A degeneration formula for quiver moduli and its GW equivalent

Transcription

1 A degeneration formula for quiver moduli and its GW equivalent CRM Bellaterra 19 June 2012 Joint work with M. Reineke and T. Weist

2 Quivers Quiver Q: an oriented graph; vertices Q 0, arrows Q 1 (α : i j). Representations of Q: vector spaces M i (i Q 0 ), linear maps M α (α Q 1 ). Same as modules over the path algebra kq. Lattice of Q: Λ = ZQ 0 ; dimension vectors Λ + = NQ 0 (d = i Q 0 d i i). Euler form: d, e = i Q 0 d i e i α:i j d id j ; {d, e} = d, e e, d. It s the Euler form of category mod kq: Ext i vanish for i > 1.

3 Classification Problem: classify rep s of Q modulo isomorphism. Find a normal form for each given representation (like Jordan form). Solved only for quivers with support a Dynkin diagram A n, D n, E 6, E 7, E 8 or an extended Dynkin diagram à n, D n, Ẽ6, Ẽ7, Ẽ8, e.g.

4 Wild quivers: examples m-loop: v ( m) m-kronecker: e 1 v 1 e 2. v 2 e m 1 e m

5 Moduli Constructed by A. King. Parameter space: R d (Q) = α:i j Hom(M i, M j ). Gauge group: G d = i Q 0 GL(M i ). G d R d via base change: (g i ) i (M α ) α = (g j M α g 1 i ) α:i j. Moduli space: quotient R d /G d.

6 (Affine) GIT Reduced gauge group: PG d = G d /scalars (to have finite stab s). Linearization: Θ : Λ Z; χ Θ ((g i ) i ) = i Q 0 det(g i ) Θ(d) dim(d)θ i. (Semi)stables: Rd sst (Q) = RΘ st R st d (Q) = RΘ sst d (Q) = (R d (Q)) χθ sst ; d (Q) = (R d (Q)) χθ st. Moduli: M Θ st d M Θ sst d (Q) = R Θ sst d (Q) = R Θ st d (Q)/PG d : iso. classes of stables; (Q)//PG d : equiv. classes of polystables.

7 GIT properties R Θ st d M Θ st d R Θ st d If M Θ st d (Q) R Θ sst d (Q) open inclusion; (Q) smooth; (Q) M Θ st d (Q) principal PG d -bundle; (Q) then dim = 1 d, d ; No oriented cycles M Θ sst d (Q) projective; Slope: µ(d) = Θ(d) dim(d). (Semi)stable iff µ(u) < ( )µ(m) for all nontrivial subrep s U M; If d is Θ-coprime M Θ st d (Q) = M Θ sst d (Q) is smooth, projective.

8 Topology From now on work over C. Let d = (d 1,..., d s ) a tuple of dim vectors, with d = d d s ; d k 0 for k = 1,..., s; µ(d d k ) > µ(d) for k < s. Define P d (q) = d ( 1) s 1 q k l d l,d k Theorem (Reineke) If d is Θ-coprime, then s d k k=1 i Q 0 j=1 i (1 q j ) 1. (q 1)P d (q) = i dim H i (M Θ st d, C)q i/2. Corollary: no odd cohomology.

9 Drawbacks All summands are only rational function of q cannot specialize to q = 1 to get χ; It s not a "positive" formula: summands have signs.

10 MPS formula Discovered by physicists J. Manschot, B. Pioline and Sen motivated by string-theoretic arguments. Terminology: abelian quivers have d i 1 for all i. Bose-Fermi statistics: P(t) or χ of non-abelian quivers. Maxwell-Boltzmann statistics: P(t) or χ of abelian quivers. Physical slogan: can trade Bose-Fermi statistics for Maxwell-Boltzmann, provided BPS state count Ω(γ) (roughly χ) is replaced by "effective index" Ω = m γ Ω(γ/m)/m2. Same holds for refined BPS counts Ω ref (γ, t) (roughly P(t)).

