AbstractWe 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 $$\mathrm{{D3/D7}}$$
D
3
/
D
7
-branes system on a non-compact Calabi–Yau threefold X. For $$X=T^2\times T^*{{\mathcal {C}}}_{g,k}$$
X
=
T
2
×
T
∗
C
g
,
k
, the product of a two torus $$T^2$$
T
2
times the cotangent bundle over a Riemann surface $${{\mathcal {C}}}_{g,k}$$
C
g
,
k
with marked points, we propose an effective theory in the limit of small volume of $${\mathcal C}_{g,k}$$
C
g
,
k
given as a comet-shaped quiver gauge theory on $$T^2$$
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 $$\mathrm{{D7}}$$
D
7
-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.