Nekrasov Conjecture I
Introduction
The Nekrasov Conjecture(theorem) basically predicts a relation between the SUSY \(N=2\) Yang-Mills and the Seiberg-Witten prepotential. So in 1994, Seiberg and Witten discovered an exact description of the low-energy behavior of certain supersymmetric gauge theories. Nekrasov later proposed a kind of regularized partition function \(Z\) to be defined on this.
Certain consequences of supersymmetry make this partition function also the partition function of a gas of instantons. Instantons are self-dual solutions of the Yang-Mills equations in the compactified euclidean 4-space. An interesting thing is that these instanton solutions of the Yang-Mills equations absolutely minimize the energy functional within their topological type. Indeed, in a standard quantum mechanics course, you may have come across the double-well potential where tunelling takes place, this is an example of the instanton effect. So this motivates some ideas for the QFT scenario, that is, a naive vacuum may not be the true vacuum of a field theory.
The Seibeg-Witten potential is defined via period integrals of certain hyperelliptic curves. Nekrasov himself eventually resolved this problem in 2003 using physical intuition and ‘comapring the infrared and ultraviolet limits’ of the gauge theories. On the other, Nakajima who also provided a solution for 4D pure gauge theory with gauge group \(U(r)\) also provided a radically different solution using certain moduli spaces of blowups. We will actually be discussing Nakajima’s result in this post.
Physical description
One can calculate vacuum expectation values of certain gauge theory observables. These observables are annihilated by a combination of the supercharges, and their expectation value is not sensitive to various parameters, the energy scale in particular. So, one can do the calculation either in the ultraviolet region(where the theory is weakly coupled and the instantons dominate), or in the infrared region, and then we can possibly relate the solution to the Seiberg-Witten prepotential of the effective low-energy theory.
Nekrasov Partition Function
The Nekrasov partition function is the generating function of the integral of the equivariant cohomology class $$ 1 \in H_{\tilde{T}}^{*}(M(r,n)):
\[Z(\epsilon_{1},\epsilon_{2},\vec{a}; \mathfrak{q}) = \sum\limits_{n=0}^{\infty} \int_{M(r,n)} 1\]where \(\epsilon_{1},\epsilon_{2},\vec{a}\) are generators of \(H_{\tilde{T}}^{*}(pt)\).
Nekrasov’s conjecture states that \(F^{inst}(\epsilon_{1},\epsilon_{2},\vec{a};\mathfrak{q}) = \epsilon_{1} \epsilon_{2} logZ(\epsilon_{1},\epsilon_{2},\vec{a};\mathfrak{q})\) is regular at \(\epsilon_{1} = \epsilon_{2} = 0\) and \(F^{inst}(0,0,\vec{a};\mathfrak{q})\) is the instanton part of the Seiberg-Witten prepotential for $$
Seiberg-Witten Prepotential
We’ll review the definition of the prepotential via the standard method, that is, via a family of hyperelliptic curves parametrized by \(\vec{u} = (u_{2},\cdots,u_{r})\)
\(C_{\vec{u}} : \Lambda^{r}(w+ \frac{1}{w}) = P(z) = z^{r}+u_{2}z^{r-2}+u_{3}z^{r-3}+ \cdots +u_{r}\) . These are the Seiberg-Witten curves. The projection \(C_{\vec{a}} \mapsto \mathbb{P}^{1}\) via \((w,z) \mapsto z\)
Let \(z_{1},\cdots,z_{r}\) be the solutions of \(P(z)=0\). Then, we can find \(z_{\alpha}^{\pm} = \pm 2\Lambda^{r}\) where \(\lvert u \rvert \gg \lvert \Lambda \rvert\). These are the \(2r-\) branched points of the projection above.
The hyperelliptic curve \(C_{\vec{u}}\) is made up of two copies of the Riemann sphere glued along the r-cuts between \(z_{\alpha}^{-}\) and \(z_{\alpha}^{+}\). Let \(A_{\alpha}\) be the cycle encircling the cut between the two points and then we have \(\sum_{\alpha} A_{\alpha} = 0\). We consider cycles \(B_{\alpha}\) so that \(\{A_{\alpha},B_{\alpha} \vert \alpha = 2,\cdots,r \}\) form a symplectic basis of \(H_{1}(C_{\vec{u}}, \mathbb{Z})\).
Then, we can finally define the Seiberg-Witten differential by
\[dS:= \frac{-1}{2\pi} z \frac{dw}{w} = \frac{-1}{2\pi} \frac{zP'(z)dz}{\sqrt{P(z)^{2} - 4 \Lambda^{2r}}}\]this is a meromorphic differential having poles at \(\pm \infty\). We then define functions \(a_{\alpha}, a_{\beta}^{D}\) on the u-plane such that
\[a_{\beta}^{D} = 2\pi i \int_{B_{\beta}} dS\] \[a_{\alpha} = \int_{A_{\alpha}} dS\]Again, we have that \(\sum a_{\alpha} = 0\)
Thus, there exists a locally defined function \(\mathcal{F(\vec{a};\Lambda)}\) on the \(\vec{u}-\) plane such that,
\[a_{\alpha}^{D} = -\frac{\partial \mathcal{F}}{\partial a_{\alpha}}\]This is called the Seiberg-Witten prepotential.
