Quantum Cohomology of Slices of the Affine Grassmannian Ivan Danilenko Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2020 c 2020 Ivan Danilenko All Rights Reserved Abstract Quantum Cohomology of Slices of the Affine Grassmannian Ivan Danilenko The affine Grassmannian associated to a reductive group G is an affine analogue of the usual flag varieties. It is a rich source of Poisson varieties and their symplectic resolutions. These spaces are examples of conical symplectic resolutions dual to the Nakajima quiver varieties. In this work, we study their quantum connection. We use the stable envelopes of D. Maulik and A. Okounkov[MO2] to write an explicit formula for this connection. In order to do this, we construct a recursive relation for the stable envelopes in the G = PSL2 case and compute the first-order correction in the general case. The computation of the purely quantum part of the multiplication is done based on the deformation approach of A. Braverman, D. Maulik and A. Okounkov[BMO]. For the case of simply-laced G, we identify the quantum connection with the trigonometric Knizhnik-Zamolodchikov equation for the Langlands dual group G_. Contents List of Figuresv Acknowledgments vi 1 Introduction1 1.1 Overview...................................1 1.2 Overview of the main result........................3 1.2.1 Stable basis and purely quantum multiplication.........3 1.2.2 Classical multiplication.......................6 1.2.3 Identification............................7 1.3 Structure of the thesis...........................9 2 Slices of the affine Grassmannian 10 2.1 Representation-theoretic notation..................... 10 2.2 Classical constructions........................... 11 2.2.1 Affine Grassmannian........................ 11 2.2.2 Transversal slices.......................... 15 2.3 Poisson structures and symplectic resolutions............... 16 2.3.1 Poisson structure.......................... 16 2.3.2 Symplectic resolutions....................... 19 2.3.3 Resolutions of slices......................... 22 2.4 Group action on resolutions........................ 24 i 2.5 Beilinson-Drinfeld Grassmannian..................... 25 2.6 Homology of resolutions.......................... 31 3 T-equivariant geometry 36 3.1 T-fixed points................................ 36 3.2 Tangent weights at T-fixed points..................... 38 3.2.1 Tangent spaces of the affine Grassmannian............ 38 3.2.2 Tangent spaces of the G (O)-orbits of the affine Grassmannian. 40 3.2.3 Tangent spaces of the resolutions of slices............. 41 3.2.4 Classical examples......................... 49 3.2.5 Combinatorial descripltion of weight multiplicities........ 53 3.3 Behavior on walls in T ........................... 58 3.3.1 Walls................................. 58 3.3.2 Affine root subgroups........................ 60 3.3.3 Invariant curves in Gr........................ 61 3.3.4 Some T-invariant curves in the resolutions of slices....... 62 3.3.5 All T-invariant curves in the resolutions of slices......... 65 3.4 T-equivariant line bundles......................... 67 3.5 More on Beilinson-Drinfeld Grassmannian................ 77 3.6 Behavior on walls in A ........................... 79 4 Stable Envelopes 84 4.1 Preliminaries on Equivariant Cohomology................ 84 4.2 Stable Envelopes.............................. 86 4.2.1 Assumptions............................. 86 4.2.2 Chambers.............................. 87 4.2.3 Attracting manifolds........................ 87 4.2.4 Partial order............................. 88 4.2.5 Support and degree in A ...................... 88 ii 4.2.6 Polarizations............................ 89 4.2.7 Definition of stable envelopes................... 90 4.2.8 First properties........................... 91 4.2.9 Steinberg correspondences..................... 92 4.2.10 Torus restriction.......................... 94 2 4.3 Stable Envelopes restrictions modulo ~ : A1 type............ 95 4.3.1 Steinberg correspondences in type A1 ............... 95 4.3.2 Recursion.............................. 101 4.3.3 Formulas for stable envelopes restrictions: type A1 ........ 104 4.4 Stable Envelopes restrictions modulo ~2: general case.......... 110 4.4.1 Wall-crossing for stable envelopes................. 110 4.4.2 Formulas for stable envelopes restrictions: general case..... 113 4.5 Classical Multiplication........................... 115 4.5.1 Stable envelopes and classical multiplication........... 115 4.5.2 Main result on classical multiplication.............. 118 5 Quantum Cohomology 123 5.1 Preliminaries on quantum cohomology.................. 124 5.2 Reduction to Picard rank 1........................ 126 5.2.1 Two approaches to the reduced virtual class........... 