Curve counting on K3 x E, the Igusa cusp form chi_{10}, and descendent integration
aa r X i v : . [ m a t h . AG ] J u l CURVE COUNTING ON K × E , THE IGUSA CUSPFORM χ , AND DESCENDENT INTEGRATION G. OBERDIECK AND R. PANDHARIPANDE
Abstract.
Let S be a nonsingular projective K S , we conjecture a formula for the Gromov-Witten theory (in all curve classes) of the Calabi-Yau 3-fold S × E where E is an elliptic curve. In the primitive case, our conjectureis expressed in terms of the Igusa cusp form χ and matches aprediction via heterotic duality by Katz, Klemm, and Vafa. Inimprimitive cases, our conjecture suggests a new structure for thecomplete theory of descendent integration for K S × E is also presented. Speculationsabout the motivic stable pairs theory of S × E are made.The reduced Gromov-Witten theory of the Hilbert scheme ofpoints of S is much richer than S × E . The 2-point function ofHilb d ( S ) determines a matrix with trace equal to the partitionfunction of S × E . A conjectural form for the full matrix is given. Contents
0. Introduction 21. Rubber geometry 52. The Igusa cusp form χ
73. Hilbert schemes of points 104. Conjectures 125. The full matrix 246. Motivic theory 33References 35
Date : June 2015. Introduction
Let S be a nonsingular projective K E be a non-singular elliptic curve. The 3-fold X = S × E has trivial canonical bundle, and hence is Calabi-Yau. Let π : X → S , π : X → E denote the projections on the respective factors. Let ι S,e : S → X, ι
E,s : E → X be inclusions of the fibers of π and π over points e ∈ E and s ∈ S respectively. We will often drop the subscripts e and s .Let β ∈ Pic( S ) ⊂ H ( S, Z ) be a class which is positive (with respectto any ample polarization), and let d ≥ β, d )determines a class in H ( X, Z ) by( β, d ) = ι S ∗ ( β ) + ι E ∗ ( d [ E ]) . The moduli space of stable maps M g (cid:0) X, ( β, d ) (cid:1) from connected genus g curves to X representing the class ( β, d ) is of virtual dimension 0.However, because S is holomorphic symplectic, the virtual class van-ishes , h M g (cid:0) X, ( β, d ) (cid:1)i vir = 0 . The Gromov-Witten theory of X is only interesting after reduction .The reduced class (cid:2) M g (cid:0) X, ( β, d ) (cid:1)(cid:3) red is of dimension 1. The ellipticcurve E acts on M g (cid:0) X, ( β, d ) (cid:1) with orbits of dimension 1. The modulispace M g (cid:0) X, ( β, d ) (cid:1) may be viewed virtually as a finite union of E -orbits. The basic enumerative question here is to count the number ofthese E -orbits.The counting of the E -orbits is defined mathematically by the fol-lowing construction. Let β ∨ ∈ H ( S, Q ) be any class satisfying(1) h β, β ∨ i = 1with respect to the intersection pairing on S . For g ≥
0, we define(2) N Xg,β,d = Z [ M g, ( X, ( β,d ))] red ev ∗ (cid:16) π ∗ ( β ∨ ) ∪ π ∗ ([0]) (cid:17) , See [24] for discussion of the virtual class for stable maps to K URVE COUNTING ON K × E where 0 ∈ E is the zero of the group law. The invariant N Xg,β,d is thevirtual count of E -orbits discussed above. Because of orbifold issuesand the possible non-integrality of β ∨ , N Xg,β,d ∈ Q . We will prove N Xg,β,d does not depend upon the choice of β ∨ satisfying(1). The count N Xg,β,d is invariant under deformations of S for which β remains algebraic. By standard arguments [24, 35], N Xg,β,d then dependsupon S and β only via the norm square2 h − h β, β i and the divisibility m ( β ). The count N Xg,β,d is independent of the com-plex structure of E . The notation(3) N Xg,m,h,d = N Xg,β,d will be used.We conjecture here four basic properties of the reduced Gromov-Witten counts N Xg,β,d :(i) a closed formula for their generating series in term of the Igusacusp form χ in case β is primitive,(ii) a reduction rule expressing the invariants for imprimitive β interms of the primitive cases determined by (i),(iii) a Gromov-Witten/Pairs correspondence governing the reducedstable pairs invariants of X ,(iv) a precise formula relating N Xg,β,d to the reduced genus 0 Gromov-Witten invariants of the Hilbert scheme Hilb d ( S ) of d points ofthe K S in case β is primitive.In the d = 0 case, the counts N Xg,β, specialize to the basic integrals(4) N Xg,β, = Z [ M g ( S,β )] red ( − g λ g of the reduced Gromov-Witten theory of K The integrals(4) are governed by the Katz-Klemm-Vafa conjecture proven in [34].Formula (i) specializes to the Jacobi form of the KKV formula. For-mulas for BPS counts of S × E associated to primitive curve classes β ∈ H ( S, Z ) are predicted in [15, Section 6.2] via heterotic duality.After suitable interpretation of the Gromov-Witten theory (2), ourformulas (i) match those of [15]. Here, λ g is the top Chern class of the Hodge bundle. G. OBERDIECK AND R. PANDHARIPANDE
The imprimitive structure (ii) is new and takes a surprisingly dif-ferent form from the standard 3-fold Gromov-Witten multiple covertheory. In fact, conjecture (ii) suggests a new structure for the com-plete theory of descendent integration for K * n Y i =1 τ α i ( γ i ) + Sg,β = Z [ M g,n ( S,β )] red n Y i =1 ψ α i i ∪ ev ∗ i ( γ i ) , γ i ∈ H ∗ ( S, Q )for imprimitive β in terms of the primitive cases.By [25, Theorem 4], the descendent integrals in the primitive cases areknown to be coefficients of quasi-modular forms.The GW/P correspondence (iii) for X is straightforward to conjec-ture. Because reduced theories are considered, the correspondence hereis not directly a special case of the standard GW/P correspondence for3-folds [20, 33].In [6, 29, 30], a triangle of parallel equivalences relating the Gromov-Witten and Donaldson-Thomas theory of C × P to the quantum coho-mology of Hilb d ( C ) was established. Equivalences relating the count-ing theories of S × P and the quantum cohomology of Hilb d ( S ) wereexpected. However, the conjectured formula (iv) relating N Xg,β,d to thereduced genus 0 Gromov-Witten invariants of Hilb d ( S ) is subtle: aninteresting correction term appears.The 2-point function in the reduced genus 0 Gromov-Witten theoryof Hilb d ( S ) studied in [27] underlies (iv) and motivates the entire paper.An interesting speculation which emerges concerns the 3-fold geometry(5) S × P / { S ∪ S ∞ } relative to the K , ∞ ∈ P :(vi) For primitive β ∈ Pic( S ), we conjecture a form for the matrixof relative invariants of the geometry (5).The reduced Gromov-Witten invariants of S × E arise as the trace ofthe matrix (vi).The precise statements of the above conjectures are given in Sections4 and 5. Conjecture A of Section 4.1 covers both (i) and (iv). Conjec-tures B, C, and D of Sections 4.2-4.4 correspond to (ii), (v), and (iii)respectively. Conjectures E, F, and G of Section 5 address (vi) via the URVE COUNTING ON K × E reduced Gromov-Witten theory of Hilb d ( S ). Conjectures E and F werefirst proposed in [27] in a different but equivalent form. Conjecture Gis a direct Hilbert scheme / stable pairs correspondence (again with acorrection term).We conclude the paper with speculations about the motivic stablepairs invariants of S × E . The theory should simultaneously refine theIgusa cusp form χ and generalize the formula of [14]. Acknowledgements . We thank J. Bryan, S. Katz, A. Klemm, D. Maulik,A. Pixton, R. Thomas, and B. Szendroi for many conversations overthe years about the Gromov-Witten theory of K χ . The paper was partially writtenwhile both authors were attending the summer school Modern trends inGromov-Witten theory at the Leibniz Universit¨at Hannover organizedby O. Dumitrescu and N. Pagani in September 2014.G.O. was supported by the grant SNF-200021-143274. R.P. waspartially supported by grants SNF-200021-143274 and ERC-2012-AdG-320368-MCSK. 1.
Rubber geometry
Definition.
Let R be the 1-dimensional rubber target obtainedfrom the relative geometry P / { , ∞} after quotienting by the scaling action. Let Y be the straight rubberover the K S , Y = S × R .