11 MPS formula: maths MPS allows to express P(t) or even motives of nonabelian moduli spaces in terms of abelian ones, at the cost of increasing arrow and twisting the stability condition. Fix i Q 0. "Blow i up": define Q 0 = Q 0 \ {i} {i k,l : k, l 1}. α : i j (resp. α : j i) in Q for j i induce arrows α p : i k,l j (resp. α p : j i k,l ) for k, l 1 and p = 1,..., l. Loop α : i i in Q induce arrows α p,q : i k,l i k,l for k, l, k, l 1 and p = 1,..., l, q = 1,..., l in Q.

12 Induced dim vectors and slopes Pick d NQ 0. Multiplicity vector m d i : a partition l lm l = d i, m l 0. Induced dim vector: Level function: l(i k,l ) = l. Induced slope: Θ ik,l = lθ i. Twist: κ(d) = j l(j)d j. Twisted slope: ˆµ = ˆΘ(d) κ(d). d(m ) ik,l = { 1, k ml 0, k > m l.

13 MPS formula Grothendieck ring of varieties: work in K = (K 0 (Var/C) Q)[[L] 1, ([L] n 1) 1 ] n 1. Theorem (motivic MPS) For arbitrary Q, d, Θ and i as above, the following identity holds in K: [L] (d i ) [Rsst 2 d (Q)] [G d ] = m d i l 1 1 m l! ( ( 1) l 1 l[p l 1 ] ) ml [R sst d(m ) ( Q)] [G d(m ) ].

14 Coprime case Recall P(X, t) := i dim Hi (X, Q)t i. Recall "q numbers" [n] q := qn 1 q 1. Corollary If d is Θ-coprime, we have t d i (d i 1) P(Md sst 1 (Q), t) = m l! m d i l 1 ( ( 1) l 1 l[l] t 2 ) ml P(M sst d(m ) ( Q), t) and χ(m sst d (Q)) = m d i l 1 1 m l! ( ( 1) l 1 l 2 ) ml χ(m sst d(m ) ( Q)).

15 MPS proof (sketch) Identification: for all m d i we have a G d(m ) -equivariant isomorphism between R d (Q) and R d(m ) ( Q). Furthermore, we have µ( d(m )) = µ(d). MPS for trivial stability Θ = 0: [L] (d i ) [R d(q)] 2 = 1 [G d ] m l! m d i l 1 ( ( 1) l 1 l[p l 1 ] ) ml [R d(m ) ( Q)] [G d(m ) ]. This is a clever rearrangement using the combinatorics of symmetric functions. It makes precise the idea of passing from B.-F. statistics to M.-B. one. But we still have to incorporate stability!

16 MPS proof (sketch) Idea: use Harder-Narasimhan stratification of R d (Q). Fix decomposition d = d d s into non-zero dimension vectors with µ(d 1 ) >... > µ(d s ). This is a HN type for d, d = (d 1,..., d s ) = d. Let R d d (Q) R d(q) the locus of rep s with HN type d. Then R d d (Q) G d P d V d, where P d < G d parabolic with Levi s k=1 G d k and V d s k=1 Rsst (Q) vector bundle of rank d k k<l p q d pd l q k.

17 MPS proof (sketch) So [R d (Q)] [G d ] = [L] k<l d l,d k d =d s k=1 [R sst d k (Q)] [G d k ] Now induction on dim(d). If dim(d) = 1 we re done. Otherwise compute [L] (d i 2 ) [R d (Q)]/[G d ] in two ways..

18 MPS proof (sketch) First way: MPS for Θ = 0 followed by HN recursion on smaller pieces. Get [L] (d i ) 2 [L] k<l d l,d k d =d s [L] (dk i 2 ) k=1 Second way: Just HN. Get m d k i l 1 1 m l! [L] (d i ) [R d(q)] 2 = [L] (d i ) 2 [L] k<l d l,d k [G d ] d =d ( ( 1) l 1 l[p l 1 ] s k=1 ) ml [R sst d k (m k ) ( Q)] [R sst d k (Q)] [G d k ] Summands with d d match by induction, so we re done. [G d k (m k ) ]

