$ invariant under $B_{1,2}$ and containing $\operatorname{Im}(I)$, i.e., $$K = \mathbb{C}[B_1,B_2]I$$ Co-stability condition is similarly given by $$ K = \mathbb{C}[B_1^\dagger, B_2^\dagger] J^\dagger $$ **Lemma** : The real moment map equation $\mu_\mathbb{R} = \zeta 1_K$ is equivalent to the (co-)stability condition when $\zeta > 0$ ($<0$). *Proof* : We write $K' = \mathbb{C}[B_1,B_2] I$ and $K' = \mathbb{C}[B_1^\dagger,B_2^\dagger] J^\dagger$ and define $K'_\perp$ and $K'_\perp$ as complements of $K'$ and $K''$ in $K$. We also define the projectors $P'$, $P''$ on $K_\perp'$ and $K_\perp''$. $P'$ annihilates all elements in a form of $B_1^a B_2^b I$, from which we have $[P',B_{1,2}] = 0$ and $P' I = 0$. Then, we have $$ \zeta 1_{K_\perp'} = P' \mu_\mathbb{R} P' = P' ([B_1,B_1^\dagger] + [B_2,B_2^\dagger] - J^\dagger J) P' $$ When $\zeta > 0$, taking a trace yields $$ 0 \le \operatorname{tr} \zeta_{>0} 1_{K_\perp'} = - \operatorname{tr} P' J^\dagger J P' \le 0 $$ Therefore, $K_\perp' = 0$. The same discussion applies to $K''$. $\square$ **Theorem** [Kempf--Ness] : The stable moduli space of framed rank-$n$ torsion-free sheaves on $\mathbb{P}^2$ is isomorphic to the $\zeta_{>0}$-deformed framed moduli space of $\mathrm{SU}(n)$-ASD connections on $\mathbb{S}^4$, i.e., $$ \widetilde{\mathscr{M}}_{n,k} = \mu_\mathbb{C}^{-1}(0)/^\text{st}\mathrm{GL}(K) \cong \mu_\mathbb{R}^{-1}(\zeta_{>0} 1_K) \cap \mu_\mathbb{C}^{-1}(0)/\mathrm{U}(K) = \mathscr{M}^\zeta_{n,k} $$ The left one is understood as the GIT quotient with the stability condition. In particular, $\widetilde{\mathscr{M}}_{n=1,k}$ agrees with the Hilbert scheme of $k$ points on $\mathbb{C}^2$ denoted by $\operatorname{Hilb}^k(\mathbb{C}^2)$. ### Nakajima quiver variety The ADHM moduli space is now known to be the first example of **Nakajima's quiver variety** associated with $\widehat{A}_0$ quiver, which consists of only a single affine node. Let us briefly discuss a generalization of the ADHM construction by Kronheimer and Nakajima for *Asymptotically Locally Euclidean (ALE) spaces* to see its quiver description. Let $\Gamma$ be a finite subgroup of $\mathrm{SU}(2)$. The ALE space associated with $\Gamma$ is the minimal resolution of the quotient singularity given by $\widetilde{\mathbb{C}^2/\Gamma} \to \mathbb{C}^2/\Gamma$, which has a deep connection with the affine Dynkin diagram of type ADE via the *McKay correspondence*. Let us focus on the cyclic group $\Gamma = \mathbb{Z}_m = \mathbb{Z} / m \mathbb{Z}$, which corresponds to the Dynkin diagram of type $\widehat{A}_{m-1}$. We define $K = \oplus_{i=0}^{m-1} K_i$, $N = \oplus_{i=0}^{m-1} N_i$ and the linear maps, $B_i \in \operatorname{Hom}(K_i,K_{i+1})$, $\overline{B}_i \in \operatorname{Hom}(K_{i+1},K_i)$, $I_i \in \operatorname{Hom}(N_i,K_i)$, $J_i \in \operatorname{Hom}(K_i,N_i)$. We write $\mathrm{U}(K) = \prod_{i=0}^{m-1} \mathrm{U}(K_i)$. Then, we define the moduli space as follows, $$ \mathscr{M}_{A_{m-1};n,k} = \mu^{-1}(0) / \mathrm{U}(K) $$ together with the moment maps given by \begin{align} \mu_\mathbb{R} & = (B_{i-1} B_{i-1}^\dagger - B_i^\dagger B_i + \overline{B}_i \overline{B}_i^\dagger - \overline{B}_{i+1}^\dagger \overline{B}_{i+1} + I_i I^\dagger - J_i^\dagger