The moduli space of stable maps to rubber, M • g (cid:0) Y, ( β, d ) (cid:1) ∼ ν,ν ∨ , has reduced virtual dimension 0. Here:(i) the superscript • indicates the domain curve may be discon-nected (but no connected components are mapped to points),(ii) β ∈ Pic( S ) and d ≥ R ,(iii) the relative conditions over 0 and ∞ of the rubber are specifiedby partitions of d weighted by the cohomology of S , G. OBERDIECK AND R. PANDHARIPANDE (iv) ν and ν ∨ are dual cohomology weighted partitions. We define(6) e N Y • g,β,d ( ν, ν ∨ ) = Z [ M • g ( Y, ( β,d )) ν,ν ∨ ] red . The definition of e N Y • g,β,d ( ν, ν ∨ ) requires no insertion as in (2).1.2. Disconnected invariants of S × E . In order to relate the inte-grals (2) and (6), we must consider the disconnected Gromov-Wittentheory of X = S × E .
Let M • g, (cid:0) X, ( β, d ) (cid:1) be the moduli space of stable maps from from pos-sibily disconnected genus g curves to X (with no connected componentsmapped to points) representing the class ( β, d ). After reduction, themoduli space is of dimension 2. For β ∨ ∈ H ( S, Z ) satisfing (1), wedefine(7) N X • g,β,d = Z [ M • g, ( X, ( β,d ))] red ev ∗ (cid:16) π ∗ ( β ∨ ) ∪ π ∗ ([0]) (cid:17) , where 0 ∈ E is the zero of the group law as before.Because of the holomorphic symplectic form of S , the stable mapswith two connected components mapping nontrivially to S contribute 0to (7). Hence, the only nontrivial contibutions to (7) come from stablemaps with a single marked connected component mapping nontriv-ially to S and possibly other connected components contracted over S . By standard vanishing considerations, all connected componentscontracted over S must be of genus 1. After evaluating the contractedcontributions, we obtain the following relation: Proposition 1.
For all g ≥ and β ∈ Pic( S ) , the disconnected andconnected counts for X satisfy X d ≥ N X • g,β,d ˜ q d = P d ≥ N Xg,β,d ˜ q d Q n> (1 − ˜ q n ) . Let { γ i } be a basis of H ∗ ( S, Z ), let and { γ ∨ i } be the dual basis. If ν = { ( ν j , γ i j ) } ,then ν ∨ = { ( ν j , γ ∨ i j ) } . URVE COUNTING ON K × E Relating X and Y . Consider the degeneration of E to a nodalrational curve C . The degeneration, X = S × E S × C , leads to a formula for N X • g,β,d in terms of the relative geometry S × P / { S ∪ S ∞ } . Then, using standard rigidification of the rubber and the divisor axiom,we obtain the relation:(8) N X • g,β,d = h X ν ∈P ( d ) z ( ν ) e N Y • β,d ( ν, ν ∨ ) u ℓ ( ν ) i u g − . Here, P ( d ) is the set of all cohomology weighted partitions of d withrespect to a fixed basis { γ i } of H ∗ ( S, Z ). The rubber series on the rightside of (8) is e N Y • β,d ( ν, ν ∨ ) = X g ∈ Z u g − e N Y • g,β,d ( ν, ν ∨ ) . Finally, z ( ν ) = | Aut( ν ) | Q i ν i is the usual combinatorial factor. For-mula (8) and Proposition 1 together imply the following result. Proposition 2.
Definition (2) for N Xg,β,d is independent of the choiceof β ∨ satisfying (1) . The Igusa cusp form χ Let H denote the Siegel upper half space. The standard coordinatesare Ω = (cid:18) τ zz e τ (cid:19) ∈ H , where τ, e τ ∈ H lie in the Siegel upper half plane, z ∈ C , andIm( z ) < Im( τ )Im(˜ τ ) . We denote the exponentials of the coordinates by p = exp(2 πiz ) , q = exp(2 πiτ ) , ˜ q = exp(2 πi ˜ τ ) . For us, the variable p is related to the genus parameter u of Gromov-Witten theory and the Euler characteristic parameter y of stable pairstheory: p = exp( iu ) , y = − p . More precisely, we have u = 2 πz and y = exp(2 πi ( z + 1 / G. OBERDIECK AND R. PANDHARIPANDE
In the partition functions, the variable q indexes classes of S , q h − ←→ a primitive class β h on S satifying 2 h − h β h , β h i , and the variable ˜ q indexes classes of E ,˜ q d − ←→ d times the class [ E ] . We will require several special functions. Let C k ( τ ) = − B k k (2 k )! E k ( τ )be renormalized Eisenstein series: C = − E , C = 12880 E , . . . . Define the Jacobi theta function by F ( z, τ ) = ϑ ( z, τ ) η ( τ )= − i ( p / − p − / ) Y m ≥ (1 − pq m )(1 − p − q m )(1 − q m ) = u exp (cid:16) − X k ≥ ( − k C k u k (cid:17) , where we have choosen the normalization (9) F = u + O ( u ) , u = 2 πz . Define the Weierstrass ℘ function by ℘ ( z, τ ) = − u − X k ≥ ( − k (2 k − kC k u k − = 112 + p (1 − p ) + X k,r ≥ k ( p k − p − k ) q kr .F ( z, τ ) and ℘ ( z, τ ) are related by the following construction. Let(10) G = F ∂ z ( F ) − ∂ z ( F ) = F ∂ z log( F ) , where ∂ z = πi ∂∂z = i ∂∂u = p ddp . Then we have the basic relation ℘ ( z, τ ) = − ∂ z (log( F ( z, τ ))) − C ( τ )(11) = − GF + 112 E . From the point of Gromov-Witten theory, the leading term u k for the specialfunctions is more natural. However, the usual convention in the literature is totake leading term (2 πiz ) k . We follow the usual convention for most of the classicalfunctions. Our convention for F is an exception which allows for fewer signs inthe statement of the Gromov-Witten and pairs results, but results in sign changeswhen comparing with classical function (see Conjecture A). URVE COUNTING ON K × E Define the coefficients c ( m ) by the expansion Z ( z, τ ) = − ℘ ( z, τ ) F ( z, τ ) = X n ≥ X k ∈ Z c (4 n − k ) p k q n . The Igusa cusp form χ (Ω) may be expressed by a result of Gritsenkoand Nikulin [13] as(12) χ (Ω) = pq ˜ q Y ( k,h,d ) (1 − p k q h ˜ q d ) c (4 hd − k ) , where the product is over all k ∈ Z and h, d ≥ • h > d > • h = d = 0 and k < χ is symmetric in the variables q and ˜ q ,(13) χ ( q, ˜ q ) = χ (˜ q, q ) . Let φ | k,m V l denote the action of the l th Hecke operator on a Jacobiform φ of index m and weight k , see [8, page 41]. The definition (12)is equivalent to(14) χ (Ω) = − ˜ q · F ( z, τ ) ∆( τ ) · exp (cid:16) − X l ≥ ˜ q l · ( Z | , V l )( z, τ ) (cid:17) , where ∆( τ ) = q Y n> (1 − q n ) . Alternatively, we may define χ (Ω) as the additive lift, χ (Ω) = − X ℓ ≥ ˜ q ℓ · (cid:0) F ∆ (cid:12)(cid:12) , V ℓ (cid:1) ( z, τ ) . Our main interest is in the inverse of the Igusa cusp form,1 χ (Ω) . By (9) and (14), χ has a pole of order 2 at z = 0 and its translates.Hence, the Fourier expansion of χ depends on the location in Ω. Wewill always assume the parameters ( z, τ ) to be in the region0 < | q | < | p | < . The above choice determines the Fourier expansion of F ∆ and thereforealso of χ . Consider the expansion in ˜ q ,1 χ (Ω) = X n ≥− ˜ q n ψ n . For the first few terms (see [16, page 27]), we have ψ − = − F ∆ ψ = 24 ℘ ∆ ψ = − (cid:18) ℘ + 34 E (cid:19) F ∆ ψ = (cid:18) ℘ + 643 E ℘ + 1027 E (cid:19) F ∆ . In particular, the leading coefficient (with p = − y ) is ψ − = − y + 2 + y − q Y m ≥ y − q m ) (1 − q m ) (1 + yq m ) . It is related to the Katz-Klemm-Vafa formula for K − ψ − (cid:12)(cid:12)(cid:12) − y =exp( − iu ) = X h ≥ g ≥ u g − q h − Z M g ( S,β h ) ( − g λ g = 1 u ∆( τ ) exp (cid:16) X k ≥ u k | B k | k · (2 k )! E k ( τ ) (cid:17) . The functions ψ d are meromorphic Jacobi forms with poles of order 2at z = 0 and its translates. The principal part of ψ d at z = 0 equals(15) a ( d )∆( τ ) 1(2 πiz ) where a ( d ) is the q d coefficient of .3. Hilbert schemes of points
Curves classes.
