Holomorphic Anomaly Equations for the Formal Quintic

Transcription

1 Peking Mathematical Journal ORIGINAL ARTICLE Holomorphic Anomaly Equations for the Formal Quintic Hyenho Lho Rahul Pandharipande Received: 4 April 208 / Revised: 20 November 208 / Accepted: 8 December 208 Peking University 209 Abstract We define a formal Gromov Witten theory of the quintic threefold via localization on P 4 Our main result is a direct geometric proof of holomorphic anomaly equations for the formal quintic in precisely the same form as predicted by B-model physics for the true Gromov Witten theory of the quintic threefold The results suggest that the formal quintic and the true quintic theories should be related by transformations which respect the holomorphic anomaly equations Such a relationship has been recently found by Q Chen S Guo F Janda and Y Ruan via the geometry of new moduli spaces Keywords Gromov Witten invariants Holomorphic anomaly equations Quintic threefold Introduction GW/SQ Let X 5 P 4 be a nonsingular quintic Calabi Yau threefold The moduli space of stable maps to the quintic of genus g and degree d M g (X 5 d M g (P 4 d has virtual dimension 0 The Gromov Witten invariants N GW gd = GW gd = [Mg (X 5 d] vir ( * Hyenho Lho hyenholho@mathethzch Rahul Pandharipande rahul@mathethzch Department of Mathematics ETH Zürich Zurich Switzerland 3 Vol:(

2 H Lho R Pandharipande have been studied for more than 20 years; see [2 3 9] for an introduction to the subject The theory of stable quotients developed in [24] was partially inspired by the question of finding a geometric approach to a higher genus linear sigma model The moduli space of stable quotients for the quintic Q g (X 5 d Q g (P 4 d was defined in [24 Section 9] The existence of a natural obstruction theory on Q g (X 5 d and a virtual fundamental class [Q g (X 5 d] vir is easily seen in genus 0 and A proposal in higher genus for the obstruction theory and virtual class was made in [24] and was carried out in significantly greater generality in the setting of quasimaps in [9] The associated integral theory is defined by N SQ gd = SQ gd = [Qg (X 5 d] vir (2 In genus 0 and the invariants (2 were calculated in [] and [8] respectively The answers on the stable quotient side exactly match the string theoretic B-model for the quintic in genus 0 and A relationship in every genus between the Gromov Witten and stable quotient invariants of the quintic has been proven by Ciocan-Fontanine and Kim [7] 2 Let H H 2 (X 5 Z be the hyperplane class of the quintic and let GW gn SQ gn (Q = H H n (q = H H n GW gn SQ = Q d [Mgn(X 5 d] vir d=0 gn = q d [Qgn(X 5 d] vir d=0 n ev i (H i= n ev i (H i= be the Gromov Witten and stable quotient series respectively (involving the pointed moduli spaces and the evaluation morphisms at the markings Let ( 5d I 0 (q = I (q =log(qi 0 (q+5 r The mirror map is defined by 3 d=0 Q(q =exp q d (5d! (d! 5 ( I (q I 0 (q = q exp d= q d (5d! (d! 5 5 d= qd (5d! (d! 5 ( 5d r=d+ (5d! qd d=0 (d! 5 r=d+ For stability marked points are required in genus 0 and positive degree is required in genus 2 A second proof (in most cases can be found in [0] r

3 Holomorphic Anomaly Equations for the Formal Quintic The relationship between the Gromov Witten and stable quotient invariants of the quintic in case is given by the following result [7]: 2g 2 + n > 0 GW gn (Q(q = I 0 (q2g 2+n SQ gn (q (3 The transformation (3 shows the stable quotient theory matches the string theoretic B-model series for the quintic X 5 2 Formal Quintic Invariants The (conjectural holomorphic anomaly equation is a beautiful property of the string theoretic B-model series which has been used effectively since [3] Since the stable quotients invariants provide a geometric proposal for the B-model series we should look for the geometry of the holomorphic anomaly equation in the moduli space of stable quotients A particular twisted theory on P 4 is related to the quintic threefold Let the algebraic torus T =(C 5 act with the standard linearization on P 4 with weights λ 0 λ 4 on the vector space H 0 (P 4 P 4( Let C M g (P 4 d f C P 4 S = f P 4( C (4 be the universal curve the universal map and the universal bundle over the moduli space of stable maps all equipped with canonical T-actions We define the formal quintic invariants following [22] by 3 Ñ GW gd = [Mg (P 4 d] vir e(rπ (S 5 (5 where e(rπ (S 5 is the equivariant Euler class defined after localization More precisely on each T-fixed locus of M g (P 4 d both R 0 π (S 5 and R π (S 5 are vector bundles with moving weights so e(rπ (S 5 = c top (R0 π (S 5 c top (R π (S 5 is well-defined The integral (5 is homogeneous of degree 0 in localized equivariant cohomology 3 The negative exponent denotes the dual: S is a line bundle and S 5 =(S 5 3

4 H Lho R Pandharipande and defines a rational number Ñ GW gd [Mg (P 4 d] vir e(rπ (S 5 Q(λ 0 λ 4 Q after the specialization4 λ i = ζ i for a primitive fifth root of unity ζ 5 = Our main result here is that the holomorphic anomaly equations conjectured for the true quintic theory ( are satisfied by the formal quintic theory (5 In particular the formal quintic theory and the true quintic theory should be related by transformations which respect the holomorphic anomaly equations In a recent breakthrough by Q Chen S Guo F Janda and Y Ruan precisely such a transformation was found via the virtual geometry of new moduli spaces intertwining the formal and true theories of the quintic 3 Holomorphic Anomaly for the Formal Quintic We state here the precise form of the holomorphic anomaly equations for the formal quintic Let H H 2 (P 4 Z be the hyperplane class on P 4 and let GW g (Q = SQ g Q d e(rπ (S 5 [Mg (P 4 d] vir d=0 (Q = Q d e(rπ (S 5 [Qg (P 4 d] vir d=0 be the formal Gromov Witten and formal stable quotient series respectively (involving the evaluation morphisms at the markings The relationship between the formal Gromov Witten and formal stable quotient invariants of quintic in case of 2g 2 + n > 0 follows from [8]: GW g with respect to the true quintic mirror map Q(q =exp ( I (q I 0 (q (Q(q = I 0 (q 2g 2 SQ g (q = q exp 5 d= qd (5d! (d! 5 ( 5d r=d+ (5d! qd d=0 (d! 5 r (6 4 Since the formal quintic theory is homogeneous of degree 0 the specialization λ i = ζ i λ 0 could also be taken as in [22] However for other specializations we expect the finite generation of Theorem and the holomorphic anomaly equation of Theorem 2 to take different forms A study of the dependence on specialization will appear in [2] For local P 2 considered in [22] and C 3 Z 3 considered in [23] the theories are independent of specialization 3

