This is joint work in progress with Mina Aganagic, Spencer Tamagni and Peng Zhou.

 

ADE type gauge groups appear in string theory via compactifying on the Calabi-Yau 3-folds which are ADE surfaces times the complex line. This is realized by the Mckay correspondence, in which the root system of ADE Lie algebras can be recovered from the minimal resolution of ADE surface singularities. The geometric Langlands duality and 3D N=4 mirror symmetry can be proposed physically from S duality after this construction. In the vast world of related proposals, there is a 2D homological mirror symmetry between Coulomb branches of certain 3D N=4 quiver gauge theories, as stated in Aganagic's ICM talk and partially proved in https://arxiv.org/abs/2406.04258. As an example, the Coulomb branch of the 3D N=4 GL(k) gauge theory with n copies of fundamental representations as matter content is related to the k-th horizontal Hilbert scheme of the A_{n-1} surface.

 

In this talk we are interested in extending this picture to the Lie supergroups GL(m|n). We will first discuss how the GL(m|n) root system can be similarly recovered from minimal resolutions of the 3-fold singularity xy = z^mw^n. Then we propose a 2D mirror symmetry parallel to the result mentioned above. It is an equivalence between the Fukaya category of symmetric product of an (m+n+2)-punctured sphere and the category of coherent sheaves on the Coulomb branch of a gauge theory associated to the 'GL(1|1) quiver'. The geometry on the coherent sheaf side is related to the horizontal Hilbert scheme of a minimal resolution of the 3-fold singularity xy = z^mw^n. It is also particularly interesting to study the braid group action on the two sides induced by braiding the punctures on the sphere, for which I hope the audience will provide good advice.