Moment map flows and the Hecke correspondence for quivers

Transcription

1 and the Hecke correspondence for quivers International Workshop on Geometry and Representation Theory Hong Kong University, 6 November 2013

2 The Grassmannian and flag varieties Correspondences in Morse theory Two ways to think of the Grassmannian Gr(k, n) denotes the Grassmannian of k-dimensional planes in C n modulo equivalence. GL(k, C) and U(k) act on Hom(C k, C n ) by g A = Ag 1. { } Gr(k, n) = A Hom(C k, C n ) : A is injective / GL(k, C) { } = A Hom(C k, C n ) : A A = id C k /U(k). The first is a Geometric Invariant Theory (GIT) quotient of Hom(C k, C n ) by GL(k, C) (A injective A stable). The second is a symplectic quotient of Hom(C k, C n ) by U(k) ( 1A A is the moment map for the U(k) action).

3 The Grassmannian and flag varieties Correspondences in Morse theory Irreducible representations of sl(2, C) Recall that sl(2, C) is generated by ( ) ( ) ( ) H =, E =, F = Relations [H, E] = 2E, [H, F] = 2F, [E, F] = H. An irreducible rep. ρ : sl(2, C) End(C n+1 ) defines a decomposition into weight spaces for H C n+1 = n C (2k n), where ρ(h)v = λv for v C (λ). k=0 E : C (λ) C (λ+2) ( raising operator ) F : C (λ) C (λ 2) ( lowering operator )

4 Examples of Correspondence Varieties The Grassmannian and flag varieties Correspondences in Morse theory Representations of sl(2, C) on the cohomology of the Grassmannian Now consider C n+1 = n H top (Gr(k, n)). k=0 H top (Gr(k, n)) will be the weight space for the action of H with weight 2k n. Construct the operators E and F using the flag variety { F(k, k + 1, n) = C k C k+1 n} C /. The flag variety is a correspondence variety for the two Grassmannians. F(k, k + 1, n) p 1 p 2 Gr(k, n) Gr(k + 1, n)

5 Examples of Correspondence Varieties The Grassmannian and flag varieties Correspondences in Morse theory The raising and lowering operators are constructed via pullback and pushforward (in Borel-Moore homology) in the diagram below. T Gr(k, n) T Gr(k + 1, n) π 1 π 2 T Gr(k, n) F(k, k + 1, n) T p 1 p 2 Gr(k + 1, n) Gr(k, n) Gr(k + 1, n) E(ω) = (π 2 ) (π1 (ω) [F(k, k + 1, n)]) F(ω) = (π 1 ) (π2 (ω) [F(k, k + 1, n)]) (ω is the fundamental class of the Grassmannian)

6 Examples of Correspondence Varieties for bundles The Grassmannian and flag varieties Correspondences in Morse theory Let Σ be a compact Riemann surface. Given a holomorphic bundle E Σ, choose p Σ and v p E p. Evaluation E(U) s v p (s(p)) C gives an exact sequence of sheaves 0 E E Cp 0 The pair (E, E ) gives the data of a quasi-parabolic bundle. deg E = d 1. Choose weights to get a parabolic structure. M par (r, d) p 1 p 2 M(r, d 1) M(r, d)

7 Examples of Correspondence Varieties for quivers The Grassmannian and flag varieties Correspondences in Morse theory Motivation: Hecke modifications of bundles on hyperkähler ALE (Asymptotically Locally Euclidean) 4-manifolds Γ a finite subgroup of SU(2). Construct minimal resolution p : C 2 /Γ C 2 /Γ. Exceptional divisor: p 1 (0) = CP 1. Given an instanton E C 2 /Γ, construct Hecke modifications over the exceptional divisor. B k (Q, v, w) p 1 p 2 M HK (Q, v e k, w) M HK (Q, v, w)

8 Flow line correspondence The Grassmannian and flag varieties Correspondences in Morse theory Let M be a manifold and f : M R a smooth function (plus extra conditions, e.g. M compact, f Morse-Bott). Label the critical sets {C j } n j=1. Define the gradient flow γ : M R M γ(x, 0) = x, t γ(x, t) = grad f (x) t=0 Upwards limit: γ(x, ) = lim t γ(x, t) Downwards limit: γ(x, ) = lim t γ(x, t) Given two critical sets C j, C k, define the space of flow lines F(C j, C k ) := {x M γ(x, ) C j, γ(x, ) C k } F(C j, C k ) := F(C j, C k )/R (R-action defined by flow)

