Yangians and quantizations of slices in the affine Grassmannian

We study quantizations of transverse slices to Schubert varieties in the affine Grassmannian. The quantization is constructed using quantum groups called shifted Yangians --- these are subalgebras of the Yangian we introduce which generalize the Brundan-Kleshchev shifted Yangian to arbitrary type. Building on ideas of Gerasimov-Kharchev-Lebedev-Oblezin, we prove that a quotient of the shifted Yangian quantizes a scheme supported on the transverse slices, and we formulate a conjectural description of the defining ideal of these slices which implies that the scheme is reduced. This conjecture also implies the conjectural quantization of the Zastava spaces for PGL(n) of Finkelberg-Rybnykov.Comment: 37 pages; v2, slightly strengthened Theorem 2.

Vanishing cycles on Poisson varieties

We extend slightly the results of Evens-Mirkovi\'c, and "compute" the characteristic cycles of Intersection Cohomology sheaves on the transversal slices in the double affine Grassmannian and on the hypertoric varieties. We propose a conjecture relating the hyperbolic stalks and the microlocalization at a torus-fixed point in a Poisson variety.Comment: 7 page

Multiplicative slices, relativistic Toda and shifted quantum affine algebras

We introduce the shifted quantum affine algebras. They map homomorphically into the quantized $K$-theoretic Coulomb branches of $3d\ {\mathcal N}=4$ SUSY quiver gauge theories. In type $A$, they are endowed with a coproduct, and they act on the equivariant $K$-theory of parabolic Laumon spaces. In type $A_1$, they are closely related to the open relativistic quantum Toda lattice of type $A$.Comment: 125 pages. v2: references updated; in section 11 the third local Lax matrix is introduced. v3: references updated. v4=v5: 131 pages, minor corrections, table of contents added, Conjecture 10.25 is now replaced by Theorem 10.25 (whose proof is based on the shuffle approach and is presented in a new Appendix). v6: Final version as published, references updated, footnote 4 adde