19 MPS and complete bipartite quivers Q bipartite: Q 0 = I J, I = sources, J = sinks. Natural linear form: Θ i = 1 for i I, Θ j = 0 for j J. If we have level l : Q 0 N + can twist as usual, so µ = Θ l /κ. Complete bip. quivers: K (l 1, l 2 ). Vertices: {i 1,..., i l1 } {j 1,..., j l2 }. Arrows: α k,l : i k j l for all k, l (just all possible arrows). i 1 j 1 j 2 i 2 j 3

20 Complete bipartite quivers Dim. vectors: same as (P 1, P 2 ) = ( l 1 i=1 p 1i, l 2 j=1 p 2j ). From now on assume P 1, P 2 are coprime. Write M Θ st (P 1, P 2 ) = M Θ st (P 1,P 2 ) (K (l 1, l 2 )) Universal MPS quiver (for complete bip): infinite N with level structure. N 0 = {i (w,m) (w, m) N 2 } {j (w,m) (w, m) N 2 }, N 1 = {α 1,..., α w w : i (w,m) j (w,m ), w, w, m, m N}. l(q (w,m) ) = w, q {i, j}, m N

21 Complete bipartite quivers MPS dim vectors same as refinements (k 1, k 2 ) = ({k 1 wi }, {k 2 wj }), with p 1i = w wk wi 1, p 2j = w wk wj 2. Let m w(k i ) = l i j=1 k wj i. Induced dim vector: { 1 for m = 1,..., mw (k d q(w,m) = p ), 0 for m > m w (k p ), for q {i, j}, and p = 1, 2 for q = i, j. MPS formula becomes Lemma (bip MPS) χ(m Θ st (P 1, P 2 )) = k P χ(m Θ l st (k 1,k 2 ) (N )) 2 l i i=1 j=1 w ( 1) k i w,j (w 1) k i w,j!w 2k i w,j.

22 Application: χ by summing over trees Idea: first MPS to get rid of d i > 1 (but increasing arrows). Then use localization wrt (C ) N 1 and reduce to just trees. May have to localize many times: theory worked out by Reineke, Weist. N (k 1, k 2 ) := supp(k 1, k 2 ) N, T (k 1, k 2 ) := {connected subtrees of N (k 1, k 2 )}. For I T (I) (sources) let σ I (T ) = j N l(j), I I = i I l(i). Define { 1 if σi (T ) > e w(t ) = d I for all I } T (I) 0 otherwise where d := l(i) and e := l(j) i T (I) j T (J)

23 Application: χ by summing over trees Theorem Corollary χ(m Θ l st (k 1,k 2 ) (N )) = T T (k 1,k 2 ) w(t ). χ(m Θ st (P 1, P 2 )) can be computed as a sum over trees. Remark Same method also works for more quivers, e.g. m-kronecker.

24 Application: asymptotics Theorem (Okada) For a, b coprime, for m 1 (depending on a, b), we have Theorem (Weist) log(χ(m Θ st K (a, b))) (a + b 1) log(m). m For a, b coprime we have 1 lim a,b/a k a log(χ(mθ st K (a, b))) m (k + 1)(log(m) + log(2) + 1) (k 1) log(k). Remark Douglas and Weist have a precise conjecture for the limit.

25 MPS as a degeneration formula in GW theory Key idea: χ of moduli spaces of some quivers are very closely related (sometimes equal) to certain Gromov-Witten invariants enumerating rational curves on algebraic surfaces. Gross, Pandharipande, Siebert: "The tropical vertex"; Kontsevich, Soibelman: "Stability structures..."; Reineke, "Poisson automorphisms..."; Gross, Pandharipande: "Quivers, curves..."; Reineke, Weist: "Refined GW/Kronecker correspondence".

26 MPS and GW Which quivers? K m ; complete bipartite quivers. Which surfaces? Weighted projective planes P(a, b, 1) for coprime a, b. Which curves? Let D 1, D 2, D out denote toric (i.e. boundary) divisors; D o i, D o out non torus-fixed part. Fix l i pts on D i. Fix partitions (P 1, P 2 ) of lenghts l 1, l 2 with P 1, P 2 coprime. We look at curves s.t.: are rational; pass through the l i pts on D o i, with multiplicity p ij (i.e. they are singular pts with prescribed multiplicity); pass through a pt on D o out. Theorem (GPS) There is a well-defined virtual count N[(P 1, P 2 )].