9 Flow line correspondence (cont.) The flow defines projection maps C j F(C j, C k ) u l The Grassmannian and flag varieties Correspondences in Morse theory C k When M is compact and f is Morse-Bott-Smale, the projection maps give F the structure of a sphere bundle. Construct a complex MB from H (C j ) for each crit. set C j. The differentials are u l : H (C k ) H λ jk +1 (C j ). Theorem. (Austin-Braam) H (MB ) = H (M). (Cup product has a similar push-pull construction)

10 Morse correspondence The Grassmannian and flag varieties Correspondences in Morse theory Given two critical sets C j, C k, define the Morse correspondence M(C j, C k ) := {(x 1, x 2 ) C j C k x M s.t. γ(x, ) = x 1, γ(x, ) = x 2 } C j M(C j, C k ) u l Goal of the talk. In the setting of Nakajima quiver varieties, the Morse correspondence is homeomorphic to the Hecke correspondence. C k

11 Definition Examples (definition) Let Q be a directed graph ( quiver = bunch of arrows ). Edges E, vertices I, head/tail maps h, t : E I. A framed complex representation of a quiver consists of Complex vector spaces V k and W k at each vertex k I, and C-linear homomorphisms A a : V t(a) V h(a) for each edge a E and i k : V k W k for each k I.

12 Examples of Correspondence Varieties definition (cont.) Definition Examples Define v = (dim C V k ) k I and w = (dim C W k ) k I to be the dimension vectors of the representation. Let Rep(Q, v, w) be the space of all representations with fixed dimension vectors v and w. The groups G v = GL(V k, C) and K v = U(V k ) act on k I k I Rep(Q, v, w). Examples: A V V V1 V2 i i W W W1 W2 i 1 A i 2

13 Definition Examples Hyperkähler quotients (Nakajima quiver varieties) Double the edges of the quiver. Rep(Q, v, w) becomes T Rep(Q, v, w). Induced action of G v and K v. Quaternionic structure on T Rep(Q, v, w). Complex structures I, J, K. There are three moment maps µ I, µ J, µ K for action of K v. Stability parameter α v Z (k v). The Nakajima quiver variety is M HK (Q, v, w) := µ 1 I (α v ) µ 1 J (0) µ 1 K (0)/K v ( ) α stable = µ 1 J (0) µ 1 K (0) /Gv.

14 Definition Examples Energy minimising representations M HK (Q, v, w) := µ 1 I (α v ) µ 1 J (0) µ 1 K (0)/K v. Consider the function f : µ 1 J (0) µ 1 K (0) R given by f (x) = µ I (x) α v 2 Then M HK (Q, v, w) = f 1 (0)/K v. Question. What are the critical sets of f? At a critical point, the representation splits into two subrepresentations x = x + x. v = v + v. x is stable for α v and G v x is closed. Both x and x are energy minimising. Modulo K v we get M HK (Q, v, w) M HK 0 (Q, v, 0).

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16 Examples of Correspondence Varieties Definition Examples Construction of hyperkähler ALE spaces (Kronheimer) Consider the case where Q has the structure of an affine Dynkin diagram of ADE type and w = 0. v is determined by the kernel of the Cartan matrix C. Γ SU(2) is determined by the McKay correspondence. Example. C C C Theorem (Kronheimer, 1989) Type Â3, Γ = Z/3Z C = M HK (Q, v, 0) is a hyperkähler asymptotically locally Euclidean 4-manifold. Moreover, M HK (Q, v, 0) = C 2 /Γ.

17 Definition Examples Instantons on ALE spaces (Kronheimer & Nakajima) Kronheimer showed that all ALE hyperkähler four manifolds can be constructed in this way. Given a bundle E over an ALE space M HK (Q, v, 0) = C 2 /Γ with a fixed framing at infinity, can construct new dimension vectors v and w determined by the Chern classes of E and the framing at infinity. Theorem (Kronheimer & Nakjima, 1990) M HK (Q, v, w ) is the moduli space of framed instantons on E. (Generalisation of the ADHM construction of instantons on S 4.)

18 Nakajima s constructions Definition Examples Theorem (Nakajima) Let ĝ be an affine Kac-Moody algebra, and let Q be the quiver associated to the Dynkin diagram of ĝ. Then there is a representation of U(ĝ) on H top (M HK (Q, v, w) x ), and this is v the irreducible representation with highest weight w. The representation is constructed via the Hecke correspondence B k (Q, v, w), consisting of pairs (x 1, x 2 ) M HK (Q, v e k, w) M HK (Q, v, w) related by an injective intertwining homomorphism. Later constructions of Nakajima use similar ideas for other algebras (quantum affine algebras, etc.)

