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Equivariant Quantum Cohomology and the Satake Equivalence MASSACHUSETS INSTITUTE Geometric OF TECHNOLOGY by JUN 16 2016 Michael Viscardi LIBRARIES Submitted to the Department of Mathematics ARCIME in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

June 2016

Sl7 I Signature redacted A u th o r ...... Department of Mathematics May 18, 2016

Certified by Signature redacted Roman Bezrukavnikov f/ Professor of Mathematics, MIT Thesis Supervisor

Accepted by... Signature redacted Alexei Borodin Chairman, Department Committee on Graduate Theses 2 Equivariant Quantum Cohomology and the Geometric Satake Equivalence by Michael Viscardi

Submitted to the Department of Mathematics on May 18, 2016, in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Abstract

Recent work on equivariant aspects of mirror symmetry has discovered relations between the equivariant quantum cohomology of symplectic resolutions and Casimir- type connections (among many other objects). We provide a new example of this theory in the setting of the affine Grassmannian, a fundamental space in the geometric Langlands program. More precisely, we identify the equivariant quantum connection of certain symplectic resolutions of slices in the affine Grassmannian of a semisimple group G with a trigonometric Knizhnik-Zamolodchikov (KZ)-type connection of the Langlands dual group of G. These symplectic resolutions are expected to be symplectic duals of Nakajima quiver varieties, and thus our result is an analogue of (part of) the work of Maulik and Okounkov in the symplectic dual setting.

Thesis Supervisor: Roman Bezrukavnikov Title: Professor of Mathematics, MIT

3 4 Acknowledgments

I would like to thank my advisor, Roman Bezrukavnikov, for his incredible generosity, knowledge, and support. I would also like to thank Davesh Maulik and Paul Seidel for agreeing to serve on my thesis committee.

I would like to thank Joel Kamnitzer, Davesh Maulik, Michael McBreen, David

Nadler, Andrei Okounkov, Paul Seidel, and Valerio Toledano Laredo for helpful discussions and comments on this project, and Thomas Haines and Simon Riche for helpful correspondence on their work.

Thanks to Gus Lonergan, Yi Sun, and Mitka Vaintrob for wide-ranging discussions on this thesis and other topics, and for making my life at MIT brighter.

Finally, an enormous thank you to my parents for their love and support.

5 6 Contents

1 Introduction 9

2 Recollections on quantum cohomology 13

2.1 The equivariant small quantum connection ...... 13

2.2 Symplectic resolutions ...... 14

2.3 Quantum cohomology of symplectic resolutions ...... 15

2.4 Reduced virtual fundamental class ...... 17

2.5 Unbroken maps ...... 17

3 Symplectic resolutions of slices in the affine Grassmannian 19

3.1 Geometric Satake equivalence ...... 19

3.2 Minuscule cells ...... 20 3.3 G radings ...... 21

3.4 Convolution ...... 23 3.5 Transversal slices ...... 24

3.6 Picard group ...... 26

3.7 Equivariant cohomology ...... 27

4 Calculation of the equivariant quantum connection 31

4.1 Torus-invariant curves ...... 31

4.2 Unbroken maps ...... 32 4.3 The tangent bundle on fixed points ...... 33

7 4.4 The tangent bundle on fixed curves ...... 34

4.5 Localization ...... 35

4.6 Trigonometric KZ connection ...... 38

4.7 Reduction to Picard rank 1 ...... 39

8 Chapter 1

Introduction

The problem considered in this thesis lies at the intersection of three major dualities of modern mathematics and physics. In roughly chronological order, the three dualities are:

* Geometric Langlands duality. The Langlands program, a vast program connecting number theory and representation theory, was initiated by Langlands in

the 1960s '[401. Its analogue in the geometry, the geometric Langlands program, was initiated by Laumon, Beilinson, Drinfeld, Ginzburg, and others [41, 4, 26], and is intensely studied today both in mathematics and in physics [39, 24, 371. The basic objects of study are a reductive algebraic group G (such as. GL, or SOn) and its Langlands dual group GV (obtained by interchanging the systems of roots and -coroots).

* (Equivariant) mirror symmetry. Originally a duality in string theory, mirror symmetry was set on mathematical footing in the 1990s, and is a fundamental duality in modern mathematical physics [31]. An equivariant version is currently being developed [56, 44, 7]. In its original form, mirror symmetry pre- dicts an equality between the quantum connection (or A-model) of a Calabi-Yau 3-fold M, and the Gauss-Manin connection (or B-model) of its mirror Mv.

9 * Symplectic duality. Also originall a duality in physics [321, a large part of

this duality has been recently set on mathematical footing in [101 and [91, and

is continuing to be developed in current work of Braverman, Finkelberg, and

Nakajima [491. It replaces the theory of semisimple Lie algebras g with a more

general and geometric theory of symplectic resolutions X and their symplectic

duals X'.

We now describe the problem more precisely. Let G be a connected semisimple algebraic group over C. A fundamental object in the geometric Langlands program is the affine Grassmannian GrG, a certain ind-scheme that geometrically "encodes" the

Langlands dual group G'. In [351, the authors identify certain symplectic slices Xo inside of GrG, and construct T x C*-equivariant symplectic resolutions X -+ Xo of these slices. In types ADE, these symplectic resolutions are conjectured in [101 to be symplectic dual to certain Nakajima quiver varieties (see Remark 3.12).

