Algebras of differential operators on homogeneous spaces can often be "\(q\)-deformed" to give algebras of difference operators, where the differentiation operators are replaced by some discrete shifts, using the group action. The most well-known examples of these are probably the double affine Hecke algebras, introduced by Cherednik. The DAHA is a certain algebra of Weyl-group equivariant difference operators on the Cartan subgroup of \(GL_N\), which deforms the double affine Weyl group.
In this talk, I'll explain how DAHA's arise naturally in a seemingly distant context: using the quantum group \(U_q(gl_N)\), we have defined a certain fully extended \((3+1)\)-dimensional topological field theory -- the "big sister" to Reshetikhin-Turaev theory -- whose value on a two-torus with a single marked point recovers the category modules for the DAHA in type \(A\). This talk is based on joint work with David Ben-Zvi and Adrien Brochier; the main technical tool is called "factorization homology", a notion introduced by Lurie and Ayala-Francis in recent years.