19 Examples of Correspondence Varieties Definition Examples example Let Q be the quiver with one vertex and no edges. For k = 1, 2, choose framed representations of Q, consisting of vector spaces V k and W k with homomorphisms i k : V k W k and j k : W k V k. A homomorphism A : V 1 V 2 intertwines the representations if the following diagram commutes. V1 i 1 j 1 W A id V2 i 2 j 2 W Choose dim V 1 = dim V 2 1. The consists of all pairs ((i 1, j 1 ), (i 2, j 2 )) for which there exists such an intertwining homomorphism.

20 Morse theory setup Flow lines/approximate flow lines Critical points for the square of the moment map The critical sets are classified by dimension vectors 0 v v. Modulo K v we have C v := M HK (Q, v, w) M HK 0 (Q, v v, 0). M HK 0 (Q, v v, 0) is contractible. Each critical set deformation retracts onto the subset C 0 v := M HK (Q, v, w) {0}. Questions. What do the spaces of flow lines look like? What about the Morse correspondence? Want to answer these questions for the subset C 0 v C v.

21 Morse theory setup Flow lines/approximate flow lines Algebraic description of gradient flow convergence Theorem (Harada, W., 2011) The downwards gradient flow of µ α 2 converges to a critical point. The limit of the flow is isomorphic to the graded object of the Harder-Narasimhan-Jordan-Hölder filtration of the initial condition. In other words, the limit of the flow is determined (up to isomorphism) by the algebraic geometry of the initial condition. This is an analog of theorems for the Yang-Mills functional (Daskalopoulos 92, Daskalopoulos-Wentworth 04, Sibley 13) and Yang-Mills-Higgs functional (W. 08, Li-Zhang 11).

22 Flow lines for quivers Morse theory setup Flow lines/approximate flow lines Given a critical point x, let Sx denote the exponential image of the negative eigenspace of the Hessian. (S x is the tangent space to the unstable manifold) Definition. Two critical points x 1 and x 2 are connected by an approximate flow line if and only if there exists δx S x 1 such that x 1 + δx flows down to x 2. M(Q, v 1, v 2, v, w) denotes the space of pairs x 1 C 0 v 1 and x 2 C 0 v 2 s.t. x 1 and x 2 are connected by an approximate flow line. (M is the Morse correspondence for this example)

23 Critical points connected by a flow line Flow construction of the Lagrangian subvariety Flow construction of Kashiwara s operators Questions Critical points connected by a flow line Combining this with the earlier theorem (algebraic description of the limit of the downwards gradient flow), we now have a relationship between Nakajima s constructions and the gradient flow picture. Theorem (W.) A pair [x 1, x 2 ] is in the B k (Q, v, w) iff [x 1, x 2 ] M(Q, v e k, v, w). (Also true for handsaw quiver varieties)

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25 Critical points connected by a flow line Flow construction of the Lagrangian subvariety Flow construction of Kashiwara s operators Questions Lagrangian subvariety Let 0 v 1, v 2 v. Nakajima defines a Lagrangian subvariety L(Q, v 1, v 2, w) M HK (Q, v 1, w) M HK (Q, v 2, w) consisting of pairs whose orbit closures intersect in µ 1 C (0) T Rep(Q, v, w). Theorem (W.) A pair [x 1, x 2 ] is in the Lagrangian subvariety iff there exists a critical point x such that x 1 and x 2 are connected by (possibly broken) flow lines to x. (x is isomorphic to the graded object of the Jordan-Hölder filtration of x 1 and x 2.)

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27 Critical points connected by a flow line Flow construction of the Lagrangian subvariety Flow construction of Kashiwara s operators Questions Kashiwara s operators Given x M HK 0 (Q, v, w), Nakajima also studies the subvariety M HK (Q, v, w) x M HK (Q, v, w) consisting of points whose orbit closures contain x. He defines an operator Ẽk mapping irreducible components of M HK (Q, v, w) x to irreducible components of M HK (Q, v e k, w) x. Another operator F k maps in the opposite direction. Kashiwara and Saito prove that the set of all irreducible components with these operators is isomorphic to the crystal of the highest weight module of the modified universal enveloping algebra of ĝ. The methods of the previous two theorems give a flow-up, flow-down construction of Kashiwara s operators. (See picture)

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29 Critical points connected by a flow line Flow construction of the Lagrangian subvariety Flow construction of Kashiwara s operators Questions Questions Can we see the for holomorphic bundles or Higgs bundles via the Yang-Mills flow? Can we prove that approximate flow lines are the same as flow lines? Nakajima constructs representations of affine Kac-Moody algebras, etc. using the and a push-pull operation. Is there a Morse-theoretic interpretation of this?