The main result of this thesis, Theorem 4.6, computes the T x C*-equivariant small quantum connection of these symplectic resolutions (modulo certain parameters) in the case that they are Picard rank 1. (Following the strategy of [131, we expect to show that the general case can be derived from this one; see section 4.7.)

In particular, we observe that this connection is closely related to the trigonometric

Knizhnik-Zamolodchikov (KZ) connection of GV, a fundamental system of differential equations with regular singularities arising in conformal field theory [181. The proof strategy is classical: we show that our spaces have finitely many torus-invariant curves, and identify each of their contributions to the quantum corrections using virtual localization.

Our result provides a new example in the program of Bezrukavnikov, Braverman, Etingof, Maulik, Okounkov, Toledano-Laredo, and others [2, 6] describing the relation between equivariant quantum cohomology of symplectic resolutions and Casimir-type connections (among many other objects). Previously, this relation has been studied in the settings of the Springer resolution [131, Nakajima quiver varieties [431, and

10 hypertoric varieties [441.

Due to the conjectural symplectic duality between quiver varieties and slices in the affine Grassmannian, our work can also be thought of as a symplectic dual analogue of (part of) the work of [431 on quiver varieties. We expect that there is a precise relation in the context of equivariant quantum K-theory [531. Equivariant quantum

K-theory associates two difference equations to a conical symplectic resolution, one

in the Kdhler parameters and the other in the equivariant parameters. In the recent

paper [11, it is conjectured, and proven in the case of quiver varieties, that symplectic

duality interchanges these two difference equations. We expect that the full picture

can be summarized as follows:

Quantum differential/difference equations for quiver varieties

Khhler parameters Equivariant parameters

Quantum cohomology Trigonometric Casimir [431 Rational qKZ [431

Quantum K-theory Trigonometric dynamical difference [551 Trigonometric qKZ [551

Quantum differential/difference equations for the affine Grassmannian

Kdhler parameters Equivariant parameters

Quantum cohomology Trigonometric KZ Rational dynamical difference

Quantum K-theory Trigonometric qKZ Trigonometric dynamical difference

Thus, we conjecture that the quantum K-theoretic difference equations for our sym-

plectic resolutions in the Kdhler and equivariant parameters are, respectively, the

trigonometric qKZ equations and the trigonometric dynamical difference equations.

These are the same difference equations appearing in the quiver variety picture, but

with the Kdhler and equivariant parameters interchanged. The first part of this

conjecture is consistent with Theorem 4.6 since the trigonometric qKZ equations nat-

urally degenerate to the trigonometric KZ equations. Thus, the result in this thesis

can be viewed as a step toward completing the understanding of quantum K-theory

11 in the context of symplectic duality for quiver varieties and affine Grassmannians.

The paper is organized as follows. In Chapter 2, we recall some basic results on quantum cohomology we will need. In Chapter 3, we recall from [35] how to construct symplectic resolutions from the affine Grassmannian, and identify those which are

Picard rank 1. Finally, in Chapter 4, we compute the quantum connections of these

Picard 1 spaces (modulo certain parameters), and relate them to a trigonometric

KZ-type connection.

12 Chapter 2

Recollections on quantum cohomology

2.1 The equivariant small quantum connection

All varieties and cohomology are over C. Let G be a reductive algebraic group and X a smooth quasi-projective G-variety such that the fixed locus XG is compact. By definition, the operator of quantum multiplication by a class 7y E H (X) has matrix elements

(Y * 71,12)= q(7, 71,72),

#8EH2 (X,Z) where (-,-) denotes the standard inner product on H (X) and the angle brackets denote 3-point, genus 0, degree 3, G-equivariant Gromov-Witten invariants on X.

Consider the trivial bundle with fiber H (X) and base H2 (X). Define the G- equivariant quantum connection of X to be the connection Vt""r" on this bundle defined, for any D E H2 (X), by

Vuantumn =dD-De,

13 where dD denotes the derivative in the direction of D. Assume that H (X) is a free module over H (pt). Then we have a natural H (pt)- module isomorphism H (X) ~ H* (X) 0 H(pt). Thus, we can view V as a family of connections, parametrized by equivariant parameters in H (pt), on the trivial bundle with fiber H*(X) and base H2 (X). It is well-known that each connection in this family is flat [31].

2.2 Symplectic resolutions

We recall some standard facts on symplectic singularities and symplectic resolutions; see [33, 91 for further details. Recall that a smooth variety X is said to be a symplectic resolution if X is equipped with an algebraic symplectic form w and the map to its affinization X0 =

Spec(Ox) is proper and birational. A symplectic resolution X -+ X0 is said to be conical if there is a C* which acts compatibly on the resolution and base, contract- ing the base to a point, and which acts with positive weight on the symplectic form w. Conical symplectic resolutions include several examples of spaces that commonly arise in geometric representation theory [91:

Example 2.1. For a reductive algebraic group G with a Borel subgroup B and nilpotent cone K C g, the Springer resolution T* (G/B) -+ K is a conical symplectic resolution, where C* acts fiberwise on T*(G/B).

Example 2.2. The Hilbert scheme Hilbl(C 2 ) of n points in C2 with its Hilbert- Chow morphism to Sym"(C 2 ) forms a conical symplectic resolution, with C*-action on Hilb.(C 2 ) induced from the standard one on C2 . More generally, the Hilbert scheme N(k,n) of n points on a crepant resolution of C2 /(Z/kZ), where Z/kZ acts symplectically on C 2 , is a conical symplectic resolution.