5 Holomorphic Anomaly Equations for the Formal Quintic In order to state the holomorphic anomaly equations we require several series in q First let L(q =( 5 5 q 5 = + 625q + 785q 2 + Let D = q d and let dq C 0 (q =I 0 C (q =D ( I I 0 where I 0 and I are the hypergeometric series appearing in the mirror map for the true quintic theory We define K 2 (q = DC 0 L 5 C 0 A 2 (q = ( DC L 5 5 A 4 (q = ( L 0 25 A 6 (q = + 25 D ( DC0 3250L 5 Let T be the standard coordinate mirror to t = log(q Then Q(q =exp(t is the mirror map Define a new series motivated by (6 The superscript B here is for the B-model Let be the free polynomial ring over C[L ± ] 2 DC 0 3 C 5 C 0 25 ( 2 DC0 ( ( DC0 DC 25 C 0 C ( D C 0 C 0 C ( ( ( ( DC0 DC0 DC D ( ( 2 5L 5 DC0 DC0 q D d C dq 0 C 0 C 0 C 0 ( + 25D 2 DC0 25 C 0 C 0 ( 2 (( DC0 DC C 0 T = I (q I 0 (q B g = I2g 2 SQ 0 g C C[L ± ][A 2 A 4 A 6 C ± 0 C K 2 ] C 0 3

6 H Lho R Pandharipande Theorem For the series g B associated to the formal quintic (i B g (q C[L± ][A 2 A 4 A 6 C ± 0 C K 2 ] for g 2 (ii k g B (q T k C[L± ][A 2 A 4 A 6 C ± 0 C K 2 ] for g k (iii k g B is homogeneous with respect to C of degree k Tk We follow here the canonical lift convention of [22 Section 04] When we write we mean that the series B (q g has a canonical lift to the free algebra The question of uniqueness of the lift has to do with the algebraic independence of the series which we do not address nor require B g (q C[L± ][A 2 A 4 A 6 C ± 0 C K 2 ] A 2 (q A 4 (q A 6 (q C ± (q C 0 (q K 2 (q Theorem 2 The holomorphic anomaly equations for the series g B formal quintic hold for g 2: associated to the C 2 0 C2 B g A 2 B g K 2 = 0 5C 2 0 C2 B g A 4 K C 2 0 C2 g B K 2 2 A = g B g i B i 6 2 T T + 2 B g 2 T i= 2 The equality of Theorem 2 holds in the ring C[L ± ][A 2 A 4 A 6 C ± 0 C K 2 ] Theorem 2 exactly matches 5 the conjectural holomorphic anomaly equations [ (252] for the true quintic theory I 2g 2 SQ 0 g The first holomorphic anomaly equation of Theorem 2 was announced in our paper [22] in February 207 where a parallel study of the toric Calabi Yau KP 2 was developed In January 208 at the Workshop on higher genus at ETH Zürich Shuai Guo of Peking University informed us that our same argument also yields the second holomorphic anomaly equation B g K 2 = 0 (7 5 Our functions K 2 and A 2k are normalized differently with respect to C 0 and C The dictionary to exactly match the notation of [ (252] is to multiply our K 2 by (C 0 C 2 and our A 2k by (C 0 C 2k 3

7 Holomorphic Anomaly Equations for the Formal Quintic In fact we had incorrectly thought (7 would fail in the formal theory and would not have included (7 without communication with Guo so Guo should be credited with the first proof of (7 4 Constants Theorem 2 and a few observations determine g B from the lower genus data { h < g } B h and finitely many constants of integration The additional observations 6 required are: (i The proof of part (i of Theorem shows that B g C[L± ][A 2 A 4 A 6 C ± 0 C K 2 ] does not depend on C Hence every term (both on the left and right in the first holomorphic anomaly equation of Theorem 2 is of degree 2 in C After multiplying by C 2 no C dependence remains (ii The proof of Theorem shows that all terms in the first equation are homogeneous of degree 2g 4 with respect to C 0 After dividing by C 2g 4 no C 0 0 dependence remains Therefore the first holomorphic anomaly equation of Theorem 2 may be viewed as holding in C[L ± ][A 2 A 4 A 6 K 2 ] Since the second holomorphic anomaly equation (7 implies B g has no K 2 dependence the first holomorphic anomaly equation determines the each of the three derivatives B g A 2 B g A 4 B g A 6 Hence Theorem 2 determines C 2 2g 0 B g uniquely as a polynomial in A 2 A 4 A 6 up to a constant term in C[L ± ] In fact the degree of the constant term can be bounded (as will be seen in Sect 75 So Theorem 2 determines g B from lower genus data together with finitely many constants of integration 6 See Remark 27 3

8 H Lho R Pandharipande The constants of integration for the formal quintic can be effectively computed via the localization formula but whether there exists a closed formula determining the constants is an interesting open question 2 Localization Graphs 2 Torus Action Let T =(C m+ act diagonally on the vector space C m+ with weights λ 0 λ m Denote the T-fixed points of the induced T-action on P m by p 0 p m The weights of T on the tangent space T pj (P m are λ j λ 0 λ j λ j λ j λ m There is an induced T-action on the moduli space Q gn (P m d The localization formula of [7] applied to the virtual fundamental class [Q gn (P m d] vir will play a fundamental role in our paper The T-fixed loci are represented in terms of dual graphs and the contributions of the T-fixed loci are given by tautological classes The formulas here are standard We precisely follow the notation of [22 Section 2] 22 Graphs Let the genus g and the number of markings n for the moduli space be in the stable range 2g 2 + n > 0 (8 We can organize the T-fixed loci of Q gn (P m d according to decorated graphs A decorated graph Γ G gn (P m consists of the data (V E N g p where (i V is the vertex set (ii E is the edge set (including possible self-edges (iii N { 2 n} V is the marking assignment (iv g V Z 0 is a genus assignment satisfying g = v V g(v+h (Γ and for which (V E N g is stable graph 7 7 Corresponding to a stratum of the moduli space of stable curves M gn 3

9 Holomorphic Anomaly Equations for the Formal Quintic (v p V (P m T is an assignment of a T-fixed point p(v to each vertex v V The markings L ={ n} are often called legs To each decorated graph Γ G gn (P m we associate the set of T-fixed loci of with elements described as follows: (a If {v i v ik }={v p(v =p i } then f (p i is a disjoint union of connected stable curves of genera g(v i g(v ik and finitely many points (b There is a bijective correspondence between the connected components of C D and the set of edges 8 and legs of Γ respecting vertex incidence where C is domain curve and D is union of all subcurves of C which appear in (a We write the localization formula as d 0 d 0 [ ] virq Q gn (P m d d [ ] virq Q gn (P m d d = Γ G gn (P m Cont Γ While G gn (P m is a finite set each contribution Cont Γ is a series in q obtained from an infinite sum over all edge possibilities (b 23 Unstable Graphs The moduli spaces of stable quotients Q 02 (P m d and Q 0 (P m d for d > 0 are the only 9 cases where the pair (g n does not satisfy the Deligne Mumford stability condition (8 An appropriate set of decorated graphs G 02 (P m is easily defined: the graphs Γ G 02 (P m all have 2 vertices connected by a single edge Each vertex carries a marking All of the conditions (i (v of Sect 22 are satisfied except for the stability of (V E N γ The localization formula holds d [ ] virq Q 02 (P m d d = Γ G 02 (P m Cont Γ (9 8 Self-edges correspond to loops of T-invariant rational curves 9 The moduli spaces Q 00 (P m d and Q 0 (P m d are empty by the definition of a stable quotient 3

