Block-Göttsche invariants from wall-crossing

Transcription

1 Block-Göttsche invariants from wall-crossing Sara Angela Filippini 6 giugno 2014

2 Outline GW theory of GPS Quiver representations GW invariants q-deformation q-deformed GW invariants

3 The GW theory of GPS Ppa, b, 1q C 3 zt0, 0, 0u{pC a,b,1q weighted projective plane Assume: gcdpa, bq 1 Toric picture: x 2l2 D 0 out D 0 2 x 22 x 21 x 11 x 12 x 1l1 D 0 1

4 Fix partitions P 1, P 2 ; P i pp ij q. Suppose gcdpp 1, P 2 q 1. Invariants: N pa,bq rpp 1, P 2 qs 7 vir $ & % rational curves with x ij prescribed singularities with multiplicities p ij and tangent to order k to D out at some point,. -

5 Examples 1 N p1,3q rp1, 1 1 1qs 1 given by pu : vq ÞÑ pu : y 1 x 1 x 2 x 3 pu x 1 vqpu x 2 vqpu x 3 vq : vq 2 N p1,1q rp1 1, 1 1qs 2 given by pu : vq ÞÑ pupu vq : pu 2vqpu 4vq : v 2 q pu : vq ÞÑ pupu 5? 3 vq : pu 2? 3vqpu 4? 3 vq : v 2 q

6 Conjectural BPS structure Define a series N Ppa,b,1q : 8 k1 N pa,bq rpkp 1, kp 2 qsτ k where gcdp P 1, P 2 q 1 (start with coprime pair of partitions). Then rewrite formally N Ppa,b,1q : 8 k1 n pa,bq rpkp 1, kp 2 qs 8 d1 1 dpk 1q 1 d 2 τ dk d 1 The n pa,bq rpkp 1, kp 2 qs are the BPS invariants underlying the GW invariants N pa,bq rpkp 1, kp 2 qs.

7 Conjecture (GPS) n pa,bq rpkp 1, kp 2 qs P Z for every a, b, k, P 1, P 2. Remark. When k 1 n pa,bq rpp 1, P 2 qs N pa,bq rpp 1, P 2 qs. Vague expectation: In great generality people expect BPS invariants to be integers because they are the Euler characteristic χ of some suitable moduli space. This is true for N pa,bq rpp 1, P 2 qs in the coprime case!

8 Reineke Weist Theorem If gcdp P 1, P 2 q 1, then N pa,bq rpp 1, P 2 qs χ pmpp loooooomoooooon 1, P 2 qq moduli space of stable representations of complete bipartite quiver l 1 l 2

9 Vague expectation: In great generality, BPS invariants should admit a natural q-deformation or quantization. In our case the Reineke Weist Theorem provides a natural candidate: pn 1 rpp 1, P 2 qs PpMpP1 p, P 2 qqpqq : q 1 2 dim MpP 1,P 2 q PpMpP 1, P 2 qqpqq, where p PpMpP1, P 2 qqpqq is the symmetrized Poincaré polynomial.

10 Tropical vertex group Fix integers a, b and a function f pa,bq P Crx, x 1, y, y 1 srrtss of the form f pa,bq 1 t x a y b gpx loooomoooon a y b, tq gpcrzsrrtss Define θ pa,bq,fpa,bq P Aut Crrtss Crx, x 1, y, y 1 srrtss by # b θpa,bq,fpa,bq pxq x f pa,bq, θ pa,bq,fpa,bq pyq y fpa,bq a.

11 Definition (KS, GS) The tropical vertex group V Aut Crrtss Crx, x 1, y, y 1 srrtss is the t-adic comletion of the subgroup generated by all θ pa,bq,fpa,bq. Remark. Elements of V are formal 1-parameter families of holomorphic symplectomorphisms of C C : they preserve the form dx x ^ dy y.

12 Example Fix l 1, l 2 P N. Then # θp1,0q,p1 txq l1 pxq x, θ p1,0q,p1 txq l 1 pyq yp1 txq l 1. # θ p0,1q,p1 tyq l 2 pxq xp1 txq l 2, θ p0,1q,p1 tyq l 2 pyq y. Basic question: compute commutators in V. More precisely, compute rθ pa,bq,f, θ pa1,b 1 q,f 1s θ 1 pa 1,b 1 q,f θ 1 pa,bq,f θ pa1,b 1 q,f 1θ 1 pa,bq,f as some expression involving the generators θ pa2,b 2 q,f 2.

13 Fundamental result: In principle, this is always possible. Suppose that a, b, a 1, b 1 0, and that µpa, bq µpa 1, b 1 q (pa, bq follows pa 1, b 1 q in clockwise order). Then D! collection of vectors pa 2, b 2 q with positive entries, and attached functions f pa2,b 2 q such that with gcdpa 2, b 2 q 1. rθ pa,bq,f, θ pa1,b 1 q,f 1s ѹ θ pa2,b 2 q,f pa 2,b 2 q pa 2,b looooooooooomooooooooooon 2 q decreasing slopes of rays (from L to R), Question: How do we compute tpa 2, b 2 q, f pa2,b 2 qu?

14 Example For l 1 l 2 2 a closed formula is known: ѹ rθ p1,0q,p1 txq 2, θ p0,1q,p1 tyq 2s k θ pk,k 1q,fpk,k 1q θ p1,1q,fp1,1q θ pk 1,kq,fpk 1,kq, where $ & % f 1,1 p1 t 2 xyq 4 f k,k 1 p1 t 2k 1 x k y k 1 q 2 f k 1,k p1 t 2k 1 x k 1 y k q 2.