Example 2.3. Let M(k, n) denote the moduli space of torsion-free sheaves E on P2 with rank S = k and c 2(S) = n together with a framing Eipi - P, . Then M(k, n)

14 mapping to its affinization, together with a suitable C* action, is a conical symplectic resolution.

Example 2.4. Nakajima quiver varieties mapping to their affinizations are known to be conical symplectic resolutions. When the quiver is affine type A, these generalize the previous two examples.

Example 2.5. The spaces considered in this paper, resolutions of slices in the affine

Grassmannian, admit the structure of a conical symplectic resolution (see Section

3.5).

In [10, 9], the notion of symplectic duality between two conical symplectic resolu-

tions is introduced. One associates a Koszul "category 0" to any symplectic resolution

(e.g. the standard category 0 for the Springer resolution), and roughly, two symplectic resolutions are said to be symplectic dual if their corresponding categories 0 are

Koszul dual.

Example 2.6. By [5], the Springer resolution is symplectic self-dual (or more canon-

ically, T*(G/B) is symplectic dual to T*(Gv/Bv), where G' is the Langlands dual

group of G).

Example 2.7. By [9, Corollary 10.111, the Hilbert schemes J-(k, n) and framed in-

stanton moduli spaces M(k, n) in Examples 2.2 and 2.3 are symplectic dual. For

k = 1 they are isomorphic.

Example 2.8. The spaces considered in this paper (Example 2.5) are expected to

be symplectic dual to certain Nakajima quiver varieties; see Remark 3.12.

2.3 Quantum cohomology of symplectic resolutions

Any conical symplectic resolution X may be deformed to an affine variety (namely, the generic fiber of its universal Poisson deformation), so that the ordinary Gromov-

15 Witten theory of X is trivial. However, such a deformation is not in general C*- equivariant, so the equivariant Gromov-Witten theory of X can be highly non-trivial.

In the notation of Section 2.1, we will take X to be a conical symplectic resolution and G = T x C* for a Hamiltonian torus T (i.e. T stabilizes the symplectic form w).

By [10, Prop. 2.51, a symplectic resolution X has no odd-dimensional cohomology, so the assumption that H (X) is a free module over H4(pt) is satisfied [291.

In [131, it is shown that the equivariant quantum connection of a symplectic resolution X -+ Xo can be expressed via Lagrangian Steinberg correspondences, i.e.

Lagrangian components of the Steinberg variety Z := X x x0 X. It is also shown how the general computation can be reduced to one for symplectic resolutions whose Pi- card group has rank 1. We simply summarize here the fundamental ideas: the divisor equation reduces us to the study of 2-point Gromov-Witten invariants. Any curve in

X must lie in a fiber of X -+ Xo since Xo is affine, so the evaluation map from the space of stable 2-pointed maps to X x X factors through the Steinberg variety. Finally, a dimension count yields that the pushforward of the reduced virtual fundamental class (see the next section) to Z can be expressed as a non-equivariantrational linear combination of Lagrangian Steinberg components. One may then apply deformation invariance of ordinary Gromov-Witten invariants to reduce the computation of these non-equivariant constants to the Picard rank 1 case.

For the Springer resolution, the equivariant quantum connection is determined in

[131 to be equivalent to the trigonometric Dunkl/affine KZ connection [171. For quiver varieties, the equivariant quantum connection is determined in [431 to be equivalent to a trigonometric Casimir-type connection [58J. These two results encompass Examples

2.1 - 2.4. The main result of this thesis identifies the answer in Example 2.5 with a trigonometric KZ-type connection.

16 2.4 Reduced virtual fundamental class

Suppose X -+ Xo is a conical symplectic resolution. Let # be an effective curve class on X, and let Mo,2 (X, /) be the moduli space of 2-pointed stable maps to X of class /. Then it is well-known that there is a so-called reduced virtual fundamental class

[o 0 ,2 (X, r)]red of dimension dim X which satisfies

[{O,2(X, 3)]red = -h[O,2 (X, 3)],i,

where [MO, 2 (X, 0)]'i' is the usual virtual fundamental class, and h is the weight of the symplectic form under the C*-action.

2.5 Unbroken maps

For convenience, we recall the notion of unbroken maps here from [43, Section 7.31.

Let X be a conical symplectic resolution with an action of a torus T that preserves the symplectic form. Let f : C -* X be a T-fixed point of Mo, 2 (X, #) such that the domain C is a chain of rational curves

C = C1 U ... U Ck

with the two marked points lying on C1 and Ck, respectively. Note that all nodes are fixed by T. We say that f is an unbroken chain if at every node of C, the tangent weights of the two branches are opposite and non-zero.

More generally, we say that f is an unbroken map if it satisfies one of the following three conditions:

1. f arises from a map C -+ XT,

2. f is an unbroken chain, or

17 3. The domain C is a chain of rational curves

C = Co U C 1 U ... U Ck

such that Co is contracted by f, the two marked points lie on Co, and the remaining curves form an unbroken chain.

A map is said to be broken if it does not satisfy any of the above conditions. The following is proven in [54, Section 3.8.31:

Lemma 2.9. Every map in a given connected component of M o,2 (X, 3)T is either unbroken or broken. Only unbroken components contribute to the T-equivariant localization of the reduced virtual fundamental class.

18 Chapter 3

Symplectic resolutions of slices in the affine Grassmannian

Let G be a connected semisimple algebraic group over C. We choose a maximal torus

T and a Borel subgroup B containing it. Let r be the rank of G, and let a1 , ... , a, denote the simple roots. Let AV denote the coweight lattice of G, and let Av+ denote the dominant coweights. Let QV denote the coroot lattice of G. Finally, let A denote the set of roots of G, A+ the set of positive roots, and E the set of simple roots.