10 H Lho R Pandharipande For Q 0 (P m d the matter is more problematic usually a marking is introduced to break the symmetry 3 Basic Correlators 3 Overview We review here basic generating series in q which arise in the genus 0 theory of quasimap invariants The series will play a fundamental role in the calculations of Sects 4 7 related to the holomorphic anomaly equation for formal quintic invariants We fix a torus action T =(C 5 on P 4 with weights 0 λ 0 λ λ 2 λ 3 λ 4 on the vector space C 5 The T-weight on the fiber over p i of the canonical bundle P 4(5 P 4 (0 is 5λ i For our formal quintic theory we will use the specialization λ i = ζ i ( where ζ is the primitive fifth root of unity Of course we then have λ 0 + λ + λ 2 + λ 3 + λ 4 = 0 λ i λ j = 0 i j λ i λ j λ k = 0 i j k λ i λ j λ k λ l = 0 i j k l 32 First Correlators We will require several correlators defined via the Euler class e(obs =e(rπ (S 5 (2 associated to the formal quintic geometry on the moduli space Q gn (P 4 d The first two are obtained from standard stable quotient invariants For γ i H T (P4 let 0 The associated weights on H 0 (P 4 P 4( are λ 0 λ λ 2 λ 3 λ 4 and so match the conventions of Sect 2 Equation (2 is the definition of e(obs The right side of (2 is defined after localization as explained in Sect 2 3

11 Holomorphic Anomaly Equations for the Formal Quintic SQ γ ψ a γn ψ a n = e(obs gnd [Qgn(P 4 d] vir γ ψ a γn ψ a n 33 Light Markings SQ 0n = d 0 Moduli of quasimaps can be considered with n ordinary (weight markings and k light (weight ε markings 2 Let γ i H T (P4 be equivariant cohomology classes and let k 0 be classes on the stack quotient Following the notation of [8] we define series for the formal quintic geometry 0+0+ γ ψ a γn ψ a n ;δ δ k n i= ev i (γ i ψ a i i q d SQ γ k! ψ a γn ψ a n t t 0n+kd where in the second series t H T (P4 We will systematically use the quasimap notation 0+ for stable quotients 0+ SQ γ ψ a γn ψ a n = γ ψ a γn ψ a n γ ψ a γn ψ a n gnd 0+ 0n = γ ψ a γn ψ a n Q 0+0+ gn k (P4 d δ j H T ([C5 C ] gn kd gnd SQ n k = e(obs ev [Q 0+0+ i (γ i ψ a i êv i j (δ j gn k (P4 d] vir i= j= 0+0+ γ ψ a γn ψ a n 0n = q d 0+0+ γ k! ψ a γn ψ a n ; t t 0n kd d 0 k 0 0n where in the second series t H T ([C5 C ] For each T-fixed point p i P 4 let 2 See Sections 2 and 5 of [6] 3

12 H Lho R Pandharipande e i = e(t p i (P 4 5λ i be the equivariant Euler class of the tangent space of P 4 at p i with twist by P 4(5 Let j i φ i = (H λ j φ i = e 5λ i e i φ i H T (P4 i be cycle classes Crucial for us are the series S i (γ =e i V ij = φi 0+0+ z ψ γ φi x ψ φ j y ψ 02 Unstable degree 0 terms are included by hand in the above formulas For S i (γ the unstable degree 0 term is γ pi For V ij the unstable degree 0 term is We also write δ ij e i (x+y S(γ = 4 φ i S i (γ i=0 The series S i and V ij satisfy the basic relation 4 k=0 e i V ij (x ye j = S i(φ k z=x S j (φ k z=y x + y (3 proven 3 in [8] Associated to each T-fixed point p i P 4 there is a special T-fixed point locus Q k m (P4 d Tp i Q k m (P4 d (4 where all markings lie on a single connected genus 0 domain component contracted to p i Let Nor denote the equivariant normal bundle of Q n k (P4 d Tp i with respect to the embedding (4 Define 3 In Gromov Witten theory a parallel relation is obtained immediately from the WDDV equation and the string equation Since the map forgetting a point is not always well-defined for quasimaps a different argument is needed here [8] 3

13 Holomorphic Anomaly Equations for the Formal Quintic γ ψ a γn ψ a n ;δ δ k 0+0+pi 0n kd e(obs n e(nor = [Q n k (P4 d Tp i ] i= 0+0+pi γ ψ a γn ψ a n 0n ev i (γ iψ a i i k êv j (δ j j= = q d γ k! ψ a γn ψ a 0+0+pi n ;t t 0n kβ d 0 k 0 34 Graph Spaces and I Functions 34 Graph Spaces The big I-function is defined in [6] via the geometry of weighted quasimap graph spaces We briefly summarize the constructions of [6] in the special case of (0+0+ -stability The more general weightings discussed in [6] will not be needed here As in Sect 33 we consider the quotient C 5 C associated to P 4 Following [6] there is a (0+0+-stable quasimap graph space QG 0+0+ gn kd ([C5 C ] (5 A C-point of the graph space is described by data ((C x y (f φ C [C 5 C ] [C 2 C ] By the definition of stability φ is a regular map to P = C 2 C of class Hence the domain curve C has a distinguished irreducible component C 0 canonically isomorphic to P via φ The standard C -action t [ξ 0 ξ ]=[tξ 0 ξ ] for t C [ξ 0 ξ ] P (6 induces a C -action on the graph space The C -equivariant cohomology of a point is a free algebra with generator z H (Spec(C = Q[z] C Our convention is to define z as the C -equivariant first Chern class of the tangent line T 0 P at 0 P with respect to the action (6 z = c (T 0 P The T-action on C 5 lifts to a T-action on the graph space (5 which commutes with the C -action obtained from the distinguished domain component As a result we have a T C -action on the graph space and T C -equivariant evaluation morphisms 3

14 H Lho R Pandharipande Since a morphism ev i QG 0+0+ gn kβ ([C5 C ] P 4 i = n êv j QG 0+0+ gn kβ ([C5 C ] [C 5 C ] j = k f C [C 5 C ] is equivalent to the data of a principal G-bundle P on C and a section u of P C C 5 there is a natural morphism and hence a pull-back map C EC C C 5 The above construction applied to the universal curve over the moduli space and the universal morphism to [C 5 C ] is T-equivariant Hence we obtain a pull-back map associated to the evaluation map êv j f H C ([C 5 C ] H (C êv j H T (C5 Q Q Q[z] H T C (QG 0+0+ gn kβ ([C5 C ] Q 342 I Functions The description of the fixed loci for the C -action on is parallel to the description in [5 Sect 4] for the unweighted case In particular there is a distinguished subset M kd of the C -fixed locus for which all the markings and the entire curve class d lie over 0 P The locus M kd comes with a natural proper evaluation map ev obtained from the generic point of P : We can explicitly write QG 0+0+ g0 kd ([C5 C ] ev M kd C 5 C = P 4 M kd M d 0 k M d (P k where M d is the C -fixed locus in QG 0+ 00d ([C5 C ] for which the class d is concentrated over 0 P The locus M d parameterizes quasimaps of class d f P [C 5 C ] with a base-point of length d at 0 P The restriction of f to P {0} is a constant map to P 4 defining the evaluation map ev As in [4 5 9] we define the big I-function as the generating function for the push-forward via ev of localization residue contributions of M kd For t H T ([C5 C ] Q Q Q[z] let 3