27 MPS and GW Theorem (Reineke, Weist; refined GW/Kronecker, coprime case) Remark N[(P 1, P 2 )] = χ(m Θ st (P 1, P 2 )). We re also interested in P 1 = ka, P 2 = kb with a, b coprime, and curves which are tangent to Dout o to order k. But the relation to χ is much more complicated.

28 MPS and GW Claim: the MPS formula for quiver rep s has a natural geometric interpretation. Under identification N[(P 1, P 2 )] = χ(m Θ st (P 1, P 2 )) MPS becomes a standard degeneration formula in GW theory.

29 GW degeneration Deg formula expresses N[(P 1, P 2 )] in terms of relative GW invariants, i.e. with tangency conditions. Weight vectors: w i = (w i1,..., w iti ) with 0 < w i1 w i2 w iti. Relative GW: N rel [(w 1, w 2 )], virtually enumerating rational curves in P( w 1, w 2, 1) which are tangent to D i at specified points (not fixed by the torus), with order of tangency specified by w i. Suppose w i = P i. Set partition I of w i : I 1 I li = {1,..., t i }, l i disjoint. Compatible if for all j, p ij = r I j w ir.

30 GW degeneration Ramification factor: R Pi w i = I ti j=1 ( 1) w ij 1 w 2 ij. Theorem (very special case of GW deg theory, lots of names) N[(P 1, P 2 )] = 2 ti N rel j=1 [(w 1, w 2 )] w ij Aut(w i ) R P i w i. (w 1,w 2 ) Claim: this is just the same as MPS formula for quivers! i=1

31 Comparison First step. By (rather easy) combinatorics we can rewrite GW deg as a sum over refinements (k 1, k 2 ) rather than weight vectors (w 1, w 2 ). k i induces a weight vector w(k i ) = (w i1,..., w iti ) of length t i = w m w(k i ), by w ij = w for all j = w 1 r=1 Then can rewrite GW deg as N[(P 1, P 2 )] = (k 1,k 2 ) (P 1,P 2 ) m r (k i ) + 1,..., N rel [(w(k 1 ), w(k 2 ))] w m r (k i ). r=1 2 l i i=1 j=1 w ( 1) k i w,j (w 1) k i w,j!w k i w,j.

32 Comparison Now compare RHS of MPS k P and GW degeneration (k 1,k 2 ) (P 1,P 2 ) χ(m Θ l st (k 1,k 2 ) (N )) 2 l i i=1 j=1 w N rel [(w(k 1 ), w(k 2 ))] 2 ( 1) k i w,j (w 1) k i w,j!w 2k i w,j l i i=1 j=1 w ( 1) k i w,j (w 1) k i w,j!w k i w,j We see that the two are equivalent iff χ(m Θ l st (k 1,k 2 ) (N )) = Nrel [(w(k 1 ), w(k 2 ))] 2 li i=1 j=1 w w k w,j i..

33 Comparison: tropical counts li i=1 j=1 w w k i w,j Claim: the RHS N rel [(w(k 1 ), w(k 2 ))] 2 nutural geometric meaning as a tropical count. Parametrized tropical curves in R 2 : proper maps h : Γ R 2 where Γ is weighted 3 valent graph with unbounded ends, such that has a restriction of h to an edge is an embedding whose image is contained in an affine line of rational slope, and balancing condition 3 i=1 w Γ(E i )m i = 0. Tropical curves: equivalence classes under reparametrization. Multiplicities: Mult V (h) = w Γ (E 1 )w Γ (E 2 ) m 1 m 2, Mult(h) = V Mult V (h).

34 Comparison: tropical counts Weight vectors (w 1, w 2 ) encode a tropical count of curves such that: the unbounded edges of Γ are E ij for 1 i 2, 1 j t i, plus a single outgoing" edge E out. We require that h(e ij ) is contained in a line e ij + Re i for some prescribed vectors e ij, and its unbounded direction is e i, w Γ (E ij ) = w ij. Counting with multiplicity we get a well-defined invariant (Mikhalkin,...) Theorem (GPS) N trop [(w 1, w 2 )] = N rel [(w 1, w 2 )] 2 i=1 ti j=1 w ij.

