The Local Gromov Witten Invariants of Configurations of Rational Curves

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1 Claremont Colleges Claremont All HMC Faculty Publications and Research HMC Faculty Scholarship The Local Gromov Witten Invariants of Configurations of Rational Curves Dagan Karp Harvey Mudd College Chiu-Chu Melissa Liu Northwestern University Marcos Mariño CERN Recommended Citation Karp, Dagan, Chiu-Chu Melissa Liu and Marcos Mariño. "The local Gromov-Witten invariants of configurations of rational curves." Geometry & Topology. 10 (2006) This Article is brought to you for free and open access by the HMC Faculty Scholarship at Claremont. It has been accepted for inclusion in All HMC Faculty Publications and Research by an authorized administrator of Claremont. For more information, please contact scholarship@cuc.claremont.edu.

2 Geometry & Topology 10 (2006) The local Gromov Witten invariants of configurations of rational curves DAGAN KARP CHIU-CHU MELISSA LIU MARCOS MARIÑO We compute the local Gromov Witten invariants of certain configurations of rational curves in a Calabi Yau threefold. These configurations are connected subcurves of the minimal trivalent configuration, which is a particular tree of 1 s with specified formal neighborhood. We show that these local invariants are equal to certain global or ordinary Gromov Witten invariants of a blowup of 3 at points, and we compute these ordinary invariants using the geometry of the Cremona transform. We also realize the configurations in question as formal toric schemes and compute their formal Gromov Witten invariants using the mathematical and physical theories of the topological vertex. In particular, we provide further evidence equating the vertex amplitudes derived from physical and mathematical theories of the topological vertex. 14N35; 53D45 1 Introduction Let Z be a closed subvariety of a smooth projective threefold such that is a local Calabi Yau threefold near Z. In some cases, the contribution to the Gromov Witten invariants of by maps to Z can be isolated and defines local Gromov Witten invariants of Z in. Information obtained from the study of local Gromov Witten theory can be used to gain insight into Gromov Witten theory in general. This has led to a great amount of interest in the subject. The study of the local invariants of curves in a Calabi Yau threefold has a particularly rich history. Their study goes back to the famous Aspinwall Morrison formula for the local invariants of a single 1 smoothly embedded in a Calabi Yau threefold with normal bundle O. 1/ O. 1/; this result is studied by Aspinwall and Morrison [3], Cox and Katz [9], Faber and Pandharipande [10], Kontsevich [18], Lian, Liu and Yau [23], Manin [27], Pandharipande [31], and Voisin [32]. The local invariants of nonsingular curves of any genus have been completely determined by Bryan and Pandharipande [6; 7; 8]. In [5], Bryan, Katz and Leung computed local invariants of Published: 7 March 2006 DOI: /gt

3 116 Karp, Liu and Mariño certain rational curves with nodal singularities, and in particular, contractible ADE configurations of rational curves. The local invariants of the closed topological vertex, which is a configuration of three 1 s meeting in a single triple point, were computed by Bryan and the first author [4]. In this paper, we will compute local invariants of certain configurations of rational curves. The configurations considered in this paper are all connected subtrees of the minimal trivalent configuration, which is a configuration of three chains of 1 s meeting in a triple point (see Figure 1 below). A precise description of the formal neighborhood will be given in Section 3. C 2 C 1 B 1 B 2 A 1 A 2 Figure 1: The minimal trivalent configuration Y N D S N A i [ B i [ C i. The normal bundles of A 1 ; B 1 ; C 1 are isomorphic to O. 1/ O. 1/; the normal bundle of any other irreducible component is isomorphic to O O. 2/. 1.1 Local Gromov Witten invariants Let Z be a closed subvariety of a smooth projective Calabi Yau threefold. Let M g.; d/ denote the stack of genus g stable maps to representing d 2 H 2.; /. It is a Deligne Mumford stack with a perfect obstruction theory of virtual dimension zero which defines a virtual fundamental zero-cycle ŒM g.; d/ vir. Whenever the substack M g.z/ consisting of stable maps whose image lies in Z is a union of path connected components of M g.; d/, it inherits a degree-zero virtual class. The genus-g local Gromov Witten invariant of Z in is defined to be the degree of this virtual class, and is denoted by N g d.z /. We write N g d.z/ when the formal neighborhood is understood. We will consider genus g, degree d local Gromov Witten invariants N g d.y N /, where d D N jd1 d 1;j ŒA j C d 2;j ŒB j C d 3;j ŒC j 2 H 2.Y N I /:

4 Gromov Witten invariants of configurations of curves 117 For simplicity, we write d D.d 1 ; d 2 ; d 3 / where d i D.d i;1 ; : : : ; d i;n /. In this paper, we always assume d is effective in the sense that d i;j 0. We will show that the local invariants N g d.y N / are well defined in the following cases: (i) (The minimal trivalent configuration) d 1;1 D d 2;1 D d 3;1 D 1. (ii) (A chain of rational curves) d 1;1 > 0, d 2;j D d 3;j D 0 for 1 j N. We will see in Section 3 that the formal neighborhood of Y N has a cyclic symmetry, so one can cyclically permute d 1 ; d 2 ; d 3 in Case (ii). We show that in the above cases the local invariants N g d.y N / are equal to certain global or ordinary Gromov Witten invariants of a blowup of 3 at points (Section 4), and we compute these ordinary invariants using the geometry of the Cremona transform (Section 2). To state our results, define constants C g by (1) Theorem 1 1 gd0 t=2 2 C g t 2g D D sin.t =2/ 1 gd0 jb 2g.2g 1/j t 2g :.2g/! (The minimal trivalent configuration) Suppose that d 1;1 D d 2;1 D d 3;1 D 1: Then N g d.y N Cg if 1 D d / D i;1 d i;n 0 for i D 1; 2; 3; 0 otherwise. Theorem 2 (A chain of rational curves) Suppose that d 1 D.d 1 ; : : : ; d N /; d 2 D d 3 D.0; : : : ; 0/; where d 1 > 0. Then 8 ˆ< N g d.y N C g d 2g 3 if d 1 D d 2 D D d k D d > 0 and / D d kc1 D d kc2 D D d N D 0 for some 1 k N ˆ: 0 otherwise. Our results are new and add to the list of configurations of rational curves for which the local Gromov Witten invariants are known. The configuration in Theorem 2 is an A N curve. It is interesting to compare Theorem 2 with the result for a generic contractible A N curve E D E 1 [ [ E N from Bryan Katz Leung [5, Proposition 2.10]:

5 118 Karp, Liu and Mariño Fact 1 (A generic contractible A N curve [5]) Assume d i > 0 for i D 1; : : : ; N. Let N g.d 1 ; : : : ; d N / denote genus g local Gromov Witten invariants of E in the class P N jd1 d j ŒE j. Then Cg d N g.d 1 ; : : : ; d N / D 2g 3 d 1 D D d N D d > 0; 0 otherwise. Note that Y 1 is the closed topological vertex. By the results in Faber Pandharipande [10] and Bryan Karp [4], N g d 1 ;d 2 ;d 3.Y 1 / is defined in the following cases: (iii) (Super-rigid 1 ) d 1 > 0; d 2 D d 3 D 0 (and its cyclic permutation). (iv) (The closed topological vertex) d 1 ; d 2 ; d 3 > 0. Fact 2 (Super-rigid 1 [10]) Suppose that d > 0. Then N g d;0;0.y 1 / D N g 0;d;0.Y 1 / D N g 0;0;d.Y 1 / D C g d 2g 3 : Fact 3 (The closed topological vertex [4]) Suppose that d 1 ; d 2 ; d 3 > 0. Then N g.y 1 Cg d / D 2g 3 d 1 D d 2 D d 3 D d > 0; d 1 ;d 2 ;d 3 0 otherwise. 1.2 Formal Gromov Witten invariants The minimal trivalent configuration Y N together with its formal neighborhood is a nonsingular formal toric Calabi Yau (FTCY) scheme yy N. The formal Gromov Witten invariants zn g d. yy N / of yy N are defined for all nonzero effective classes (see Section 5.1 and Bryan Pandharipande [8, Section 2.1]). Moreover, zn g d. yy N / D N g d.y N / in all the above cases (i) (iv). Introduce formal variables ; t i;j and define zz N.I t/ D exp 2 zn g;d. yy N /e dt g0 d 2g where d runs over all nonzero effective classes, and t D.t 1 ; t 2 ; t 3 /; t i D.t i;1 ; : : : ; t i;n /; d t D 3 jd1 N d i;j t i;j : We call zz N.I t/ the partition function of formal Gromov Witten invariants of yy N. It is the generating function of disconnected formal Gromov Witten invariants of yy N.