Let S be a nonsingular projective K S [ d ] = Hilb d ( S )denote the the Hilbert scheme of d points of S . The Hilbert scheme S [ d ] is a nonsingular projective variety of dimension 2 d . Moreover, S [ d ] carries a holomorphic symplectic form, see [1, 26]. URVE COUNTING ON K × E We follow standard notation for the Nakajima operators [26]. For α ∈ H ∗ ( S ; Q ) and i >
0, let p − i ( α ) : H ∗ ( S [ d ] , Q ) −→ H ∗ ( S [ d + i ] , Q ) , γ p − i ( α ) γ be the Nakajima creation operator defined by adding length i punctualsubschemes incident to a cycle Poincare dual to α . The cohomology of S [ d ] can be completely described by the cohomology of S via the actionof the operators p − i ( α ) on the vacuum vector1 S ∈ H ∗ ( S [0] , Q ) = Q . Let p be the class of a point on S . For β ∈ H ( S, Z ), define the class C ( β ) = p − ( β ) p − ( p ) d − S ∈ H ( S [ d ] , Z ) . If β = [ C ] for a curve C ⊂ S , then C ( β ) is the class of the curve givenby fixing d − C and letting a single pointmove on C . For d ≥
2, let A = p − ( p ) p − ( p ) d − S be the class of an exceptional curve – the locus of spinning doublepoints centered at a point s ∈ S plus d − s .For d ≥ H ( S [ d ] , Z ) = (cid:8) C ( β ) + kA (cid:12)(cid:12) β ∈ H ( S, Z ) , k ∈ Z (cid:9) . The moduli space of stable maps M , ( S [ d ] , C ( β ) + kA ) carries areduced virtual class of dimension 2 d .3.2. Elliptic fibration.
Let S be an elliptic K π : S −→ P with a section, and let F ∈ H ( S, Z ) be the class of a fiber. The genericfiber of the induced fibration π [ d ] : Hilb d ( S ) −→ Hilb d ( P ) = P d , is a nonsingular Lagrangian torus. Let L z ⊂ Hilb d ( S )denote the the fiber of π [ d ] over z ∈ P d .Let β h be a primitive curve class on S with h β h , F i = 1 and square h β h , β h i = 2 h − . Here, the maps are required to have connected domains. No superscript • appears in the notation. For z , z ∈ P d , define the invariant N Hilb k,h,d = (cid:10) L z , L z (cid:11) Hilb d ( S ) β h ,k = Z [ M , ( S [ d ] ,C ( β h )+ kA )] red ev ∗ ( L z ) ∪ ev ∗ ( L z )which (virtually) counts the number of rational curves incident to theLagrangians L z and L z .A central result of [27] is the following complete evaluation of N Hilb k,h,d . Theorem 3.
For d > , we have X k ∈ Z X h ≥ N Hilb k,h,d y k q h − = F ( z, τ ) d − ∆( τ ) where y = − e πiz and q = e πiτ . In the d = 1 case, the class A vanishes on S [1] = S . By convention,only the k = 0 term in the sum on the left is taken. Then, Theorem 3specializes in d = 1 to the Yau-Zaslow formula [36] for rational curvecounts in primitive classes of K d = 0, we obtain X k ∈ Z X h ≥ N Hilb k,h, y k q h − = F ( z, τ ) − ∆( τ ) = 1 F ( z, τ ) ∆ . The result is exactly the Katz-Klemm-Vafa formula as discussed inSection 2. While the d = 0 specialization is not geometrically well-defined from the point of view of the Hilbert scheme, the result stronglysuggests a correspondence between the Gromov-Witten theory of K d ( S ). Precise conjectures willbe formulated in the next Section.4. Conjectures
Primitive case.
Let β h ∈ Pic( S ) ⊂ H ( S, Z ) be a primitive classwhich is positive (with respect to any ample polarization) and satsifies h β h , β h i = 2 h − . Let ( E,
0) be a nonsingular elliptic curve with origin 0 ∈ E . For d > H d ( y, q ) = X k ∈ Z X h ≥ y k q h − Z [ M ( E, ( S [ d ] ,C ( β h )+ kA )] red ev ∗ ( β ∨ h,k ) . URVE COUNTING ON K × E The moduli space (16) is of stable maps with 1-pointed domains withcomplex structure fixed after stabilization to be ( E, M ( E, ( S [ d ] , C ( β h ) + kA ) is 1. The divisor class β ∨ h,k ∈ H ( S [ d ] , Q ) is chosen to satisfy(17) Z C ( β h )+ kA β ∨ h,k = 1 . The integral (16) is well-defined.Following the perspective of [6, 29, 30], a connection between thedisconnected Gromov-Witten invariants N • g,β h ,d of K × E and the series(16) obtained from the geometry of S [ d ] is natural to expect.We may rewrite H d ( y, q ) by degenerating ( E,
0) to the nodal ellipticcurve (and using the divisor equation) as(18) H d ( y, q ) = X k ∈ Z X h ≥ y k q h − Z [ M , ( S [ d ] ,C ( β h )+ kA )] red (ev × ev ) ∗ [∆ [ d ] ] , where [∆ [ d ] ] ∈ H d ( S [ d ] × S [ d ] , Q ) is the diagonal class. Equation (18)shows the integral (16) is independent of the choice of β ∨ h,k satisfying(17). By convention, H ( q ) = X h ≥ q h − Z [ M , ( S [1] ,C ( β h ))] red (ev × ev ) ∗ [∆ [1] ]= 2 q ddq ∆ − = − E ∆ . For the second equality, we have used the Yau-Zaslow formula.We define a generating series over all d > H ( y, q, ˜ q ) = X d> H d ( y, q ) ˜ q d − . The analogous generating series over all d for the 3-fold geometry X = S × E is defined by(19) N X • ( u, q, ˜ q ) = X g ∈ Z X h ≥ X d ≥ N X • g,β h ,d u g − q h − ˜ q d − . The main conjecture in the primitive case is the following.
Conjecture A.
Under y = − exp( iu ) , N X • ( u, q, ˜ q ) = H ( y, q, ˜ q ) + 1 F ∆ · q Y n ≥ − (˜ q · G ) n ) = − χ (Ω) . The Igusa cusp form χ (Ω) and the functions F ( z, τ ), ∆( τ ), and G ( z, τ ) are as defined in Section 2.The second factor in the correction term added to H ( y, q, ˜ q ) can beexpanded as1 e q Y n ≥ − (˜ q · G ) n ) = G · τ ) (cid:12)(cid:12)(cid:12) ˜ q = G · ˜ q = ˜ q − + 24 G + 324 G ˜ q + 3200 G ˜ q + · · · . From definition (10) of G and property (9), G = 1 + O ( q ) . Hence the full correction term has ˜ q − coefficient F ∆ which is theKatz-Klemm-Vafa formula (required since H ( y, q, ˜ q ) has no ˜ q − term).The ˜ q term yields the identity − E ∆ + 24 GF ∆ = − ℘ ∆which is equivalent to (11).We do not at present have a geometric explanation for the full cor-rection term(20) 1 F ∆ · q Y n ≥ − (˜ q · G ) n ) . Denote the ˜ q d coefficient of (20) by φ d = a ( d )∆( τ ) · G d +1 F . Here, a ( d ) is the q d coefficient of . Then φ d is a meromorphic Jacobiform with poles of order 2 at z = 0 and its translates. The principalpart of φ d at z = 0 equals a ( d )∆( τ ) 1(2 πz ) . Comparing with (15), we see φ d accounts for all the poles in − ψ d .The second equality in Conjecture A therefore determines a natural splitting(21) − ψ d = H d + φ d URVE COUNTING ON K × E of − ψ d into a finite (holomorphic) quasi-Jacobi form H d and a polarpart φ d . In particular, the Fourier expansion of H d is independent of themoduli τ . Hence, all wall-crossings are related to φ d . The splitting of ψ d into a finite and polar part has been studied by Dabholkar, Murthy,and Zagier [7] and has a direct interpretation in a physical model ofquantum black holes. In fact, up to the E summand in (11) oursplitting matches their simplest choice, see [7, Equations 1.5 and 9.1].The two equalities of Conjecture A are independent claims. The firstis a correspondence result (up to correction). We have made verifica-tions by partially evaluating both sides. The second equality, whichevalutes the series, has already been seen to hold for the coefficients of˜ q − and ˜ q . The second equality has been proven for the coefficient of˜ q in [27]. The conjecture(22) N X • ( u, q, ˜ q ) = − χ (Ω)is directly related to the predictions of Section 6.2 of [15]. J. Bryan [4]has verified conjecture (22) for the coefficients q − and q .The conjectural equality (22) may be viewed as a mathematicallyprecise formulation of [15, Section 6.2]. The Igusa cusp form χ ap-pears in [15] via the elliptic genera of the symmetric products of a K χ terminology is not used in [15]). The development ofthe reduced virtual class occurred in the years following [15]. Sincethe K × E geometry carries a free E -action, a further step (beyondreduction) must be taken to avoid a trivial theory. Definition (2) withan insertion is a straightforward solution. Finally, the Igusa cusp form χ is related to the disconnected reduced Gromov-Witten theory of K × E . With these foundations, the prediction of [15] may be inter-preted to exactly match (22).By the symmetry (13) of the Igusa cusp form χ , Conjecture Apredicts a surprising symmetry for disconnected Gromov-Witten theoryof X , N X • g,β h ,d = N X • g,β d ,h , for all primitive classes β h and β d . In the notation (3), the symmetrycan be written as N X • g, ,h,d = N X • g, ,d,h for all h, d ≥ Bryan’s calculation is on the sheaf theory side, see Conjecture D below.