15 For now we restrict to the simplest case: rθ p1,0q,p1 txq l 1, θ p0,1q,p1 tyq l 2 s ѹ pa,bq θ pa,bq,fpa,bq. Even this is already very hard: Closed formulae are not known for l 1 l 2 4. However, there are very interesting theoretical results on computing tpa, bq, f pa,bq u: Theorem (Theorem A (GPS 10)) Consider the formal power series log f pa,bq c pa,bq k ptxq ak ptyq bk. Then c pa,bq k k k 0 P a ka P b kb N pa,bq rpp a, P b qs, where P a, P b ordered partitions, and lenp a l 1, lenp b l 2.

16 Tropical significance The GPS Theorem is based on a tropical computation together with some nice correspondence results. The tropical technique is called factor{defo and leads to: Theorem (Theorem A (GPS)) c pa,bq k k 2¹ P a ka P b kb w i1 R Pi w i Autpw i q Ntrop pa,bq pwq, where w pw 1, w 2 q is a pair of weight vectors of arbitrary length parametrizing a family of tropical counts tn trop pa,bq pwqu. R Pi w i, Autpw i q are some ramification and automorphism factors.

17 Geometric meaning: rational plane tropical curves with w 1 w 2 incoming ends and a single outgoing end. Example N trop pp1, 1q, p1, 2qq Fact: These counts are well-defined, and depend only on w.

18 Refinement We can actually work over Crrs 1,..., s l1, t 1,..., t l2 ss, and consider r l 1 ¹ i1 θ p1,0q,1 si x, l 2 ¹ j1 θ p0,1q,1 tj y s. Then again r l 1 ¹ θ p1,0q,1 si x, l 2 ¹ θ p0,1q,1 tj y s ѹ i1 j1 pa,bq θ pa,bq,fpa,bq, () where log f pa,bq k P a ka P b kb N pa,bq rpp 1, P 2 qss P 1 t P 2 x ka y kb. Corollary The invariants N pa,bq rpp 1, P 2 qs are determined by the factorization pq.

19 Natural q-deformation Basic idea: Some of the factorizations admit a natural q-deformation. This can be used to q-deform the GW invariants. To see the q-deformation we need a different point of view on the θ s. Let pγ, x, yq be a lattice with antisymmetric, bilinear form. Consider the Lie algebra with ñ g becomes a Poisson algebra. g generated by e α, α P Γ, re α, e β s xα, βye α β, e α e β e α β.

20 Let R be a complete local or Artin C-algebra. Then pg g bc p R lim g b C R{m k Ñ R. Let f α P pg be an element of the form f α P 1 m R re α se α. (1.1) Then we introduce θ α,fα automorphisms of the R-algebra pg by Write: θ Ω α,f α θ α,f Ω α for Ω P Q. θ α,fα pe β q e β f xα,βy α.

21 The Wall-crossing Group Definition The wall-crossing group VΓ,R r V Γ,R is the completion of the subgroup generated by automorphisms of the form θα,1 Ω σe α for α P Γ, σ P m R and Ω P Q. Dilogarithm: Li 2 pσe α q k 1 σ k e kα k 2. Fact: θ α,1 σemα expp 1 m adpli 2p σe mα qqq.

22 q-deformed algebra We replace g with the associative, noncommutative algebra over Cpq 1 2 q: g q generated by ê α, α P Γ, with ê α ê β q 1 2 xα,βy ê α β. 1 Classical limit: lim q 1 q 1 rê α, ê β s xα, βyê α β. 2 Ñ1 Fixing a local complete or Artin C-algebra R as usual, we define pg q g q p bc R. (fundamental case: g q rrtss, where t is a central variable.)

23 q-dilogarithm: Epσê α q n 0 p q 1 2 σê α q n p1 qqp1 q 2 q p1 q n q. For Ω P Q we introduce automorphisms ˆθ Ω rσê α s of pg q acting by ˆθ Ω rσê α spê β q Ad E Ω pσê α qpê β q E Ω pσê α qê β E Ω pσê α q. Definition U Γ,R is the completion of the subgroup of Aut Cpq 12 qb C R pg q generated by automorphisms of the form ˆθ Ω rp q 1 2 q n σê α qs (where α P Γ, σ P m R, Ω P Q, n P Z), with respect to the m R -adic topology.

24 Refinement: as in the numerical case, we can work over Crrs 1,..., s l1, t 1,..., t l2 ss, and look at Lemma (Stoppa-F.) rˆθ l 1 rσ 1 ê α1 s, ˆθ l 2 rσ 2 ê α2 ss ˆθ a1 α 1 a 2 α 2 Ad exp P 1 ka 1 P 2 ka 2 w 2¹ i1 pr Pi w i Autpw i q pn trop pα 1,α 2 q pwqsp 1 t P 2 êkpa 1 α 1 a 2 α 2 q q 1 2 q 1 2. where pr Pi w i,q ¹ j p 1q w ij 1 w ij rw ij s q #ti i,, P i w i u, and N p trop pα 1,α 2 qpwq is the Block-Göttsche invariant obtained by replacing m Υ pv q with rm Υ pv qs q for every V P Υ.

25 Main theorem Corollary A natural candidate for the q-deformed GW invariant is pnrpp 1, P 2 qs w 2¹ i1 pr Pi w i Autpw i q p N trop pα 1,α 2 q pwq. Theorem (Stoppa-F.) Suppose pp 1, P 2 q is primitive. Then the two choices of quantization coincide: pn 1 rpp 1, P 2 qs p NrpP1, P 2 qs.