6 Gromov Witten invariants of configurations of curves 119 In Section 5, we will compute zz N.I t/ by the mathematical theory of the topological vertex (see Li Liu Liu Zhou [22]) and get the following expression (Proposition 17): 1 (2) zz N.I t/ D exp 1 nœn 2 3 2k 1 k 2 N E zw E.q/ e n.t i;k 1 CCt i;k2 / 3Y. 1/ ji j e ji jt i;1 s. i / t.ui.q; t i //: where E D. 1 ; 2 ; 3 / is a triple of partitions, q D ep 1, Œn D q n=2 q n=2. The precise definitions of zw E.q/ and s. i / t.ui.q; t i // will be given in Section 1.3. In particular, we will show that (3) zz 1.I t/ D E zw E.q/ 3Y 1. 1/ ji j e ji jt i W. i / t.q/ D exp where t D.t 1 ; t 2 ; t 3 /, W.q/ is defined by (9) in Section 1.3, and! Q n.t/ nœn 2 (4) Q n.t/ D e nt 1 C e nt 2 C e nt 3 e n.t 1Ct 2 / e n.t 2Ct 3 / e n.t 3Ct 1 / C e n.t 1Ct 2 Ct 3 / : In Section 6, we will compute zz N.I t/ by the physical theory of the topological vertex (see Aganagic Klemm Mariño Vafa [2]) and get the following expression (Proposition 20): 1 (5) Z N.I t/ D exp 1 nœn 2 3 2k 1 k 2 N E W E.q/ e n.t i;k 1 CCt i;k2 / 3Y. 1/ ji j e ji jt i;1 s. i / t.ui.q; t i // where W E.q/ is defined by (8) in Section 1.3. In particular, we will show that (Proposition 19): (6) Z 1.I t/ D E W E.q/ 3Y 1. 1/ ji j e ji jt i W. i / t.q/ D exp! Q n.t/ nœn 2

7 120 Karp, Liu and Mariño The equivalence of the physical and mathematical theories of the topological vertex boils down to the following combinatorial identity: (7) W 1 ; 2 ; 3.q/ D zw 1 ; 2 ; 3.q/: It is known that (7) holds when one of the three partitions is empty (see the work of Li, C-C M Liu, K Liu and Zhou [24; 22]). When none of the partitions is empty, Klemm has checked all the cases where j i j 6 by computer. Up to now, a mathematical proof of (7) in full generality is not available. Equations (3) and (6) imply the following result. Theorem 3 3Y W E.q/ 1/ E. ji j e ji jti W. i / t.q/ D E 3Y zw E.q/. 1/ ji j e ji jt i W. i / t.q/: Theorem 3 provides further evidence of (7) equating the vertex amplitudes derived from physical and mathematical theories of the topological vertex. 1.3 The topological vertex In [2], Aganagic, Klemm, Mariño, and Vafa proposed that Gromov Witten invariants of any toric Calabi Yau threefold can be expressed in terms of certain relative invariants of its 3 charts, called the topological vertex. They suggested that these local relative invariants should count holomorphic maps from bordered Riemann surfaces to 3 where the boundary circles are mapped to three explicitly specified Lagrangian submanifolds L 1 ; L 2 ; L 3. The topological vertex depends on three partitions E D. 1 ; 2 ; 3 /, where i corresponds to the winding numbers (the homology classes of boundary circles) in L i. There is a symmetry on 3 cyclically permuting L 1 ; L 2 ; L 3, so one expects the topological vertex to be symmetric under a cyclic permutation of the three partitions 1 ; 2 ; 3. In [2] the topological vertex was computed by using the conjectural relation between open Gromov Witten invariants on toric Calabi Yau threefolds and Chern Simons invariants of knots and links. It has the following form: (8) W E.q/ D q 2 =2C 3 =2 c 1 c.3 / t 1 3 ; 1 ; 3 W. 2 / t 1.q/W 2 3.q/ : W 2.q/

8 Gromov Witten invariants of configurations of curves 121 In (8), t denotes the partition transposed to. The expression (8) involves various quantities that we now define. is given by D i i. i 2i C 1/: The coefficients c are Littlewood Richardson coefficients. They can be defined in terms of Schur functions as follows s s D c s : Here, Schur functions are regarded as a basis for the ring ƒ of symmetric polynomials in an infinite number of variables. The quantity W.q/ can be also defined in terms of Schur functions as follows: (9) W.q/ D s x i D q ic 1 2 : One can show that (10) W t.q/ D q =2 W.q/: We also define, in analogy to skew Schur functions, (11) W =.q/ D c W.q/: Finally, W.q/ is defined by (12) W.q/ D q =2C =2 W t =.q/w t =.q/: This expression for W.q/ is different from the one used originally in [2]. The fact that both agree follows from cyclicity of the vertex, and it has been proved in detail by Zhou [33]. The expression for the vertex in terms of Schur functions is given in Okounkov Reshetikhin Vafa [30] where the cyclicity of the vertex is also proved. In Li Liu Liu Zhou [22] the topological vertex was interpreted and defined as local relative invariants of a configuration C 1 [C 2 [C 3 of three 1 s meeting at a point p 0 in a relative Calabi Yau threefold.z; D 1 ; D 2 ; D 3 /, where K Z CD 1 CD 2 CD 3 Š O Z, C i intersects D i at a point p i p 0, and C i \D j is empty for i j. The partition i corresponds to the ramification pattern over p i. It is shown in [22] that Gromov Witten invariants of any toric Calabi Yau threefold (or more generally, formal Gromov Witten invariants of formal toric Calabi Yau threefolds) can be expressed in terms of local relative invariants as described above, and the gluing rules coincide with those stated

9 122 Karp, Liu and Mariño in Aganagic Klemm Mariño Vafa [2]. The following expression of the vertex was derived in [22]: (13) zw E.q/ D q /=2 c C c 1 c 3. 1 / t 2. 1 / t / t C ; 1 ; 3 ; 1 ; 3 q. 2 C 3 2 /=2 W C 3.q/ Here 2 D / if D. 1 2 /. Recall that z D Y i1 i m i m i! 1 z 1./ 3.2/: where m i D m i./ is the number of parts of the partition equal to i (see Macdonald [26, p.17]). It is expected that the two different enumerative interpretations in [2] and in [22] of the vertex give rise to equivalent counting problems, in the spirit of the following simple example: counting ramified covers of a disc by bordered Riemann surfaces with prescribed winding numbers is equivalent to counting ramified covers of a sphere by closed Riemann surfaces with prescribed ramification pattern over 1. Finally, we introduce some notation which will arise in computations in Section 6. For any positive integer n, define (14) u i n.q; t i/ D 1 1 C Œn N e n.t i;2cct i;k / : kd2 Given a partition D. 1 2 ` > 0/, define (15) u i.q; t Ỳ i/ D u i j.q; t i /: and jd1 (16) s.u i.q; t i // D In particular, when N D 1, we have jjdjj./ u i z.q; t i/: u i n D 1 Œn D q ic1=2 i>0

