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 \(X=T^2\times T^*{{\mathcal {C}}}_{g,k}\), 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 \(\mathrm{{D7}}\)-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.

我们在支持嵌套实例的三维量规理论中引入并研究了表面缺陷，该缺陷涉及量规在缺陷处的抛物线形化。这是通过非紧凑型Calabi–Yau三倍X上的\（\ mathrm {{D3 / D7}} \）-脑系统设计的。对于\（X = T ^ 2 \乘以T ^ * {{\数学{C}}} _ {g，k} \），两个圆环的乘积\（T ^ 2 \）乘以一个余弦束带有标记点的黎曼曲面\（{{\ mathcal {C}}} _ {g，k} \），我们提出了在\（{\ mathcal C} _ {{g，k} \）作为\（T ^ 2 \）上的彗星状颤动量规理论给出彗星的尾部由每个标记点的旗颤组成，头部描述由属g引起的自由度。在数学上，我们针对仿射平面上点的嵌套Hilbert方案的模空间上某个束的虚拟等变椭圆类的单个\（\ mathrm {{D7}} \）- brane猜想显式，得到该方程组。自然会出现与字符变体的椭圆同调和椭圆形修改的Macdonald多项式有关的联系。