J_i)_{i=0}^{m-1} \\ \mu_\mathbb{C} & = (B_{i-1} \overline{B}_{i-1} - \overline{B}_i B_i + I_i J_i)_{i=0}^{m-1} \end{align} The index $i$ is understood as modulo $m$, $i \equiv i + m$ (mod $m$). We may consider the deformation, $\mu_{\mathbb{R},i} = \zeta_i 1_{K_i}$, leading to the corresponding stability condition. The resulting moduli space is identified with the Nakajima quiver variety of type $A_{m-1}$. This construction applies to generic quivers. ## Lecture 4 The goal of this lecture is to derive the **Nekrasov's formula** for the equivariant volume of the instanton moduli space. **Definition** : Nekrasov partition function is a generating function of the equivariant volume of the instanton moduli space, $$ Z_n = \sum_{k=0}^\infty \mathfrak{q}^k Z_{n,k} , \qquad Z_{n,k} = \int_{\widetilde{\mathscr{M}}_{n,k}} 1$$ **Remark** : For the rank one case, we have $\widetilde{\mathscr{M}}_{n=1,k} = \operatorname{Hilb}^k(\mathbb{C}^2)$. Hence, the integral is given by $$ Z_k = Z_{n=1,k} = \int_{\operatorname{Hilb}^k(\mathbb{C}^2)} 1 $$ which is analogous to the Donaldson--Thomas invariant of $\mathbb{C}^3$, given by an integral over the virtual fundamental class of the corresponding Hilbert scheme of points, $$ \mathrm{DT}_k[\mathbb{C}^3] = \int_{[\operatorname{Hilb}^k(\mathbb{C}^3)]^\text{vir}} 1 $$ Since $\operatorname{vdim} \operatorname{Hilb}^k(\mathbb{C}^3) = 0$, this integral makes sense in the non-equivariant setting as well. In the following, we simply write $\mathscr{M} = \widetilde{\mathscr{M}}_{n,k}$. ### Equivariant localization formula The key idea to compute the integral is the so-called equivariant localization formula. **Theorem** [Berline--Vergne, Atiyah--Bott] : Let $X$ be a smooth manifold, which admits an action of a compact Lie group $G$. We denote by $F$ the fixed point loci of the $G$-action on $X$ and the inclusion $i : F \hookrightarrow X$. Let $N_F$ be the normal bundle to $F$ and the equivariant Euler class $e_\mathsf{T}(N_F)$. Then, integral of an equivariant close form $\alpha$ over $X$ localizes on $F$, $$ \int_X \alpha = \sum_F \int_{F} \frac{i^* \alpha}{e_\mathsf{T}(N_F)} $$ When $F$ is a point, $F = \{p\}$, we have $$ \int_X \alpha = \sum_p \frac{i^* \alpha}{e_\mathsf{T}(T_p X)} $$ Let us study the equivariant action on the moduli space, $$ \mathscr{M} = \{ (B_{1,2},I,J) \mid \mu_{\mathbb{C}} = 0, \text{stability condition} \}/\mathrm{GL}(K) $$ We have \begin{align} \mathrm{GL}(N) : & (B_{1,2},I,J) \mapsto (B_{1,2},I h^{-1},h J) \\ \mathrm{GL}(K) : & (B_{1,2},I,J) \mapsto (g B_{1,2} g^{-1},gI,Jg^{-1}) \\ \mathrm{GL}(\mathbb{C}^2) : & (B_1,B_2,I,J) \mapsto ( q_1^{-1} B_1, q_2^{-1} B_2,I , q_{12}^{-1} J) , \quad q_{12} = q_1 q_2 \end{align} We parametrize $g = e^\phi$, $h = e^a$, $q_i = e^{\epsilon_i} \in \mathbb{C}^\times$ ($i=1,2$), where $\phi \in \mathfrak{gl}(K)$, $a \in \mathfrak{gl}(N)$. We may assume $a = \oplus_{\alpha=1}^n a_\alpha$, $I = \oplus_{\alpha=1}^n I_\alpha$, $J = \oplus_{\alpha=1}^n J_\alpha$, such that $I_\alpha a = a_\alpha I_\alpha$, $a J_\alpha = a_\alpha J_\alpha$, without loss of generality. Then, the fixed point equations are given as follows, $$ [\phi,B_i] - \epsilon_i B_i = 0 , \quad \phi I - I a = 0 , \quad - J \phi + a