Cherednik algebras, \(W\)-algebras and the equivariant cohomology of the moduli space of instantons on \(\mathbb A^2\)
The authors observe a striking relation between \(N=2\) supersymmetric gauge theory in dimension four and two-dimensional Conformal Field theory. On one hand, there is the conformal blocks of Liouville theory associated with a punctured Riemann surface and the group \(U\), and on the other hand, there is the instanton part of the Nekrasov partition function for a suitable four-dimentional gauge theory associated with the group \(U_r\). Parts of this correspondence are obtained in the physics literature. In mathematical terms, a consequence of the AGT conjecture is the existence of a representation of the affine \(W\)-algebra of \(G\) on the equivariant intersection cohomology of the moduli space of \(G^L\)-instantons on \(\mathbb R^4\) satisfying some extra properties. \(G\) and \(G^L\) denotes a pair of complex reductive groups which are dual to each other in the sense of Langlands. This paper studies this action in detail for the gauge group \(G=G^L=\text{GL}_r\). A detailed construction is given, based on degenerate double affine Hecke algebras. The main result is described more precisely in what follows: let \(M_r=\bigsqcup_{n>0}M_{r,n}\) be the moduli space of rank \(r\) torsion free sheaves on \(\mathbb P^2\), equipped with a framing along \(\mathbb P^1_\infty\subset\mathbb P^2\). For fixed \(n\), \(M_{r,n}\) is a smooth symplectic variety, \(\dim M_{r,n}=2rn\), acted upon by a \(r+2\)-dimensional torus \(\tilde D=(\mathbb C^\times)^2\times D\) where \((\mathbb C^\times)^2\) acts on \(\mathbb P^2\) and \(D=(\mathbb C^\times)^r\) acts on the framing. In the particular case when \(r=1\), the moduli space \(M_{1,n}\) is isomorphic to the Hilbert scheme \(\text{Hilb}_n\) of \(n\) points on \(\mathbb C^2\). This article is in some sense guided by Nakajima's construction of a representation of the rank one Heisenberg algebra on the space \(\tilde L^{(1)}=\bigoplus_{n\geq 0} H_\ast(\text{Hilb}_n)\). The construction is by geometric methods, identifying it with the standard level one Fock space. For \(r\geq 1\) there is a representation of rank one Heisenberg algebra on the space \(L^{r}=\bigoplus_{n\geq 0}H_\ast^{\tilde D}(M_{r,n})\). This is however neither irreducible nor cyclic. Let \(R_r=\mathbb C[x,y,e_1,\dots,e_r]\), \(K_r=\mathbb C(x,y,e_1,\dots,e_r)\), be the cohomology ring of the classifying space of \(\tilde D\) and its fraction field respectively. \(L^{(r)}\) is then an \(R_r\)-module, and so it makes sense to let \(L_K^{(r)}=L^{(r)}\otimes_{R_r}K_r\) and \(\kappa=-y/x,\;\varepsilon_i=e_i/x,\;\xi=1-\kappa,\;i\in[1,r].\) For the level \(k\) affine \(W\)-algebra of \(\mathfrak{gl}_r\), \(W_k(\mathfrak{gl}_r)\), the cup product in equivariant cohomology yields a bilinear map \((\bullet,\bullet):L_K^{(r)}\times L_K^{(r)}\rightarrow K_r\) called the intersection pairing. Let \(\overset\rightarrow e=(e_1,e_2,\dots,e_r)\), \(\overset\rightarrow\varepsilon = \overset\rightarrow e/x\), and \(\rho=(0,-1,-2,\dots,1-r)\). The main theorem of this paper is the following: \noindent (a) There is a representation of \(W_k(\mathfrak{gl}_r)\) of level \(k=\kappa-r\) on \(L_K^{r}\), identifying it with the Verma module \(M_\beta\) of highest weight \(\beta=-(\overset\rightarrow\varepsilon+\xi\rho)/\kappa\). \noindent (b) The action is quasi-unitary with respect to the intersection pairing on \(L_K^{r}\). \noindent (c) The Gaiotto state \(G=\sum_{n\geq 0}G_n\), \(G_n=[M_{r,n}]\) is a Whittaker vector of \(M_\beta\). As \(W_k(\mathfrak{gl}_1)\) is a Heisenberg algebra of rank one, the result is a generalization to higher ranks of the representation of the Heisenberg algebra on the equivariant cohomology of the Hilbert scheme. For \(r=2\), the theorem implies an action of the Virasoro algebra on the cohomology of the moduli space of \(U_2\) instantons on \(\mathbb R^4\). Of utmost interest is the relation with the AGT conjecture for the pure \(N=2\) supersymmetric gauge theory: Nekrasov's partition function is the generating function of the integral of the equivariant cohomology class \(1\in H^D_\ast(M_{r,n})\), which gives \(Z(x,y,\overset\rightarrow e;q)=\underset{n\geq 0}\sum