Essentially, what is happening here is that the low energy effective Wilsonian action for the massless vector multiplets \((a_{\alpha})\) is governed by the prepotential \(\mathcal{F}(\vec{a};\Lambda)\) which received one-loop perturbative and instanton non-perturbative corrections
\[\mathcal{F}(\vec{a};\Lambda) = \mathcal{F}^{pert}(\vec{a};\Lambda) + \mathcal{F}^{inst}(\vec{a};\Lambda)\]where the latter term is also a power series in \(\Lambda\) (see ahead for some details)
The instanton measure tends to get very complicated with the growth of the instanton charge, adn the integrals are hard to evaluate. Nekrasov attempted to solve this problem via the localiation technique.
that is, one should study \(G \times \mathbb{T}^{2}\) equivariant cohomology of the moduli space \(\tilde{\mathcal{M}_{k}}\) where \(G\) acts by rotating the guage orientation of the instantion at infinity, and \(\mathbb{T}^{2}\) si the maximal torus of \(SO(4)\) which also acts naturally on the above moduli space.
Consider \(p : \tilde{\mathcal{M}_{k}} \to pt\) the collapse map and the quantity,
\[Z(\epsilon_{1},\epsilon_{2},\vec{a};q) = \sum\limits_{k=0}^{\infty} q^{k} \int_{\tilde{\mathcal{M}_{k}}} 1\]where \(\int 1\) denotes the localization of the pushforward \(p_{*}(1)\) of \(1 \in H_{G \times \mathbb{T}^{2}}(\tilde{\mathcal{M}_{k}})\) in \(H_{G \times \mathbb{T}^{2}}(pt) = \mathbb{C}[\epsilon_{1},\epsilon_{2}, \mathcal{U}]\)
One can show that \(\mathcal{F}\) has the following behavior at \(\Lambda \to 0\):
\[\mathcal{F} = \sum\limits_{\alpha < \beta}[(a_{\alpha}-a_{\beta})^{2}log(\frac{i (a_{\alpha}-a_{\beta})}{\Lambda} - \frac{3}{2}(a_{\alpha} - a_{\beta})^{2})] + \Lambda^{2r}.O(\Lambda^{2r}\]The first part here is the perturbative part and the second part is the instanton part. To understand this derivation, see this.
Now, consider the period matrix of \(C_{\vec{u}}\), \(\tau_{\alpha \beta}\)
Note that here \(\tau_{\alpha \beta} = -\frac{1}{2 \pi i} \frac{\partial^{2} \mathcal{F}}{\partial a_{\alpha} \partial a_{\beta}}\)
Consider \(\mathfrak{sl}_{r}\) and its Cartan subalgebra \(\mathfrak{h}\). Let \(\Delta \subset \mathfrak{h}^{*}\) be the set of roots, then consider the simple root \(\alpha_{i} \in \mathfrak{h}^{*}\) for \(i=1,\cdots,r-1\), that is, \(\alpha_{i} = (0,\cdots,1,-1,0,\cdots,0)\) where the \(1\) is in the \(ith\) position and \(-1\) is obviously in the \((i+1)th\) position. Accordingly, we also have the set of positive roots, \(\Delta_{+} = \{e_{\alpha \beta} \}\). Given \(\vec{a} = (a_{1},\cdots,a_{r})\) then \(\langle \vec{a},e_{\alpha \beta} \rangle = a_{\alpha} - a_{\beta}\)
Let \(Q\) be the coroot lattice of \(\mathfrak{h}\).
The perturbative part can be rewritten as:
\[\sum_{\Delta_{+}} [\langle \vec{a},\alpha \rangle log(\frac{i \langle \vec{a}, \alpha \rangle}{\Delta} - \frac{3}{2}\langle \vec{a},\alpha \rangle^{2})]\]The period matrix here is:
\[\tau_{ij} = \frac{i}{\pi} \langle alpha_{i}^{\vee},\alpha \rangle \langle \alpha_{j}^{\vee}, \alpha \rangle log(\frac{i \langle \vec{a},\alpha \rangle}{\Lambda}) + \Lambda^{2r}.O(\Lambda^{2r})\]There are also 2 formulas that will be needed ahead.
\[\frac{\partial \mathcal{F}}{\partial log \Lambda} = -2ru_{2}\]This is known as the renormalization group equation.
\[\frac{\partial u_{2}}{\partial log \Lambda} = - \frac{2r}{\pi i} \frac{\partial u_{2}}{\partial a^{i}} \frac{\partial u_{2}}{\partial a^{j}} \frac{\partial}{\partial \tau_{ij}} log\Theta_{E}(\vec{0} \vert \tau)\]where \(\Theta_{E}(\vec{\eta} \vert \tau) = \sum_{\vec{k} \in Q} exp(\pi i \sum_{i,j}\tau_{i,j} k^{i}k^{j} + 2\pi i \sum_{i} k^{i}(\eta^{i} + \frac{1}{2}))\)
This one is the contact term equation .
Unlike Nekrasov’s original proof, this proof by Nakajima relies on some clever algebraic geometry, using moduli spaces on \(\mathbb{P}^{2}\)
Setting up the Proof
Moduli spaces
In particular, we will be dealing with framed moduli spaces on \(\mathbb{P}^{2}\), where we will prove analogous results as the Hilbert scheme calculations for \(\mathbb{C}^{2}\).