10 Gromov Witten invariants of configurations of curves 123 So (17) s.u i.q; t i // D s.x i D q ic 1 2 / D W.q/: Acknowledgements The authors give warm thanks to Jim Bryan for helpful conversations. The first author thanks NSERC for its support. The second author thanks Jun Li, Kefeng Liu, Jian Zhou for collaboration [22] and Shing-Tung Yau for encouragement. Finally, the authors thank the referee for the detailed comments and numerous valuable suggestions on the presentation. 2 Cremona In this section we prove Theorems 1 and 2 using the geometry of the Cremona transform. We assume that the formal neighborhood Y N is as constructed in Section 3. We also assume that the local invariants of Y N are equal to certain ordinary invariants of, which we prove in Section The blowup of 3 at points We briefly review the properties of the blowup of 3 at points used here for completeness and to set notation. This material can be found in much greater detail in, for instance, Griffiths Harris [13]. Let! 3 be the blowup of 3 along M distinct points fp 1 ; : : : ; p M g. We describe the homology of. All (co)homology is taken with integer coefficients. Note that we may identify homology and cohomology as rings via Poincaré duality, where cup product is dual to intersection product. Let H be the total transform of a hyperplane in 3, and let E i be the exceptional divisor over p i. Then H 4.; / has a basis H 4. / D hh; E 1 ; : : : ; E M i : Furthermore, let h 2 H 2. / be the class of a line in H, and let e i be the class of a line in E i. The collection of all such classes form a basis of H 2. /. H 2. / D hh; e 1 ; : : : ; e M i The intersection ring structure is given as follows. Let pt 2 H 0. / denote the class of a point. Two general hyperplanes meet in a line, so H H D h. A general hyperplane

11 124 Karp, Liu and Mariño and line intersect in a point, so H h D pt. Also, a general hyperplane is far from the center of a blowup, so all other products involving H or h vanish. The restriction of O.E i / to E i Š 2 is the dual of the bundle O 2.1/, so E i E i is represented by minus a hyperplane in E i, i.e. E i E i D e i, and E 3 i D. 1/3 1 pt D pt (see Fulton [11]). Furthermore, the centers of the blowups are far away from each other, so all other intersections vanish. In summary, the following are the only non-zero intersection products. H H D h H h D pt E i E i D e i E i e i D pt Also, we point out the that the canonical bundle K is easy to describe in this basis: M K D 4H C 2 Finally, we introduce a notational convenience for the Gromov Witten invariants of 3 blown up at points in a Calabi Yau class. Any curve class is of the form E i ˇ D dh M a i e i for some integers d; a i where d is non-negative. Thus K ˇ D 0 if and only if 2d D P M a i. In that case, the virtual dimension of M g.; ˇ/ is zero, and Z h i g;ˇ D ŒM g.;ˇ/ vir 1 is determined by the discrete data fd; a i ; : : : ; a M g. Then, we may use the shorthand notation For example, h i g;ˇ D hdi a 1; : : : ; a M i g : h i g;5h e 1 e 2 2e 3 3e 5 3e 6 D h5i 1; 1; 2; 0; 3; 3i g : Furthermore, the Gromov Witten invariants of do not depend on ordering of the points p i, and thus for any permutation of M points, hdi a 1 ; : : : ; a M i g D di a.1/ ; : : : ; a.m / g :

12 Gromov Witten invariants of configurations of curves Properties of the invariants of the blowup of 3 at points First, we use the fact, shown in Bryan Karp [4], that the Gromov Witten invariants of the blowup of 3 along points have a symmetry which arises from the geometry of the Cremona transformation. P Theorem 4 (Bryan Karp [4]) Let ˇ D dh M a ie i with 2d D P M a i and assume that a i 0 for some i > 4. Then we have the following equality of Gromov Witten invariants: h i g;ˇ D h i g;ˇ0 where ˇ0 D d 0 h P M a0 i e i has coefficients given by d 0 D 3d 2.a 1 C a 2 C a 3 C a 4 / a 0 1 D d.a 2 C a 3 C a 4 / a 0 2 D d.a 1 C a 3 C a 4 / a 0 3 D d.a 1 C a 2 C a 4 / a 0 4 D d.a 1 C a 2 C a 3 / a 0 5 D a 5 : am 0 D a M : We also use the following vanishing lemma, and a few of its corollaries. Lemma 5 Let be the blowup of 3 at M distinct generic points fx 1 ; : : : ; x M g, and ˇ D dh P M a ie i with 2d D P M a i, and assume that d > 0 and a i < 0 for some i. Then M g.; ˇ/ D : Corollary 6 For any M points fx 1 ; : : : ; x M g and and ˇ as above the corresponding invariant vanishes; h i g;ˇ D 0: This follows immediately from the deformation invariance of Gromov Witten invariants and Lemma 5.

13 126 Karp, Liu and Mariño Proof In genus zero, Lemma 5 follows from a vanishing theorem of Gathmann [12, Section 3]. In order to prove Lemma 5, for arbitrary genus, it suffices to show that the result holds for a specific choice of points, as if the moduli space is empty for a specific choice, then it is empty for the generic choice. By choosing some of the points to be coplanar, and the rest to also be coplanar on a second plane, the result follows. For further details, see Karp [17]. Corollary 7 Let be the blowup of 3 along M points and define ˇ D dh P M a ie i where 2d D P M a i and d > 0. Also define 0! to be the blowup of at a generic point p, so that 0 is deformation equivalent to the blowup of 3 at M C 1 distinct points. Let fh 0 ; e1 0 ; : : : ; e0 M C1 g be a basis of H 2. 0 /, and let ˇ0 D dh 0 P M a iei 0. Then hdi a 1 ; : : : a M ; 0i 0 g D hdi a 1; : : : ; a M i g Proof This result follows from the more general results of Hu [14]. An independent proof using Lemma 5 can be found in Karp [17]. 2.3 Proof of Theorem 1 Let the blowup space N C1 and the minimal trivalent configuration Y N be as constructed in Section 3 on page 128. By Proposition 8 on page 133 we have Assume that the invariant is non-zero: N g d.y N / D h i N C1 g;d : h i N C1 g;d D 3I 1; 1 d 1;2 ; : : : ; d 1;N 1 d 1;N ; d 1;N ; 1; 1 d 2;2 ; : : : ; d 2;N 1 d 2;N ; d 2;N ; 1; 1 d 3;2 ; : : : ; d 3;N 1 d 3;N ; d 3;N N C1 0 Then, by Corollary 6, the coefficient of each e i ; f i ; g i is non-negative. Thus, for i D 1; 2; 3, (18) 1 d i;2 d i;n 0: g

14 Gromov Witten invariants of configurations of curves 127 Therefore we compute h i N C1 g;d D 3I 1; 0; : : : ; 0; 1; 1; 0; : : : ; 0; 1; 1; 0; : : : ; 0; 1 N C1 g D h3i 1; 1; 1; 1; 1; 1i 2 g ; where the last equality follows from Corollary 7. So when (18) holds, we have N g d.y N / D N g 1;1;1.Y 1 / D C g : The last equality follows from Fact 3 (see Bryan Karp [4]). 2.4 Proof of Theorem 2 Let the blowup space z N C1 and the chain of rational curves YA N Section 3 on page 130. By Proposition 10 on page 137 we have be as constructed in where Assume that the invariant is non-zero: N g d.y N / D N g d 1.Y N A / D h i z N C1 g;d d 1 D.d 1 ; : : : ; d N /; d 2 D d 3 D.0; : : : ; 0/: h i z N C1 g;d D hd 1 I d 1 ; d 1 d 2 ; : : : ; d N 1 d N ; d N i z N C1 g 0: By Corollary 6 the multiplicities are decreasing: d 1 d 2 d N 0 Therefore, as d 1 > 0, there exists some 1 j N such that Then, using Corollary 7, we compute d 1 d j > 0; d jc1 D D d N D 0: N g d.y N / D d 1 I d 1 ; d 1 d 2 ; : : : ; d j 1 d j ; 0; : : : 0 z N C1 D d 1 I d 1 ; d 1 d 2 ; : : : ; d j 1 d j ; d j z j C1 g : g

