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.
Highlights
Introduction and discussionThe study of defects can be used to characterize the behavior of physical theories and the related mathematical structures
In this paper we are interested in surface defects in four-dimensional supersymmetric gauge theories; namely, real codimension two submanifolds were a specific reduction of the gauge connection taking place
In this paper we introduce and study surface defects supporting nested instantons with respect to the parabolic reduction of the gauge group at the defect
Summary
The study of defects can be used to characterize the behavior of physical theories and the related mathematical structures. The effective theory describing the dynamics of such surface defects is obtained in the limit of small area of C and turns out to be a quiver gauged linear sigma model which flows in the infrared to a nonlinear sigma model of maps from T 2 to the moduli space of nested instantons. We study the circle reduction of this system, which leads to a T-dual D6/D2 quantum mechanics In this case, we find that the generating function of the defects, obtained by summing over all possible decompositions of the connection at the puncture, or in other terms over all possible tails of the quiver, displays a very nice polynomial structure in the equivariant parameters.
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