# Defects, nested instantons and comet shaped quivers

## TL;DR

This paper introduces a surface defect in 4D gauge theories with nested instantons, proposing a comet-shaped quiver gauge theory on a torus and connecting it to elliptic cohomology and Macdonald polynomials.

## Contribution

It develops a new effective theory for nested instantons on a Riemann surface, providing explicit formulas and conjectural links to elliptic cohomology and symmetric functions.

## Key findings

- Proposes a comet-shaped quiver gauge theory on T^2.
- Derives explicit formulas for the virtual equivariant elliptic genus.
- Establishes connections with elliptic cohomology and modified Macdonald polynomials.

## 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 $X=T^2\times T^*{\mathcal C}_{g,k}$, 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 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.