15 128 Karp, Liu and Mariño Note that for any 1 i j C 1 we may reorder z d1 I d 1 ; d 1 d 2 ; : : : ; d j 1 d j ; d j C1 j D g hd 1 I d 1 ; d 1 ; d i d ic1 ; 0; 0; d 1 d 2 ; : : : Applying Cremona invariance (Theorem 4) we compute : : : ; d i 2 d i 1 ; d ic1 d ic2 ; : : : ; d j 1 d j ; d j z j C1 h i z N C1 g;d D d 1 2.d i d ic1 /I d 1.d i d ic1 /; 0; d ic1 d i ; d ic1 d i ; d 1 d 2 ; : : : ; d j 1 d j ; d j z j C3 g : Then, by Corollary 6, d ic1 d i. Since this inequality holds for every 1 i j we have d 1 d j. Therefore d 1 D D d j D d: g Thus we have h i z N C1 g;d D hdi d; 0; : : : ; 0; di z j C1 g D hdi d; di z 2 g D N g d;0;0.y 1 / D C g d 2g 3 The last equality follows from Faber Pandharipande [10]. 3 Construction We construct these configurations as subvarieties of a locally Calabi Yau space N C1, which is obtained via a sequence of toric blowups of 3 : N C1 N C1! N N! 2! 1 1! 0 D 3 In fact, ic1 will be the blowup of i along three points. Our rational curves will be labeled by A i ; B i ; C i, where 1 i N, reflecting the nature of the configuration. Curves in intermediary spaces will have super-scripts, and their corresponding proper transforms in will not. The standard torus D. / 3 action on 3 is given by.t 1 ; t 2 ; t 3 /.x 0 W x 1 W x 2 W x 3 / 7!.x 0 W t 1 x 1 W t 2 x 2 W t 3 x 3 /:

16 Gromov Witten invariants of configurations of curves 129 There are four fixed points in 0 W D 3 ; we label them p 0 D.1W 0W 0W 0/, q 0 D.0W 1W 0W 0/, r 0 D.0W 0W 1W 0/ and s 0 D.0W 0W 0W 1/. Let A 0, B 0 and C 0 denote the (unique, invariant) line in 0 through the two points fp 0 ; s 0 g, fq 0 ; s 0 g and fr 0 ; s 0 g, respectively. Define 1 1! 0 to be the blowup of 0 at the three points fp 0 ; q 0 ; r 0 g, and let A 1 ; B 1 ; C 1 1 be the proper transforms of A 0 ; B 0 and C 0. The exceptional divisor in 1 over p 0 intersects A 1 in a unique fixed point; call it p Similarly, the exceptional divisor in 1 also intersects each of B 1 and C 1 in unique fixed points; call them q 1 and r 1. q 2 B 2 2 C 2 1 r 2 C 2 2 B 2 1 A 2 1 A 2 2 p 2 Figure 2: The invariant curves in 2 Now define 2 2! 1 to be the blowup of 1 at the three points fp 1 ; q 1 ; r 1 g, and let A 2 1 ; B2 1 ; C be the proper transforms of A 1 ; B 1 ; C 1. The exceptional divisor over p 1 contains two fixed points disjoint from A 2 1. Choose one of them, and call it p 2; this choice is arbitrary. Similarly, there are two fixed points in the exceptional divisors above q 1 ; r 1 disjoint from B1 2; C 1 2. Choose one in each pair identical to the choice of p 2 and call them q 2 and r 2 (identical makes sense here as the configuration of curves in Figure 2 is rotationally symmetric). This choice is indicated in Figure 2. Let A 2 2 denote the (unique, invariant) line intersecting A 2 1 and p 2. Define B2 2; C 2 2 analogously.

17 130 Karp, Liu and Mariño Clearly 2 is deformation equivalent to a blowup of 3 at six distinct points. The invariant curves in 2 are depicted in Figure 2, where each edge corresponds to a invariant curve in 2, and each vertex corresponds to a fixed point. q 3 B 3 3 B 3 2 C 3 1 r 3 C 3 3 C 3 2 B A A 3 2 A 3 3 p 3 Figure 3: The invariant curves in 3 We now define a sequence of blowups beginning with 2. Fix an integer N 2. For each 1 < i N, define ic1 ic1! i to be the blowup of i along the three points p i ; q i ; r i. Let A ic1 j ic1 denote the proper transform of Aj i for each 1 j i. The exceptional divisor in ic1 above p i contains two fixed points, choose one of them and call it p ic1. Similarly choose q ic1 ; r ic1, and define A ic1 ic1 ic1 to be the line intersecting A ic1 i and p ic1, with B ic1 ic1 ; C ic1 ic1 defined similarly. The invariant curves in 3 are shown in Figure 3. Finally, we define the minimal trivalent configuration Y N N C1 by Y N D [ A j [ B j [ C j ; where 1jN A j D A N C1 j ; B j D B N C1 j ; C j D C N C1 j : The configuration Y N is shown in Figure 4, along with all other invariant curves in N C1. It contains a chain of rational curves: Y N A D A 1 [ [ A N :

18 Gromov Witten invariants of configurations of curves 131 h f 1 g 1 g 1 g 2 f 1 f N C1 g 2 g 3 f N C1 f N C1 ŒB N D f N f N C1 f N C1 g 3 g 4 f N f N C1 g 2 g1 g 5 g 6 g 3 ŒB 4 D f 4 f 5 f 1 g 4 ŒB 3 D f 3 f 4 f 4 f 5 f 6 g N g N C1 g N C1 g g N C1 1 gn C1 g N C1 ŒCN D g N g N C1 ŒC 4 D g 4 g 5 ŒC 3 D g 3 g 4 ŒB 2 D f 2 f 3 ŒC 1 Dh ŒC2 g 1 g 2 D g 2 g 3 ŒB 1 D h f 1 f 2 ŒA 1 D h e 1 e 2 ŒA 2 D e 2 e 3 f 3 f 2 f 3 f 4 f 2 f 3 f 1 f 2 ŒA 3 D e 3 e 4 h e1 g 1 e 2 e 3 ŒA 4 D e 4 e 5 h e 1 f 1 e 4 ŒA N D e N e N C1 e N C1 e N C1 e 2 e 3 e 3 e 4 e 5 e 6 e N e N C1 e N C1 e 1 e 2 e 1 e N C1 e 1 Figure 4: The invariant curves in N C1 3.1 Homology We now compute H. N C1 ; / and identify the class of the configuration ŒY N 2 H 2. N C1 ; /. All (co)homology will be taken with integer coefficients. We denote divisors by upper case letters, and curve classes with the lower case. In addition, we decorate homology classes in intermediary spaces with a tilde, and their total transforms in N are undecorated. Let ze 1 ; zf 1 ; zg 1 2 H 4. 1 / denote the exceptional divisors in 1! 0 over the points p 0 ; q 0 and r 0, and let E 1 ; F 1 ; G 1 2 H 4. / denote their total transforms. Continuing,

