Instanton counting, Donaldson invariants, line bundles on moduli spaces of sheaves on rational surfaces. Lothar Göttsche ICTP, Trieste

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1 Instanton counting, Donaldson invariants, line bundles on moduli spaces of sheaves on rational surfaces Lothar Göttsche ICTP, Trieste joint work with Hiraku Nakajima and Kota Yoshioka AMS Summer Institute in Algebraic Geometry July 25 August 12, 2005, University of Washington, Seattle 1

2 0) Introduction Donaldson invariants: C -inv. of compact 4-manifolds For X proj. surface: intersection number M µl)d on moduli space M X H c 1, c 2 ) of H-stable rk 2 sheaves on X Nekrasov partition function Zɛ 1, ɛ 2, a, Λ): generating function of equiv. Donaldson inv. of A 2 equivariant intersection number of moduli of instantons Mn). X proj. toric surface = X glued together from A 2 s Aim A: Compute Donaldson invariants of X in terms of Nekrasov partition function 2

3 K-theory Nekrasov partition function Z K ɛ 1, ɛ 2, a, Λ): generating function for chh Mn), O)) K-theoretic Donaldson invariants of A 2 L line bundle on X L := OµL)) Donaldson l. b. on M X H c 1, c 2 ) Aim B: Compute χm H X c 1, c 2 ), L) in terms of Z K Note: D-inv φ H X,c 1 c 1 L) d ) = c 1L) d M Xc 1,c 2 ) Riemann-Roch = χm, L) = φ H X,c c 1 1 L) d )/d! + l.o.t 3

4 Motivation: 1) Nekrasov partition function is closely related to relation Seiberg Witten-invariants Donaldson invariants 2) Formula can be viewed as analogue of topological vertex formula 3) χm, L) should be K-theoretic Donaldson invariants still not constructed). Want to understand analogues of all basic properties of Donaldson-invariants 4

5 1) Nekrasov Partition function Instanton moduli space P 2 = A 2 l, { framed coh. sheaves E, φ) on P 2 Mn) := rke) = 2, c 2 E) = n, φ : E l Ol 2 smooth quasiproj, dim 4n Torus Action: C C acts on P 2, l ): e ɛ 1, e ɛ 2)z 1, z 2 ) = e ɛ 1z 1, e ɛ 2z 2 ) C acts on Mn) by change of framing e a E, φ) = E, diage a, e a ) φ) Fixpoint set of C ) 3 is finite: { IZ1 I Z2, id) Z i Hilb n ia 2, 0), ideal gen. by monomials } } 5

6 Let X variety over C with action of T = C ) k = {e w 1,..., e w k)} and X T = {p 1,..., p e } Equiv. cohom. H T X): module over H T pt) = C[w 1,..., w k ] Equiv. integration: X compact, α equiv. lift of α H X) α = α pi X p i c top T pi X) Note α pi, c top T pi X) C[w 1,..., w k ] w1 =...=w k =0 Nekrasov Partition function Mn) with action of C ) 3 = {e ɛ 1, e ɛ 2, e a )} Z inst ɛ 1, ɛ 2, a, Λ) = n 0 Λ 4n Mn) 1 Cɛ 1, ɛ 2, a)[[λ]] Zɛ 1, ɛ 2, a, Λ) = Z inst Z per 6

7 . Nekrasov conjecture Nekrasov-Okounkov, Yoshioka-Nakajima) 1) Z = exp F ɛ 1,ɛ 2,a,Λ) ɛ 1 ɛ 2 ), F regular at ɛ 1 = ɛ 2 = 0 2) F 0 = F ɛ1 =ɛ 2 =0 is Seiberg-Witten prepotential periods of SW-curve, a family of elliptic curves). K-theory Nekrasov Partition function ZK inst ɛ 1, ɛ 2, a, Λ) = Λ 4n e ɛ 1+ɛ 2 )n n 0 i 1) i chh i Mn), O)) Cɛ 1, ɛ 2, a)[[λ]] in equivariant K-theory. Eigenspaces of are finite dim. Z K = ZK inst Zper K Yoshioka-Nakajima): Similar result for Z K Same statement, different family of elliptic curves Know also next two orders in ɛ 1, ɛ 2 of F and F K 7

8 2) Review of Donaldson invariants for alg. surfaces X, H) projective surface M X H c 1, c 2 ) = {H-stable rank 2 sheaves} E X M universal sheaf L H 2 X) µl) = 2c 2 E) 1 2 c 1E) 2 )/L H 2 M) φ X c 1,H expl)) = n M X H c 1,n) expµl))λdimm) p g X) > 0: independent of H p g X) = 0: depends on H via system of walls and chambers in ample cone C X 8

9 Walls: ξ H 2 X, Z) defines wall of type c 1, c 2 ) if ξ c 1 mod 2H 2 X, Z) and 4c 2 c ξ2 0 Wall W ξ := {H C X H ξ = 0} Chambers=connected components of C X \ walls M H X c 1, c 2 ) and invariants constant on chambers, change when H crosses wall i.e. H H + with H ξ < 0 < H + ξ) Kotschick-Morgan conj.: wallcrossing for Donaldson inv. is polynomial in ξ, L, L 2, coefficients depend only on c 2, ξ 2 and topology of X Using K-M conjecture [G] determined gen. function for wallcrossing in terms of modular forms See also Moore-Witten, Marino-Moore 9

