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 D3/D7-branes system on a non compact Calabi-Yau threefold X. For X = T 2 × T * C g,k , the product of a two torus T 2 times the cotangent bundle over a Riemann surface C g,k with marked points, we propose an effective theory in the limit of small volume of 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 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.

Flags of sheaves, quivers and symmetric polynomials

“…,n,µ (with the notations of [1]) will be addressed to as the moduli space of stable representations of (1.1) or, equivalently, as the moduli space of nested instantons. Altogether, the previous considerations prove the following theorem.…”

“…Let us remark that virtual invariants of V g have a connection to the cohomology of character varieties of punctured Riemann surfaces, and in particular to the conjecture proposed in [19] whose physical interpretation was provided in [7]. The interested reader can find the details in [1].…”

“…In [1] we introduced the moduli space of nested instantons as the moduli space of stable representations of a suitable quiver. This arises in the study of surface defects in supersymmetric gauge theory on T 2 × C g,k , where T 2 is a real two torus and C g,k a genus g complex projective curve with k marked points.…”

“…String theory motivations: The D-brane set-up engineering the surface defect is described in [1], and its analysis naturally led to a description in terms of representations of a quiver in the category of vector spaces, the D-branes being the objects and the open strings being the morphisms. Let us briefly resume the D-brane geometry and its relation with the relevant mathematical problems.…”

“…

We study the representation theory of the nested instantons quiver presented in [1], which describes a particular class of surface defects in four-dimensional supersymmetric gauge theories. We show that the moduli space of its stable representations provides an ADHM-like construction for nested Hilbert schemes of points on C 2 , for rank one, and for the moduli space of flags of framed torsion-free sheaves on P 2 , for higher rank.

…”