19 132 Karp, Liu and Mariño for each 1 i N C 1, let ze i ; zf i ; zg i 2 H 4. i / denote the exceptional divisors over the points p i 1 ; q i 1 ; r i 1 and let E i ; F i ; G i 2 H 4. / denote their total transforms. Finally, let H denote the total transform of the hyperplane in 0 D 3. The collection of all such classes fh; E i ; F i ; G i g, where 1 i N C 1, spans H 4. N C1 /. Similarly, for each 1 i N C 1, let ze i ; z f i ; zg i 2 H 2. ic1 / denote the class of a line in ze i ; zf i ; zg i and let e i ; f i ; g i 2 H 2. / denote their total transforms. In addition, let h 2 H 2. N C1 / denote the class of a line in H. Then H 2. N C1 / has a basis given by fh; e i ; f i ; g i g. The intersection product ring structure is given as follows. Note that N C1 is deformation equivalent to the blowup of 3 at 3N distinct points. Therefore, these H H D h H h D pt E i E i D e i E i e i D pt F i F i D f i F i f i D pt G i G i D g i G i g i D pt are all of the nonzero intersection products in H. N C1 /. In this basis, the classes of the components of Y N are given as follows. ( h e1 e 2 if i D 1 ŒA i D e i e ic1 otherwise ( h f1 f 2 if i D 1 ŒB i D f i f ic1 otherwise ( h g1 g 2 if i D 1 ŒC i D g i g ic1 otherwise To see this, recall that A 1 is the proper transform of a line through two points which are centers of a blowup, and that A i, for i > 1, is the proper transform of a line in an exceptional divisor containing a center of a blowup. B i and C i are similar. 4 Local to global In this section, we will show that the local invariants N g d.y / are equal to the ordinary invariants h i N C1 g;d in case Y is either the minimal trivalent configuration Y N or the chain of rational curves YA N defined in Section 3.

20 Gromov Witten invariants of configurations of curves The minimal trivalent configuration Proposition 8 Let f W! N C1 represent a point in M g. N C1 ; d/, where 1 D d i;1 d i;n 0: Then the image of f is contained in the minimal trivalent configuration Y N D [ A j [ B j [ C j : Proof map 1jN We use the toric nature of the construction. Assume that there exists a stable Œf W! N C1 2 M g. N C1 ; d/ such that Im.f / 6 Y N. Then there exists a point p 2 Im.f / such that p 62 Y N. Recall that invariant subvarieties of a toric variety are given precisely by orbit closures of one-parameter subgroups of. So in particular the limit of p under the action of a one-parameter subgroup is a fixed point. Moreover, since p 62 Y N, there exists a one-parameter subgroup W! such that lim t!0.t/ p D q where q is fixed and q 62 Y N. The limit of acting on Œf is a stable map f 0 such that q 2 Im.f 0 /. It follows that q is in the image of all stable maps in the orbit closure of Œf 0. Thus, there must exist a stable map Œf 00 W! N C1 2 M g. N C1 ; d/ such that Im.f 00 / is invariant and Im.f 00 / 62 Y N. We show that this leads to a contradiction. Let F denote the union of the invariant curves in N C1 ; it is shown above in Figure 4. We study the possible components of F contained in the image of f 00. Note that the push forward of the class of is given by f 00 Œ D 3h e 1 N 1.d 1;j d 1;jC1 /e jc1 d 1;N e N C1 jd1 f 1 N 1.d 2;j d 2;jC1 /f jc1 d 2;N f N C1 jd1 g 1 N 1.d 3;j d 3;jC1 /g jc1 d 3;N g N C1 : jd1

21 e 2 e Karp, Liu and Mariño g 2 g 3 f N C1 f N C1 ŒB N D f N f N C1 f N C1 g 3 g 4 f N f N C1 g 2 g 5 g 6 ŒB 4 D f 4 f 5 g 3 g 4 ŒB 3 D f 3 f 4 f 4 f 5 f 6 g N g N C1 g N C1 g N C1 g N C1 ŒCN D g N g N C1 ŒC 4 D g 4 g 5 ŒC 3 D g 3 g 4 ŒB 2 D f 2 f 3 ŒC 1 Dh ŒC2 g 1 g 2 D g 2 g 3 ŒB 1 D h f 1 f 2 ŒA 1 D h e 1 e 2 ŒA 2 D e 2 e 3 f 3 f 2 f 3 f 4 f 2 f 3 ŒA 3 D e 3 e 4 ŒA 4 D e 4 e 5 e 4 ŒA N D e N e N C1 e N C1 e N C1 e 2 e 3 e 3 e 4 e 5 e 6 e N e N C1 e N C1 Figure 5: The possible curves in Im.f 00 / Suppose that A 1 [ B 1 [ C 1 Im.f 00 /. Then f 00 Œ contains (at least) 3h. Note that ŒF has no h terms. Therefore Im.f 00 / does not contain any of the curves h e 1 f 1 ; h e 1 g 1 ; h f 1 g 1. And furthermore each of A 1, B 1 and C 1 must have multiplicity one. There are no remaining terms that contain e 1 ; f 1 or g 1. Also, since the image of f 00 contains precisely one of A 1 ; B 1 ; C 1, we conclude that the multiplicity of terms contain positive e 1 ; f 1 ; g 1 must be zero. Thus, Im.f 00 / is contained in the configuration shown in Figure 5. Now, note that in d the sum of the multiplicities of the e i s is -2. This is true of the curve A 1 as well. Therefore the total multiplicity of all other e terms must vanish. But all other e terms are of the form e i e ic1 or e j. Since the former contribute nothing to the total multiplicity, we conclude that there are no e j terms in the image of f 00. Therefore Im.f 00 / must be contained in the configuration shown in Figure 6.

22 Gromov Witten invariants of configurations of curves 135 g 2 g 3 ŒB N D f N f N C1 g 3 g 4 f N f N C1 g 5 g 6 ŒB 4 D f 4 f 5 ŒB 3 D f 3 f 4 f 5 f 6 ŒB 2 D f 2 f 3 g N g N C1 ŒCN ŒC 4 D g 4 g 5 ŒC 3 D g 3 g 4 ŒC 2 D g 2 g 3 ŒB 1 D h f 1 f 2 ŒA 1 D h e 1 e 2 f 3 f 4 f 2 f 3 D g N g N C1 ŒA 2 D e 2 e 3 ŒA 3 D e 3 e 4 ŒA 4 D e 4 e 5 ŒA N D e N e N C1 e 2 e 3 e 3 e 4 e 5 e 6 e N e N C1 Figure 6: The remaining possible curves in Im.f 00 / But Im.f 00 / is connected, and contains h terms. Therefore it can not contain nor be contained in any of the three outer parts of Figure 6. Therefore Im.f 00 / Y N. This contradicts our assumption, and therefore at least one of A 1 ; B 1 ; C 1 is not in Im.f 00 /. Without loss of generality, suppose A 1 6 Im.f 00 /. Let d e;f ; d e;g ; d f;g denote the degree of f 00 on the components h e 1 f 1 ; h e 1 g 1 ; h f 1 g 1 respectively. Since A 1 is not contained in the image of f 00, we must have as these are the only multiplicities of h. 0 < d e;f C d e;g 3 e 1 terms, and there are no terms containing Furthermore, in order for Im.f 00 / to simultaneously be connected and contain terms for i > 1, it must be the case that Im.f 00 / contains two of fe 1 ; e 1 e 2 ; e 1 e N C1 g: e i