126 5.2.2 Choice of basis........................... 129 5.2.3 Reduction to walls......................... 132 5.3 Computation of wall contributions..................... 136 5.3.1 Wall deformations.......................... 136 5.3.2 Unbroken maps........................... 138 5.3.3 Off-diagonal case.......................... 140 5.3.4 Diagonal case............................ 150 5.3.5 Quantum multiplication...................... 154 iii 6 Representation-Theoretic Reformulation 158 6.1 Trigonometric Knizhnik-Zamolodchikov connection........... 158 6.2 Quantum connection............................ 163 6.3 Identification................................ 165 iv List of Figures 3.1 A typical path associated to a fixed point................. 38 3.2 Tangent spaces............................... 47 3.3 Left and Right Conditions......................... 48 3.4 Incidence conditions on T α_ and T α_ ................... 49 Σi Σi+1 α_ 3.5 A typical picture of Tp X at a fixed point p................ 49 α_ −α_ 3.6 A typical picture of Tp X ⊕ Tp X at a fixed point p.......... 50 3.7 Tangent spaces at the T-fixed points of the resolution of type A4 .... 51 ∗ − 3.8 Tangent spaces at the T-fixed points of T G=Pν ............ 52 3.9 Tangent spaces at the T-fixed points of T ∗G=B ............. 53 1 _ 3.10 Typical pictures for four possible cases of k = n + intersecting with Γα . 54 2 p 3.11 Weight multiplicities in TpX for p represented by the red path...... 57 3.12 An example of two T-fixed points connected by a curve......... 64 v Acknowledgments First of all, I would like to thank my advisor Andrei Okounkov for his guidance in the last five years. He was the center of an incredible group of students and postdocs at Columbia, and his enthusiasm and knowledge was the driving force for our community. I would like to thank my fellow students at Columbia Noah Arbesfeld, Micah Gay, Yakov Kononov, Dmitrii Korb, Henry Liu, Renata Picciotto, Dmitrii Pirozhkov, Petr Pushkar, Andrey Smirnov, Shuai Wang and many others. I can't overestimate the impact of such a welcoming environment on my life. My deepest gratitude goes to all people whose support and guidance helped me to love mathematics and go to a graduate school, especially Natalia Soprunova, Elena Bunina, Petr Sergeev, Evgeni Vyrodov, Alexei Morozov and Ivan Cherednik. I am thankful to Mina Aganagic, Joel Kamnitzer, Davesh Maulik, Andrei Negut, Alex Weekes and Changjian Su for the discussions related to this work. Last but not least, I would like to thank my family for being supportive. vi to my son Kirill vii Chapter 1 Introduction 1.1 Overview The central object in this work is the moduli space called the affine Grassmannian Gr. It is associated to a complex connected reductive group G, and one can think of it as an analogue of flag varieties for a corresponding Kac-Moody group. It is known to have deep connections with representation theory and Langlands duality [G, MV1, MV2]. More precisely, the geometric objects we study are transversal slices in affine Grass- mannians. To give an idea of what these slices are, recall that the affine Grassmannian Gr has a cell structure similar to the Schubert cells in the ordinary flag variety. A λ λ transversal slice Grµ describes how one orbit is attached to another. The slices Grµ first appeared in mathematical physics as Coulomb branches in supersymmetic theo- ries [SW]. They are naturally algebraic Poisson varieties and, in some cases, admit a λ λ smooth symplectic resolution Grµ ! Grµ, which is a notion of independent mathemat- ical interest [K2,K3,BK]. The goal of this work is to study the quantum cohomologies of these resolutions. One of our motivations is 3d mirror symmetry. It is known to exchange the Coulomb (X) and the Higgs branch (X_) of vacuua in a 3d supersymmetric field theory [IS,SW]. In [MO2] D. Maulik and A. Okounkov studied the quantum cohomology for Nakajima quiver varieties [N2,N1,N3], symplectic resolutions which naturally arise as the Higgs 1 branch of vacua X_. Mirror symmetry is a source of conjectural relations between the enumerative geometry of X and X_, and understanding the Coulomb side makes it possible to verify this relation directly. In this project the main algebraic structure which captures the enumerative geom- ∗ etry of X is the (equivariant) quantum cohomology QHT (X). It is a deformation of usual (equivariant) cohomology by curve counts called Gromov-Witten invariants. The ∗ work of Maulik and Okounkov [MO2] showed that QHT (X) of Nakajima quiver vari- eties admit a natural action of a quantum group called the Yangian. The crucial step in this construction is a special basis of geometric origin in the equivariant cohomology ∗ HT (X), called stable envelopes. It is defined for