Donaldson Thomas invariants for A-type square product quivers

Transcription

1 Donaldson Thomas invariants for A-type square product quivers Justin Allman (Joint work with Richárd Rimányi 2 ) US Naval Academy 2 UNC Chapel Hill 4th Conference on Geometric Methods in Representation Theory University of Missouri, 9 November 26 /8

2 Quantum dilogarithm series and pentagon identity Definition For a variable z, the quantum dilogarithm series in Qpq {2 qrrzss is Epzq ` Theorem (Pentagon identity) 8ÿ n p zq n q n2 {2 ś n i p qi q In the algebra Qpq {2 qrry, y 2 ss{py 2 y qy y 2 q we have Epy q Epy 2 q Epy 2 q Ep q {2 y 2 y q Epy q This identity is often credited to Schützenberger (953) but appeared more or less in the form above in the work of Faddeev Kashaev (994) as a quantum mechanical generalization of a dilogarithm function defined first by Euler, and then refined by Rogers (97) We seek generalizations of this identity 2/8

3 Quivers Let Q pq, Q q be a quiver with vertex set Q and arrow set Q For a P Q let ta, ha P Q respectively denote its head and tail (target and source) vertex For any dimension vector γ we have the representation space M γ à apq HompC γptaq, C γphaq q with action of the algebraic group G γ ś ipq GLpC γpiq q by base-change at each vertex For dimension vectors γ, γ 2 P N Q let χ denote the Euler form: χpγ, γ 2 q ÿ ÿ γ piqγ 2 piq γ ptaqγ 2 phaq ipq apq Let λ denote its opposite anti-symmetrization λpγ, γ 2 q χpγ 2, γ q χpγ, γ 2 q 3/8

4 Quantum algebra of Q Let q {2 be an indeterminate and q denote its square The quantum algebra A Q of the quiver is the Qpq {2 q-algebra generated by the symbols y γ, one for each dimension vector γ; subject to the relation y γ`γ 2 q 2 λpγ,γ2q y γ y γ2 Remark The elements y γ form a Qpq {2 q-vector space basis The elements y ei form a set of algebraic generators (Where e i is the dimension vector with at the i-th vertex and zeroes elsewhere) Observe we that the relation above also implies that y γ y γ2 q λpγ,γ2q y γ2 y γ 4/8

5 Example: A 2 Remark Notice that λpe i, e j q #tarrows i Ñ ju #tarrows j Ñ iu Consider the quiver ÐÝ 2 and let y ei y i Then y 2 y q y y 2 y e`e 2 q {2 y 2 y Thus the pentagon identity says that Epy qepy 2 q Epy 2 qepy e`e 2 qepy q The left-hand side gives an ordering of the simple roots of A 2 ; the right-hand side gives an ordering for the positive roots of A 2 5/8

6 Generalizing the pentagon identity Definition 2 A Dynkin quiver is an orientation of a type A, D, or E Dynkin diagram By Gabriel s Theorem, these are exactly the representation finite quivers, ie for which there are only finitely many G γ -orbits in M γ For each i P Q, there is a simple root α i, which is identified with the dimension vector e i Since each positive root β ř i d β i α i for some positive integers d β these are also identified with dimension vectors i, Theorem (Reineke (2), Rimányi (23)) For Dynkin quivers Q there exist orderings on the simple and positive roots such that ñź ñź Epy α q Epy β q α simple β positive where ñ indicates the products are taken in the specified orders 6/8

7 Donaldson Thomas invariant Theorem (Reineke (2), Rimányi (23)) For Dynkin quivers Q there exist orderings on the simple and positive roots such that Ñź Ñź Epy α q Epy β q α simple β positive where the arrows indicate the products are taken in the specified orders The common value of both sides above is the Donaldson Thomas invariant E Q of the quiver Q It is known that the identity above is a consequence of the Pentagon Identity 7/8

8 Square products The square product of two Dynkin quivers is formed by the process below: (Here we do the example A 3 D 4 ) Assign alternating orientations to A 3 and D 4, eg 3 A 3 : 2 3 and D 4 : 2 4 make a grid of vertices A 3 ˆ D 4 (use matrix notation to name locations) reverse the arrows in the full sub-quivers tiu ˆ D 4 and A 3 ˆ tju whenever i is a sink in A 3 and j is a source in D 4 The result is the diagram of oriented squares: The nodes are called odd, the nodes are called even 8/8