15 Holomorphic Anomaly Equations for the Formal Quintic Res Mkd (t k = k j= êv j (t Res M kd [QG 0+0+ g0 kd ([C5 C ]] vir k j= = êv j (t [M kd] vir e(nor vir M kd where Nor vir M kd is the virtual normal bundle Definition 3 The big I-function for the (0+0+-stability condition as a formal function in t is 343 Evaluations I(q t z = d 0 Let H H T ([C5 C ] and H H T (P4 denote the respective hyperplane classes The I-function of Definition 3 is evaluated in [6] Proposition 4 For the restriction t = t H H T ([C5 C ] Q I(t = k 0 q d ( k! ev Res Mkd (t k q d e t(h+dz z k=0 (5H + kz 4 d i=0 k= (H λ i + kz d=0 5d We return now to the functions S i (γ defined in Sect 33 Using Birkhoff factorization an evaluation of the series S(H j can be obtained from the I-function see [8]: S( = I I t=0h=z= z d dt S(H = S( z d S( dt t=0h=z= z d S(H 2 dt = S(H z d S(H dt t=0h=z= z d S(H 3 dt = S(H2 z d dt S(H2 t=0h=z= S(H 4 = S( = z d dt S(H3 z d dt S(H3 t=0h=z= z d dt S(H4 z d dt S(H4 t=0h=z= (7 3

16 H Lho R Pandharipande For a series F C[[ z ]] the specialization F z= denotes constant term of F with respect to z 344 Further Calculations Define small I-function by the restriction I(q H T (P4 Q[[q]] Define differential operators I(q =I(q t t=0 D = q d dq M = H + zd Applying z d to I and then restricting to t = 0 has same effect as applying M to I dt [ ( z d k I] dq t=0 = M k I The function I satisfies the following Picard Fuchs equation ( M 5 q(5m + z(5m + 2z(5M + 3z(5M + 4z(5M + 5z I = 0 implied by the Picard Fuchs equation for I ( ( z d 5 5 ( ( q 5 z d + kz I = 0 dt dt k= The restriction I H=λi admits the following asymptotic form ( ( 2 = e μλ i z z (R 0 + R + R 2 + zλi λ i I H=λi (8 with series μ R k C[[q]] A derivation of (8 is obtained in [27] via the Picard Fuchs equation for I H=λi The series μ and R k are found by solving differential equations obtained from the coefficient of z k For example + Dμ = L R 0 = L 3 R = 3 20 (L L5 R 2 = 9L 800 ( L4 2

17 Holomorphic Anomaly Equations for the Formal Quintic where L(q =( 5 5 q 5 The specialization ( is used for these results Define the series C i by the equations C 0 = I z= t=0h= C i = z d dt S(Hi z= t=0h= for i = (9 (20 The following relations were proven in [27] C 0 C C 2 C 3 C 4 = L 5 C i = C 4 i for i = From Eqs (7 and (8 we can show the series S i (H k =S(H k H=λi t=0 have the following asymptotic expansion: S i ( =e μλ i z S i (H =e μλ i z S i (H 2 =e μλ i z S i (H 3 =e μλ i z S i (H 4 =e μλ i z ( ( z ( z 2 R C 00 + R 0 + R λ i λ i Lλ ( i ( z ( z 2 R C 0 C 0 + R + R2 + λ i λ i L 2 λ 2 ( i ( z ( z 2 R C 0 C C 20 + R 2 + R λ i λ i L 3 λ 3 ( i ( z ( z 2 R C 0 C C 2 C 30 + R 3 + R λ i λ i L 4 λ 4 ( i ( z ( z 2 R C 0 C C 2 C 3 C 40 + R 4 + R λ i λ i (2 We follow here the normalization of [27] Note R 0k = R k As in [27 Theorem 4] we obtain the following constraints Proposition 5 (Zagier Zinger [27] For all k 0 we have Define generators R k C[L ± ] = DC 0 C 0 = D 2 = D = DC C 3

18 H Lho R Pandharipande From (7 we obtain the following result Lemma 6 For k 0 we have R k+ = R 0 k+ + DR 0k L L R 0k R 2 k+ = R k+ + DR k L R 3 k+ = R 2 k+ + DR 2k L R 4 k+ = R 3 k+ + DR 3k L R 0 k+ = R 4 k+ + DR 4k L L R k L R k + DL L 2 R k + L R 2k + L R 2k 3DL L 2 R 2k + L R 3k 2DL L 2 R 3k DL L 2 R 4k (22 Applying Lemma 6 for k = 0 we obtain the following two equations among above generators which were also proven in [26 Section 3] First D = 2 5 (L5 +2(L (L (23 For the second equation define 4 B = 5 B 2 = 5 2 ( + 2 B 3 = 5 3 ( B 4 = 5 4 (D Then we have B 4 = (L 5 (0B 3 35B B 24 (24 For the proof of first holomorphic anomaly equation we will require the following generalization of Proposition 5 Proposition 7 For all k 0 we have (i R k C[L ± ][ ] (ii R 2k = Q 2k R k with Q L 2k C[L± ][ ] (iii R 3k R 4k C[L ± ][ 2 ] 4 We follow here the notation of [26] for B k 3

19 Holomorphic Anomaly Equations for the Formal Quintic Proof (i Using Lemma 6 we can calculate R k+ = DR 0k L + R 0 k+ R 0k L (ii Using Lemma 6 and relations (23 we can calculate R 2 k+2 = D2 R 0k R 0 k+ L 2 5L + L4 R 0 k+ + 2DR 0 k+ + R 5 L 0 k+2 2DR 0k L 2 (iii We can also explicitly calculate R 3k and R 4k in terms of using Lemma 6 and relations (23 and (24 We can check (iii using these explicit calculations and Proposition 5 We leave the details to the reader For the proof of second holomorphic anomaly equation we will require the following result Proposition 8 For all k 0 we have 2R 0 k+ L + DR 0k LR 0 k+ + R 0k L 2 + R 0k 2 R 0k L 2 L 2 R 0k R 0 k R 0 k 2 2 (i R k = P 0k R 0 k with P L 0k C[L± ] (ii R 2k C[L ± ][A 2 A 4 ] (iii R 3k = P 3k R 2 k with P L 3k C[L± ][A 2 A 4 A 6 ] (iv R 4k C[L ± ][A 2 A 4 A 6 ] Proof The proof follows from the explicit calculations in the proof of Proposition 7 and the definition of A 2 A 4 A 6 4 Higher Genus Series on M gn 4 Intersection Theory on M gn We review here the now standard method used by Givental [5 6 20] to express genus g descendent correlators in terms of genus 0 data We refer the reader to [22 Section 4] for a more leisurely treatment Let t 0 t t 2 be formal variables The series T(c =t 0 + t c + t 2 c 2 + 3