J - \epsilon_{12} J = 0 , \quad \epsilon_{12} = \epsilon_1 + \epsilon_2 $$ **Lemma** : The left and right eigenvectors of $\phi$ are given by $B_1^i B_2^j I_\alpha$ and $J_\alpha B_1^i B_2^j$ with the eigenvalues, $$ a_\alpha + i \epsilon_1 + j \epsilon_2, \quad a_\alpha - (i+1) \epsilon_1 - (j+1) \epsilon_2 $$ *Proof* : It follows from the fixed point equations. $\square$ We then apply the stability condition, $K = \mathbb{C}[B_1,B_2]I = \oplus_{\alpha=1}^n \mathbb{C}[B_1,B_2]I_\alpha =: \oplus_{\alpha=1}^n K_\alpha$, and hence $J_\alpha B_1^i B_2^j = 0$ for any $i$, $j$, leading to $J = 0$. Since $\dim K = k < \infty$, $B_1^i B_2^j I_\alpha = 0$ for sufficiently large $i$, $j$. Indeed, this corresponds to an $n$-tuple monomial ideal parametrized by partitions, $\lambda = \{\lambda_\alpha\}_{\alpha=1,\ldots,n}$ with $|\lambda_\alpha| = \dim K_\alpha$. ### Instanton partition function We have the complex on the moduli space (ADHM complex), $C^\bullet : C^0 \to C^1 \to C^2$, where \begin{align} C^0 & = \operatorname{End}(K) \\ C^1 & = \operatorname{End}(K) \otimes \mathbb{C}^2 \oplus \operatorname{Hom}(N,K) \oplus \operatorname{Hom}(K,N) \otimes \det \mathbb{C}^2 \\ C^2 & = \operatorname{End}(K) \otimes \det \mathbb{C}^2 \end{align} together with maps, \begin{align} d_0(\phi) & = ([\phi,B_{1,2}],\phi I, - J \phi) \\ d_1(b_{1,2},i,j) & = [b_1,B_2] + [B_1,b_2] + i J + I j \end{align} We see that $d_1 \circ d_0(\phi) = [\phi,\mu_\mathbb{C}]$. Then, we have the cohomologies, $$ H^0(C^\bullet) = \ker d_0 , \quad H^1(C^\bullet) = \ker d_1 / \operatorname{im} d_0 , \quad H^2(C^\bullet) = \operatorname{coker} d_1 $$ corresponding to the spaces of symmetry, deformation, and obstruction. In this case, $H^0 = 0$, $H^2 = 0$. Hence, as a K-theory class, we have \begin{align} [T\mathscr{M}] & = [C^1] - [C^0] - [C^1] \\ & = N^\vee K + q_{12} K^\vee N - (1-q_1)(1-q_2) K^\vee K \end{align} under the abuse of notation, $N^\vee K = [N^\vee \otimes K]$, etc. **Remark** : From this expression, we see the (virtual) dimension of the tangent space, $\dim_\mathbb{C} T\mathscr{M} = 2 nk = \dim_\mathbb{C} \mathscr{M}$. Let $\lambda^\vee$ be the transposition of $\lambda$. We define the *arm length* and the *leg length*, $$ a_\alpha(i,j) = \lambda_{\alpha,i} - j , \quad \ell_\alpha(i,j) = \lambda^\vee_{\alpha,j} - i $$ Then, we have the following. **Lemma** : The equivariant Chern character of the tangent bundle at the fixed point $\lambda$ is given as follows, $$ \operatorname{ch} T_\lambda \mathscr{M} = \sum_{\alpha,\beta=1}^n e^{a_\beta - a_\alpha} \left( \sum_{s \in \lambda_\alpha} q_1^{\ell_\beta(s)} q_2^{-a_\alpha(s)-1} + \sum_{s \in \lambda_\beta} q_1^{-\ell_\alpha(s)-1} q_2^{a_\beta(s)} \right) $$ *Proof* : Recall $$ \operatorname{ch} N = \sum_{\alpha=1}^n e^{a_\alpha} \quad \operatorname{ch} K = \sum_{\alpha=1}^n \sum_{(i,j) \in \lambda_\alpha} e^{a_\alpha} q_1^{i-1} q_2^{j-1} $$ Then, the remaining is a combinatorial computation. See, e.g., Nakajima, *[Lectures on Hilbert Schemes of Points on Surfaces](https://doi.org/10.1090/ulect/018)* for details. $\square$ **Theorem** [Nekrasov's formula] : $$ Z_{n,k} = \sum_{\lambda \mid |\lambda| = k} \prod_{\alpha,\beta=1}^n \left[\prod_{s \in \lambda_\alpha} (a_\beta - a_\alpha + \epsilon_1 \ell_\beta(s) - \epsilon_2 a_\alpha(s) - \epsilon_2)^{-1} \prod_{s \in \lambda_\beta} (a_\beta - a_\alpha - \ell_\alpha(s) - \epsilon_1 + \epsilon_2 a_\beta(s))^{-1} \right] $$ *Proof* : We may use the equivariant localization formula. From the previous Lemma, we read off the Chern roots of the tangent bundle at $\lambda$, $\operatorname{ch} T_\lambda \mathscr{M} = \sum_x e^x$, and the Euler class is given by the product of them, $e_\mathsf{T}(T_\lambda \mathscr{M}) = \prod x$. $\square$ We are in particular interested in the asymptotic behavior of the partition function, $$ \mathscr{F} := \lim_{\epsilon_{1,2} \to 0} \epsilon_1 \epsilon_2 \log Z $$ which is identified with the so-called **Seiberg--Witten prepotential**. For $n = 1$, corresponding to the Hilbert scheme of points on $\mathbb{C}^2$, it depends only on $\epsilon_{1,2}$. The generating function has a closed formula, $$ Z_{n=1} = \sum_{\lambda} \mathfrak{q}^{|\lambda|} \prod_{s \in \lambda} (\epsilon_1 \ell(s) - \epsilon_2 a(s) - \epsilon_2)^{-1} (- \ell(s) - \epsilon_1 + \epsilon_2 a(s))^{-1} = \exp \left( \frac{\mathfrak{q}}{\epsilon_1 \epsilon_2} \right) $$ Hence, in this case, the prepotential is simply given by $\mathscr{F} = \mathfrak{q}$. ### Further studies #### Tautological insertion We use the same notation $K$ for the tautological bundle over $\mathscr{M}$. Let $\alpha$ be a characteristic class associated with $K$. In general, we may consider the integral, $$ \int_{\mathscr{M}} \alpha $$ - $\alpha = \operatorname{ch}(K)$ : generating function of chiral rings - $\alpha = e_\mathsf{T}(M \otimes K)$ : hypermultiplet contribution (fundamental represetation), $\dim M \le 2n$ - $\alpha = e_\mathsf{T}(M \otimes T\mathscr{M})$ : hypermultiplet contribution (adjoint represetation), $\dim M = 1$. In the limit $\operatorname{ch} M = 1$, it computes the Euler characteristic of $\mathscr{M}$ (Vafa--Witten theory). #### Equivariant K-theory The K-theoretic Nekrasov partition function is given by the equivariant holomorphic Euler characteristic of the structure sheaf of the moduli space, $$ Z_{n,k}^{(K)} = \chi(\mathscr{M},\mathscr{O}) = \int_{\mathscr{M}} \operatorname{td}(T\mathscr{M}) $$ We remark that $\operatorname{ch}(O)=1$. We can in general consider the insertion, $$ \chi(\mathscr{M},\mathscr{E}) = \int_{\mathscr{M}} \operatorname{ch} (\mathscr{E}) \operatorname{td}(T\mathscr{M}) $$ Physically, the K-theoretic integral computes a gauge theory partition function on $\mathbb{R}^4 \times \mathbb{S}^1$. There is a further uplift to the equivariant elliptic cohomology corresponding to a six-dimensional theory compactified on an elliptic curve. In this case, the partition function is identified with the equivariant elliptic genus of the moduli space. #### Higher-dimensions As seen before, the rank-one instanton moduli space is identified with the Hilbert scheme of points on $\mathbb{C}^2$. This point of view allows us to study higher-dimensional generalizations. The equivariant integral over $\operatorname{Hilb}^k(X)$ computes the (equivariant) **Donaldson--Thomas invariants** of $X$ for $X = \mathbb{C}^3$ and $\mathbb{C}^4$. The higher-rank version is based on the so-called Quot-schemes.