Conjectures for the motivic generalization of the d = 0 case arepresented in [14]. An interesting connection to the Mathieu M moon-shine phenomena appears there in the data. Since the Gromov-Wittentheory of X is related via − χ by Conjecture A to the elliptic generaof the symmetric products of K M moonshinemust also arise here.4.2. Imprimitive classes.
The generating series N X • ( u, q, ˜ q ) definedby (19) concerns only the primitive classes β h ∈ Pic( S ). To study theimprimitive case, we define(23) N Xβ ( u, ˜ q ) = X g ∈ Z X d ≥ N Xg,β,d u g − ˜ q d − for any β ∈ Pic( S ). The coefficents of N Xβ ( u, ˜ q ) are connected invari-ants. We may write (23) in the notation (3) as N Xmβ h ( u, ˜ q ) = X g ≥ X d ≥ N Xg,m,m ( h − ,d u g − ˜ q d − for primitive β h ∈ Pic( S ) satisfying h β h , β h i = 2 h − . In the primitive ( m =1) case, instead of writing N Xβ h , we write N Xh ( u, ˜ q ) = X g ≥ X d ≥ N Xg, ,h,d u g − ˜ q d − Conjecture B.
For all m > , (24) N Xmβ h ( u, ˜ q ) = X k | m k N X ( mk ) ( h − ( ku, ˜ q ) , for the primitive class β h . Conjecture B expresses the series N Xmβ h in terms of series for primitiveclasses corresponding to the divisors k of m . To such a divisor k , weassociate the class mk β h with square D mk β h , mk β h E = (cid:16) mk (cid:17) (2 h −
2) = 2 (cid:18)(cid:16) mk (cid:17) ( h −
1) + 1 (cid:19) − . The term in the sum on the right side of (24) corresponding to k may be viewed as the contribution of the primitive class of square By Proposition 1, there is no difficulty in moving back and forth between con-nected and disconnected invariants.
URVE COUNTING ON K × E equal to h mk β h , mk β h i . The primitive contribution of the divisor k to N Xg,m,m ( h − ,d is(25) k g − · N Xg, , ( mk ) ( h − ,d . The scaling factor k g − is independent of d . In fact, the variable ˜ q plays no role in formula (24). To emphasize the point, the contributionof the divisor k geometrically is a contribution of the class (cid:16) mk β h , d (cid:17) = ι S ∗ (cid:16) mk β (cid:17) + ι E ∗ ( d [ E ])to ( mβ h , d ) in the 3-fold S × E . Unless d = 0, such a contribution cannot be viewed as a multiple cover contribution in the usual Gopakumar-Vafa perspective of Calabi-Yau 3-fold invariants.In the d = 0 case, Conjecture B specializes to the multiple coverstructure of the KKV conjecture proven in [34] which is usually formu-lated in terms of BPS counts. We could rewrite Conjecture B in termsof nonstandard 3-fold BPS counts which do not interact with the curveclass [ E ] associated to ˜ q . Instead, we have written Conjecture B in themost straightforward Gromov-Witten form. In fact, the simple formof Conjecture B suggests a much more general underlying structure for K h = 0. Lo-calization arguments (with respect to the C ∗ acting on the − N Xmβ ( u, ˜ q ) = 1 m N X ( mu, ˜ q ) . Hence, Conjecture B predicts the primitive contributions correspondingto k = m all vanish in the h = 0 case. Such vanishing is correct: thereduced Gromov-Witten invariants of X vanish for classes ( β, d ) where β is primitive and h β, β i < − . Finally, an elementary analysis leads to the proof of Conjecture B inall cases for g = 1. Both sides of (24) are easily calculated.4.3. Descendent theory for K surfaces. Let S be a nonsingularprojective K β ∈ Pic( S ) be a positive class. We define The localization required here is parallel to the proof of the scaling in [9, The-orem 3]. the (reduced) descendent Gromov-Witten invariants by * n Y i =1 τ α i ( γ i ) + Sg,β = Z [ M g,n ( S,β )] red n Y i =1 ψ α i i ∪ ev ∗ i ( γ i ) , γ i ∈ H ∗ ( S, Q ) . A potential function for the descendent theory of K F g (cid:0) τ k ( γ l ) · · · τ k r ( γ l r ) (cid:1) = ∞ X h =0 D τ k ( γ l ) · · · τ k r ( γ l r ) E Sg,β h q h − for g ≥ Q [ E ( q ) , E ( q ) , E ( q )]of holomorphic quasimodular forms (of level 1) is the Q -algebra gen-erated by Eisenstein series E k , see [3]. The ring QMod is naturallygraded by weight (where E k has weight 2 k ) and inherits an increasingfiltration QMod ≤ k ⊂ QModgiven by forms of weight ≤ k . The precise result proven in [25] is thefollowing. Theorem 4.
The descendent potential is the Fourier expansion in q ofa quasimodular form F g ( τ k ( γ ) · · · τ k r ( γ r )) ∈ q ) QMod ≤ g +2 r with pole at q = 0 of order at most . Conjectures C1 and C2 below will reduce all descendent invariants tothe primitive case.Conjecture C1 is an invariance property. Let S and e S be two K ϕ : (cid:16) H ( S, R ) , h , i (cid:17) → (cid:16) H ( e S , R ) , h , i (cid:17) be a real isometry sending a effective curve class β ∈ H ( S, Z ) to aneffective curve class e β ∈ H ( e S, Z ), ϕ ( β ) = e β . Since there is a canonical isomorphism H ( S, Z ) ∼ = H ( S, Z ) , we may consider β also as a cohomology class. URVE COUNTING ON K × E It is convenient to extend ϕ to all of H ∗ ( S, R ) by ϕ ( ) = , ϕ ( p ) = p where and p are the identity and point classes respectively. Conjecture C1. If β ∈ H ( S, Z ) and e β ∈ H ( S, Z ) have the samedivisibility, * r Y i =1 τ α i ( γ i ) + Sg,β = * n Y i =1 τ α i ( ϕ ( γ i )) + e Sg, e β . Let δ i be the (complex) codimension of γ i , γ i ∈ H δ i ( S, Q ) . Conjecture C1 implies the invariant (cid:10) Q ri =1 τ α i ( γ i ) (cid:11) Sg,β depends onlyupon g , the divisibility of β , and all the pairings h γ i , γ j i , h γ i , β i , h β, β i for δ i = δ j = 1. For the Gromov-Witten theory of curves, a similarinvariance statement has been proven in [28].Conjecture C2 expresses descendent invariants in imprimitive classesin term of primitive classes. Let β h be a primitive curve class on S .Since all invariants vanish if h <
0, we assume h ≥
0. Let m be apositive integer. For every divisor k of m , let S k be a K ϕ k : (cid:16) H ( S, R ) , h , i (cid:17) → (cid:16) H ( S k , R ) , h , i (cid:17) for which ϕ ( mk β h ) is a primitive and effective curve class on S k . • If h >
0, such S k are easily found. • If h = 0, such S k exist only in the k = m case. Conjecture C2.