23 136 Karp, Liu and Mariño Thus d e;f C d e;g D 3; d f;g D 0 and B 1 ; C 1 6 Im.f 00 /. This forces Im.f 00 / to be contained in the configuration shown in Figure 7. g 1 g 2 f 1 f N C1 g 2 g 3 f N C1 f N C1 ŒB N D f N f N C1 f N C1 g 3 g 4 f N f N C1 g 2 g1 g 5 g 6 g 3 ŒB 4 D f 4 f 5 f 1 g 4 ŒB 3 D f 3 f 4 f 4 f 5 f 6 ŒB 2 D f 2 f 3 f 3 g N g N C1 g N C1 g g N C1 1 gn C1 g N C1 ŒCN D g N g N C1 ŒC 4 D g 4 g 5 ŒC 3 D g 3 g 4 ŒC 2 D g 2 g 3 ŒA 2 D e 2 e 3 f 2 f 3 f 4 f 2 f 3 f 1 f 2 ŒA 3 D e 3 e 4 h e1 g 1 e 2 e 3 ŒA 4 D e 4 e 5 h e 1 f 1 e 4 ŒA N D e N e N C1 e N C1 e N C1 e 2 e 3 e 3 e 4 e 5 e 6 e N e N C1 e N C1 e 1 e 2 e 1 e N C1 e 1 Figure 7: The other possibility for curves in Im.f 00 / Again we have that Im.f 00 / is connected and contains f i ; g j for some i; j > 1. Therefore Im.f 00 / contains at least one of f 1 f 2 ; f 1 f N C1 and also at least one of g 1 g 2 ; g 1 g N C1. But the multiplicity of f 1 and g 1 in d is 1. Therefore d e;f ; d e;g 2:

24 Gromov Witten invariants of configurations of curves 137 This contradictions shows that our assumption A 1 6 Im.f 00 / is incorrect. Therefore A 1 Im.f 00 /. An identical argument also shows that B 1 ; C 1 Im.f 00 /. However we showed above that A 1 ; B 1 ; C 1 6 Im.f 00 /. This contradiction shows that our original assumption is incorrect. Therefore there does not exist a point p 2 Im.f 00 / such that p 62 Y N. Thus Im.f 00 / Y N, and the result holds. Remark 9 Note that this argument does not hold for general a 1 ; b 1 ; c 1. For instance, it is a fun exercise to show that there is more than one invariant configuration of curves in in the following classes. ˇ1 D2.h e 1 e 2 / C.e 2 e 3 / C 2.h f 1 f 2 / C.f 2 f 3 / C 2.h g 1 g 2 / C.g 2 g 3 / ˇ2 D4.h e 1 e 2 / C.e 2 e 3 / C 2.h f 1 f 2 / C 2.h g 1 g 2 / C.g 2 g 3 / ˇ3 D4.h e 1 e 2 / C 4.e 2 e 3 / C 4.h f 1 f 2 / C 4.f 2 f 3 / C 4.h g 1 g 2 / C 4.g 2 g 3 /: 4.2 A chain of rational curves Proposition 10 Let f W C! N C1 represent a point in M g. N C1 ; d/, where d 1;1 > 0; d 2;j D d 3;j D 0; j D 1; : : : ; N: Then the image of f is contained in the chain of rational curves Y N A D A 1 [ [ A N defined in Section 3. Since YA N does not contain any of the curves B i; C i, the blowups with centers p i and q i in the construction of N C1 are extraneous. In order to simplify the argument in this case, consider the space N C2! z N C1 N C1! z N N! 1! 3 ; where the construction of z N C1 follows that of N C1, without the extraneous blowups. So z ic1! z i is the blowup of z i along the point p i, where p i is defined in Section 3. Thus, z N C1 is deformation equivalent to the blowup of 3 at N C 1

25 e 2 e Karp, Liu and Mariño points. Since YA N does not contain the curves B i; C i, clearly the formal neighborhood of YA N in z N C1 agrees with the construction in N C1. We continue to let E i be the total transform of the exceptional divisor over p i, and e i be the class of a line in E i. Furthermore, we continue to let H denote the pullback of the class of a hyperplane in 3, and h be the class of a line in H. Then, fh; E i g is a basis for H 4. z N C1 / and fh; e i g is a basis for H 2. z N C1 /. The non-zero intersection pairings are given as follows. H H D h H h D pt E i E i D e i E i e i D pt The invariant curves in z N C1 are shown together with their homology classes in Figure 8. h h h ŒA 1 D h e 1 e 2 ŒA 2 D e 2 e 3 h e 1 ŒA 3 D e 3 e 4 ŒA 4 D e 4 e 5 h e 1 e 4 ŒA N D e N e N C1 e N C1 e N C1 e 2 e 3 e 3 e 4 e 5 e 6 e N e N C1 e N C1 e 1 e 2 e 1 e N C1 e 1 Figure 8: The invariant curves in z N C1 Proof of Proposition 10 As shown in above, we may use the toric nature of z N C1 to construct a stable map Œf 00 W! z N C1 2 M g. z N C1 ; d/ such that Im.f 00 / is invariant, but Im.f 00 / 6 YA N. We show that this leads to a contradiction. We study the class f 00 Œ D d. Note that the multiplicity of the e 1 term is the same as that of h. Furthermore, each e 1 occurs along with h, and there are no h terms.

26 e 2 e 3 Gromov Witten invariants of configurations of curves 139 Therefore Im.f 00 / can not contain any terms containing positive e 1, nor can it contain any of the curves in class h. Thus, the image of f 00 is contained in the configuration of curves shown in Figure 9. ŒA 1 D h e 1 e 2 ŒA 2 D e 2 e 3 h e 1 ŒA 3 D e 3 e 4 ŒA 4 D e 4 e 5 h e 1 e 4 ŒA N D e N e N C1 e N C1 e N C1 e 2 e 3 e 3 e 4 e 5 e 6 e N e N C1 e N C1 Figure 9: The possible curves in Im.f 00 / Since a 1 > 0, it must be that f 00 Œ contains at least one e i term with non-zero multiplicity for i > 1. Also, Im.f 00 / is connected and so we conclude that the image of f must not contain either of the curves of class h e 1 in Figure 9. Now, note that the total multiplicity of the e terms is 2a 1, and that the curve A 1 must also have this property. Therefore the sum of all other e terms must be zero. Since the other e terms are of the form e i e ic1 or e j, we conclude that Im.f 00 / does not contain any of the curves e j. Thus Im.f 00 / is contained in the configuration depicted in Figure 10. However, since z N C1 is connected and contains h, we conclude that Im.f 00 / Y N A. This contradiction shows that our original assumption is incorrect, and that the result holds. 5 Mathematical theory of the topological vertex Let (19) N C1 N C1! N N! 2! 1 1! 0 D 3

27 140 Karp, Liu and Mariño ŒA 1 D h e 1 e 2 ŒA 2 D e 2 e 3 ŒA 3 D e 3 e 4 ŒA 4 D e 4 e 5 ŒA N D e N e N C1 e 2 e 3 e 3 e 4 e 5 e 6 e N e N C1 Figure 10: The remaining possible curves in Im.f 00 / be the toric blowups constructed in Section 3. Let Y N N C1 be the minimal trivalent configuration, and let yy N be the formal completion of N C1 along Y N. Then yy N is a nonsingular formal scheme, and M g. yy N ; d/ is a separated formal Deligne Mumford stack with a perfect obstruction theory of virtual dimension zero. It has a virtual fundamental class when it is proper, which is not true in general. 5.1 Formal Gromov Witten invariants of yy N In (19) D. / 3 acts on j and the projections j are equivariant, so yy N is a formal scheme together with a action. The point s 0 D A 1 \ B 1 \ C 1 is fixed by the action, so acts on T s0 0 and ƒ 3 T s0 0. Let be the rank 2 subtorus of which acts trivially on ƒ 3 T s0 0. The union of one dimensional orbit closures of the action on j is a configuration of rational curves, which corresponds to a graph (see Figure 11). The action on j can be read off from the slopes of the edges of the graph associated to j.more precisely, let ƒ D Hom. ; / be the group of irreducible characters of. If we fix an identification Š. / 2 then an element in ƒ is of the form s p 1 sq 2 where.s 1 ; s 2 / are coordinates on. / 2 and p; q 2. The line segment associated to C Š 1 is tangent to.p; q/ 2 if the irreducible characters of the actions on T x C and T y C are s p 1 sq 2 and s p 1 s q 2 (see Figure 12). Similarly, the action on yy N can be read off from Figure 13 in Section 5.5. Let u 1 ; u 2 be a basis of H 2.pt; / so that H 2.pt; / D u 1 u 2. For any nonzero effective class d (d i;j 0), define Z zn g d. yy N 1 / D ŒM g. yy N ;d/ vir e.n vir / :