Let \(M(r,n)\) be the framed moduli space of torsion-free sheaves on \(\mathbb{P}^{2}\) with rank \(r\) and \(c_{2}=n\) which parametrizes isomorphism classes of \((E,\phi)\) such that
(1) \(E\) is torsion-free sheaf of rank \(r\), \(\langle c_{2}(E),[\mathbb{P}^{2}] \rangle = n\) which is locally free in a neighborhood of \(l_{\infty}\).
(2) \(\phi:E \vert_{l_{\infty}} \to \mathcal{O}_{l_{\infty}}^{\oplus r}\) is an isomorphism called framing at infinity.
\[l_{\infty} = \{ [0:z_{1}:z_{2}] \in \mathbb{P}^{2} \} \subset \mathbb{P}^{2}\]is the line at infinity.
Firstly, we have a quick \(Ext\) calculation
Lemma 1
\(Hom(E,E(-l_{\infty})) = Ext^{2}(E,E(-l_{\infty})) = 0\)
Proof of Lemma 1
This can easily done using the Grothendieck-Serre duality, using the multiplication by \(z_{0}\) map and finally the Serre vanishing theorem.
Corollary 1.1
\(M(r,n)\) is a nonsingular variety of dimension \(2nr\)
Proof of Lemma 2
(Riemann-Roch + Lemma 1)
Reframing our framed moduli spaces
(sorry for the cheap pun)
Anyways, some of these details are quite complex and they will somewhat dealt with in Appendix 1.
But basically, the idea is that \(M(r,n)\) is isomorphic to special kind of quotient space, that is, the quotient space of all 4 of \(B_{1},B_{2} \in End(\mathbb{C^{n}}),i \in Hom(\mathbb{C}^{r},\mathbb{C}^{n}),j \in Hom(\mathbb{C}^{n},\mathbb{C}^{r})\) satisfying:
(1) \([B_{1},B_{2}] + ij = 0\) (2) there exists no proper subspace \(S \subset \mathbb{C}^{n}\) such that \(B_{t}(S) \subset S(t=1,2)\) and \(im(i) \subset S\)
modulo the action of \(GL_{n}(\mathbb{C})\) given by, \(g.(B_{1},B_{2},i,j) = (gB_{1}g^{-1},gB_{2}g^{-1},gi,jg^{-1})\)
It can be shown that (1) is surjective and the action is free on stable points, this proves that \(M(r,n)\) is smooth.
Also, let \(M_{0}(r,n)\) be the framed moduli space of ideal instantons of \(S^{4}\), which is given by
\[M_{0}(r,n) = \bigcup\limits_{k=0}^{n} M_{0}^{reg}(r,k) \times Sym^{n-k}(\mathbb{C}^{2})\]where \(M_{0}^{reg}(r,k)\) is the framed moduli space of genuine instantons and the other factor is the symmetric product.
From Lemma 2 of Appendix 2, \(M_{0}^{reg}(r,n)\) can be identified with the framed moduli space of locally-free sheaves on \(\mathbb{P}^{2}\) and also with the same space of linear data \((B_{1},B_{2},i,j)\) as above but with the extra condition that (2) is also satisfied by the transposes \(B_{1}\mathsf{T},B_{2}\mathsf{T},j\mathsf{T}\). Then, from Lemma 3 of Appendix 2,
\(M_{0}(r,n)\) can be identified(as a top space) with the GIT quotient
\[\{(B_{1},B_{2},i,j) \vert [B_{1},B_{2}] + ij = 0} // GL_{n}(\mathbb{C})\]By Lemma 4 of Appendix 2, \(M(r,n)\) also has the structure of a hyper-Kahler manifold of dimension \(4nr\). Indeed, \(M(r,n)\) is isomorphic to the quotient
\[\{(B_{1},B_{2},i,j) \vert [B_{1},B_{2}] + ij=0 , [B_{1},B_{2}^{*}] + ii^{*}-j^{*}j = \zeta id \} / U(n)\]By this new reframing of these spaces, we get a projective morphism \(\pi: M(r,n) \to M_{0}(r,n)\) where \(M_{0}(r,n)\) borrows its scheme structure from the GIT quotient above.
Then finally, from Lemma 5 from Appendix 2, in terms of the original definition as framed moduli spaces, the correpsonding map between closed points can be identified with
\((E, \phi) \mapsto ((E^{\vee \vee}, \phi), Supp(E^{\vee \vee}/E)) \in M_{0}^{reg}(r,k) \times Sym^{n-k}(\mathbb{C}^{2})\) where the support is counted with multiplicities. \(E^{\vee \vee}\) is a locally-free sheaf.