For primitive classes β h and m > , * r Y i =1 τ α i ( γ i ) + Sg,mβ h = X k | m k g − P ni =1 δ i * n Y i =1 τ α i ( ϕ k ( γ i )) + S k g,ϕ k ( mk β h ) . In the h = 0 case, the k = m terms on the right side of the equalityin Conjecture C2 are defined to vanish. By Conjecture C1, the rightside is independent of the choices of S k and ϕ k . The first evidence: the KKV formula interpreted as the Hodge inte-gral (4) exactly satisfies Conjecture C2 with the integrand viewed ashaving no descendent insertions. In fact, ( − g λ g can be expanded interms of descendent integrands on strata — applying Conjecture C2 tosuch an expansion exactly yields the multiple cover scaling of the KKVformula. In particular, Conjecture C2 together with the KKV formulain the primitive case implies the full KKV formula.Conjecture B, when fully expanded, has a scaling factor of k g − which corresponds to Conjecture C2 with no insertions. In fact, Con-jecture B follows from Conjecture C2 via the product formula [2] forvirtual classes in Gromov-Witten theory. Conjecture C2 was motivedfor us by Conjecture B.The second evidence: Maulik in [21, Theorem 1.1] calculated descen-dents for the A singularity. We may interpret the calculation of [21] asverifiying Conjecture C2 in case h = 0. The scaling of Conjecture C2appears in [21, Theorem 1.1] as the final result because the primitivecontributions corresponding to k = m all vanish in the h = 0 case. Ofcourse, the A singularity only captures codimensions 0 and 1 for δ .A simple example not covered by the two above cases is the integral(27) (cid:10) τ ( p ) (cid:11) S ,mβ where p ∈ H ( S, Q ) is the point class. The primitive class β may betaken to be the fiber F of an elliptically fibered K π : S → P . The primitive invariant is immediate: (cid:10) τ ( p ) (cid:11) K ,β = 1Hence, Conjecture C2 yields the prediction h τ ( p ) i S ,mβ h = X k | m k − h τ ( p ) i K ,β = X k | m k . We can evaluate (27) directly from the geometry of stable maps in theclass mF of S . The integral equals the number of connected degree m covers of an elliptic curve by an elliptic curve (times m for theinsertion), m X k | m k = X k | m k , URVE COUNTING ON K × E which agrees with the prediction.A much more interesting example is the genus 2 invariant (cid:10) τ ( p ) , τ ( p ) (cid:11) S , β in twice the primitive class β . Via standard geometry, β may be takento be the hyperplane section of a K S with a degree 2 cover ǫ : S → P branched along a nonsingular sextic C ⊂ P . Conjecture C2 predicts the following equation: (cid:10) τ ( p ) , τ ( p ) (cid:11) S , β = (cid:10) τ ( p ) , τ ( p ) (cid:11) K ,β + 2 · − (cid:10) τ ( p ) , τ ( p ) (cid:11) K ,β . The primitive counts can be found in [5, Theorem 1.1], (cid:10) τ ( p ) , τ ( p ) (cid:11) K ,β = 1 , (cid:10) τ ( p ) , τ ( p ) (cid:11) K ,β = 8728 . So we obtain the prediction(28) (cid:10) τ ( p ) , τ ( p ) (cid:11) S , β = 8728 + 2 · . The verification of (28) is more subtle than the primitive calculation.We study the geometry of curves in class 2 β on the branched K S . The two point insertions on S determine two points p, q ∈ P .There are 3 contributions to the invariant (28):(i) genus 2 curves in the series 2 β arising as ǫ − ( Q ) where Q ⊂ P is a conic passing through p and q and tangent to the branchdivisor C at 3 distinct points,(ii) genus 2 curves which are the union of two genus 1 curves arisingas ǫ inverse images of a tangent line of C through p and atangent line of C through q ,(iii) genus 2 curves which are the union of genus 2 and genus 0curves arising as the ǫ inverse images of the unique line passingthrough p and q and a bitangent line of C .The most difficult count of the three is the first. An analysis showsthere are no excess issues, hence (i) is equal to the corresponding genus0 relative invariant of P /C ,(29) Z [ M , ( P /C , (1)6(2)3 ] vir ev − ( p ) ∪ ev − ( q ) = 6312 , where (1) (2) indicates the (unordered) relative boundary conditionof 3-fold tangency. For (ii), there are 30 tangent lines of C through p and another 30through q . Since we have a choice of node over the intersection of thetwo lines, the contribution (ii) is2 · = 1800 . Since the number of bitangent to C is 324, the contribution (iii) is2 ·
324 = 648remembering again the factor 2 for the choice of node. Hence, wecalculate (cid:10) τ ( p ) , τ ( p ) (cid:11) S , β = 6312 + 1800 + 648 = 8760in perfect (and nontrivial) agreement with the prediction (28).4.4. Gromov-Witten/Pairs correspondence.
Let S be a nonsin-gular projective K X = S × E . A stable pair ( F, s ) is a coherent sheaf F with dimension 1 support in X and a section s ∈ H ( X, F ) satisfying the following stability condition: • F is pure , and • the section s has zero dimensional cokernel.To a stable pair, we associate the Euler characteristic and the class ofthe support C of F , χ ( F ) = n ∈ Z and [ C ] = ( β, d ) ∈ H ( X, Z ) . For fixed n and ( β, d ), there is a projective moduli space of stable pairs P n ( X, ( β, d )), see [33, Lemma 1.3].The moduli space P n ( X, ( β, d )) has a perfect obstruction theory ofvirtual dimension 0 which yields a vanishing virtual fundamental class.If β ∈ Pic( S ) is a positive class, then the obstruction theory can be To calculate the relative invariant (29), we have used the program GROWIwritten by A. Gathmann and available on his webpage [10] at TU Kaiserslautern.The submission line to GROWI isgrowi N = 1 , G = 0 , D = 2 , E = 6 , H : 2 , [1 ,
2] : 3 , and the output is 37872 = 3! · (cid:10) τ ( p ) , τ ( p ) (cid:11) S , β by his imprimitive genus 0 Yau-Zaslow calculation in [11]. URVE COUNTING ON K × E reduced to obtain virtual dimension 1. Let β ∨ ∈ H ( S, Q ) be any classsatisfying(30) h β, β ∨ i = 1with respect to the intersection pairing on S . For n ∈ Z , we define(31) P Xn,β,d = Z [ P n ( X, ( β,d ))] red τ (cid:16) π ∗ ( β ∨ ) ∪ π ∗ ([0]) (cid:17) . We follow here the notation of Section 0 for the projections π and π . The insertions in stable pairs theory are defined in [33]. Definition(31) is parallel to (2). As in the Gromov-Witten case, definition (31)is independent of β ∨ satisfying (30) by degeneration and the study ofthe stable pairs theory of the rubber geometry Y .Define the generating series of stable pairs invariants for X is class( β, d ) by P Xβ,d ( y ) = X n ∈ Z P Xn,β,d y n . Elementary arguments show the moduli spaces P n ( X, ( β, d )) are emptyfor sufficiently negative n , so P Xβ,d is a Laurent series in y . Let N X • β,d ( u ) = X g ∈ Z N X • g,β,d u g − be the corresponding Gromov-Witten series for disconnected invariants. Conjecture D.
For a positive class β ∈ Pic( S ) and all d , the series P Xβ,d ( y ) is the Laurent expansion of a rational function in y and N X • β,d ( u ) = P Xβ,d ( y ) after the variable change y = − exp( iu ) . The d = 0 case of Conjecture D is exactly the Gromov-Witten/Pairscorrespondence established in [34] for all β as a step in the proof of theKKV conjecture. The following result is further evidence for Conjec-ture D. Proposition 5.
For primitive β h ∈ Pic( S ) and all d , the series P Xβ h ,d ( y ) is the Laurent expansion of a rational function in y and N X • β,d ( u ) = P Xβ,d ( y ) after the variable change y = − exp( iu ) . Proof.
We may assume S is elliptically fibered as in Section 3.2. Thereduced virtual class of the moduli spaces of stable maps and stablepairs under the degeneration(32) S × C R × C ∪ R × C was studied in [25]. Here, R is a rational elliptic surface. The twocomponents of the degeneration (32) meet along along F × C where F ⊂ R is a nonsingular fiber of π : R → P . The crucial observation is that the reduced virtual class of the modulispaces associated to S × C may be expressed in terms of the standardvirtual classes of the relative geometries (32) of the degeneration. Theabove argument is valid also for the degeneration(33) X = S × E R × E ∪ R × E .
Since the GW/Pairs correspondence for the relative geometry R × E / F × E follows from the results of [31, 32], we obtain the reduced correspon-dence for S × E . (cid:3) The full matrix
The Fock space.