20 H Lho R Pandharipande in the additional variable c plays a basic role The variable c will later be replaced by the first Chern class ψ i of a cotangent line over M gn T(ψ i =t 0 + t ψ i + t 2 ψ 2 + i with the index i depending on the position of the series T in the correlator Let 2g 2 + n > 0 For a i Z 0 and γ H (M gn define the correlator ψ a ψ a n γ gn = k 0 γψ a k! ψ a n n Mgn+k Here γ also denotes the pull-back of γ via the morphism k T(ψ n+i i= M gn+k M gn defined by forgetting the last k points In the above summation the k = 0 term is Mgn γψ a ψ a n n We also need the following correlator defined for the unstable case 02 = k T(ψ k! 2+i k>0 M02+k i= For formal variables x x n we also define the correlator x ψ x n ψ γ gn (25 in the standard way by expanding as a geometric series x i ψ Denote by L the differential operator L = t t i = t 0 t i t t 0 t 2 0 t The string equation yields the following result 3 i= Lemma 9 For 2g 2 + n > 0 and γ H (M gn we have L γ gn = 0 L x ψ x n ψ γ gn ( = + + x x n x ψ x n ψ γ gn

21 Holomorphic Anomaly Equations for the Formal Quintic We consider C(t [t 2 t 3 ] as Z-graded ring over C(t with Define a subspace of homogeneous elements by [ ] C [t t 2 t 3 ] Hom C(t [t 2 t 3 ] After the restriction t 0 = 0 and application of the dilaton equation the correlators are expressed in terms of finitely many integrals (by the dimension constraints From this we easily see [ ] ψ a ψ a n γ gn t0 =0 C [t t 2 t 3 ] Hom Using the leading terms (of lowest degree in t we obtain the following result Lemma 0 The set of genus 0 correlators { } 0n t0 =0 freely generate the ring C(t [t 2 t 3 ] over C(t Definition For γ H (M gk let deg(t i =i for i 2 n 4 P a a n γ gn (s 0 s s 2 Q(s 0 s be the unique rational function satisfying the condition ψ a ψ a n γ gn t0 =0 = Pa a 2 a n γ gn si = 0i+3 t0 =0 By applying Lemma 9 we obtain the two following results; see [22 Section 4] Proposition 2 For 2g 2 + n > 0 we have γ gn = P 0 0γ gn si = 0i+3 Proposition 3 For 2g 2 + n > 0 γ x ψ x n ψ n = e 02( i x i gn P a a n γ gn si = 0i+3 The definition given in (25 of the correlator is valid in the stable range a a n x a + x a n+ n L 02 = 02 t0 =0 = 0 3

22 H Lho R Pandharipande The unstable case (g n =(0 2 plays a special role We define by adding the degenerate term to the terms obtained by the expansion of as a geometric series The degenerate x i ψ i term is associated to the (unstable moduli space of genus 0 with 2 markings By [22 Section 42] we have Proposition 4 x ψ x 2 ψ 2 42 Local Invariants and Wall Crossing The torus T acts on the moduli spaces M gn (P 4 d and Q gn (P 4 d We consider here special localization contributions associated to the fixed points p i P 4 Consider first the moduli of stable maps Let be the union of T-fixed loci which parameterize stable maps obtained by attaching T-fixed rational tails to a genus g n-pointed Deligne Mumford stable curve contracted to the point p i P 4 Similarly let be the parallel T-fixed locus parameterizing stable quotients obtained by attaching base points to a genus g n-pointed Deligne Mumford stable curve contracted to the point p i P 4 Let Λ i denote the localization of the ring at the five tangent weights at p i P 4 Using the virtual localization formula [7] there exist unique series for which the localization contribution of the T-fixed locus M gn (P 4 d Tp i to the equivariant Gromov Witten invariants of formal quintic can be written as 3 2g 2 + n > 0 x ψ x 2 ψ 2 x + x 2 02 = e M gn (P 4 d Tp i M gn (P 4 d Q gn (P 4 d Tp i Q gn (P 4 d C[λ ± 0 λ± 4 ] S pi Λ i [ψ][[q]] ( ( + x x 2 x + x 2

23 Holomorphic Anomaly Equations for the Formal Quintic Q d [Mgn(P 4 d Tp i ] vir d=0 Here H p i g is the standard vertex class = k=0 e(obs e(nor ψ a ψ a n n H p i k! g ψ a ψ a n n Mgn+k k S pi (ψ n+j j= e(e g T p i (P 4 (5λ i e(t pi (P 4 e(e g (5λ i (26 obtained from the Hodge bundle E g M gn+k Similarly the application of the virtual localization formula to the moduli of stable quotients yields classes F pi k H (M gn k C Λ i for which the contribution of Q gn (P 4 d Tp i is given by q d e(obs d=0 [Qgn(P 4 d Tp i ] e(nor ψ a ψ a n n vir q k = H p i k! g ψ a ψ a n n F p i k Mgn k k=0 Here M gn k is the moduli space of genus g curves with markings satisfying the conditions: (i the points p i are distinct (ii the points p j are distinct from the points p i with stability given by the ampleness of {p p n } { p p k } C ns C ( m ω C p i + ε i= k p j for every strictly positive ε Q The Hodge class H p i g is given again by formula (26 using the Hodge bundle j= E g M gn k Definition 5 For γ H (M gn let 3

24 H Lho R Pandharipande ψ a ψ a n n γ p i = gn ψ a ψ a n n γ p i0+ gn = Proposition 6 (Ciocan-Fontanine and Kim [8] For 2g 2 + n > 0 we have the wall-crossing relation where Q(q is the mirror map k=0 k=0 γψ a k! ψ a n n Mgn+k k S pi (ψ n+j j= q k γψ a k! ψ a n n F p i k Mgn k ψ a ψ a n n γ p i gn (Q(q=(IQ 0 2g 2+n ψ a ψ a n n ( I Q Q(q =exp (q I Q 0 (q γ p i0+ (q gn Proposition 6 is a consequence of [8 Lemma 55] The mirror map here is the mirror map for quintic discussed in Sect Propositions 2 and 6 together yield γ p i = P 0 0γ ( p gn gn i 03 p i 04 γ p i0+ gn = P 0 0γ ( p gn i 0+ p i Similarly using Propositions 3 and 6 we obtain pi x ψ x n ψ γ gn = e ( i p i 02 x i a a n P a a n γ gn pi0+ x ψ x n ψ γ gn = e ( i0+ p i 02 x i a a n P a a n γ gn ( p i 03 p i 04 x a + x a n+ n ( p i p i0+ 04 x a + x a n+ n (27 3