35 Comparison: tropical vertex So we are left with proving, Theorem N trop [(w(k 1 ), w(k 2 ))] = χ(m Θ l st (N )). (k 1,k 2 ) The proof is based on the results of Gross, Pandharipande, Siebert and Reineke on the tropical vertex group. Work on Q = N (k 1, k 2 ). Introduce Poisson algebra with bracket R = C[[x j(w,m ), y i(w,m) w, w, m, m N]] {x j(w,m ), y i(w,m) } = {j (w,m ), i (w,m) } x j(w,m ) y i(w,m) = ww x j(w,m ) y i(w,m).

36 Comparison: tropical vertex Kontsevich-Soibelman Poisson automorphisms: T j(w,m) (x j(w,m ) ) = x j(w,m ), T j(w,m) (y i(w,m ) ) = y i(w,m ) ( 1 + x j(w,m) ) ww, and T i(w,m) (x j(w,m ) ) = x j(w,m ) ( 1 + y i(w,m) ) ww, T i(w,m) (y i(w,m ) ) = y i(w,m ).

37 Idea: compute the product j (w,m) Q 0 T j(w,m) i (w,m ) Q 0 T i(w,m ) in two different ways. First way: Reineke s theorem. The above product equals µ Q T µ, where e.g. T µ (y i(w,m) ) = y i(w,m) (Q µ,j(w,m ) ) ww, j (w,m ) Q 0 and Q µ,j(w,m ) is generating series of Euler characteristics for moduli spaces of stable representations of Q with slope µ and a 1-dimensional framing at j (w,m ).

38 Comparison: tropical vertex Corollary The coefficient of the monomial j (w,m ) Q 0 x j(w,m ) in the series y 1 i (w,m) T µ (y i(w,m) ) is given by i (w,m ) Q 0 y i(w,m ) w w w m w (k 2 ) χ(m Θ l st (N )) (k 1,k 2 )

39 Comparison: tropical vertex Second way: scattering diagrams. Sketch: identify T i(w,m), T j(w,m ) with operators acting on the ring R = C[x ±1, y ±1 ][[ξ j(w,m), η i(w,m ) w, w, m, m N]]. Using GPS notation, we identify T j(w,m) = θ ( ( ) w ) w (1,0), 1+ ξ j(w,m) x with a standard element of the tropical vertex group over R, V R, and similarly T i(w,m) = θ ( ( ) w ) w. (0,1), 1+ η i(w,m) y

40 Comparison: tropical vertex So consider saturated scattering diagram for the product θ (1,0),(1+(ξj(w,m) x) w ) w θ (0,1),(1+(ηi(w,m y) w ). (1) w ) j (w,m) Q 0 i (w,m ) Q 0 (Essentially, add operators along rays to make the whole product around loops trivial). As we are only interested in coeff of j (w,m ) Q 0 x j(w,m ) i (w,m) Q 0 y i(w,m), work over truncated ring C[x ±1, y ±1 ][[ξ j(w,m), η i(w,m ) w, w, m, m N]]/(ξ 2w j (w,m), η 2w i (w,m ) ).

41 Comparison: tropical vertex GPS theory: 1-1 correspondence between rays of saturated scattering diagram and rational tropical curves with suitable unbounded edges and weights. With a bit of work, this implies that the coefficient of j (w,m ) Q 0 x j(w,m ) i (w,m) Q 0 y i(w,m) in the operator attached to ray of slope µ in saturated scattering diagram is w w w m w (k 2 )N trop [(w(k 1 ), w(k 2 ))]. We conclude by essential uniqueness of ordered product factorizations.

42 Final question We have proved the equality N trop [(w(k 1 ), w(k 2 ))] = χ(m Θ l st (N )) (k 1,k 2 ) But is there actually a natural 1-1 correspondence between curves and representations? At least for generic choices? In our work we made some progress in this direction, in particular constructing the 1-1 correspondence in some examples. But we don t have a general answer. Personally I believe that in general there is no natural correspondence. Pandharipande observed that this would make the equality of invariants more interesting.