The Fock space of the K3 surface S ,(34) F ( S ) = M d ≥ F d ( S ) = M d ≥ H ∗ ( S [ d ] , Q ) , is naturally bigraded with the ( d, k )-th summand given by F kd ( S ) = H k + d ) ( S [ d ] , Q )For a bihomogeneous element µ ∈ F kd ( S ), we let | µ | = d, k ( µ ) = k. The Fock space F ( S ) carries a natural scalar product (cid:10) · | · (cid:11) definedby declaring the direct sum (34) orthogonal and setting (cid:10) µ | ν (cid:11) = Z S [ d ] µ ∪ ν for every µ, ν ∈ H ∗ ( S [ d ] , Q ). For α, α ′ ∈ H ∗ ( S, Q ), we also write h α, α ′ i = Z S α ∪ α ′ . URVE COUNTING ON K × E If µ, ν are bihomogeneous, then h µ | ν i is nonvanishing only in the case | µ | = | ν | and k ( µ ) + k ( ν ) = 0.For all α ∈ H ∗ ( S, Q ) and m = 0, the Nakajima operators p m ( α ) acton F ( S ) bihomogeneously of bidegree ( − m, k ( α )), p m ( α ) : F kd −→ F k + k ( α ) d − m . The commutation relations(35) [ p m ( α ) , p m ′ ( α ′ )] = − mδ m + m ′ , (cid:10) α, α ′ (cid:11) id F ( S ) , are satisfied for all α, α ′ ∈ H ∗ ( S ) and all m, m ′ ∈ Z \ X ⊂ X m induces a map τ ∗ m : H ∗ ( X, Q ) −→ H ∗ ( X m , Q ) ∼ = H ∗ ( X, Q ) ⊗ m . For τ ∗ = τ ∗ , we have τ ∗ ( α ) = X i,j g ij ( α ∪ γ i ) ⊗ γ j , where { γ i } is a basis of H ∗ ( X ) and g ij is the inverse of the intersectionmatrix g ij = (cid:10) γ i , γ j (cid:11) .For γ ∈ H ∗ ( S, Q ), define the degree zero Virasoro operator L ( γ ) = − X k ∈ Z \ : p k p − k : τ ∗ ( γ ) = − X k ≥ X i,j g ij p − k ( γ i ∪ γ ) p k ( γ j ) , where : −− : is the normal ordered product, see [18]. For α ∈ H ∗ ( S, Q ),we have then [ p k ( α ) , L ( γ )] = k p k ( α ∪ γ ) . Let ∈ H ∗ ( S ) denote the unit. The restriction of L ( γ ) to F d ( S ), L ( γ ) | F d ( S ) : H ∗ ( S [ d ] , Q ) −→ H ∗ ( S [ d ] , Q )is the cup product by the class D d ( γ ) = 1( d − p − ( γ ) p − ( ) d − ∈ H ∗ ( S [ d ] , Q )of subschemes incident to γ , see [19]. In the special case, γ = , L = L ( ) is the energy operator, L ( ) | F d ( S ) = d · id F d ( S ) . Finally, define Lehn’s diagonal operator [19]: ∂ = − X i,j ≥ ( p − i p − j p i + j + p i p j p − ( i + j ) ) τ ∗ ([ X ]) . For d ≥ ∂ acts on F d ( S ) by the cup product with − ∆ S [ d ] , where∆ S [ d ] = 1( n − p − ( ) p − ( ) n − denotes the class of the diagonal in S [ d ] .5.2. Quantum multiplication.
Let S be an elliptic K B and fiber class F . For h ≥
0, let β h = B + hF . We will define quantum multiplication on F ( S ) with respect to theclasses β h .For α , . . . , α m ∈ H ∗ ( S [ d ] , Q ), define the quantum bracket(36) (cid:10) α , . . . , α m (cid:11) S [ d ] q = X h ≥ X k ∈ Z y k q h − Z [ M ,m ( S [ d ] ,C ( β h )+ kA )] red ev ∗ ( α ) · · · ev ∗ m ( α m )as an element of Q (( y ))(( q )). Because d is determined by the α i , weoften omit S [ d ] . The multilinear pairing h· · · i extends naturally to theFock space by declaring the pairing orthogonal with respect to (34).Let ǫ be a formal parameter with ǫ = 0. For a, b, c ∈ H ∗ ( S [ d ] , Q ) , define the (primitive) quantum product ∗ by h a | b ∗ c (cid:11) = (cid:10) a | b ∪ c (cid:11) + ǫ · (cid:10) a, b, c (cid:11) q . As (cid:10) · · · (cid:11) q takes values in Q (( y ))(( q )), the product ∗ is defined over thering H ∗ ( S [ d ] , Q ) ⊗ Q (( y ))(( q )) ⊗ Q [ ǫ ] /ǫ . By the WDVV equation in the reduced case (see [27, Appendix 1]), ∗ is associative. We extend ∗ to an associative product on F ( S ) by b ∗ c = 0 whenever b and c are in different summands of (34).The parameter ǫ has to be introduced since we use reduced Gromov-Witten theory to define the bracket (36). It can be thought of as aninfinitesimal virtual weight on the canonical class K S [ n ] and correspondsin the toric case (see [23, 29]) to the equivariant parameter ( t + t )mod ( t + t ) . By standard arguments [27], the moduli space M ,m ( S [ d ] , C ( β h )+ kA ) is emptyfor k sufficiently negative. URVE COUNTING ON K × E We are mainly interested in the 2-point quantum operator E Hilb : F ( S ) ⊗ Q (( y ))(( q )) −→ F ( S ) ⊗ Q (( y ))(( q ))defined by the bracket (cid:10) a | E Hilb b (cid:11) = (cid:10) a, b (cid:11) q and extended q and y linearly. Because M , ( S [ d ] , α ) has reduced vir-tual dimension 2 d , E Hilb is a self-adjoint operator of bidegree (0 , D , D ∈ H ( S [ d ] , Q ) be divisor classes. By associativity andcommutativity,(37) D ∗ ( D ∗ a ) = D ∗ ( D ∗ a )for all a . By the divisor axiom, we have D d ( γ ) ∗ · (cid:12)(cid:12)(cid:12) F d ( S ) = (cid:16) L ( γ ) + ǫ p ( γ ) E Hilb (cid:17)(cid:12)(cid:12)(cid:12) F d ( S ) −
12 ∆ S [ d ] ∗ · (cid:12)(cid:12)(cid:12) F d ( S ) = (cid:16) ∂ + ǫ y ddy E Hilb (cid:17)(cid:12)(cid:12)(cid:12) F d ( S ) . for every γ ∈ H ( S, Q ). Here, ddy is formal differentiation with respectto the variable y , and p ( γ ) for γ ∈ H ∗ ( S ) is the degree 0 Nakajimaoperator defined by the following condition :[ p ( γ ) , p m ( γ ′ )] = 0for all γ ′ ∈ H ∗ ( S ) , m ∈ Z and p ( γ ) q h − y k S = (cid:10) γ, β h (cid:11) q h − y k S . After specializing D i , we obtain the main commutator relations for E Hilb on F ( S ),(38) p ( γ ) [ E Hilb , L ( γ ′ )] = p ( γ ′ ) [ E Hilb , L ( γ )] p ( γ ) [ E Hilb , ∂ ] = y ddy [ E Hilb , L ( γ )] , for all γ, γ ′ ∈ H ( S, Q ). The equalities (38) are true only after restrict-ing to F ( S ), and not on all of F ( S ) ⊗ Q (( y ))(( q )) by definition of p ( γ )and y ddy .Equation (38) shows the commutator of E Hilb with a divisor inter-section operator to be essentially independent of the divisor. This definition precisely matches the action of the extended Heisenberg algebra (cid:10) p k ( γ ) (cid:11) , k ∈ Z on the full Fock space F ( S ) ⊗ Q [ H ∗ ( S, Q )] under the embedding q h − q B + hF , see [17, section 6.1]. The operators E ( r ) . Let(39) ϕ m,ℓ ( y, q ) ∈ C (( y / ))[[ q ]]be fixed power series that satisfy the symmetries(40) ϕ m,ℓ = − ϕ − m, − ℓ ℓϕ m,ℓ = mϕ ℓ,m . for all ( m, ℓ ) ∈ Z \
0. Depending on the functions (39), define for all r ∈ Z operators E ( r ) : F ( S ) ⊗ C (( y / ))(( q )) −→ F ( S ) ⊗ C (( y / ))(( q ))by the following recursion relations: Step 1.
For all r ≥ E ( r ) (cid:12)(cid:12)(cid:12) F ( S ) ⊗ C (( y / ))(( q )) = δ r F ( y, q ) ∆( q ) · id F ( S ) ⊗ C (( y / ))(( q )) , where F ( y, q ) and ∆( q ) are the functions defined in section 2 consideredas formal expansions in the variables y and q . Step 2.
For all m = 0 , r ∈ Z , [ p m ( γ ) , E ( r ) ] = X ℓ ∈ Z ℓ k ( γ ) m k ( γ ) : p ℓ ( γ ) E ( r + m − ℓ ) : ϕ m,l ( y, q ) Here k ( γ ) denotes the shifted complex cohomological degree of γ , γ ∈ H k ( γ )+1) ( S ; Q ) , and : −− : is a variant of the normal ordered product defined by: p ℓ ( γ ) E ( k ) := ( p ℓ ( γ ) E ( k ) if ℓ ≤ E ( k ) p ℓ ( γ ) if ℓ > . The two steps uniquely determine the operators E ( r ) . It follows fromthe symmetries (40), that E ( r ) respects the Nakajima commutator re-lations (35). Hence E ( r ) acts on F ( S ) and is therefore well defined. Bydefinition, it is an operator of bidegree ( − r, y -linear, but not q linear. Conjecture E.
