Defects, nested instantons and comet-shaped quivers

Abstract

We introduce and study a surface defect in four-dimensional gauge theories supporting nested instantons with respect to the parabolic reduction of the gauge group at the defect. This is engineered from a mathrm{{D3/D7}}-branes system on a non-compact Calabi–Yau threefold X. For X=T^2times T^*{{mathcal {C}}}_{g,k}, the product of a two torus T^2 times the cotangent bundle over a Riemann surface {{mathcal {C}}}_{g,k} with marked points, we propose an effective theory in the limit of small volume of {mathcal C}_{g,k} given as a comet-shaped quiver gauge theory on T^2, the tail of the comet being made of a flag quiver for each marked point and the head describing the degrees of freedom due to the genus g. Mathematically, we obtain for a single mathrm{{D7}}-brane conjectural explicit formulae for the virtual equivariant elliptic genus of a certain bundle over the moduli space of the nested Hilbert scheme of points on the affine plane. A connection with elliptic cohomology of character varieties and an elliptic version of modified Macdonald polynomials naturally arises.