DonaldsonThomas invariants for A-type square product quivers Justin - PowerPoint PPT Presentation

Donaldson–Thomas invariants for A-type square product quivers Justin Allman 1 anyi 2 ) (Joint work with Rich´ ard Rim´ 1 US Naval Academy 2 UNC Chapel Hill 4th Conference on Geometric Methods in Representation Theory University of Missouri, 19 November 2016 1/18 Allman, J. DT for Square Products

Quantum dilogarithm series and pentagon identity Definition 1 For a variable z , the quantum dilogarithm series in Q p q 1 { 2 qrr z ss is 8 p´ z q n q n 2 { 2 ÿ E p z q “ 1 ` i “ 1 p 1 ´ q i q . ś n n “ 1 Theorem (Pentagon identity) In the algebra Q p q 1 { 2 qrr y 1 , y 2 ss{p y 2 y 1 ´ qy 1 y 2 q we have E p y 1 q E p y 2 q “ E p y 2 q E p´ q ´ 1 { 2 y 2 y 1 q E p y 1 q . This identity is often credited to Sch¨ utzenberger (1953) but appeared more or less in the form above in the work of Faddeev–Kashaev (1994) as a quantum mechanical generalization of a dilogarithm function defined first by Euler, and then refined by Rogers (1907). We seek generalizations of this identity. 2/18 Allman, J. DT for Square Products

Quivers Let Q “ p Q 0 , Q 1 q be a quiver with vertex set Q 0 and arrow set Q 1 . For a P Q 1 let ta , ha P Q 0 respectively denote its head and tail (target and source) vertex. For any dimension vector γ we have the representation space à Hom p C γ p ta q , C γ p ha q q M γ “ a P Q 1 i P Q 0 GL p C γ p i q q by with action of the algebraic group G γ “ ś base-change at each vertex. For dimension vectors γ 1 , γ 2 P N Q 0 let χ denote the Euler form : ÿ ÿ χ p γ 1 , γ 2 q “ γ 1 p i q γ 2 p i q ´ γ 1 p ta q γ 2 p ha q . i P Q 0 a P Q 1 Let λ denote its opposite anti-symmetrization λ p γ 1 , γ 2 q “ χ p γ 2 , γ 1 q ´ χ p γ 1 , γ 2 q . 3/18 Allman, J. DT for Square Products

Quantum algebra of Q Let q 1 { 2 be an indeterminate and q denote its square. The quantum algebra A Q of the quiver is the Q p q 1 { 2 q -algebra generated by the symbols y γ , one for each dimension vector γ ; subject to the relation y γ 1 ` γ 2 “ ´ q ´ 1 2 λ p γ 1 ,γ 2 q y γ 1 y γ 2 . Remark The elements y γ form a Q p q 1 { 2 q -vector space basis. The elements y e i form a set of algebraic generators. (Where e i is the dimension vector with 1 at the i -th vertex and zeroes elsewhere) Observe we that the relation above also implies that y γ 1 y γ 2 “ q λ p γ 1 ,γ 2 q y γ 2 y γ 1 . 4/18 Allman, J. DT for Square Products

Example: A 2 Remark Notice that λ p e i , e j q “ # t arrows i Ñ j u ´ # t arrows j Ñ i u . Consider the quiver 1 Ð Ý 2 and let y e i “ y i . Then y e 1 ` e 2 “ ´ q ´ 1 { 2 y 2 y 1 y 2 y 1 “ q y 1 y 2 Thus the pentagon identity says that E p y 1 q E p y 2 q “ E p y 2 q E p y e 1 ` e 2 q E p y 1 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 . 5/18 Allman, J. DT for Square Products

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, i.e. for which there are only finitely many G γ -orbits in M γ . For each i P Q 0 , there is a simple root α i , which is identified with the dimension vector e i . i d β i α i for some positive integers d β Since each positive root β “ ř i , these are also identified with dimension vectors. Theorem (Reineke (2010), Rim´ anyi (2013)) For Dynkin quivers Q there exist orderings on the simple and positive roots such that ñ ñ ź ź E p y α q “ E p y β q . α simple β positive where “ ñ ” indicates the products are taken in the specified orders. 6/18 Allman, J. DT for Square Products