10 3) Donaldson inv. and χm, L) versus Nekrasov part. fctn X smooth toric surface, has C C = {e ɛ 1, e ɛ 2)}-action fixpoints {p 1,..., p e }, wx i ), wy i ) weights of action on T pi X Fix F nef with M X F c 1, c 2 ) = for all c 2. For H ample: Donaldson invariants: φ X c 1,HexpL)) = ξ lim Coeff ɛ 1,ɛ 2 0 t 0 t Λ e i=1 Z wx i ), wy i ), t ξ p i 2, Λe L 2 p i ) ) Here ξ H 2 X, Z) with ξ c 1 mod 2 and ξh > 0 > ξf. Holomorphic Euler Characteristic: χmh X c 1, n), L)Λ dimm n = ξ lim Coeff ɛ 1,ɛ 2 0 e t ) 0 1 Λ e Z K wx i ), wy i ), t ξ p i 2 i=1, Λe 2L K 4 pi ) ) 10

11 . 3) Method of proof X toric surface = M X H c 1, c 2 ) and Don. inv. depend on chamber of H. In one chamber moduli space is empty = everything determined by wallcrossing Aim: Relate wallcrossing to partition function. Let ξ define wall W ξ. When H crosses wall, sheaves E in 0 I Z ξ + c 1 )/2) E I W ξ + c 1 )/2) 0, Z, W ) X [l] X [m] l + m = 4c 2 c ξ2 ) are replaced by extensions the other way round Geometrically: Start with M X H c 1, c 2 ). Successively for l = 0, 1..., 4c 2 c ξ2 : blowup bundle PExt 1 I W, I Z ξ))) over X [l] X [m], blow down exceptional divisor D to PExt 1 I Z, I W ξ))). Finally arrive at M X H + c 1, c 2 ). 11

12 Fix l, m, let p : D X [l] X [m] projection [ ] wallcrossing for D-inv = D Ap B)) = X [l] X [m]p A)B [ ] wallcrossing for χl) = χd, A p B )) = χx [l] X [m], p A )B ) Now apply Bott formula on X [l] X [m] : Action of C C on X lifts to X [l] X [m] l,m X [l] X [m] ) C ) 2 = n 1,...,n e Mn 1 ) C )3... Mne ) C )3 Show: contribution for both sides is the same at every fixpoint T Z,W ) X [n] X [m] = Ext 1 I Z, I Z ) Ext 1 I W, I W ) T Z,W ) Mn) = Ext 1 I Z, I Z ) Ext 1 I W, I W ) Ext 1 I Z, I W )e 2a Ext 1 I W, I Z )e 2a 12

13 4) Explicit formulas in modular forms and elliptic functions. Develop F = ɛ 1 ɛ 2 log Z, F K = ɛ 1 ɛ 2 log Z K : F ɛ 1, ɛ 2, a, Λ) = F 0 + ɛ 1 + ɛ 2 )H + ɛ 1 ɛ 2 F 1 + ɛ 1 + ɛ 2 ) 2 G + h.o.t Similarly for F K. Then e i=1 Z wx i ), wy i ), t ξ p i 2 1 = exp = exp wx i i )wy i ) 2 F 0 ξ 2 a) F 0 log Λ a ) e, Λe L 1 2 p i = exp wx i=1 i )wy i ) F wx i ), wy i ), t ξ p i 2 ) 2 F 0 a) 2t/2, Λ)ξ p i ) ξ, L F 0 L, L log Λ) 2 4 ) +... by Bott formula on X. Similarly for Z K. Put τ := 1 2 F 0 2 period of SW elliptic curve. 2πi a) 2 F 0 2 F 0 h := 1 8 log Λ a, Q := 16 1 log Λ) 2, similarly for h K, Q K., Λe L 2 p i ) ) 13

14 q := e πiτ, θ µν z, τ) := n Z 1) n+ µ 2 )ν q n+ µ 2 )2 e 2πin+ µ 2 )z, θ µν τ) := θ µν 0, τ), µ, ν = 0, 1 Donaldson invariants φ X c 1,HexpL)) = ξ ± Coeff q ξ/2)2 exp q 0. Holomorphic Euler characteristic χmh X c 1, n), L)Λ dimm) n Z = ξ ) ) ξ, 2L hλ + 2L) 2 QΛ 2 θ 01 τ) σx) B ) ± Coeff q ξ/2)2 exp ξ, 2L K h K + 2L K) 2 Q K )θ 01 τ) σx) B K q 0 h K Λ) = 2πi i θ 11, τ) ) 1 = hλ + OΛ 3 ) θ 01, τ) θ01 h K 2πi Q K = log, τ) ) = QΛ 2 + OΛ 4 ) θ 01 τ) q d 1 Λ 2 ) 2 + 4Λ2 θ 00 τ/2) 4 dq θ 10 τ/2) B K = 4 = B + OΛ 2 ) 4Λ 2 θ 10 τ/2) 2 14