Five dimensional gauge theories and vertex operators (Q2876634)

This paper was written and narrowly circulated back in 2008. It studies gauge theories with gauge groups being products of unitary groups \(U(r)\times U(r')\) on five dimensional spaces \(X\times S^1\), where \(X\) is a complex surface. The moduli space of \(U(r)\) instantons is partly compactified to the space \(\mathcal{M}(r)\) of torsion free rank \(r\) sheaves on a compactification of \(X\), and the matter in the bifundamental representation is described by a natural sheaf formed by \(\mathrm{Ext}\) groups on \(\mathcal{M}(r)\times\mathcal{M}(r')\). It defines a Fourier-Moukai operator between \(K\)--groups \(\Phi_{\mathrm{Ext}}\!\!: K_{G\times T}(\mathcal{M}(r))\to K_{G\times T}(\mathcal{M}(r'))\), where \(G\) is the group of constant gauge transformations and \(T\) is a maximal torus of \(\mathrm{GL}(2,\mathbb{C})\).NEWLINENEWLINEFor \(r=r'=1\) the correct \(G\times T\) equivariant cohomology was computed by Carlsson and Okounkov, this paper generalizes the computation to \(K\)--theory. It is in the spirit of BPS/CFT correspondence, which reduces BPS protected observables to correlation functions in a \(q\)--deformed two dimensional CFT. The operator \(\Phi_{\mathrm{Ext}}\) is computed through the Macdonald-Mehta-Cherednik identity, which expresses the values of of the Hermitian form \(\langle f,\Theta g\rangle_{\Delta}\) in the basis of the Macdonald polynomials. Here \(\langle\cdot,\cdot\rangle_{\Delta}\) is the Macdonald's Hermitian inner product for the root system of \(\mathrm{GL}(N)\), and \(\Theta\) is the theta function of its weight lattice. As an application the authors compute the partition functions for \(A_r\)--type quiver \(U(1)^{r+1}\) theories by reducing instanton sums to the trace of a product of vertex operators.NEWLINENEWLINEAs a side issue it is observed that the Wiener-Hopf factorization of \(\Theta\) produces an operator that takes the orthogonal Macdonald polynomials to the interpolation Macdonald polynomials. For general lattices the support of the factorization is a half-space of the lattice, but for \(\mathrm{GL}(N)\) it is much smaller, the positive orthant of \(\mathbb{Z}^N\). This explains why the theory of interpolation Macdonald polynomials is much richer for \(\mathrm{GL}(N)\) than say for \(\mathrm{SL}(N)\).