3.1 Geometric Satake equivalence

Let IC = C((t)) and 0 = C[[t]]. Recall that the affine Grassmannian Grc is the ind-scheme defined by G(K)/G(O).

The Langlands dual group Gv, defined by interchanging the root and coroot systems of G, is of fundamental importance in the Langlands program and geometric representation theory. By the Tannakian formalism, this group can be reconstructed from its tensor category of finite-dimensional representations (Rep(GV), 0). The geometric Satake equivalence [42, 26, 451 describes this category in terms of the geometry

19 of the affine Grassmannian of G: namely, it gives an isomorphism of tensor categories

PervG(o)(GrG) ~ Rep(Gv), where the left-hand side denotes the category of left G(O)-equivariant perverse sheaves on GrG, with tensor structure given by the convolution product (see section 3.4).

It is well-known that the G(O)-orbits on Gr0 are indexed by the dominant coweights A' of G, which can be identified with the dominant weights of GV. For

A E Av, we denote the corresponding G(O)-orbit by GrA. If we let tA denote the image of t under the map

C*((t)) = Gm(C) -+ G(K) -+ GrG, then GrA = G(O) _ tA. Under the geometric Satake equivalence, the intersection cohomology sheaf ICA := IC(GrA) is sent to the irreducible highest-weight representation

V of Gv.

Let s E C* act on Gr by "loop rotation" t F-+ st. This C* action contracts each

G(O)-orbit GrA to the G-orbit G/PA, where PA = Stab(tA) is the parabolic subgroup of G spanned by the root subgroups U, for a such that (a, A) < 0.

3.2 Minuscule cells

Of particular importance to us will be the orbits GrA with A minuscule, that is, minimal in A' with respect to the standard partial ordering. It is well-known that this is equivalent to saying that the pairing of A with any root of G is -1, 0, or 1.

Since Gr = U,,< Gr", it follows that Ur Gr) is smooth for A minuscule; by the previous section, we have GrA ~ G/P.

It follows that every non-zero minuscule coweight is fundamental. By definition, minuscule coweights are indexed by Av/Qv. In particular, there are no non-zero

20 minuscule coweights in types E8 , F4 , or G2 . In the remaining types, the non-zero minuscule coweights, and the corresponding representations of GV, are as follows, where the wi are fundamental coweights [8, Section 7.3}:

Type A. : w1 , .. . , w. (exterior powers of the vector representation)

Type B, : w, (the vector representation)

Type C, : w, (the spin representation)

Type D : Wi, LJn_1, W (the vector and half-spin representations)

Type E6 : w1 , w6 (the two 27-dimensional representations)

Type E7 : w7 (the 56-dimensional representation)

We now give some examples of the corresponding minuscule Schubert cells:

Example 3.1. The minuscule coweights for G = PGL, are 0, w 1, w2 ,... , (An-, where wi is the ith fundamental weight of SLn = (PGLn)v. The corresponding Schubert cells are Gro = pt and Grl = Gr(i, n), 1 < i < n -1.

Example 3.2. For G of type B, and of adjoint type, the minuscule Schubert cell

G/P,, is isomorphic to Gr,(n, 2n), the Grassmannian of maximal isotropic subspaces in a 2n-dimensional symplectic vector space.

Example 3.3. For G of type E6 and of adjoint type, the minuscule Schubert cell G/P,, is isomorphic to OP2 , the 16-dimensional projective plane over the complexified octonions.

For a complete description of minuscule Schubert cells in all types, see [161.

3.3 Gradings

For any A E A', the geometric Satake equivalence gives an isomorphism of vector spaces H*(ICA) = VA. This vector space has two natural gradings: one by cohomo-

21 logical degree in H*(ICA), and one by weight spaces of V. We describe how each of these gradings is transported to the other side of the equivalence.

Fix a regular nilpotent element e E gv, and complete it to an s12-triple e, f, h using the Jacobson-Morozov theorem [191. For example, one can take e = ea; then h = a. Then:

Lemma 3.4. [26] Under the geometric Satake equivalence,

H'(ICA) 2{v G V : h.v =iv}, i E Z.

Remark 3.5. While the right-hand side of the lemma depends on the choice of h and the left-hand side does not, the isomorphism in the lemma may also be conjugated by h via its action on R.ep(GV).

Example 3.6. For g' = Sin(C), we may take e to be the matrix with 1's above the diagonal and O's elsewhere, and h the diagonal matrix with diagonal entries n - 1,n - 3,..., -n + 1. Let A = wi, so that GrA ~ P" and V ~ C'. Then

ICX is the constant perverse sheaf Cpn-1[n - 1], which has non-zero cohomology in degrees n - 1,n - 3,..., -n+1.

Choose y < A. Observe that, since each Gr" is simply connected, ICIGr is a constant complex, i.e. a graded vector space. Define the Brylinski filtration on V\ associated to e by

F(VA) = {x EVA :e-zx=0}.

This filtration is independent of e since all such e are conjugate under T. Then we have:

Lemma 3.7. [261 There is a canonical isomorphism

(ICA IGr') -2i- (2p,p) 9 Zr(VA [A]),

22 where V,[p] denotes the A weight space in V\. In particular, ICA Gr and V\[p] are isomorphic as ordinary vector spaces.