There exist unique functions ϕ m,ℓ for ( m, ℓ ) ∈ Z \ that satisfy: (i) Initial conditions: ϕ , = G ( y, q ) − , ϕ , = − iF ( y, q ) , ϕ , − = − q ddq ( F ( y, q ) ) . URVE COUNTING ON K × E (ii) E (0) satisfies the WDVV equations: p ( γ ) [ E (0) , L ( γ ′ )] = p ( γ ′ ) [ E (0) , L ( γ )] p ( γ ) [ E (0) , ∂ ] = y ddy [ E (0) , L ( γ )] on F ( S ) for all γ, γ ′ ∈ H ( S, Q ) . Conjecture E has been checked numerically on F d ( S ) for d ≤
5. Thefunctions ϕ m,ℓ + sgn( m ) δ ml are expected to be quasi Jacobi forms withweights and index for all non-vanishing cases given by the followingtable: index weight m = 0 , ℓ = 0 | m | + | ℓ | m = 0 , ℓ = 0 | m | − ϕ m,l are: ϕ , + 1 = 2 K · (cid:18) J ℘ ( z ) − J E + 32 ℘ ( z ) + J ∂ z ( ℘ ( z )) − E (cid:19) ϕ , = 2 K · (cid:18) J ℘ ( z ) − J E + 12 ∂ z ( ℘ ( z )) (cid:19) ϕ , = − · J · K ϕ , − = − · K · (cid:18) J − J ℘ ( z ) − J E − ∂ z ( ℘ ( z )) (cid:19) ϕ , − = 2 J · K · (cid:18) J − J ℘ ( z ) − J E − ∂ z ( ℘ ( z )) (cid:19) , where K = iF and J = ∂ z (log( F )).Conjectures E and F (below) were first proposed in different butequivalent forms in [27].5.4. Further conjectures.
Let L be the energy operator on F ( S ).We define the operator G L : F ( S ) ⊗ Q (( y ))(( q )) −→ F ( S ) ⊗ Q (( y ))(( q ))by G L ( µ ) = G ( y, q ) | µ | · µ for any homogeneous µ ∈ F ( S ). Conjecture F.
For S an elliptic K3 surface we have on F ( S ) E Hilb = E (0) − F ∆ G L . We stated the Conjecture for the elliptic K S with respectto the classes β h = B + hF . By extracting the q h − -coefficient and deforming the K S ′ , β ′ ) of a K S ′ and a primitive curve class β ′ of square 2 h − F ∆ G L on the Fock space F ( S ) isTr F ( S ) F ∆ ˜ q L − G L = 1 F ∆ X d ≥ G d χ ( S [ d ] )˜ q d − . By G¨ottsche’s formula [12], we obtain precisely the correction term(20). Hence Conjectures A and F together implyTr F ( S ) ˜ q L − E (0) = − χ (Ω) . The above equation is a purely algebraic statement about the operator E (0) .Let P n ( Y, ( β h , d )) be the moduli space of stable pairs on the straightrubber geometry Y = S × R defined in Section 1. The reduced virtual dimension of the modulispace P n ( Y, ( β h , d )) is 2 d . Letev i : P n ( Y, ( β h , d )) → S [ d ] , i = 0 , ∞ be the boundary maps .Define the bidegree (0 ,
0) operator E Pairs : F ( S ) ⊗ Q (( y ))(( q )) −→ F ( S ) ⊗ Q (( y ))(( q ))on F d ( S ) by (cid:10) µ (cid:12)(cid:12) E Pairs ν (cid:11) = X h ≥ X n ∈ Z y n q h − Z [ P n ( Y, ( β h ,d ))] red ev ∗ ( µ ) ∪ ev ∗∞ ( ν ) . For d = 0, we take S [0] to be a point. URVE COUNTING ON K × E Conjecture G.
On the elliptic K S , E Hilb + 1 F ∆ G L = y − L E Pairs . We have stated Conjecture G as relating E Hilb and E Pairs . CombiningConjectures F and G leads to the direct prediction on F ( S ): E Pairs = y L E (0) . A conjecture relating the stable pairs theory of S × R to the Gromov-Witten side is formulated exactly as in Conjecture D. We can expressthe conjectural relationship between the different theories by the tri-angle: Gromov-Wittentheoryof K × P Quantum cohomologyof Hilb d ( K K × P Three examples. (i)
Let F be the fiber of the elliptic fibration.Then, we have (cid:10) p − ( F ) d S | E (0) p − ( F ) d S (cid:11) = ( − d (cid:10) S | p ( F ) d E (0) p − ( F ) d S (cid:11) = ( − d (cid:10) S | p ( F ) d E ( d ) ϕ d , p − ( F ) d S (cid:11) = ( − d (cid:10) S | p ( F ) d E (0) ( − d ϕ d , ϕ d − , S (cid:11) = ϕ d , ϕ d − , F ( y, q ) ∆( q )= F ( y, q ) d − ∆( q )in agreement with Theorem 3. We have used p ( F ) = 1 above. (ii) Let W = B + F . Then W = 0 and (cid:10) W, β h (cid:11) = h −
1. In particular p ( W ) acts as ∂ τ = q ddq . We have (cid:10) p − ( W ) d S | E (0) p − ( W ) d S (cid:11) = ( − d (cid:10) S | p ( W ) d E ( d ) ϕ d , p − ( W ) d S (cid:11) = (cid:10) S | p ( W ) d E (0) ϕ d , ϕ d − , S (cid:11) = ∂ dτ ϕ d , ϕ d − , F ( y, q ) ∆( q ) ! = ∂ dτ (cid:18) F ( y, q ) d − ∆( q ) (cid:19) . (iii) Let p ∈ H ( S ; Z ) be the class of a point. For all d ≥
1, let C ( F ) = p − ( F ) p − ( p ) d − S ∈ H ( S [ d ] , Z ) . Then, assuming Conjecture F, (cid:10) C ( F ) (cid:11) q = (cid:10) C ( F ) , D d ( F ) (cid:11) q = 1( d − (cid:10) p − ( F ) p − ( p ) d − S | E (0) p − ( F ) p − ( e ) d − S (cid:11) = 1( d − (cid:10) p − ( p ) d − S | E (0) ϕ , ϕ − , p − ( e ) d − S (cid:11) = ( − d − ( d − (cid:10) S | E (0) ϕ , ϕ − , ( ϕ , + 1) d − p ( p ) d − p − ( e ) d − S (cid:11) = ϕ , ϕ − , ( ϕ , + 1) d − F ( y, q ) ∆( q )= G ( y, q ) d − ∆( q )in full agreement with the first part of Theorem 2 in [27].5.6. The A resolution. Let E Pairs B = [ q − ] E Pairs and E Hilb B = [ q − ] E Hilb be the restriction of E Hilb and E Pairs to the case of the class β = B . The corresponding local case was considered before in [22, 23]. Defineoperators E ( r ) B by (cid:10) S | E ( r ) B S (cid:11) = y (1 + y ) δ r [ p m ( γ ) , E ( r ) B ] = (cid:10) γ, B (cid:11) (cid:0) ( − y ) − m/ − ( − y ) m/ (cid:1) E ( r + m ) B We denote with [ q − ] the operator that extracts the q − coefficient. URVE COUNTING ON K × E for all m = 0 and γ ∈ H ∗ ( S ), see [23, Section 5.1]. Translating theresults of [22, 23] to the K Theorem 6.