Consider \(T\), the maximal torus of \(GL_{n}(\mathbb{C})\) which consists of diagonal matrices, and also let \(\tilde{T} = \mathbb{C}^{*} \times \mathbb{C}^{*} \times T . We can then define an action of\) \tilde{T} \(on\)M(r,n)$$ as follows,
given \((t_{1},t_{2}) \in \mathbb{C}^{*} \times \mathbb{C}^{*}\), we get the automorphism of \(\mathbb{P}^{2}\), \(F_{t_{1},t_{2}}\) defined by,
\[F_{t_{1},t_{2}}([z_{0}:z_{1}:z_{2}]) = [z_{0}:t_{1}z_{1}:t_{2}z_{2}]\]For \(\vec{e} = diag(e_{1},\cdots,e_{r}) \in T\), let \(G_{\vec{e}}\) denote the isomorphism on \(\mathcal{O}_{l_{\infty}}^{\oplus r}\) given by
\[(s_{1},\cdots,s_{r}) \in \mathcal{O}_{l_{\infty}}^{\oplus r} \mapsto (e_{1}s_{1},\cdots,e_{r}s_{r})\]Then, for \((E,\phi) \in M(r,n)\), we define the action
\((t_{1},t_{2},\vec{e}).(E,\phi) = ((F_{t_{1},t_{2}}^{-1})^{*}E,\phi^{'})\) where \(\phi^{'}\) is the composite map
\[(F_{t_{1},t_{2}}^{-1})^{*}E\vert_{l_{\infty}} \mapsto (F_{t_{1},t_{2}}^{-1})^{*} \mathcal{O}_{l_{\infty}}^{\oplus r} \mapsto \mathcal{O}_{l_{\infty}}^{\oplus r} \mapsto \mathcal{O}_{l_{\infty}}^{\oplus r}\]where the final map is \(G_{\vec{e}}\), the middle arrow is the homomorphism given by the action. The first map is \((F_{t_{1},t_{2}}^{-1})^{*}\phi\). In a similar way, we also have an action of \(\tilde{T}\) on \(M_{0}(r,n)\). Also, this makes the above map \(\pi\) equivariant under this action.
Fixed points, Young diagrams
We can also show that these two actions can be identified with the actions on the linear data defined by,
\((B_{1},B_{2},i,j) \mapsto (t_{1}B_{1},t_{2}B_{2},ie^{-1},t_{1}t_{2}ej)\) for \(t_{1},t_{2} \in \mathbb{C}^{*},e=diag(e_{1},\cdots,e_{r}) \in (\mathbb{C}^{*})^{r}\)
One can show that this action preserves the equation \([B_{1},B_{2}]+ij=0\), preserves the stability conditions and commutes with the action of \(GL_{n}(\mathbb{C})\). Hence, it induces an action on both \(M(r,n),M_{0}(r,n)\).
This fact is proven in Lemma 1 in Appendix 3.
Now, we begin to elucidate the connection between the fixed points of this \(\tilde{T}\) action and Young diagrams via some commutative algebra.
Lemma 1
(1) \((E,\phi)\in M(r,n)\) is fixed by the \(\tilde{T}-\)action if and only if \(E\) has a decomposition \(E=I_{1} \oplus \cdots \oplus I_{r}\) satisfying the following conditions for \(a=1,\cdots,r\),
a. \(I_{a}\) is an ideal sheaf of a 0-dim subscheme \(Z_{a} \subset \mathbb{C}^{2} = \mathbb{P}^{2}l_{\infty}\) b. \(I_{a} \vert_{l_{\infty}}\) is mapped to the \(a-\)th factor of \(\mathcal{O}_{l_{\infty}}^{\oplus r}\) under \(\phi\) c. \(I_{a}\) is fixed by the action of \(\mathbb{C}^{*} \times \mathbb{C}^{*}\), coming from that on \(\mathbb{P}^{2}\)
(2) The fixed point set \(M(r,n)^{\tilde{T}}\) consists of finitely many points parametrized by r-tuple \((Y_{1},\cdots,Y_{r})\) of Young diagrams such that \(\sum_{a} \lvert Y_{a} \rvert = n\)
(3) The fixed point set in \(M_{0}(r,n)\) consists of a single point \(n[0] \in Sym^{n}(\mathbb{C}^{2}) \subset M_{0}(r,n)\)
Proof of Lemma 1
(1) Recall that \(\tilde{T} = \mathbb{C}^{*} \times \mathbb{C}^{*} \times T\), and so \(E\) is fixed by the latter \(T-\)action if and only if it decomposes as \(E=I_{1} \oplus \cdots \oplus I_{r}\) with \(I_{a} \in M(1,n_{a})\), such that \(I_{a} \vert_{l_{\infty}}\) is mapped to the \(a-\)th factor of \(\mathcal{O}_{l_{\infty}}^{\oplus r}\) under \(\phi\). Note that the double dual has \(c_{1}(I_{a}^{\vee \vee} = 0)\), so that \(I_{a}^{\vee \vee} = \mathcal{O}_{\mathbb{P}^{2}}, and so via the natural inclusion,\)I_{a}\(can be realized as an ideal shead of a 0-dim subscheme contained in\)\mathbb{C}^{2}$$. Thus a,b are satisfied.
(2) Using Lemma 2 in Appendix 3, \(I_{a}\) is fixed if and only if it is generated by monomials \(x^{i}y^{j}\). So, \(I_{a}\) corresponds to a Young diagram \(Y_{a}\) where \(x^{i-1}y^{j-1}\) corresponds to \((i,j)\). The ideal \(I_{a}\) are linearly spanned by monomials outside the Young diagram \(Y_{a}\).
(3) Consider the GIT quotient description of \(M_{0}(r,n)\) in the previous section. Assume that the equivalence class of \((B_{1},B_{2},i,j)\) is fixed by the \(T^{2}-\)action. We may also assume WLOG that \((B_{1},B_{2},i,j)\) has a closed \(GL_{n}(\mathbb{C})-\) orbit. Then, the class if fixed if and only if \((t_{1}B_{1},t_{2}B_{2},i,t_{1}t_{2}j)\) lies int he same \(GL_{n}(\mathbb{C})\) orbit.