3.4 Convolution

By the isomorphism GH x GH G Gx H\GIH in the setting G = G(K) and H = G(O), the G(K)-orbits in Gr x Gr are also indexed by A'. Given (L1 , L2 ) E Gr x Gr and A E A', we say that L1 and L2 are in relative A position A and write L1 -- + L 2 if (L1 , L 2) lies in the G(A)-orbit corresponding to A.

Let LO = Gro. Given ,, ... , A,, E A', we define the (closed) convolution diagram

-- - ;--~r " := {(L1, ... , Ln) E Gr" : Lo << Li A L2 - L}.

We will also denote this space by Grl. Define the convolution morphism

by m(L1, . . . , L.) = L,. The map m is known to be semismall [451, and when the Ai Al..A . ~ -Aj are minuscule, Gr is smooth (since each Gr is), and hence m is a resolution

of singularities.

Example 3.8. If G = PGL2 , then Grw ~ Gr' is the Hirzebruch surface F2 =

P(Op1(-2) V pi) = T*P1, and the map m: Grw x Grw -+ Gr2 is the contraction of

the zero section.

Given ICA,..., ICAn E PervG(O) (GrG), define the convolution product

ICA * ... * IC := mi(IC(Gr l)).

23 Since m is semismall, we have

ICA * .* IC. E PervG(o)(GrG), which under geometric Satake corresponds to the tensor product of representations VA ... 0VA..

3.5 Transversal slices

Consider the group G[t- 1] c G((t)). Let G1[t- 1 ] denote the kernel of the natural

1 ("evaluation at oo") homomorphism G[t- ] -+ G. For p E A', set Gr, = G 1 [[t-1]] -t. Finally, for A, p E A' with A > p, set

GrA =UrA fnGr,.

By Lemma 2.9 in [11], Gr is a transversal slice to Gr" inside of Gr, and by Theorem

2.7 in [351, GrI has symplectic singularities. We now describe an instance in which -A Gr, has a symplectic resolution.

Suppose A 1,... A, E A' are minuscule coweights, and consider the convolution morphism

For p E A' with p < A1 + . . . + A , consider the transversal slice Gr , and set

Trhe b [ T r 2

Then by [35, Theorem 2.91,

... A " :-A . A.n -Ai 7 ?n A Gr

24 is a symplectic resolution.

1 2 Example 3.9. If G = PGL 2 , then Gr' T*P , and the map Grl'' + G '" is the contraction of the zero section (or alternatively, the Springer resolution for S[2).

Example 3.10. [351 More generally, if G = PGL2 , Al = ... = A, = w, and y =

2 (n-2)w, then Gr0 " -+ Gro0 '+\ is a resolution of the A,_ 1 singularity C /(Z/n).

Note that the groups G and C* act compatibly on both the resolution and the base. It is clear that the C*-action contracts the base to the point t", hence contracts the resolution to the fiber m-1 (t). This C*-action makes m into a conical symplectic resolution (see Section 2.2).

Remark 3.11. If the coweights Ai are not all minuscule, the variety Gr' will be singular; however, it is known to have symplectic singularities [35, Theorem 2.91.

Remark 3.12. Suppose G is simply-laced. Then to A and p one can associate a quiver variety Q(A, p), whose graph is the Dynkin diagram of G with some choice of orientation, and with dimension vector v and framing vector w defined by A = wpw and p = A - E viai. Then it is conjectured [9, Example 10.27] that Gr1.' and

Q(A, M) are symplectic duals.

Remark 3.13. In type A, the space Gr, is known to be isomorphic to a quiver variety, as well as a resolution of a Slodowy slice intersected with a nilpotent orbit; for details, see [461.

In the special case n = 2, we have the following:

Lemma 3.14. The central fiber m-1(tW) of Gr,, is irreducible.

1 Proof. By [61, Cor. 5.1.5J, the irreducible components of m (t/) of dimension (2p, A1 + A2 - are in bijection with a basis for the p multiplicity space

Hom(V,, VA ® VA 2).

25 1 But since A, and A2 are minuscule, [301 gives that m- (t) is equidimensional, and [52, Lemma 10.21 gives that the multiplicity space is 1-dimensional. This proves the lemma.

3.6 Picard group

Recall that the affine Grassmannian of G is naturally equipped with an ample G(O)- equivariant line bundle C := 0(1), which generates the Picard group if G is simply connected [61, Theorem 2.4.21.

There are n natural line bundles defined on Gr , as follows. Consider the n-fold iterated convolution morphism

m : Gr = G(K) xG(O) Gr x G(O) ... XG(O) Gr -+ Gr.

Let 7r, be the projection to the first factor of Gr, and define 1 = ir*L. For 2 < i < n, the G(0)-equivariant line bundle L on the ith convolution factor induces a line bundle Li on Gr. It is then known that satisfies naturality with respect to convolution as follows (see, e.g. [591): m*L =,C1 (D ...(D L,.

We now restrict everything to the resolution of a slice rn Gr." - Gr+. Then L restricted to the affine variety Gr is trivial, so we conclude that, on

Let Di = c1 (Li) be the corresponding divisor classes, so that we have

D +.. .+ D, = 0.

Note that for G not simply connected, L may not be equivariant oil each connected

26 component of GrG, in particular on GrAI. However, a suitable multiple LON will be

G(O)-equivariant, e.g. for N = |Av/QvI. So the above construction may be applied

to LON to yield the same result.

The rank of the Picard group of Gr,A is not known in general. However, in

the case n = 2 we have the following:

-A 1 1,A2 Lemma 3.15. For p < A, + A2 , the space Gr, has Picard rank 1.