25 Holomorphic Anomaly Equations for the Formal Quintic 5 Higher Genus Series on the Formal Quintic 5 Overview We apply Givental s the localization strategy [5 6 20] for Gromov Witten theory to the stable quotient invariants of formal quintic The contribution Cont Γ (q discussed in Sect 2 of a graph Γ G g (P 4 can be separated into vertex and edge contributions We express the vertex and edge contributions in terms of the series S i and V ij of Sect 33 Our treatment here follows our study of KP 2 in [22 Section 5] 52 Edge Terms Recall the definition 5 of V ij given in Sect 33 φi V ij = (28 x ψ φ j 0+0+ y ψ 02 Let V ij denote the restriction of V ij to t = 0 Via formula (9 V ij is a summation of contributions of fixed loci indexed by a graph Γ consisting of two vertices connected by a unique edge Let w and w 2 be T-weights Denote by V w w 2 ij the summation of contributions of T-fixed loci with tangent weights precisely w and w 2 on the first rational components which exit the vertex components over p i and p j The series V w w 2 ij includes both vertex and edge contributions By definition (28 and the virtual localization formula we find the following relationship between V w w 2 ij and the corresponding pure edge contribution E w w 2 ij e i V w pi 0+ pj w 2 ij e j = w ψ E w 0+ w 2 x ψ ij 02 w 2 ψ x 2 ψ 02 p i 0+ p i 0+ = e w x E w w 2 w + x ij = p i e a a 2 + p i x w E w w 2 ( a ij +a 2 w a wa 2 2 p j w e 2 w 2 + x 2 p j x e 2 w 2 x a x a 2 2 p j p j x 2 5 We use the variables x and x 2 here instead of x and y 3

26 H Lho R Pandharipande After summing over all possible weights we obtain δ ij e i (V ij e e i (x + x 2 j = e i V w w 2 e ij j w w 2 The above calculations immediately yield the following result Lemma 7 We have p i e x e = w w 2 e p j x 2 e i (V ij δ ij e x e i (x + x 2 j a x a 2 2 p i 0+ p j E w w 2 w w e 2 ( a ij +a 2 w a wa 2 2 The notation [ ] a x x a 2 in Lemma 7 denotes the coefficient of x a x a 2 in the 2 2 series expansion of the argument 53 A Simple Graph Before treating the general case we present the localization formula for a simple graph 6 Let Γ G g (P 4 consist of two vertices and one edge v v 2 Γ(V with genus and T-fixed point assignments e Γ(E g(v i =g i p(v i =p i Let w and w 2 be tangent weights at the vertices p and p 2 respectively Denote by Cont Γw w 2 the summation of contributions to d>0 of T-fixed loci with tangent weights precisely w and w 2 on the first rational components which exit the vertex components over p and p 2 We can express the localization formula for (29 as w ψ Hp [ ] vir q (e(obs d Q g (P 4 d g p0+ g E w w 2 2 w 2 ψ p20+ Hp 2 g 2 g 2 (29 6 We follow here the notation of Sect 2 3

27 Holomorphic Anomaly Equations for the Formal Quintic which equals a a 2 e p w [ ] P ψ a H p p 0+ g w a where H p i g i is the Hodge class (26 We have used here the notation [ P ψ k ψ k n n = P k k n H p i h h g ] pi 0+ Hp i h hn E w w 2 2 e p w 2 [ ] P ψ a 2 H p p g 2 and applied (27 After summing over all possible weights w w 2 and applying Lemma 7 we obtain the following result for the full contribution w a 2 2 ( p i p i0+ 04 g 2 Cont Γ = w w 2 Cont Γw w 2 of Γ to d 0 qd (e(obs [ Q g (P 4 d ] vir Proposition 8 We have Cont Γ = [ P ψ a H p i a a 2 >0 ] pi 0+ g g ( a +a 2 e p i x e [ ] P ψ a 2 H p pj 0+ j g 2 g 2 p j x 2 e i (V ij δ ij e x e i (x + x 2 j a x a A General Graph We apply the argument of Sect 53 to obtain a contribution formula for a general graph Γ Let Γ G g0 (P 4 be a decorated graph as defined in Sect 2 The flags of Γ are the half-edges 7 Let F be the set of flags Let be a fixed assignment of T-weights to each flag We first consider the contribution Cont Γw to d 0 w F Hom(T C Z Q [ ] vir q (e(obs d Q g (P 4 d 7 Flags are either half-edges or markings 3

28 H Lho R Pandharipande of the T-fixed loci associated Γ satisfying the following property: the tangent weight on the first rational component corresponding to each f F is exactly given by w(f We have The terms on the right side of (30 require definition: The sum on the right is over the set Z F of all maps >0 A F Z >0 corresponding to the sum over a a 2 in Proposition 8 For v V with n incident flags with w-values (w w n and A-values (a a n For e E with assignments (p(v p(v 2 for the two associated vertices 8 and w -values (w w 2 for the two associated flags The localization formula then yields (30 just as in the simple case of Sect 53 By summing the contribution (30 of Γ over all the weight functions w and applying Lemma 7 we obtain the following result which generalizes Proposition 8 Proposition 9 We have where the vertex and edge contributions with incident flag A-values (a a n and (b b 2 respectively are 3 Cont Γw = Aut(Γ A Z F >0 [ P ψ a ψ a n Cont A Γw (v = n Cont Γw (e =e Cont Γ = Aut(Γ [ Cont A Γ (v =P ψ a ψ a n n Cont A Γ (e = ( b +b 2 [ 8 In case e is self-edge v = v 2 Cont A Γw (v Cont A Γw (e v V e E w a wa n n Hp(v g(v ] p(v0+ g(vn p(v 0+ p(v w w e 2 E w w 2 p(v p(v 2 A Z F >0 Hp(v g(v Cont A Γ (v Cont A Γ (e v V e E ] p(v0+ g(vn e p(v x e p(v x 2 e i (V ij ] e e i (x + x 2 j x b x b 2 2 (30

29 Holomorphic Anomaly Equations for the Formal Quintic where p(v =p i and p(v 2 =p j in the second equation 55 Legs Let Γ G gn (P 4 be a decorated graph with markings While no markings are needed to define the stable quotient invariants of formal quintic the contributions of decorated graphs with markings will appear in the proof of the holomorphic anomaly equation The formula for the contribution Cont Γ (H k H k n of Γ to q d d 0 n j=0 ] vir ev (H k j e(obs [Q g (P 4 d is given by the following result Proposition 20 We have Cont Γ (H k H k n = Cont A Γ Aut(Γ (v Cont A Γ (e Cont A Γ (l v V e E l L where the leg contribution is A Z F >0 Cont A Γ (l = ( A(l [ e p(l0+ 02 z S p(l (H k l ]z A(l The vertex and edge contributions are same as before The proof of Proposition 20 follows the vertex and edge analysis We leave the details as an exercise for the reader The parallel statement for Gromov Witten theory can be found in [5 6 20] 6 Vertices Edges and Legs 6 Overview Following the analysis of KP 2 in [22 Section 6] which uses results of Givental [5 6 20] and the wall-crossing of [8] we calculate here the vertex and edge contributions in terms of the function R k of Sect 344 3

30 H Lho R Pandharipande 62 Calculations in Genus 0 We follow the notation introduced in Sect 4 Recall the series T(c =t 0 + t c + t 2 c 2 + Proposition 2 (Givental [5 6 20] For n 3 we have p i 0n =( Δ i 2g 2+n ( k 0 where the functions Δ i Q l are defined by T(ψ k! n+ T(ψ n+k M0n+k t 0 =0t =0t j 2 =( j Q j λ j i From (2 and Proposition 6 we have p i 02 z S φi pi ( =e i i z ψ = e ( l z ( + Q l 02 Δi λ i p i 02 = μλ i Δi = C 0 R 0 Q k = R k R 0 l= Using Proposition 6 again we have proven the following result Proposition 22 For n 3 we have p i0+ 0n ( = R 2 n 0 k 0 T(ψ k! n+ T(ψ n+k M0n+k t 0 =0t =0t j 2 =( j R j λ j R i 0 Proposition 22 immediately implies the evaluation p i0+ 03 = R 0 (3 Another simple consequence of Proposition 22 is the following basic property Corollary 23 For n 3 we have p i0+ 0n C[R ± 0 R R 2 ][λ i ] 3