We have E Hilb B + y (1 + y ) Id F (S) = y − L E Pairs B = E (0) B . From numerical experiments [27], we expect the expansions ϕ m, = (cid:0) ( − y ) − m/ − ( − y ) m/ (cid:1) + O ( q ) for all m = 0 ϕ m,ℓ = O ( q ) for all ℓ = 0 , m ∈ Z . Because of [ q − ] G L F ∆ = y (1 + y ) Id F ( S ) , we find conjectures F and G to be in complete agreement with Theorem6.From Theorem 6, we obtain the interesting relationTr F ( S ) q L − E Pairs B = 1 y + 2 + y − q Y m ≥ y − q m ) (1 − q m ) (1 + yq m ) . By the symmetry of χ in the variables q and ˜ q , we obtain agreementwith Conjecture A. 6. Motivic theory
Let S be a nonsingular projective K β ∈ Pic( S )be a positive class (with respect to any ample polarization). We willassume β is irreducible (not expressible as a sum of effective classes).To unify our study with [14], we end the paper with a discussion ofthe motivic stable pairs invariants of X = S × E in class ( β, d ). Following the conjectural perspective of [14], we assumethe Betti realization of the motivic invariants of X is both well-defined and independent of deformations of S for which β remains algebraicand irreducible.We define a generating function Z of the Betti realizations of themotivic stable pairs theory of X in classes ( β h , d ) where β h is irreducibleand satisfies h β h , β h i = 2 h − . The series Z depends upon the variables y , q , ˜ q just as before and anew variable u for the virtual Poincare polynomial: Z ( u, y, q, ˜ q ) = 1 u − + 2 + u X h ≥ X d ≥ H (cid:0) P n ( S × E, ( β h , d )) (cid:1) y n q h − ˜ q d − . Here we follow the notation of [14, Section 6] for the normalized virtualPoincar´e polynomial H (cid:0) P n ( S × E, ( β h , d )) (cid:1) ∈ Z [ u, u − ] . In the definition of Z , the prefactor u − +2+ u (the reciprocal of the nor-malized Poincar´e polynomial of E ) quotients by the translation actionof E on P n ( S × E, ( β h , d )).Because of the u normalization, we have the following symmetry of Z in the variable u :(i) Z ( u, y, q, ˜ q ) = Z ( u − , y, q, ˜ q ) .Two further properties which constrain Z are:(ii) the specialization u = − must recover the stable pairs invari-ants (determined by Conjectures A and D), Z ( − , y, q, ˜ q ) = − χ , (iii) the coefficient of ˜ q − must specialize to the motivic series of [14,Section 4] , ( uy − (cid:0) u − − y − (cid:1) · Coeff ˜ q − (cid:16) Z ( u, y, q, ˜ q ) (cid:17) = ∞ Y n =1 − u − y − q n )(1 − u − yq n )(1 − q n ) (1 − uy − q n )(1 − uyq n ) , obtained from the Kawai-Yoshioka calculation [17] . To obtain further constraints, we study the virtual Poincar´e polyno-mial H (cid:16) P − h − d ( S × E, ( h, d )) /E (cid:17) ∈ Z [ u, u − ]which arises as(41) Coeff y − h − d q h − ˜ q d − (cid:0) Z (cid:1) . URVE COUNTING ON K × E For q h − ˜ q d − , the coefficient (41) corresponds to the the lowest orderterm in y . We have an isomorphism of the moduli spaces , P − h − d ( S × E, ( h, d )) /E ∼ = P − h + d ( S, h ) . Hence, we obtain a fourth constraint for Z .(iv) Coeff y − h − d q h − ˜ q d − (cid:0) Z (cid:1) equals the y − h + d q h − coefficient of uy −
1) ( u − − y − ) ∞ Y n =1 − u − y − q n )(1 − u − yq n )(1 − q n ) (1 − uy − q n )(1 − uyq n ) The function − χ has a basic symmetry in the variables q and ˜ q .As stated, condition (iv) is not symmetric in q and ˜ q . However, thesymmetry Coeff y − h − d q h − ˜ q d − (cid:0) Z (cid:1) = Coeff y − h − d q d − ˜ q h − (cid:0) Z (cid:1) can be easily verified from (iv). Unfortunately, further calculationsshow that the symmetry in the variables q and ˜ q appears not to lift tothe motivic theory.A basic question is to specify the modular properties of Z . Wehope conditions (i)-(iv) together with the modular properties of Z willuniquely determine Z . There is every reason to expect the function Z will be beautiful. References [1] A. Beauville,
Vari´et´es K¨ahleriennes dont la premi`ere classe de Chern est nulle ,J. Differential Geom. (1984), 755–782.[2] K. Behrend, The product formula for Gromov-Witten invariants , J. alg. Geom. (1999), 529–541.[3] J. Bruinier, G. van der Geer, G. Harder, and D. Zagier, The 1-2-3 of modularforms , Springer Verlag: Berlin, 2008.[4] J. Bryan,
The Donaldson-Thomas theory of K × E via the topological vertex ,arXiv:1504.02920.[5] J. Bryan and C. Leung, The enumerative geometry of K surfaces and modularforms , JAMS (2000), 549–568.[6] J. Bryan and R. Pandharipande, The local Gromov-Witten theory of curves ,J. Amer. Math. Soc. (2008), 101–136.[7] A. Dabholkar, S. Murthy, and D. Zagier, Quantum black holes, wall crossing,and mock modular forms , arXiv:1208.4074.[8] M. Eichler and D. Zagier,
The theory of Jacobi forms , volume 55 of
Progressin Mathematics , Birkh¨auser Boston Inc., Boston, MA, 1985.[9] C. Faber and R. Pandharipande.
Hodge integrals and Gromov-Witten theory ,Invent. Math., (2000), 173–199. We follow the notation of [14] for the moduli of stable pairs P n ( S, h ) on K [10] A. Gathmann, GROWI: a C++ program for Gromov-Witten invariants , .[11] A. Gathmann, The number of plane conics that are 5-fold tangent to a givencurve , Compos. Math. (2005), 487–501.[12] L. G¨ottsche, The Betti numbers of the Hilbert scheme of points on a smoothprojective surface , Math. Ann. (1990), 193–207.[13] V. A. Gritsenko and V. V. Nikulin,
Siegel automorphic form corrections ofsome Lorentzian Kac-Moody Lie algebras , Amer. J. Math. (1997), 181–224.[14] S. Katz, A. Klemm, and R. Pandharipande,
On the motivic stable pairs in-variants of K surfaces , arXiv:1407.3181.[15] S. Katz, A. Klemm, and C. Vafa, M-theory, topological strings, and spinningblack holes , Adv. Theor. Math. Phys. (1999), 1445–1537.[16] T. Kawai, K surfaces, Igusa cusp form, and string theory , hep-th/9710016.[17] T. Kawai and K Yoshioka, String partition functions and infinite products ,Adv. Theor. Math. Phys. (2000), 397–485.[18] M. Lehn, Lectures on Hilbert schemes, in Algebraic structures and modulispaces , volume 38 of
CRM Proc. Lecture Notes , pages 1–30, Amer. Math.Soc., Providence, RI, 2004.[19] M. Lehn,
Chern classes of tautological sheaves on Hilbert schemes of points onsurfaces , Invent. Math. (1999), 157–207.[20] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande,
Gromov-Wittentheory and Donaldson-Thomas theory. I,
Compos. Math., (2006), 1263–1285.[21] D. Maulik,
Gromov-Witten theory of A n -resolutions , Geom. and Top. (2009), 1729–1773.[22] D. Maulik and A. Oblomkov, Donaldson-Thomas theory of A n × P , Compos.Math. 145 (2009), no. 5, 12491276.[23] D. Maulik and A. Oblomkov, Quantum cohomology of the Hilbert scheme ofpoints on A n -resolutions , JAMS (2009), 1055–1091.[24] D. Maulik and R. Pandharipande, Gromov-Witten theory and Noether-Lefschetz theory , in
A celebration of algebraic geometry , Clay MathematicsProceedings , 469–507, AMS (2010).[25] D. Maulik, R. Pandharipande, and R. Thomas, Curves on K surfaces andmodular forms , J. of Topology (2010), 937–996.[26] H. Nakajima, Lectures on Hilbert schemes of points on surfaces , AMS: Provi-dence, RI, 1999.[27] G. Oberdieck,
Gromov-Witten invariants of the Hilbert scheme of points of a K surface , arXiv:1406.1139.[28] A. Okounkov and R. Pandharipande, Virasoro constraints for target curves ,Invent. Math. (2006), 47–108.[29] A. Okounkov and R. Pandharipande,
Quantum cohomology of the Hilbertscheme of points of the plane , Invent. Math. (2010), 523–557.[30] A. Okounkov and R. Pandharipande,
The local Donaldson-Thomas theory ofcurves , Geom. Topol. (2010), 1503–1567.[31] R. Pandharipande and A. Pixton, Gromov-Witten/Pairs descendent correspon-dence for toric 3-folds , Geom. and Top. (to appear).[32] R. Pandharipande and A. Pixton,
Gromov-Witten/Pairs correspondence forthe quintic 3-fold , arXiv:1206.5490.[33] R. Pandharipande and R. P. Thomas,
Curve counting via stable pairs in thederived category , Invent. Math. (2009), 407–447.
URVE COUNTING ON K × E [34] R. Pandharipande and R. P. Thomas, The Katz-Klemm-Vafa conjecture for K surfaces , arXiv:1404.6698.[35] C. T. C. Wall, On the orthogonal groups of unimodular quadratic forms , Math.Ann. (1962), 328–338.[36] S.-T. Yau and E. Zaslow,
BPS states, string duality, and nodal curves on K B457 (1995), 484–512.
ETH Z¨urich, Department of Mathematics
E-mail address : [email protected] ETH Z¨urich, Department of Mathematics
E-mail address ::