9 Example: A 2 A 2 Begin with p Ð 2q ˆ p Ñ 2q For u, v P Q, let y eu y u ; let y eu`e v y u`v pq p2q p2q p22q Theorem (Keller (2,23), A Rimányi (26)) We have the following identity of quantum dilogarithm series Epy p2q qepy p2q qepy pq`p2q qepy p2q`p22q qepy pq qepy p22q q Epy pq qepy p22q qepy pq`p2q qepy p2q`p22q qepy p2q qepy p2q q The common value of both sides is the Donaldson Thomas invariant E Q,W where W is the superpotential determined by traversing the oriented cycle once The left-hand side comes from an ordering on horizontal positive roots; the right-hand side comes from an ordering on vertical positive roots 9/8

10 The general statement Let ΦpA N q denote the set of positive roots of type A N ; let pa N q denote the set of simple roots (this is identified with pa N q ) Theorem (A Rimányi (26)) For the square product A n A m we have the identity ñź pi,φqp pa nqˆφpa mq Epy pi,φq q ñź pψ,jqpφpa nqˆ pa mq Epy pψ, jq q /8

11 How to prove? Theorem (A Rimányi (26)) For the square product A n A m we have the identity ñź ñź Epy pi,φq q Epy pψ, jq q pi,φqp pa nqˆφpa mq pψ,jqpφpa nqˆ pa mq Method Cluster theory and combinatorics Find a maximal green sequence of quiver mutations Keller (2, 23) describes how, from this, one can algorithmically write down the factors on each side The result must be the DT-invariant E Q,W Method 2 Topology and geometry (our method) For each γ, stratify M γ Use spectral sequence for stratification to relate Poincaré series for cohomology of each strata /8

12 Stratify the representation space Recall that by Gabriel s theorem, a Dynkin quiver with dimension vector d has finitely many G d orbits in M d In fact, each orbit corresponds to a vector pm β q βpφ such that d ř β m ββ Fix a dimension vector γ for A n A m and form strata in M γ as follows For each i P pa nq, fix a Dynkin quiver orbit along the corresponding row Allow complete freedom in the maps along vertical arrows of the quiver Call this a horizontal stratum There are finitely many of these Similarly define vertical strata by fixing orbits along columns corresponding to j P pa mq 2/8

13 Example: A 2 A 2 Fix the dimension vector γ p 2 2 q η codimpη; M γ q Table: The six horizontal strata θ codimpθ; M γ q Table: The four vertical strata 3/8

14 Equivariant cohomology spectral sequence Let G œ X and let X Ť j η j be a stratification by G-invariant subvarieties Form ď F i codim R pη j qďi and obtain a topological filtration F Ă F Ă Ă F dimr px q X η j Apply the Borel construction for equivariant cohomology to obtain B G F Ă B G F Ă Ă B G X There is an associated spectral sequence in cohomology E p,q Remark The application of this spectral sequence goes at least back to Atiyah & Bott (983), to study Yang Mills equations 4/8

15 Rapid-decay cohomology from superpotential Let X be a complex manifold/variety and f : X Ñ C a regular function For t P R, set S t tz P C : Rrzs ă tu Definition 3 The rapid-decay cohomology H px ; f q is the limit as t Ñ 8 of the cohomology of the pair H px, f ps t qq Fortunately, this stabilizes at some finite t! Andif X has a G-action, an equivariant version can be defined On M γ we have a natural choice of regular function as follows Assign the sum over oriented square paths p, W řp p as a superpotential on Q (W P CQ{rCQ, CQs) Define a regular function W γ : M γ Ñ C by pf a q apq P M γ ÞÝÑ ÿ Trpf p q where f p means the composition around the oriented square p p 5/8

16 The big idea Theorem (A Rimányi) The spectral sequence E ij (in rapid decay cohomology) converges to H G γ pm γ ; W γ q and the spectral sequence degenerates at the E page; taking the direct sum over all horizontal strata η E ij à H j à G η pη; W γ q codim R pη;m γq i codim R pη;m γq i taking the direct sum over all vertical strata θ E ij à H j à G θ pθ; W γ q codim R pθ;m γq i codim R pθ;m γq i H j wpηq pbg η q; H j wpθq pbg θ q G η (resp G θ ) is an isotropy subgroup for η (resp θ) Picture please 6/8

17 Convergence of E p,q for A 2 A 2 with γ p 2 2 q horizontal strata vertical strata w w w codimensions 7/8

18 Wrap-up Recall for A 2 A 2 the identity Epy p2q qepy p2q qepy pq`p2q qepy p2q`p22q qepy pq qepy p22q q Epy pq qepy p22q qepy pq`p2q qepy p2q`p22q qepy p2q qepy p2q q Our theorem is that the identity above encodes the picture on the previous page simultaneously for all dimension vectors Other questions/projects: Find a combinatorial Rosetta stone between stratifications and maximal green sequences Play the game above with different stratifications Complete the picture above for A n D m, A n E m, D n E m, etc 8/8