Three-Dimensional Mirror Symmetry for Characteristic Classes on Bow Varieties

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The geometric manifestation of solving systems of polynomial equations involves studying the "shape" of all solutions of such a system, which is called an algebraic variety. Most of the points of an algebraic variety are smooth in the sense that their neighborhoods look like n-dimensional linear spaces. However, certain points called “singularities” do not have this smooth property, for example the vertex of a cone. There is a plethora of complicated possible singularities, and their understanding has applications in physics, robotics, and economics. A key tool to study singularities is assigning calculable discrete invariants (for example numbers or sequences of numbers) to them in such a way that the discrete invariant encodes some of the geometric properties of the singularity. On type of these discrete invariants are what are called "characteristic classes," which come in many variants and flavors. Remarkably, a recently discovered characteristic class of singularities, called a "stable envelope," also appears in string theory in theoretical physics. The relation to string theory suggests a so-far hidden symmetry of characteristic classes. Namely, it predicts that there are pairs of seemingly unrelated algebraic varieties whose stable envelopes of singularities coincide. The main goal of the project is to prove this statement in its mathematical setting and derive geometric applications. The project will provide research training opportunities for graduate students. The PI will generalize the concept of stable envelopes from Nakajima quiver varieties to a broader class of varieties called Cherkis bow varieties. The algebraic combinatorics underlying the geometric study of bow varieties are NS5-D5 brane configurations, which are more complete than that of quiver varieties or homogeneous spaces. This fact gives rise to new operations on bow varieties: Hanany-Witten transition and combinatorial 3d mirror symmetry. The PI will utilize these operations to build a Hall algebra structure on the elliptic cohomology of bow varieties, generalizing cohomological and K theoretic Hall algebras of quivers. The elliptic Hall algebra structure complements the geometric study of singularities, and the two together will be used to organize an inductive proof of a three-dimensional mirror symmetry statement. The cohomological and K-theoretic limits provide coincidences among motivic invariants of singularities. Dimension count arguments of the same elliptic Hall algebra promise Donaldson-Thomas invariants, as well as identities among elliptic functions that generalize quantum dilogarithm identities.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.