31 Holomorphic Anomaly Equations for the Formal Quintic 63 Vertex and Edge Analysis By Proposition 9 we have decomposition of the contribution to Γ G g (P 4 to the stable quotient theory of formal quintic into vertex terms and edge terms Cont Γ = Lemma 24 We have Cont A Γ (v C(λ 0 λ 4 [L± ] Proof By Proposition 9 Aut(Γ The right side of the above formula is a polynomial in the variables with coefficients in C(λ 0 λ 4 The lemma then follows from the evaluation (3 Corollary 23 and Proposition 5 Both the positive and the negative powers of p(v0+ are required here 03 since R ± occurs in Corollary 23 0 Let e E be an edge connecting the T-fixed points p i p j P 4 Let the A-values of the respective half-edges be (k l Lemma 25 We have Cont A Γ (e C(λ 0 λ 4 [L± 2 ] and A Z F >0 [ Cont A Γ (v =P ψ a ψ a n n p(v0+ 03 and the degree of Cont A (e Γ with respect to is the coefficient of in Cont A (e Γ is Cont A Γ (v Cont A Γ (e v V e E Hp(v g(v ] p(v0+ g(vn { } p(v0+ 0n t0 =0 n 3 Proof By Proposition 9 Cont A Γ (e =( k+l [ Using also the equation ( k+l+ R k R l 5L 3 λ k 2 λ l 2 i j e μλ i x μλ j y ei (V ij ] δ ij e e i (x + y j x k y l 4 r=0 e i V ij (x ye j = S i (φ r z=x S j (φr z=y x + y 3

32 H Lho R Pandharipande we write Cont A (e Γ as [ ( k+l e μλ i μλ j x y 4 S i (φ r z=x S j (φ r z=y ]x k y l x k+ y l 2 + +( k x k+l r=0 where the subscript signifies a (signed sum of the respective coefficients If we substitute the asymptotic expansions (2 for S i ( S i (H S i (H 2 S i (H 3 S i (H 4 in the above expression the lemma follows from Proposition 7 Similarly we obtain the following result using Proposition 8 Lemma 26 We have Cont A Γ (e C(λ 0 λ 4 [L± A 2 A 4 A 6 ] and the degree of Cont A Γ with respect to is the coefficient of in Cont A Γ is 64 Legs ( ( k+l+ R0 k R 2 l 5L 3 λ k i λ l 3 j Using the contribution formula of Proposition 20 [ Cont A Γ (l = ( A(l + R 2 k R 0 l 5L 3 λ k 3 λ l i j e p(l0+ 02 z S p(l (H k l ]z A(l we easily conclude when the insertion at the marking l is H 0 C 0 Cont A Γ (l C(λ 0 λ 4 [L± ] ; when the insertion at the marking l is H C 0 C Cont A Γ (l C(λ 0 λ 4 [L± ] ; when the insertion at the marking l is H 2 C 0 C C 2 Cont A Γ (l C(λ 0 λ 4 [L± ] ; when the insertion at the marking l is H 3 C 0 C C 2 C 3 Cont A Γ (l C(λ 0 λ 4 [L± 2 ] ; 3

33 Holomorphic Anomaly Equations for the Formal Quintic when the insertion at the marking l is H 4 C 0 C C 2 C 3 C 4 Cont A Γ (l C(λ 0 λ 4 [L± 2 ] 7 Holomorphic Anomaly for the Formal Quintic 7 Proof of Theorem By definition we have K 2 (q = L 5 A 2 (q = ( L A 4 (q = ( L ( 2 A 6 (q = L ( + 0 5L 5 ( ( Hence statement (i follows from Propositions and Lemmas Since statement (ii B g (q C[L± ][C ± 0 K 2 A 2 A 4 A 6 ] T = q C q k B g T k (q C[L± ][C ± 0 C K 2 A 2 A 4 A 6 ] (32 follows since the ring C[L ± ][C ± 0 C K 2 A 2 A 4 A 6 ]=C[L± ][C ± 0 C 2 ] is closed under the action of the differential operator D = q q 3

34 H Lho R Pandharipande by (23 The degree of C in (32 is which yields statement (iii Remark 27 The proof of Theorem actually yields: C 2 2g B 0 g C2 2g C k 0 k g B C[L ± ][A T k 2 A 4 A 6 K 2 ] 72 Proof of Theorem 2: First Equation Let Γ G g (P 4 be a decorated graph Let us fix an edge f E(Γ: If Γ is connected after deleting f denote the resulting graph by If Γ is disconnected after deleting f denote the resulting two graphs by where g = g + g 2 Γ 0 f G g 2 (P 4 Γ f G g (P4 and Γ 2 f G g2 (P4 There is no canonical order for the 2 new markings We will always sum over the 2 labellings So more precisely the graph Γ 0 in case should be viewed as sum of 2 f graphs Γ 0 f (2 +Γ0 f (2 Similarly in case we will sum over the ordering of g and g 2 As usually the summation will be later compensated by a factor of in the formulas 2 By Proposition 9 we have the following formula for the contribution of the graph Γ to the theory of the formal quintic Cont Γ = Aut(Γ A Z F 0 Cont A Γ (v Cont A Γ (e v V e E Let f connect the T-fixed points p i p j P 4 Let the A-values of the respective halfedges be (k l By Lemma 25 we have Cont A Γ (f = ( k+l+ R k R l 5L 3 λ k 2 λ l 2 i j (33 3

35 Holomorphic Anomaly Equations for the Formal Quintic If Γ is connected after deleting f we have ( Cont 5L5 A Γ (f Cont A Aut(Γ C 2 0 C2 Γ (v v V A Z F 0 = Cont Γ 0(H H f e E e f Cont A Γ (e The derivation is simply by using (33 on the left and Proposition 20 on the right If Γ is disconnected after deleting f we obtain Aut(Γ A Z F 0 = Cont Γ f ( Cont 5L5 A Γ (f Cont A C 2 0 C2 Γ (v v V (H Cont Γ 2(H f e E e f Cont A Γ (e by the same method By combining the above two equations for all the edges of all the graphs Γ G g (P 4 and using the vanishing Cont A Γ (v = 0 of Lemma 24 we obtain ( L5 SQ g0 = g H SQ 2 g i H SQ 5C 2 0 C2 i= i + 2 H H SQ g 2 (34 We have followed here the notation of Sect By definition of A 2 A 4 A 6 we have following equations Since I 2g 2 chain rule SQ 0 g ( C 2 ( C 2 ( C 2 A 2 K 2 5C 2 A 2 K 2 5C 2 A 2 K 2 5C 2 A 4 + K2 2 50C 2 A 4 + K2 2 50C 2 A 4 + K2 2 50C 2 = 5L 5 A 6 A = 0 6 A 2 = 0 6 = g B the left side of (34 after multiplication by I2g 2 is by the 0 3