Proof. Recall from [10, Prop. 2.51 that a conical symplectic resolution X has no odd

cohomology groups (this is also clear in our setting since Gr is a union of even-

dimensional cells), and hence that Pic(X) 0z C ~ H2 (X, C). So we have

Pic(GrA,2) 0 C H2(Gr, , C) ~ H 2(m-1(tj), C).

Note that m-1(t") is a B-invariant irreducible (by Lemma 3.14) subvariety of Gr~

G/P1 . This maximal parabolic flag variety has an affine paving by B-orbits with a single codimension 1 stratum; thus, we obtain an affine paving of m- 1 (t") with a

single codimension 1 stratum, so H2 (m-1 (ta), C) is 1-dimensional.

In the case n = 2, we let D := D1 - D2 = 2D1 = -2D 2 -

3.7 Equivariant cohomology

We first compute the ordinary cohomology of our symplectic resolutions:

Lemma 3.16. We have

H*(Gr,", VAI 0 ...0 V, .

Proof. As a vector space, the stalk of ICA, * ... * IC,, = mAIC(Gr.An) at t" is

known [261 to be isomorphic to the weight space VA, 0 ... 0 VA, [p]; see Lemma 3.7

27 -A,...,A s for the case n =1. Since Gr,, is smooth, we conclude that

H*(Grl.)= H*(m-1(tI)) = V 1 0 . . ..0 V.

By Lemma 4.1, the space Gr is T-equivariantly formal, so

Hj(Gr.A) H*(Gr. ) Oc C[ai, ... , a, h] ~VA 1 .9... & V\. [y] ac C [ai,., ar, h], and localized equivariant cohomology is given by

H*~G ''", V, 0 . .. 0 VA. [p] Oc C (c1, ... , a. , h).

The weight spaces in each VA\ are obviously 1-dimensional, so the fixed-point basis in

H*(GrA ) may be indexed as v, . , where vi E W - A and E vj = p.

For non-localized equivariant cohomology, we have the following result (referring to [281 for more details):

Lemma 3.17.

H (Gr, ) = HomUh(My, VA, 0 ... 0 VA, 0 , where Uh(g) is the asymptotic universal enveloping algebra of g, y is an equivariant parameter, and M, and A,_, are universal Verma modules for Uh(g).

Proof. We have that H r(Gr'.An) is Poincar6 dual to H M(Gr' A) = H (M

The lemma now follows from [28, Prop. 8.1.51. l

28 For n = 2, we will be interested in the truncated Casimir operator

tuw S e. 0 e,

~A1 ,A2 which acts on H,(Gr, ) by acting factor-wise on Vx, 0 VA 2 .

29 30 Chapter 4

Calculation of the equivariant quantum connection

From now on, we restrict to the case n = 2, i.e. the resolution of a slice m Gr, -+

4.1 Torus-invariant curves

We first determine the torus-fixed points and torus-invariant curves of Gr' . For brevity, we set T T x C*.

Lemma 4.1. 1. The T-fixed points in Gr,A are indexed by pairs of weights

v2 , and v v = /. In (v1 ,v 2 ) of Gv such that cG1 W -A, E W -A 2 1 + 2 particular, there are finitely many such points.

2. Let (v1, v2 ) be a T-fixed point of GrA, , and let a' be a root of Gv such that

v1 + a E W -A and v2 -av E W - A2 . Then there is a unique T-invariant curve

CV1, ,,, connecting the points (V1, V2 ) and (v1 + av, v2 - av), and all T-invariant curves are of this form. In particular, there are finitely many such curves.

Proof. Since each Ai is minuscule, the set of weights of the representation V of Gv

31 is simply W - Ai. It is well-known that the T-fixed points of GrAi are then given by t" with v E W - Aj, and the T-fixed points of GrA1,A2 are given by (t"1, tv+v2) with vi E W - A1 and v2 E W - A2 . Hence, the T-fixed points of the fiber of the 1 convolution morphism m~- (t) are indexed by such pairs (vi, v2) with v, + v2 = y.

Finally, since the C*-action contracts GrAA2 to its central fiber m- 1 (t"), the T-fixed points of Gr, are the same as the T-fixed points of m-1(t"). This proves (1).

Likewise, the T-invariant curves in Gr, are the same as the T-invariant curves

1 in m- (t). Note that, by definition, m~'(tP) is a subvariety of GrAl ~ G/PA1 . It is well-known that this parabolic flag variety has finitely many T-invariant curves, given by (SL2 ),-orbits, where a is any root of G belonging to G/P. Such a curve connects the fixed points corresponding to v, and s,(vi), and since m-'(t") is a

T-invariant (in fact G(O)-invariant) subvariety of GrA,, this curve must lie entirely inside of m- 1 (t"). Now, we have sa(vi) = vi - (vi, a) aV, and since v1 is minuscule and the curve connects distinct points, sc(vi) = v, a'. Thus, for roots aV of G' such that v, + av E W - A, there is a unique T-invariant curve connecting t" and tvi+av in Gr'l. Since v1 + v2 = p, this curve connects the T-fixed points (VI, v2) and

(vi + av, v2 - a) in m- 1(t). This proves (2).

4.2 Unbroken maps

Recall from section 2.5 the definition of unbroken maps.

