Quiver mutation and derived equivalence

Transcription

1 U.F.R. de Mathématiques et Institut de Mathématiques Université Paris Diderot Paris 7 Amsterdam, July 16, 2008, 5ECM

2 History in a nutshell quiver mutation = elementary operation on quivers discovered in mathematics: cluster algebras (Fomin-Zelevinsky, 2000) in physics: Seiberg duality (Vafa, Berenstein-Douglas,... ) Aim: Categorify quiver mutation using recent work by Derksen-Weyman-Zelevinsky Ginzburg

3 Plan

4 A quiver is an oriented graph Definition A quiver Q is an oriented graph: It is given by a set Q 0 (the set of vertices) a set Q 1 (the set of arrows) two maps s : Q 1 Q 0 (taking an arrow to its source) t : Q 1 Q 0 (taking an arrow to its target). Remark A quiver is a category without composition.

5 A quiver can have loops, cycles, several components. Example The quiver A 3 : 1 α 2 3 diagram A 3 : β is an orientation of the Dynkin Example Q : 3 λ 1 ν µ 2 α 5 β γ 4 6 We have Q 0 = {1, 2, 3, 4, 5, 6}, Q 1 = {α, β,...}. α is a loop, (β, γ) is a 2-cycle, (λ, µ, ν) is a 3-cycle.

6 Definition of quiver mutation Let Q be a quiver without loops or 2-cycles. Definition (Fomin-Zelevinsky) Let i Q 0. The mutation µ i (Q) is the quiver obtained from Q as follows 1) for each subquiver j j [ab] l ; 2) reverse all arrows incident with i; b i a l, add a new arrow 3) remove the arrows in a maximal set of pairwise disjoint 2-cycles (e.g. yields, 2-reduction ).

7 Examples of quiver mutation A simple example: ) 2) 3)

8 More complicated examples: Google quiver mutation! Aim: Categorify these combinatorics!

9 Special case: source mutation Notation k a field, kq the path algebra: p kp, path e i the lazy path attached to a vertex i, P i = e i kq the projective right module associated with e i, Mod(kQ) the category of all right kq-modules, D(kQ) the derived category of Mod(kQ). Theorem (Bernstein-Gelfand-Ponomarev Happel 1986) Let Q = µ i (Q), where i is a source. There is an equivalence (=reflection functor) R : D(kQ ) D(kQ) such that P j P j for j i and P i (P i i j P j).

10 General case: Much more complicated What if i is neither source nor sink? An easy counter-example Q : Q = µ 1 (Q) : Easy: D(kQ) and D(kQ ) are very far from being equivalent. Remedy? Hint from physics: Study quivers with potentials!

11 Completed path algebras, potentials Notation k a field, Q a finite quiver without loops or 2-cycles kq = completed path algebra = p path kp, HH 0 = kq/[ kq, kq] = {infinite lin. comb. of cycles of Q}, each a Q 1 yields the cyclic derivative a : HH 0 kq such that path p vu. p=uav Definition A potential on Q is an element W HH 0 not involving cycles of length 0.

12 Mutation of quivers with potential Theorem (Derksen-Weyman-Zelevinsky) The mutation operation Q µ i (Q) admits a good extension to quivers with potentials (Q, W ) µ i (Q, W ) = (Q, W ), i.e. µ i (Q) is isomorphic to the quiver Q at least if W is generic (and to the 2-reduction of Q if W is arbitrary). Example b 2 1 c a W = abc W = 0

13 Ginzburg s dg algebra (Q, W ) a quiver with potential (Q may have loops and 2-cycles) Definition (Ginzburg) Q = quiver with Q 0 = Q 0 and the arrows of Q in degree 0, a new arrow a : j i of degree 1 for each a : i j of Q, a loop t i : i i of degree 2 for each vertex i of Q. Γ = Γ(Q, W ) = k Q endowed with the unique d of degree 1 such that d(a) = 0 for each arrow a of Q, d(a ) = a W for each arrow a of Q, d(t i ) = e i ( a Q 1 [a, a ])e i for each vertex i of Q.

14 Example of Ginzburg s dg algebra Quiver with potential Q : b 2 1 c a 3, W = abc, Ginzburg dg algebra t 1 t 2 2 b a a b c 1 3 c t 3, d(a ) = bc, d(t 1 ) = cc b b,...

15 H 0 Γ and the derived category of Γ Remark Γ is an enhancement of H 0 Γ = kq/( a W a Q 1 ) = Jacobi algebra of (Q, W ). Definition DΓ =derived category of Γ (objects: differential graded Γ-modules), per Γ =perfect derived category = closure of Γ Γ under shifts, extensions, direct summands, D b Γ = bounded derived category = {M DΓ dim H M < }.

16 Γ is smooth and 3-Calabi Yau Remarks Γ is homologically smooth, i.e. Γ per(γ e ), where Γ e = Γ Γ op. Therefore, we have D b Γ per Γ. Γ is 3-Calabi Yau (as a bimodule), i.e. RHom Γ e(γ, Γ e ) Γ[ 3] in D(Γ e ). Therefore, D b Γ is 3-CY as a triang. cat., i.e. D Hom(L, M) = Hom(M, L[3]) for all L, M in D b Γ, where D = Hom k (?, k).

17 The main theorem Q w/o loops or 2-cycles, i a vertex of Q, Γ = Γ(µ i (Q, W )). Theorem (-Yang) There is a canonical equivalence DΓ DΓ taking P j to P j for j i and P i to cone(p i i j P j). It induces equivalences in per and D b. Remarks This improves on a result by Jorge Vitória ( v2), cf. also Mukhopadhyay-Ray, Berenstein-Douglas,... The canonical t-structure on D b Γ yields a new t-structure on D b Γ. If W is generic, we get lots of new t-structures on D b Γ...

18 Links to cluster theory NC 3-CY variety (KS) 0 D b Γ KS (soon) coeff. alg. (FZ) X -space (FG) generalized cluster cat.: 2-CY per Γ C Γ 0 ( ) cluster alg. (FZ) A-space (FG) KS=Kontsevich-Soibelman, FZ=Fomin-Zelevinsky, FG=Fock-Goncharov

19 Link to cluster algebras, application Illustration of the link ( ) from C Γ to the cluster algebra: Theorem (-Caldero) Suppose Q does not have any oriented cycles. Then { } { rigid indecomp. cluster variables objects of C Γ Application of these techniques: Theorem (K) of the cl. alg. A Q }. The periodicity conjecture (Al. Zamolodchikov, 1991) for pairs of Dynkin diagrams (Kuniba-Nakanishi, 1992) is true.

20 Summary Quiver mutation is derived equivalence of Ginzburg algebras. The periodicity conjecture is true. Google quiver mutation!

21 The periodicity conjecture Notation and two Dynkin diagrams with vertex sets J, J, h, h their Coxeter numbers, C, C their Cartan matrices, Y i,i,t variables where i J, i J, t Z. Y -system associated with (, ) j i Y i,i,t 1Y i,i,t+1 = (1 + Y j,i,t) c ij. 1 j i (1 + Yi,j,t ) c i j Periodicity conjecture (Al. Zamolodchikov, Kuniba-Nakanishi) All solutions to this system are periodic of period dividing 2(h + h ).