However, note that \((t_{1}B_{1},t_{2}B_{2},i,t_{1}t_{2}j) \to 0\) as \(t_{1},t_{2} \to 0\), so \((0,0,i,0)\) lies in the closure of the orbit. However, the orbits are closed so \((0,0,i,0)\) must lie in the orbit. Finally, by the \(GL_{n}(\mathbb{C})\) action, we see that \((B_{1},B_{2},i,j) = (0,0,0,0)\)
Detour into Young Diagrams
We’ll need some new details about Young diagrams for the proof ahead. Let \(Y=(\lambda_{1} \geq \lambda_{2} \geq \cdots)\) be a Young diagram where \(\lambda_{i}\) is the length of the \(i\)th column. And, let \(Y'\) be the tranpose of \(Y\). Let \(l(Y)\) be the number of columns of \(Y\), that is, \(l(Y) = \lambda_{1}'\).
Let \(a_{Y}(i,j) = \lambda_{i} - j,a'(i,j) = j-1\) These are called the arm-length and arm-colength.
\(l_{Y}(i,j) = \lambda_{j}'-i,l'(i,j)=i-1\) These are called the leg-length and leg-colength. here we set \(\lambda_{i}= 0\) for \(i>l(Y)\).
If we’re given two Young diagrams, \(Y_{a},Y_{b}\), then separate \(Y_{a}\) into two regions \(Y_{a}[0],Y_{a}[1]\) as
\[Y_{a}[0]=\{(i,j)\in Y_{a} \vert j \leq l(Y_{b}') \}, Y_{a}[1] = \{(i,j)\in Y_{a} \vert j>l(Y_{b}') \}\]if \(l(Y_{a}') \leq l(Y_{b}')\), then \(Y_{a}[1]\) is empty. Note that either \(Y_{a}[1]\) or \(Y_{b}[1]\) is the empty set.
We can also construct one-dimensional \(\tilde{T}-\)modules denoted by \(e_{\alpha}\) via \((t_{1},t_{2},e_{1},\cdots,e_{r}) \in \tilde{T} \mapsto e_{\alpha}\) for every \(\alpha\). We also denote by \(t_{1},t_{2}\) one dimensional \(\tilde{T}-\)modules constructed in a similar manner
Tangent space at fixed point
Let \((E.\phi)\) be a fixed point of the \(T^{2}-\) action, then \(T_{(E,\phi)}M(r,n)\) is a module of \(T^{2}\). The Young diagrams can be used to describe the module structure.
We now have our first main calculation/theorem
Theorem 1
Let \(X = (E,\phi)\) be a fixed point of \(\tilde{T}-\) action and let \(\vec{Y}=(Y_{1},\cdots,Y_{r})\) be the corresponding Young diagram.
Then, the \(\tilde{T}-\)module structure of \(T_{X}M(r,n)\) is given by
\[T_{X}M(r,n) = \sum\limits_{\alpha,\beta = 1}^{r} N_{\alpha,\beta}(t_{1},t_{2})\]where \(N_{\alpha,\beta}(t_{1},t_{2}) = e_{\beta}e_{\alpha}^{-1}[\sum\limits_{s \in Y_{\alpha}}(t_{1}^{-l_{Y_{\beta}}(s)}t_{2}^{a_{Y_{\alpha}}(s)+1}) + \sum\limits_{t \in Y_{\beta}}(t_{1}^{l_{Y_{\alpha}}(t)+1}t_{2}^{-a_{Y_{\beta}}(t)})]\)
(Sketch of Proof of Theorem 1)
We sue the description of the linear data for this calculation. Considering this map of complexes, \(V = \mathbb{C}^{n}, W = \mathbb{C}^{r}\)
\[Hom(V,V) \to (Hom(V,V) \oplus Hom(V,V)) \oplus (Hom(W,V) \oplus Hom(V,W)) \to Hom(V,V)\]appropriately defining \(\sigma(\eta)\), the first two components via differentials \(\eta B_{i}-B_{i}\eta\) and the last two as \(\eta i,-j\eta\).
and defining the last map \(\tau(C_{1},C_{2},I,J):= [B_{1},C_{2}] + [C_{1},B_{2}] + iJ+Ij\) so that \(\sigma\) is the differential of the \(GL(V)-\)action and \(\tau\) is the differential of the map \((B_{1},B_{2},i,j) \mapsto [B_{1},B_{2}] + ij\).
Using Lemma 3 in Appendix 3, one can show that \(\sigma\) is injective,\(\tau\) is surjective and that the \(T_{Y}M(r,n)\), for \(Y=GL(V).(B_{1},B_{2},i,j)\), is isomorphic to the middle cohomology group of the above complex.
One can then use the \(GL(V)\) orbits, look at fixed points, and consider \(V\) as a \(\tilde{T}-\) module via some simple algebra. We can then modify the map of complexes above to make it \(T^{r+2}-\)equivariant. Of course, I am skipping some details here, which can all be found in Nakajima’s paper.
Then, decomposing \(W,V\) appropriately as \(T^{r+2}-\)modules, we can calculate homology of this modified map of complexes which itself decomposes into 2D components(from \(V,W\) decomposition respectively) which splits the map of complexes itself into 2D components. Now, by a previous lemma, the basis of the \(V_{\alpha}\) component terms is \(\{x^{i-1}y^{i-1} \}\) for \((i,j) \in Y_{\alpha}\). This yields the corresponding Young diagrams. Then, some non-trivial but standard calculations of the cohomology term yields the final result.
