Hall algebra multiplication for stable envelopes on bow varieties
Tommaso Maria Botta, Richard Rimanyi
TL;DR
This work delivers an explicit formula for elliptic stable envelopes on type A Cherkis bow varieties by expressing them as shuffles of simple one-tie building blocks within an elliptic cohomology Hall algebra. It develops a unified framework across cohomology, K-theory, and elliptic cohomology, including a detailed treatment of D5 resolutions, parabolic induction, and the role of dynamical (Kähler) parameters. A central contribution is the W-tilde and W functions whose ordered products encode stable envelopes, together with wheel conditions ensuring consistency of the shuffle structure. The main result, proven by reduction to partial flag varieties and maximal resolutions, yields explicit formulas and shows that 3d mirror symmetry translates into theta-function identities (and Fay-type trisecant identities) among these expressions, providing concrete, computable identities for fixed-point data encoded by 01-matrices that share row/column sums. This work thus links bow-variety geometry, elliptic Hall algebras, and explicitly computable theta-function identities with potential applications to geometric representation theory and quantum group representations.
Abstract
Elliptic stable envelopes are fundamental components in the geometric realization of quantum group representations. We present a formula for elliptic stable envelopes on type A Cherkis bow varieties, as a product of simple basic objects in an elliptic cohomology Hall algebra. Combined with the 3d mirror symmetry property of elliptic stable envelopes, our result implies theta function identities for any pair of 01-matrices sharing the same row and column sums.