Lemma 4.2. The unbroken maps f : C -+ Gr, are of the following two types:

1. C ~_P 1 and f is a multiple cover of a T-invariant curve C,, ,a branched over its two endpoints.

2. C ~- Co U C1, where CO is a rational curve contracted to a T-fixed point, the

1 two marked points lie on CO, and C1 ~ P is a multiple cover of a T-invariant curve C,,,,,a branched over its two endpoints.

32 Proof. The curve C,,,2,a has T-weight a at (vi, v2 ), since it is an (SL2)a-orbit and since the loop rotation C* acts trivially on the central fiber m- 1 (0) in which C,,, is contained. Since these are linearly independent, the only unbroken chains are P's, which by definition yields the lemma. L

As noted in 143, 7.3.11, the contribution of the second type of unbroken map is diagonal in the fixed-point basis, so we will focus on the first type of map.

4.3 The tangent bundle on fixed points

We begin with the following elementary result:

Lemma 4.3. For any coweights p ; A of G, the tangent space to GrA at t' is isomorphic to max(O,(op))-1

OEA n=Q

Proof. We have

Ttp Gra g(0)/(0(0) n Stab(p)) ~-g(O)/(9(0) n g(0)1),

where g(Q)" := adt-P g(O). For X in a root subspace go, we have adt-A X = t~fl4")X, so we can decompose

g (0)=( go (0)

Substituting these into the above yields the lemma.

We can now determine the T-weights of tangent bundle at a T-fixed point:

33 Lemma 4.4. Let (vi, v2 ) e Gr be a T-fixed point. Then the T-weights of the

tangent space TareGr, are pairs of roots /3, -3 such that (/,v 1 ) = 1 and (3, v2) = -1.

Proof. Since GrAs = G/PA,, the T-weights of T,, GrAt are all roots 13 such that (3, vi) >

0. Since each vi is minuscule, this implies (/3, vi) = 1. Hence, the T-weights of

)(-A1,A2 T(,V(Gr ) are 3 such that either (3, v1 ) = 1 or ()3, v2) = 1. There is a split exact sequence

0 -÷ T(ne)(Gr, 2 ) a T(vv 2)(Gr ) - T,(Gr") -+ 0

since v 1+v 2 = p. By Lemma 4.3, the T-weights of Th,(Gr") are roots / with (/,p) > 0.

So the above sequence gives that the T-weights of T(,,, 2)Gr, are / such that either (3, vi) = 1 or (3, v2) = 1, and (/, vj) + (/, v2 ) = (pip) <; 0. But since

(/3, vi) c {-1, 0, 1}, these conditions in fact imply that (/, i) = 0. This proves the lemma. 0-

4.4 The tangent bundle on fixed curves

We now decompose the tangent bundle T(GrA' 2 ) into T-equivariant line bundles on

each T-invariant curve ClV2,C- Define L3 as the T-equivariant line bundle on Cl,,, whose fibers at (vi, v2) and

(v+av, v2 -av) have T-weights / and s,(), respectively. Since sa(3) = '3(, av) a, the degree of L, is - (/3, av).

Lemma 4.5. The restriction of T(G4r Aj) to C ,,,a decomposes as a direct sum of T-equivariant line bundles

L#p

where 3 ranges over the T-weights of the tangent space TV,, 2 GrA , i.e. all roots satisfying (/3, v1) = k1 and (/, v2) = -F1. For / appearing in the "horizontal" direc-

34 tion T, GrAl, the degree of La is non-negative, and for / appearing in the "vertical" direction Tu, GrA2, the degree of L3 is non-positive.

Proof. The tangent bundle of Gr, is equivariant with respect to the reductive part of the stabilizer of P, in particular with respect to SL(2).. So the Borel of sl(2), acts on the tangent space at (vI, iV2 ). The element e, in the radical of that Borel must act trivially, since it cannot be that (3, v1 ) = +1 and (3 + a, vi) = 1. Thus, the tangent bundle decomposes as an SL(2),-equivariant bundle into the direct sum of L3, where 3 ranges over the T-weights of the tangent space at (v1 , v2 ) obtained in Lemma 4.4, as desired.

By the proof of Lemma 4.1, a satisfies (a, v1 ) = -1 and (a, v2 ) = 1. The horizontal roots 3 satisfy (/, v1 ) = 1, so (6 - a, vj) = 2, and thus /3 - a cannot be a root since v, is minuscule. Considering root strings, we conclude that (/, av) <; 0, so deg L3 = - (3, av) > 0. The vertical roots are simply the negatives of the horizontal ones by Lemma 4.4, so the claim for vertical roots follows. I

4.5 Localization

We can now calculate the contribution of C,, to the quantum product by the divisor class D in the fixed-point basis v(v 1 ,, 2 ). Let f : C -+ Gr, be a degree d cover of C,,,,, i.e. the first type of unbroken map on GrA identified in Lemma

4.2. There is a single quantum parameter q - q1CVII2A. We want to calculate

(D * v(,,,), v(,,v,,,_,v))

= -h jZ(D, d[CI,V2 ,a)(ev,[M0o, 2(Gr, ', d[CI,v2,a] )]red, V(. 1 ,L9 ) D v(0 +kv,v-av))- d>O

35 Note that a factor of d can be pulled out from (D, d[C,,]). By virtual localization, modulo h we have:

d(ev*[Mo, 2 (Gr, ' d[Cv, , )r"d1, V(lV 2 ) V(VI+aV,2-aV))

dA~A, _A,A2H (C, f *TGr, = e(T(,,)Gr, )e'(T(l+ovv-v)Gr ) H) c, f*TdA1,A2 H0 (C, f*T~)r, )

where the factor of 1/d comes from Aut(f) = Z/dZ, and e' denotes the product of non-zero T-weights.