Blowups of P2
Let \(\hat{\mathbb{P}^{2}}\) be the blowup of \(\mathbb{P}^{2}\) and let \(p:\hat{\mathbb{P}^{2}} \to \mathbb{P}^{2}\). By definition, \(\hat{\mathbb{P}^{2}}\) is the closed subvariety of \(\mathbb{P}^{2} \times \mathbb{P}^{1}\) defined via
\(\{([z_{0}:z_{1}:z_{2}],[z:w]) \in \mathbb{P}^{2} \times \mathbb{P}^{1} \vert z_{1}w=z_{2}z \}\). Let’s denote by \(l_{\infty}\), the equation \(z_{0}=0\) in \(\hat{\mathbb{P}^{2}}\).
Similarly as before, define \(\hat{M}(r,k,n)\) to be the framed moduli space of torsion-free sheaves \((E,\phi)\) on \(\hat{\mathbb{P}^{2}}\) with rank \(r\), \(\langle c_{1}(E),[C]\rangle=-k\) and $$\langle c_{2}(E)-\frac{r-1}{2r}c_{1}(E)^{2},[\hat{\mathbb{P}^{2}}] \rangle = n.
By a similar argument as in the previous section, we also get:
Lemma 1
\(\hat{M}(r,k,n)\) is a nonsingular variety of dimension \(2nr\) and
\(Hom(E,E(-l_{\infty})) = Ext^{2}(E,E(-l_{\infty})) = 0\)
Anyays, moving on, we can also define again, an action of \(\tilde{T}\) on \(\hat{M}(r,k,n)\) as follows:
for \((t_{1},t_{2}) \in \mathbb{C}^{*} \times \mathbb{C}^{*}\), let \(F_{t_{1},t_{2}}'\) be an automorphism of $$\hat{\mathbb{P}^{2}}$ defined by:
\(F_{t_{1},t_{2}}'([z_{0}:z_{1}:z_{2}],[z:w]) = ([z_{0}:t_{1}z_{1}:t_{2}z_{2}],[t_{z}:t_{2}w])\).
The action of the remaining \(T\) remains the same as before.
There is also another important result than we will need along the way, which relates the framed moduli space of ideal instantons \(M_{0}(r,n)\) from the previous section to out current moduli space. Specifically,
there exists a projective morphism \(\hat{\pi}:\hat{M}(r,k,n) \to M_{0}(r,n-\frac{1}{2r}k(r-k)) where\)0 \leq k<r$$ defined via
\[(E,\phi) \mapsto (((p_{*}E)^{\vee \vee}, \phi), Supp(p_{*}E^{\vee \vee}/p_{*}E) + Supp(R^{1}p_{*}E))\]Proof
(see Appendix 4) — note here that the morphism \(\hat{\pi}\) is equivariant and also the line bundle \(\mathcal{O}(C)\) is an equivariant line bundle as \(C\) is invariant under the \(\mathbb{C}^{*} \times \mathbb{C}^{*}-\) action
Fixed point result (again, generalized)
Just like before, we can characterize fixed points \((E,\phi) \in \hat{M}(r,k,n)\) in a similar manner.
Theorem 1
(1) \((E,\phi) \in \hat{M}(r,k,n)\) is a fixed point of the \(\tilde{T}-\)action if and only if \(E\) has a decomposition \(E=I_{1}(k_{1}C) \oplus \cdots \oplus I_{r}(k_{r}C)\) satisfying the following three conditions for every \(a = 1,\cdots,r\), a. \(I_{a}(k_{a}C)=I_{a} \otimes \mathcal{O}(k_{a}C)\) where \(k_{a} \in \mathbb{Z}\) and \(I_{a}\) is an ideal sheaf of a 0-dim subscheme \(Z_{a} \subset \hat{C}^{2}=\hat{\mathbb{P}^{2}} \backslash l_{\infty}\) b. \(\phi(k_{a}C) \vert_{l_{\infty}}\) is in the \(a\)th factor of \(\mathcal{O}_{l_{\infty}}^{\oplus r}\) c. \(I_{a}\) is fixed by the \(\mathbb{C}^{*} \times \mathbb{C}^{*}\) action, coming from that on \(\hat{\mathbb{P}^{2}}\)
(2) The fixed point set of \(\hat{M}(r,k,n)\) under the \(\tilde{T}-\)action consists of finitely many points parametrized by \(r-\)tuples \(((k_{i},Y_{i}^{1},Y_{i}^{2}))\) for \(i=1,\cdots,r\) where \(Y_{a}^{l}\) are Young diagrams for \(l=1,2\) such that,
\(\sum_{k_{a}} = k\) and \(\sum_{a} (\lvert Y_{a}^{1} \rvert + \lvert Y_{a}^{2} \rvert) + \frac{1}{2r} \sum\limits_{a<b>} \lvert k_{a}-k_{b} \rvert^{2} = n\)
Proof
First note that the fixed point set of \(\mathbb{C}^{*} \times \mathbb{C}^{*}\) in \(\hat{\mathbb{C}^{2}} = \hat{\mathbb{P}^{2}} \backslash l_{\infty}\) consists of 2 points \(([1:0:0],[1:0]),([1:0:0],[0:1])\), let us denote them by \(p_{1},p_{2}\) respectively.
We omit the proof of (1) or leave it as an exercise. For now, we’ll deal with the proof of the second part,
If \(E\) is fixed by the \(T^{2}-\)action, then \(Z_{a}\) is also fixed \(\Rightarrow Supp(Z_{a}) \subset \{p_{1},p_{2}\}\) which means that \(Z_{a}=Z_{a}^{1} \cup Z_{a}^{2}\) of subschemes supported at \(p_{1},p_{2}\) respectively. We’ll just deal with the \(p_{1}\) case for illustration.