q^n([M_{r,n}],[M_{r,n}])\). The Gaiotto state \(G\) is an element of the completed Verma module \(\widehat M_\beta=\underset{n\geq 0}\prod M_{\beta,n}\), where \(M_{\beta,n}=H^{\tilde D}_\ast(M_{r,n})\otimes_{R_r}K_r\). Letting \(\{W_{d,l};l\in\mathbb Z,\;d\in[1,r]\}\) be the set of the Fourier modes of the generating fields of \(W_k(\mathfrak{gl}_r)\), \(M_\beta\) has a unique bilinear form \((\bullet,\bullet)\) such that the highest weight vector has norm \(1\) and the adjoint of \(W_{d-1}\) is \(W_{d,l}\) for \(l\geq 0\). Then \(G\) is uniquely determined by the Whittaker condition which gives \(Z(x,y,\overset\rightarrow e;q)=\underset{n\geq 0}\sum q^n(G_n,G_n)\). The present article's main objective is to prove the above theorem. Since \(W\)-algebras doesn't in general possess a presentation by generators and relations, an intermediate algebra \(\text{SH}^c\), defined over the field \(F=\mathbb C(\kappa)\) is applied. This algebra does act on \(L_K^{(r)}\) by some correspondences. Lot of this rather long article is used for its definition and main properties: (a) \(\text{SH}^c\) is \(\mathbb Z\)-graded, \(\mathbb N\)-filtered, and has a triangular decomposition. (b) For \(n\geq 1\) there is a surjective algebra homomorphism \(\Psi_n:\text{SH}\rightarrow\text{SH}_n\) with \(\bigcap_n\text{Ker}\Psi_n=\{0\}\) where \(\text{SH}\) is the specialization of \(\text{SH}^c\) at \(c_0=0\) and \(c_l=-\kappa^l\omega^l\) for \(l\geq 1\). (c) The part of order \(\leq 0\) for the \(\mathbb N\)-filtration is \(\text{SH}^c[\leq 0]=\mathcal H\otimes F[c_l;l\geq 2]\). The algebra \(\text{SH}^c\) is generated by \(\text{SH}^c[\leq 0]\) and \(D_{0,2}\). (d) Letting \(\text{SH}^{(1,0,\dots)}\) be the specialization of \(\text{SH}^c\) at \(c_0=1\) and \(c_l=0\) for \(l\geq 0\), it has a faithful representation in \(\Lambda=F[p_l;l\geq 1]\) such that \(\mathcal H\) acts in the standard way and \(D_{0,2}\) acts as the Laplace-Beltrami operator. (e) The specialization of \(\text{SH}^{(1,0,\dots)}\) at \(\kappa=1\) is isomorphic to the universal algebra of the Witt algebra \(W_{1+\infty}\). These properties are all elements in the proof of the following essential theorems about \(\text{SH}\): A. Let \(\text{SH}^{r}_K\) be the specialization of \(\text{SH}^c\otimes K_r\) to \(c_0=r\) and \(c_i=p_i(\varepsilon_1,\dots,\varepsilon_r)\). Then there is a faithful representation \(\rho^{(r)}\) of \(\text{SH}^{r}_K\) on \(L^{(r)}_K\) such that \(L^{(r)}_K\) is generated by the fundamental class of \(M_{r,0}\). B. There is an embedding of graded and filtered algebras \(\Theta^{(r)}:\text{SH}^{r}_K\rightarrow\mathcal U(W_k(\mathfrak{gl}_r))\), where \(\mathcal U(W_k(\mathfrak{gl}_r))\) is a quotient of \(\mathfrak U(W_k(\mathfrak{gl}_r))\), the current algebra of \(W_k(\mathfrak{gl}_r)\). This extends to a surjective morphism \(\mathfrak U(\text{SH}_K^{(r)})\rightarrow\mathcal U(W_k(\mathfrak{gl}_r))\). The map \(\Theta^{(r)}\) induces an equivalence between the categories of admissible \(\text{SH}_K^{(r)}\) and \(U(W_k(\mathfrak{gl}_r))\) modules. C: The representation \(L^{r}_K\) of \(W_k(\mathfrak{gl}_r)\) is a Verma module. It is quasi-unitary with respect to the intersection pairing. The element \(G\) is a Whittaker vector for \(W_k(\mathfrak{gl}_r)\). In addition to the intricate definition of \(\text{SH}^c\) (which includes parts from Heisenberg and Virasoro subalgebras), there is also a need for the definition of equivariant cohomology of the Hilbert scheme, of the moduli space of torsion free sheaves, and of the commuting variety. Then comultiplication on \(\text{SH}^c\) can be defined and the relation to \(W_k(\mathfrak{gl}_r)\) is set. Finally, the authors achieve the Gaiotto state which in essence is the definition of the element \(G\) and and its Whittaker condition. A lot of detailed computations and proof of results needed in the exposition is given in a series of appendices from A to G. The article is really a good illustration of the connection between algebra and physics, given (extremely roughly) by representations as a state space. However, this is, as the authors admit, very involved and makes the article depend on a lot of advanced mathematics.
scientific article; zbMATH DE number 6233894
scientific article