Let S ={3 EA: (/3,vi) = 1 and (3,v 2) = -1}, and set

= f*L, @ O5ES

so that, by Lemma 4.5, we have f*TGr, = T o T*. Thus, by Lemma 11.1.3 in [431, we obtain that the above quantity is equal to 1; more precisely, it is given by

(-1)rank T+deg T+#z

where #z denotes the number of 0 weights in T D T*. Each of the quantities in the exponent is easily calculated; we obtain

1 ~A1 ,2 rankT=ISI= dimGr, =(p, A,+ A2 -p) 2

deg T = -d KZ3, av

#z=1.

It is also easy to check that deg T is even.

Finally, note that the effective curve class [CV 1 , ,] 0 generates H 2(G/P1 ), and

36 D = 2D1 restricts to 0(2) on G/P\, so

(D, [Cv1,V2,a]) = 2.

We conclude that

(D * v(vl,v 2 ), v(vI+cv,v2 -av)) = (--)(PA1 A2-k) 2 h q d = -(-l)(PA1+A2-1) 2 h q d>0 1q'

Summing over all torus-invariant curves, we finally obtain:

Theorem 4.6. D*=DU-2h q +. 1-q where the dots denote a scalar operator, and 2 acts on the fixed point basis V(v1 ,v2 ) E -A 1 A2 H*(Gr' )ic by

Q(V(VV2)) = ( )(IPI+A -Mx 1 2 V(Vi+aVV 2 -aV) mod h,

where the sum is taken over all a E A such that v1+av E W - and v2 -av E W-A 2 .

On the other hand, the truncated Casimir operator Qtrunc identified in Section 3.7 acts on the fixed-point basis with the same non-zero matrix elements as . So up to rescaling coefficients and modulo h, we have 2 = Qt"nc. We conjecture that the two operators are, in fact, equal.

By Theorem 4.6, the T-equivariant quantum connection of Gr. is given by

Vquantun = dD- D U +2h q +... 1 - q

37 4.6 Trigonometric KZ connection

Let GV be a semisimple algebraic group of rank r with maximal torus TV, and let gV and tv denote their Lie algebras. Choose simple roots ao and root vectors ea, E g for i = 1, ... , r. Let A+ denote the set of all positive roots. Let x 1 ,..., x,. be an orthonormal basis in tv. Let

r = xi 0 xi + E (e. 9 e-, + e-, D e.) i=1 aEA+

be the corresponding Casimir element in U(gv)O2, and decompose Q into positive and negative parts:

+ x0x+ e. e-, Q= xi 0xi + Z e-a ) e. i=1 i=1 aEA+

Define the trigonometric r-matrix

Q+eu= Q

Let V1,.. V, be representations of Gv. Consider the trivial vector bundle with fiber

V1 ® ... V, and base Cn with coordinates (Ui,. .. , u,). For a E t and h c C, define the trigonometric KZ connection VKZ(a, h) on this vector bundle by (see, e.g. [571)

VKZ(a, h) = d - 2h )(uj-uj)d(uj-u)+La(z)du) \i

where r(zj) denotes r acting on the ith and jth tensor factors, and likewise for a(.

Note that this is invariant under the action of the diagonal, so we may translate all the ui so that ul +... + u, = 0.

Note that Q+ and Q- preserve the weight decomposition of V1 0 ... V. Hence,

for any p appearing as a weight of V1 0 ... D V,, the trigonometric KZ connection

38 restricts to a connection on the subbundle with fiber V1 0 ... 0 V,, [p].

Specializing to n = 2 and setting u = u1 - u2 , we can write differentiation in the direction u as

VKZ = d_ - 2h(a(') + a 2)+ 2h (Q~eu + Q) U\ ( -eu

= du - 2h(a(1 ) + a(2) _ Q-) + 2h euQ 1 - eu

Note that the quantum connection and trigonometric KZ connection cannot match exactly, since the latter is trivial for h = 0 while the former is not (due to classical multiplication). However, identifying q = e", we see that, assuming our conjecture from the previous section, the purely quantum part matches the corresponding part of the trigonometric KZ connection, but with Q replaced by Wtu". We thus view the quantum connection as a "truncated" trigonometric KZ-type connection.

Finally, it is natural to conjecture that DU = 2(a(l) + a(2) - h-), and that the full connections are equal under the identification of equivariant parameters A = ha.

4.7 Reduction to Picard rank 1

We briefly describe how we expect the strategy of [131 allows us to reduce to the case n = 2.

The semi-universal Poisson deformation of Gr is the corresponding Beilinson-

Drinfeld Grassmannian Gr ... 'An over A = C (x,..., x.) (see [341). For the xi pairwise distinct, the fiber is the ordinary product GrA x ... x GrAn. Over a generic point of the hyperplane {xi = x3 }, i 5 j, the fiber is the convolution product Grt x A times the ordinary product of the remaining factors.

As described in [351, this restricts to a Poisson deformation of Gr . For the xi pairwise distinct, the fiber is affine. Over a generic point of the hyperplane

an affine {xi = X3 }, we expect that the fiber to be related to a Gr, -bundle over

39 variety for some p'. Applying the argument described in [131 for general symplectic resolutions, we would obtain that the quantum corrections are given by a sum over

1 K i < j n, of the quantum corrections for Gr, , yielding each of the summands in the trigonometric KZ connection.

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