28 Gromov Witten invariants of configurations of curves 141 Y Y 2 Y Figure 11: Configurations of invariant curves y. p; q/ x.p; q/ Figure 12: The action can be read off from the slope A priori zn g d. yy N / is a rational function in u 1 ; u 2 with coefficients, homogeneous of degree 0. By results in Li Liu Liu Zhou [22], zn g d. yy N / 2 is a constant function independent of u 1 ; u 2. We call zn g d. yy N / formal Gromov Witten invariants. For the

29 142 Karp, Liu and Mariño cases (i) (iv) described in Section 1, N g d.y N / D h i N C1 g;d Z D Z D ŒM g. N C1 ;d/ vir 1 e.n vir / ŒM g. N C1 ;d/ vir 1 Z 1 D ŒM g. yy N ;d/ vir e.n vir / D zn g d. yy N /: As in Section 1, introduce formal variables ; t i;j, and define a generating function (20) F N.I t/ D 2g 2 zn g d. yy N /e dt g0 d where t D.t 1 ; t 2 ; t 3 /; t i D.t i;1 ; : : : ; t i;n /; d t D 3 jd1 N d i;j t i;j : The partition function of the formal Gromov Witten invariants of yy N is defined to be zz N.I t/ D exp.f N.I t// : By connectedness and cyclic symmetry, we only need to compute zn g d. yy N / in the following cases (see Figure 13): (D1) d D.d 1 ; 0; 0/, d 1;j > 0 for j k and d 1;j D 0 for j > k, where 1 k N. (D2) (D3) (D4) d D.d 1 ; 0; 0/, d 1;j > 0 for k 1 j k 2 and d 1;j D 0 otherwise, where 2 k 1 k 2 N. d D.d 1 ; d 2 ; 0/, d 1;m > 0 for m j and d 1;m D 0 for m > j, d 2;m > 0 for m k and d 2;m D 0 for m > k, where 1 j ; k N. d D.d 1 ; d 2 ; d 3 /, d i;j > 0 for j k i and d i;j D 0 for j > k i, where 1 k 1 ; k 2 ; k 3 N. Any other zn g d. yy N / is either manifestly zero (because M g. yy N ; d/ is empty) or is equal to one of the above case. Let FN 1.I t 1/; FN 2.I t 1/; FN 3.I t 1; t 2 /; FN 4.I t/

30 Gromov Witten invariants of configurations of curves 143 (1) 0 0 d 1;1 d 1;2 d 1;k 0 (2) d 1;k 0 1 d 1;k2 d 2;k 0 0 d 2;k2 (3) d 2;1 d1;1 0 d1;j 0 (4) d 2;1 d 3;1 d 1;1 d 1;k1 0 0 d 3;k3 Figure 13: Four cases denote the contribution to F N.I t/ from (D1), (D2), (D3), (D4), respectively. Then F N.I t/ D 3 FN 1.I t i/ C 5.2 Summary of results 3 FN 2.I t i/ CFN 3.I t 1; t 2 / C FN 3.I t 2; t 3 / C FN 3.I t 3; t 1 / C FN 4.I t/: Let C g be defined by as before. Then where 1 gd0 g0 t=2 2 C g t 2g D sin.t =2/ C g n 2g 3 2g 2 D 1 nœn 2 Œn D q n=2 q n=2 ; q D ep 1 : In Section 5.5, we will compute zn g d. yy N / by the mathematical theory of the topological vertex and obtain the following results. We will do the computations by the physical theory of the topological vertex in Section 6.

31 144 Karp, Liu and Mariño Proposition 11 Suppose that d 1 D.d 1 ; : : : ; d N /; d 2 D d 3 D.0; : : : ; 0/: where d 1 > 0. Then 8 < zn g d. yy N C / D g n 2g 3 d 1 D d 2 D D d k D n > 0 and d : kc1 D d kc2 D D d N D 0 for some 1 k N 0 otherwise which is equivalent to (21) FN 1.I t 1/ D 1 nœn 2 n>0 Proposition 11 is equivalent to Theorem 2. 1kN e n.t 1;1CCt 1;k / Proposition 12 Suppose that d 1 D.d 1 ; : : : ; d N /; d 2 D d 3 D.0; : : : ; 0/: where d 1 D 0. Then 8 < zn g d. yy N C / D g n 2g 3 d j D n > 0 for k 1 j k 2 and d : j D 0 otherwise, where 2 k 1 k 2 N 0 otherwise which is equivalent to (22) FN 2.I t 1/ D 1 nœn 2 n>0 2k 1 k 2 N e n.t 1;k 1 CCt 1;k2 / Proposition 12 corresponds to a chain of.0; 2/ rational curves. Proposition 13 Suppose that d 1 D.n; 0; : : : ; 0/; d 2;1 > 0; d 3 D.0; : : : ; 0/ where n > 0. Then 8 < zn g d. yy N C / D g n 2g 3 d 2;1 D d 2;2 D D d 2;k D n and d : kc1 D d kc2 D D d N D 0 for some 1 k N 0 otherwise: Proposition 14 Suppose that d i;1 > 0. Then zn g d. yy N / D 0 unless d i;1 d i;2 d i;n :

32 Gromov Witten invariants of configurations of curves 145 Proposition 15 Suppose that where i D 1; 2; 3 and 1 k i N. Then di > 0 j k d i;j D i 0 j > k i (a) (b) zn g d 1 ;d 2 ;0. yy N Cg n / D 2g 3 d 1 D d 2 D n > 0 0 otherwise zn g d 1 ;d 2 ;d 3. yy N Cg n / D 2g 3 d 1 D d 2 D d 3 D n > 0 0 otherwise Proposition 14 and Proposition 15 are consistent with Theorem 1. Let N D 1 in Proposition 11 and Proposition 15, we get Corollary 16 where t D.t 1 ; t 2 ; t 3 / and zz 1.I t/ D exp 1! Q n.t/ nœn 2 Q n.t/ D e nt 1 Ce nt 2 Ce nt 3 e n.t 1Ct 2 / e n.t 2Ct 3 / e n.t 3Ct 1 / Ce n.t 1Ct 2 Ct 3 / : Finally, we will derive the following expression of zz N.I t/, where the notation is the same as that in Section 1: Proposition 17 3 zz N.I t/ D exp FN 2.I t i/ where F 2 N.I t i/ D 1 In particular, when N D 1 we have 1 E nœn 2 zw E.q/ 3Y. 1/ ji j e ji jt i;1 s. i / t.ui.q; t i //: 2k 1 k 2 N e n.t i;k 1 CCt i;k2 / F 2 1.I t i/ D 0; s. i / t.ui.q; t i // D W. i / t.q/; which gives the following corollary.

