Abstract: |
3d mirror symmetry is a duality in physics, where Higgs and Coulomb branches of certain pairs of 3d N=4 SUSY gauge theories are exchanged with each other. Motivated from this, M. Aganagic and A. Okounkov introduced the enumerative geometric conjecture that the vertex functions of the mirror theories are related to each other. The two sets of q-difference equations satisfied by the vertex functions, in terms of the K\"ahler and equivariant parameters, are expected to exchange with each other. The conjecture therefore leads to a nontrivial relation between their monodromy matrices, the so-called elliptic stable envelopes. In this talk, I will discuss the proof in several cases of the conjecture for both vertex functions and elliptic stable envelopes. This is based on joint works with R. Rim\'anyi, A. Smirnov, and A. Varchenko. |