Consider the coordinate system \((z_{1}/z_{0},w/z)\) around \(p_{1}\), then \(Z_{a}^{1}\) is generated by monomials \(x^{i}y^{j}\) and then \(Z_{a}^{1}\) corresponds to a Young diagram again(say \(Y_{a}^{1}\)). The same applies for \(p_{2}\), we get \(Y_{a}^{2}\).
Again, we also have a \(\tilde{T}-\)module structure for \(T_{X}\hat{M}(r,k,n)\) as well, where \(X=(E,\phi)\).
Theorem 2
Let \((E,\phi)\) be a fixed point under the \(\tilde{T}\) action, and let \((\vector{k},\vec{Y^{1}},\vec{Y^{2}})\) be the corresponding \(r-\)tuple,
Then, the \(\tilde{T}-\)module structure is given by
\[T_{X}\hat{M}(r,k,n) = \sum\limits_{a,b=1}^{r} \bigl L_{a,b}(t_{1},t_{2}) + t_{1}^{k_{b}-k_{a}}M_{a,b}^{1}(t_{1},t_{2}) + t_{2}^{k_{b}-k_{a}}M_{a,b}^{2}(t_{1},t_{2}) \bigr\]where for the case \(k_{a}>k_{b}\), we have
\(L_{a,b}(t_{1},t_{2}) = e_{b}e_{a}^{-1} \sum\limits_{i,j>0,W_{1}(i,j)=1} t_{1}^{-i}t_{2}^{-j}\) where \(W_{1}(i,j)=1\) if and \(i+j \leq k_{a}-k_{b}-1\) and is \(0\) otherwise
and for the case \(k_{a}+1<k_{b}\),
\(L_{a,b}(t_{1},t_{2}) = e_{b}e_{a}^{-1} \sum\limits_{i,j>0,W_{1}(i,j)=1} t_{1}^{i+1}t_{2}^{j+1}\) where \(W_{2}(i,j)=1\) if and \(i+j \leq k_{a}-k_{b}-2\) and is \(0\) otherwise
and \(L_{a,b}(t_{1},t_{2})=0\) for any other case.
Proof
Here, we will have to perform some \(Ext\) calculations.
We have the decomposition \(E=I_{1}(k_{1}C) \oplus \cdots \oplus I_{r}(k_{r}C)\), from the previous results, the tangent space \(T_{X}\hat{M}(r,k,n)=Ext^{1}(E,E(-l_{\infty}))\) and so this \(Ext\) can be appropriately decomposed as:
\[Ext^{1}(E,E(-l_{\infty}))= \bigoplus_{a,b}Ext^{1}(I_{a}(k_{a}C),I_{b}(k_{b}C-l_{\infty}))\]Each respective factor has weight \(e_{b}e_{a}^{-1}\) above as a \(T-\)module. The remaining task is to describe each respective factor as a \(T^{2}-\)module. Also, we will suppress the \(e_{b}e_{a}^{-1}\) from here onwards.
Let \(Ext^{*}\) be the alternating sum \((-1)^{i} Ext^{i}\). From the section (Moduli Spaces, Lemma 1), we know that the \(Ext^{2}\) term is \(0\) and so \(Ext^{*}(I_{a}(k_{a}C),I_{b}(k_{b}C-l_{\infty}))=-Ext^{1}(I_{a}(k_{a}C),I_{b}(k_{b}C-l_{\infty}))\). Using the exact sequence \(0 \to I_{a} \to \mathcal{O} \to \mathcal{O}_{Z_{a}} \to 0\), we have that
(eqn 1)
\[Ext^{*}(I_{a}(k_{a}C),I_{b}(k_{b}C-l_{\infty}))=Ext^{*}(\mathcal{O}(k_{a}C),\mathcal{O}(k_{b}C-l_{\infty})) - Ext^{*}(\mathcal{O}(k_{a}C),\mathcal{O}_{Z_{b}}(k_{b}C-l_{\infty})) - Ext^{*}(\mathcal{O}_{Z_{a}}(k_{a}C),\mathcal{O}(k_{b}C-l_{\infty})) + Ext^{*}(\mathcal{O}_{Z_{a}}(k_{a}C),\mathcal{O}_{Z_{b}}(k_{b}C-l_{\infty}))\]First, we deal with the last 3 terms here. We can rewrite the last 3 terms as follows,
(eqn 2)
\[- Ext^{*}(\mathcal{O}(k_{a}C),\mathcal{O}_{Z_{b}}(k_{b}C-l_{\infty})) - Ext^{*}(\mathcal{O}_{Z_{a}}(k_{a}C),\mathcal{O}(k_{b}C-l_{\infty})) + Ext^{*}(\mathcal{O}_{Z_{a}}(k_{a}C),\mathcal{O}_{Z_{b}}(k_{b}C-l_{\infty})) = - Ext^{*}(\mathcal{O},\mathcal{O}_{Z_{b}}((k_{b}-k_{a})C-l_{\infty})) - Ext^{*}(\mathcal{O}_{Z_{a}}((k_{a}-k_{b})C),\mathcal{O}(-l_{\infty})) + Ext^{*}(\mathcal{O}_{Z_{a}},\mathcal{O}_{Z_{b}}((k_{b}-k_{a})C-l_{\infty}))\]We have the decomposition $Z_{a}=Z_{a}^{1} \cup Z_{a}^{2}\(according to the support\)p_{1},p_{2}\(. First, note that a mixed term of the kind\)Ext^{*}(\mathcal{O}{Z{a}^{1}}(k_{a}C),\mathcal{O}{Z{b}^{2}}(k_{b}C-l_{\infty})) = 0$$.