36 H Lho R Pandharipande C 2 0 C2 g B K 2 g B + K2 2 g B C[L ± ][C ± A 2 5C 2 0 C2 A 4 50C 2 0 C2 A 0 C K 2 A 2 A 4 A 6 ] 6 On the right side of (34 we have I 2(g i 2+ 0 H SQ g i = B g i (q = GW g i (Q(q where the first equality is by definition and the second is by wall-crossing (6 Then g i (Q(q = GW g i T (Q(q = B g i T (q GW where the first equality is by the divisor equation in Gromov Witten theory and the second is again by wall-crossing (6 So we conclude I 2(g i 2+ H SQ 0 g i = B g i (q C[[q]] T (35 Similarly we obtain I 2(g i 2+ H SQ 0 i = B i (q C[[q]] T (36 The above equations transform (34 after multiplication by I 2g 2 0 to exactly the first holomorphic anomaly equation of Theorem 2 C 2 0 C2 B g A 2 2 B I 2(g 2+2 H H SQ 0 g 2 = g (q C[[q]] T 2 5C 2 0 C2 B g A 4 K 2 + as an equality in C[[q]] In order to lift the first holomorphic anomaly equation to the ring 50C 2 0 C2 g B K 2 2 A = g B g i B i 6 2 T T + 2 B g 2 T i= 2 C[L ± ][A 2 A 4 A 6 C ± 0 C K 2 ] we must lift the equalities (35 (37 The proof is identical to the parallel lifting for KP 2 given in [22 Section 73] (37 3

37 Holomorphic Anomaly Equations for the Formal Quintic 73 Proof of Theorem 2: Second Equation By Proposition 9 we have the following formula for the contribution of the graph Γ to the stable quotient theory of formal quintic Cont Γ = Aut(Γ A Z F 0 Cont A Γ (v Cont A Γ (e v V e E Let f connect the T-fixed points p i p j P 4 Let the A-values of the respective halfedges be (k l By Lemma 26 we have Cont A Γ (f ( = ( k+l+ R0 k R 2 l + R 2 k R 0 l (38 5L 3 λ k λ l 3 5L i j 3 λ k 3 λ l i j If Γ is connected after deleting f we have ( Cont 5L5 A Γ (f Cont A Aut(Γ C 2 Γ (v v V A Z F 0 = Cont Γ 0( H 2 f e E e f Cont A Γ (e The derivation is simply by using (38 on the left and Proposition 20 on the right If Γ is disconnected after deleting f we obtain Aut(Γ A Z F 0 = Cont Γ f ( 5L5 C 2 ( Cont Γ 2(H 2 f Cont A Γ (f Cont A Γ (v v V e E e f Cont A Γ (e by the same method By combining the above two equations for all the edges of all the graphs Γ G g (P 4 and using the vanishing of Lemma 24 we obtain ( 5L5 C 2 SQ g g0 = Cont A Γ (v = 0 i= SQ g i H2 SQ i + H2 SQ g 2 = 0 (39 3

38 H Lho R Pandharipande The second equality in the above equations follow from the string equation for formal stable quotient invariants Since Eq (39 is equivalent to the second holomorphic anomaly equation of Theorem 2 as an equality in C[[q]] In order to lift the second holomorphic anomaly equation to the ring we must lift the equalities K 2 = L 5 C[L ± ][A 2 A 4 A 6 C ± 0 C K 2 ] SQ g i = 0 H 2 SQ g 2 = 0 The proof follows from the properties of the unit in a CohFT Specifically the method of the proof of [25 Proposition 22] is used We leave the details to the reader 74 Genus One Invariants We do not study the genus unpointed series B (q in the paper so we take I 2 0 I 0 H SQ = B T H H SQ 2 = 2 B T 2 as definitions of the right side in the genus case There is no difficulty in calculating these series explicitly using Proposition 20 B T 2 B T 2 ( = L5 C 2 A K L 5 = ( L 5 ( D C C 2 A K L 5 75 Bounding the Degree For the holomorphic anomaly equation for KP 2 the integration constants can be bounded [22 Section 75] A parallel result hold for the formal quintic The degrees in L of the term of 3

39 Holomorphic Anomaly Equations for the Formal Quintic for formal quintic always fall in the range SQ g C[L ± ][A 2 A 4 A 6 K 2 ] [5 5g 0g 0] (40 In particular the constant (in A 2 A 4 A 6 K 2 term of g SQ missed by the holomorphic anomaly equation for formal quintic is a Laurent polynomial in L with degree in the range (40 The bound (40 is a consequence of Proposition 9 the vertex and edge analysis of Sect 6 and the following result Lemma 28 The degrees in L of R ip fall in the range [ i4p + ] Proof The proof for the functions R 0p follows from the arguments of [27] The proof for the other R ip follows from Lemma 7 Acknowledgements We thank I Ciocan-Fontanine E Clader Y Cooper B Kim A Klemm Y-P Lee A Marian M Mariño D Maulik D Oprea E Scheidegger Y Toda and A Zinger for discussions over the years about the moduli space of stable quotients and the invariants of Calabi Yau geometries The work of Q Chen F Janda S Guo and Y Ruan as presented at the Workshop on higher genus is crucial for the wider study of the formal (and true quintic We are very grateful to them for sharing their ideas with us R P was partially supported by SNF ERC-202-AdG MCSK ERC-207-AdG MACI SwissMAP and the Einstein Stiftung H L was supported by the Grants ERC-202-AdG MCSK and ERC-207-AdG MACI This project has received funding from the European Research Council (ERC under the European Union s Horizon 2020 research and innovation programme (grant agreement No References Alim M Scheidegger E Yau S-T Zhou J: Special polynomial rings quasi modular forms and duality of topological strings Adv Theor Math Phys (204 2 Behrend K Fantechi B: The intrinsic normal cone Invent Math (997 3 Bershadsky M Cecotti S Ooguri H Vafa C: Holomorphic anomalies in topological field theories Nucl Phys B (993 4 Ciocan-Fontanine I Kim B: Moduli stacks of stable toric quasimaps Adv Math (200 5 Ciocan-Fontanine I Kim B: Wall-crossing in genus zero quasimap theory and mirror maps Algebr Geom (204 6 Ciocan-Fontanine I Kim B: Big I-functions In: Development of Moduli Theory Kyoto vol 69 Advanced Studies in Pure Mathematics Mathematical Society of Japan Tokyo pp (206 7 Ciocan-Fontanine I Kim B: Quasimap Wallcrossings and Mirror Symmetry arxiv : Ciocan-Fontanine I Kim B: Higher genus quasimap wall-crossing for semi-positive targets JEMS (207 9 Ciocan-Fontanine I Kim B Maulik D: Stable quasimaps to GIT quotients J Geom Phys (204 0 Clader E Janda F Ruan Y: Higher Genus Quasimap Wall-Crossing Via Localization arxiv : Cooper Y Zinger A: Mirror symmetry for stable quotients invariants Michigan Math J (204 3

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