33 146 Karp, Liu and Mariño Corollary 18 zz 1.I t/ D E 3Y zw E.q/. 1/ ji j e ji jt i W. i / t.q/: where t D.t 1 ; t 2 ; t 3 /. Equation (6) in Section 1 follows from Corollary 16 and Corollary Three-partition Hodge integrals Three-partition Hodge integrals arise when we calculate Z zn g d. yy N 1 / D ŒM g. yy N ;d/ vir e.n vir / by virtual localization (see Li Liu Liu Zhou [22, Section 7] for such calculations). We recall their definition in this subsection. Let w 1 ; w 2 ; w 3 be formal variables, where w 3 D w 1 w 2. Let w 4 D w 1. Write w D.w 1 ; w 2 ; w 3 /. For E D. 1 ; 2 ; 3 /. ; ; / D E, define d 1 E D 0; d 2 E D `.1 /; d 3 E D `.1 / C `. 2 /; `. E/ D The three-partition Hodge integrals are defined by p. 1/`. E/ (23) G g; E.w/ D jaut. E/j where 3Y `. Y i / jd1 Q i j 1 ad1.i j w ic1 C aw i /.j i 1/!w i j 1 i 3Y Z M g;`. E/ ƒ _ g.u/ D ug 1 u g 1 C C. 1/ g g : 3 `. i /: ƒ _ E/ 1 g.wi/w`. i Q`. i / jd1.w i.w i i j d i E Cj // Note that G g; E.w 1 ; w 2 ; w 3 / has a pole along w i D 0 if i. The following cyclic symmetry is clear from the definition: (24) G g; 1 ; 2 ; 3.w 1; w 2 ; w 3 / D G g; 2 ; 3 ; 1.w 2; w 3 ; w 1 / D G g; 3 ; 1 ; 2.w 3; w 1 ; w 2 /

34 Gromov Witten invariants of configurations of curves 147 Note that p 1`. E/ Gg; E.w/ 2.w 1 ; w 2 ; w 3 / is homogeneous of degree 0, so G g; E.w 1 ; w 2 ; w 1 w 2 / D G g; E.1; w 2 w 1 ; 1 w 2 w 1 /: Introduce variables, p i D.p1 i ; pi 2 ; : : :/, i D 1; 2; 3. Given a partition, define p i D pi 1 pì./ for i D 1; 2; 3. In particular, p i D 1. Write p D.p 1 ; p 2 ; p 3 /; p E D p 1 p 2 p 3 : Define generating functions 1 G E.I w/ D gd0 2g 2C`. E/ G g; E.w/; G.I pi w/ D E E G E.I w/p E ; G.I pi w/ D exp.g.i pi w// D 1 C E E G E.I w/p E : In particular, G g;; ;.1; 0; p 1d 2g 1/ D 2 b g ; D.d/; 0; `./ > 1: where ( 1; g D 0; b g D 2g 2 RM g;1 1 g g > 0: It was proved by Faber and Pandharipande [10] that (25) So 1 gd0 (26) G.n/; ;.I 1; 0; 1/ D b g t 2g D t=2 sin.t=2/ : p 1 2n sin.n=2/ D 1 nœn :

35 148 Karp, Liu and Mariño Similarly, we have (27) G.n/; ;.I 1; 1; 0/ D. 1/n 1 : nœn The following formula of three-partition Hodge integrals was derived in Li Liu Liu Zhou [22]: (28) G.I w/ D E 3Y j i jdj i j q 1 2 i w ic1! w i i. i / z i zw E.q/; In particular, (29) (30) G ; ;.I1;; G 1 ; 2 ;.I1;; 1/ D jjdjj 1/ D j i jdj i j./ q 1 2 W.q/: z 1. 1 / z / q / W z 1. 2 / t.q/: 2 Equation (29) is equivalent to the formula of one-partition Hodge integrals conjectured in Mariño Vafa [28], which was proved in Liu Liu Zhou [25] and Okounkov Pandharipande [29]. Equation (30) is equivalent to the formula of two-partition Hodge integrals proved in Liu Liu Zhou [24]. 5.4 Relative formal GW invariants of the topological vertex Symplectic relative Gromov Witten theory was developed by Li and Ruan [19], and Ionel and Parker [15; 16]. The mathematical theory of the topological vertex in [22] is based on Jun Li s algebraic relative Gromov Witten theory [20; 21]. Given a triple of partitions E D. 1 ; 2 ; 3 / E and a triple of integers n D.n 1 ; n 2 ; n 3 /, let F.n/ 2 ; E

36 Gromov Witten invariants of configurations of curves 149 be the disconnected relative formal GW invariants of the topological vertex defined in [22]. Introduce variables ; pj i as in Section 5.3, and define generating functions F E.I n/ D C`. E/ F ; E.n/ F.I pi n/ D 1 C E E F E.I n/p E F.I pi n/ D log.f.i pi n// D E E F E.I n/p E 1 F E.I n/ D gd0 2g 2C`. E/ F g; E.n/: By virtual localization, F g; E.n/ can be expressed in terms of three-partition Hodge integrals and double Hurwitz numbers. We have (31) F.I n/ D E where. 1/ P 3.n i 1/j i j. p 3 E/ 1/`. G E.I w/ Y j i jdj i j z i ˆ i ; i p 1 ni w ic1 w i (32) ˆ;./ D H ;; C`./C`./. C `./ C `.//! D e =2././ : z z is a generation function of disconnected double Hurwitz numbers H ;;. Equations (28), (31), and (32) imply (33) F.I n/ D E 3Y j i jdj i j! q 1 2 i n i i. i / z i zw E.q/; Note that the w dependence on the right hand side of (31) cancels and the right hand side of (33) is independent of w. Since ˆ.0/ D ı =z, we have F.w 2 w 1 ; w 3 w 2 ; w 1 w 3 / D. 1/ P3.n i 1/j i j. p 1/`. E/ j i jdj i j G E.I w 1; w 2 ; w 3 /:

37 150 Karp, Liu and Mariño Also F E.I n/ is independent of n i if i is empty. (34) (35) F g;; ;.0; n 2 ; n 3 / D. 1/ jj. p 1/`./ G g;; ;.1; 0; 1/ F g;;;. 1; 0; n 3 / D. 1/ jj. p 1/`./C`./ G g;;;.1; 1; 0/ From (26), (27), (34), (35) we conclude that if then (36) (37) F ; ;.I 0; n 2 ; n 3 / D F ; ;.I 1; n 2 ; n 3 / D (. 1/ n 1 p 1 nœn D.n/ 0 `./ > 1: (. 1/ n p 1 nœn D.n/ 0 `./ > 1: If ; then (see Liu Liu Zhou [24, p7] for details) (38) F ;;.I 1; 0; n 3 / D We also have 1 n D D.n/ 0 otherwise: F.1/;.1/;.I 0; 0; 0/ D 1; F.1/;.1/;.1/.I 0; 0; 0/ D p 1Œ1 : 5.5 Computations The action on yy N can be read off from Figure 13 as explained in the first two paragraphs of Section 5.1. We now degenerate each 1 into two 1 s intersecting at a node. The total space of the normal bundle O.n/ O. n 2/ degenerates to O.a/ O. a 1/ and O.b/ O. b 1/ with a C b D n. For each node we introduce a pair of framing vectors to encode the action (see Figure 15). We refer to [22, Section 4] for details. The framing here corresponds to the framing of Lagrangian submanifolds in the article by Aganagic, Klemm, Mariño and Vafa [2] and the framing of knots and links in Chern Simons theory. In Figure 14, all the 1 s have normal bundles O. 1/ O. 1/ or O O. 2/; in Figure 15, all the 1 s have normal bundles O O. 1/. Using the connected version of the gluing formula [22, Theorem 7.5], we have FN 4.I t/ D j i j>0 F E.I 0; 0; 0/ 3Y `. Y i / N jd1 kd1 e i j.t i;1cct i;k / i j F. i j /;.i j /;. 1; 0; 0/ k 1 i j F. i j /; ;.I 0; 0; 0/

38 Gromov Witten invariants of configurations of curves 151 D F E.I 0; 0; 0/ 3Y Y `. i / j i j>0 jd1 i j. 1/ i j 1 p 1 j i Œi j N e i j.t i;1cct i;k / kd1 d 1;4 d 1;3 d 2;2 d d 1;2 d 2;1 2;3 d 1;1 d 2;4 d3;1 d 3;2 d 3;3 d 3;4 Figure 14: Graph of yy N Figure 15: Degeneration of yy N