Then, we consider the terms for \(Z_{a}^{1},Z_{b}^{1}\), consider a coordinate system \((x,y)=(z_{1}/z_{0},w/z)\). The divisor \(C\) is given by the equation \(x=0\), we also have an isomorphism \(O_{Z_{a}^{1}} \simeq O_{Z_{a}^{1}}\) of sheaves. It becomes an isomorphism of equivariant sheaves by a twist as follows \(O_{Z_{a}^{1}} \simeq t_{1}^{m}O_{Z_{a}^{1}}\). Hence, the corresponding summand above in for \(p_{1}\) in (eqn 2) is:
(eqn 3)
\[t_{a}^{k_{b}-k_{a}} \bigl - Ext^{*}(\mathcal{O},\mathcal{O}_{Z_{b}^{1}})-Ext^{*}(\mathcal{O}_{Z_{a}^{1}},\mathcal{O}(-l_{\infty})) + Ext^{*}(O_{Z_{a}^{1}},\mathcal{O}_{Z_{b}^{1}}(-l_{\infty})) \bigr\]We can consider \(Z_{a}^{1}\) as a subscheme of \(\mathbb{P}^{2}\) supported at \([1:0:0]\) where the \(T^{2}-\) action on \(\mathbb{P}^{2}\) is given by \([z_{0}:z_{1}:z_{2}] \mapsto [z_{0}:t_{1}z_{1}:t_{2}/t_{1}z_{2}]\). Let \(I_{a}^{1}\) be the correponding ideal sheaves. Using the exact sequence \(0 \to I_{a}^{1} \to \mathcal{O}_{\mathbb{P}^{2}} \to \mathcal{O}_{Z_{a}^{1}} \to 0\) ,we get that (eqn 3) is
\[t_{1}^{k_{b}-k_{a}}(Ext^{*}(I_{a}^{1}),I_{b}^{1})- Ext(\mathcal{O}_{\mathbb{P}^{2}},\mathcal{O}_{\mathbb{P}^{2}}(-l_{\infty})))\]The second term is \(0\). Thus, we reduce the problem to the previous case(Tangent Space at Fixed Point, Theorem 1) where we replace \((t_{1},t_{2})\) by \((t_{1},\frac{t_{2}}{t_{1}})\), this yields the \(M_{a,b}^{1}(t_{1},t_{2})\) term in the theorem statement. A similar argument can be employed to deal with the terms corresponding to \(Z_{a}^{1},Z_{b}^{1}\) which would yield the \(M_{a,b}^{2}(t_{1},t_{2})\) .
It remains to deal with the first term of eqn (1), first set \(n=K_{a}-k_{b}\).
Note that \(Ext^{*}(\mathcal{O}(k_{a}C),\mathcal{O}(k_{b}C-l_{\infty}))=-Ext^{1}(\mathcal{O}(k_{a}C),\mathcal{O}(k_{b}C-l_{\infty})) = -H^{1}(\mathcal{O}((k_{b}-k_{a})C-l_{\infty}))\).
For the case \(n=0\), we have \(H^{1}(\hat{\mathbb{P}^{2}},\mathcal{O}(-l_{\infty})) = 0\) by our previous lemma. This matches with \(L_{a,b}\)(that is, the third case).
For the case \(n>0\), consider the exact sequences \(0 \to \mathcal{O}(-nC) \to \mathcal{O}((-n+1)C) \to \mathcal{O}_{C}((-n+1)C) \to 0\) . Note that this equivariant under the \(T^{2}-\)action. Also note that \(C\) is a projective line \(\mathbb{P}^{1}\) with self-intersection (-1) so we have \(H^{1}(C,\mathcal{O}_{C}((-n+1)C))=H^{1}(\hat{\mathbb{P}^{2}}, \mathcal{O}_{\mathbb{P}^{1}}(n-1)) = 0\). So, from the cohomology long exact sequence, we have:
\[0 \to H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(n-1)) \to H^{1}(\hat{\mathbb{P}^{2}},\mathcal{O}(-nC-l_{\infty})) \to H^{1}(\hat{\mathbb{P}^{2}},\mathcal{O}((-n+1)C-l_{\infty})) \to 0\]Using induction, we also have
\[H^{1}(\hat{\mathbb{P}^{2}},\mathcal{O}(-nC-l_{\infty})) = \bigoplus_{d=0}^{n-1} H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(d))\]Since the \(H^{0}\) term is the space of homogeneous polynomials in \(z,w\) of degree \(d\), it is equal to \(\sum\limits_{i=0}^{d} t_{1}^{-i}t_{2}^{-d+i}\) in the ring \(R[T^{2}]\). This matches with the first case of the \(L_{a,b}\) expression.
We leave the remaining \(n<0\) case as an exercise.
(the remaining part of the proof will be discussed in the next part of this post along with some discuss about Nekrasov’s solution itself, which is very different from Nakajima’s. There will also be a deeper discussion on some of the physics-related ideas)