Donaldson–Thomas invariant Theorem (Reineke (2010), Rim´ anyi (2013)) For Dynkin quivers Q there exist orderings on the simple and positive roots such that Ñ Ñ ź ź E p y α q “ E p y 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. 7/18 Allman, J. DT for Square Products

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 , e.g. 3 A 3 : D 4 : 1 2 3 and 1 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 t i u ˆ D 4 and A 3 ˆ t j u 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 . 8/18 Allman, J. DT for Square Products

Example: A 2 ˝ A 2 Begin with p 1 Ð 2 q ˆ p 1 Ñ 2 q . p 11 q p 12 q For u , v P Q 0 , let y e u “ y u ; let y e u ` e v “ y u ` v . p 21 q p 22 q Theorem (Keller (2011,2013), A.–Rim´ anyi (2016)) We have the following identity of quantum dilogarithm series E p y p 12 q q E p y p 21 q q E p y p 11 q`p 12 q q E p y p 21 q`p 22 q q E p y p 11 q q E p y p 22 q q “ E p y p 11 q q E p y p 22 q q E p y p 11 q`p 21 q q E p y p 12 q`p 22 q q E p y p 12 q q E p y p 21 q 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 . 9/18 Allman, J. DT for Square Products

The general statement Let Φ p A N q denote the set of positive roots of type A N ; let ∆ p A N q denote the set of simple roots (this is identified with p A N q 0 ). Theorem (A.–Rim´ anyi (2016)) For the square product A n ˝ A m we have the identity ñ ñ ź ź E p y p i ,φ q q “ E p y p ψ, j q q p i ,φ qP ∆ p A n qˆ Φ p A m q p ψ, j qP Φ p A n qˆ ∆ p A m q 10/18 Allman, J. DT for Square Products

How to prove? Theorem (A.–Rim´ anyi (2016)) For the square product A n ˝ A m we have the identity ñ ñ ź ź E p y p i ,φ q q “ E p y p ψ, j q q p i ,φ qP ∆ p A n qˆ Φ p A m q p ψ, j qP Φ p A n qˆ ∆ p A m q Method 1. Cluster theory and combinatorics Find a maximal green sequence of quiver mutations Keller (2011, 2013) 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 Poincar´ e series for cohomology of each strata. 11/18 Allman, J. DT for Square Products

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 p m β q β P Φ such that d “ ř β m β β. Fix a dimension vector γ for A n ˝ A m and form strata in M γ as follows. For each i P ∆ p A n q , 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 ∆ p A m q . 12/18 Allman, J. DT for Square Products

Example: A 2 ˝ A 2 Fix the dimension vector γ “ p 2 2 1 1 q 2 1 2 1 0 0 0 0 1 1 0 0 1 1 2 2 2 2 η 0 0 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 codim p η ; M γ q 0 1 1 2 4 5 Table: The six horizontal strata. 1 1 1 2 2 1 2 2 θ 1 1 1 0 0 1 0 0 0 0 0 1 1 0 1 1 codim p θ ; M γ q 0 2 2 4 Table: The four vertical strata. 13/18 Allman, J. DT for Square Products

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 “ η j codim R p η j qď i and obtain a topological filtration F 0 Ă F 1 Ă ¨ ¨ ¨ Ă F dim R p X q “ X . Apply the Borel construction for equivariant cohomology to obtain B G F 0 Ă B G F 1 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 (1983), to study Yang–Mills equations. 14/18 Allman, J. DT for Square Products

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 “ t z P C : ℜ r z s ă t u . Definition 3 The rapid-decay cohomology H ˚ p X ; f q is the limit as t Ñ ´8 of the cohomology of the pair H ˚ p X , f ´ 1 p S t qq . Fortunately, this stabilizes at some finite t 0 ! 0. And...if 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 “ ´ ř p p as a superpotential on Q . ( W P C Q {r C Q , C Q s ) Define a regular function W γ : M γ Ñ C by ÿ p f a q a P Q 1 P M γ ÞÝ Ñ ´ Tr p f p q p where f p means the composition around the oriented square p . 15/18 Allman, J. DT for Square Products