Gromov-Witten Invariants and Modular Forms

Transcription

1 Gromov-Witten Invariants and Modular Forms Jie Zhou Harvard University Joint work with Murad Alim, Emanuel Scheidegger and Shing-Tung Yau arxiv:

2 Overview - Background and motivation - Solving topological string amplitudes in terms of quasi modular forms - Example: K P 2 - Special geometry polynomial ring - Conclusions and discussions

3 Background Given a CY 3fold Y, one of the most interesting problems to count the Gromov-Witten invariants of Y and consider the generating functions of genus g Gromov-Witten invariants of Y F g GW (Y, t) = ω i1 ω ik g,β = β H 2 (Y,Z) k j=1 e ω(t) g,β = β H 2 (Y,Z) ev j ω ij [M g,k (Y, β)] vir, N GW g,β e β ω(t), here ω(t) = h 1,1 (Y ) i=1 t i ω i, where ω i, i = 1, 2 h 1,1 (Y ) are the generators for the Kahler cone of Y. For some special CY 3-folds, the generating functions F g GW (Y, t) could be computed by localization technique, topological vertex, etc. For general CY 3-folds, they are very dicult to compute.

4 Background - Physics (topological string theory)tells that F g GW (Y, t) is the holomorphic limit of some non-holomorphic quantity called the A model genus g topological string partition function of Y F g GW (Y, t) = lim F g (Y, t, t) LVL The above expression lim LVL means the holomorphic limit based at the large volume limit t = i : think of t, t as independent coordinates, x t, send t to i. - Mirror symmetry predicts the existence of the mirror manifold X of Y in the sense that F g (Y ) (and its holomorphic limit F g (Y )) is identical to some quantity F g (X ) (and the holomorphic limit F g (X )) called the B model genus g topological string partition function on X, under the mirror map.

5 Background - The genus zero topological string partition function was studied intensively since the celerated work Candelas, de La Ossa, Green & Parkes (1991) - The partition functions F g (X ), g 1 satisfy some dierential equations called the holomorphic anomaly equations Bershadsky, Cecotti, Ooguri & Vafa (1993), and are easier to compute than F g (Y ). - Thanks to mirror symmetry, one can try to extract Gromov-Witten invariants of Y by studying properties of (the moduli space of) X and solving F g (X ) from the equations.

6 Motivation In some nicest cases, F g (Y ) (and its holomorphic limit F g GW (Y )) are expected to have some modular properties. Some examples include - Y = elliptic curve Rudd (1994), Dijkgraaf (1995), Kaneko & Zagier (1995)... FGW 1 (t) = log η(q), q = exp 2πit FGW 2 (t) = (10E 3 2 6E 2 E 4 4E 6 ) (t) is a quasi modular form of weight 6g 6... F g GW - STU model: Y = a special K 3 bration - FHSV model: Y = K 3 T 2 /Z 2. IIA HE duality tells that F g (Y ) have nice modular properties Kachru & Vafa (1995), Marino & Moore (1998), Klemm & Marino (2005), Maulik & Pandharipande (2006)...

7 Overview In this talk, we shall work only on the B model of X. We shall - solve F g (X ), g 0 from the holomorphic anomaly equations for certain noncompact CY 3folds X and express them in terms of the generators of the ring of almost-holomorphic modular forms. The results we obtain predict the correct GW invariants of the mirror manifold Y under the mirror map. - explore the duality of F g (X ) for these particular noncompact CY 3folds - construct the analogue of the ring of almost-holomorphic modular forms for general CY 3folds by using quantities constructed out of the special Kahler geometry on the moduli space M complex (X ) of complex structures of X

8 Example Here is an example we could compute : Y = K P 2 Aganagic, Bouchard & Klemm (2006), [ASYZ] FGW 1 (Y, t) = 1 2 log η(q)η(q3 ), q = exp 2πiτ, t τ FGW 2 (Y, t) = E(6A4 9A 2 E + 5E 2 ) 1728B A A3 B χ 10 B B 6 where A, B, C, E are explicit quasi modular forms (with multiplier systems) of weights 1, 1, 1, 2 respectively, with respect to the modular group Γ 0 (3). I will explain in detail how this modular group comes out.

9 Example FGW 3 (Y, t) ( 2532A A 7 B A 4 B AB 9) E = ( B A A 5 B A 2 B 6) E 2 + ( B A A 3 B 3 120B 6) E 3 + ( B A 4 420AB 3) E A2 E 5 6 5E B B B A A 9 B A 6 B 6 496A 3 B 9 + 2(8 3χ)B B 12

10 Solving F g : Special Kahler metric Consider the family of CY 3-folds π : X M, where M is some deformation space (of complex structures) of the CY 3fold X. The base M is equipped with the Weil-Petersson metric whose Kahler potential is given by e K = i Ω Ω (1) where Ω is a holomorphic section of the Hodge line bundle L = R 0 π Ω 3 M X. The curvature of the Weil-Petersson metric G i j satises the so-called special geometry relation Strominger (1990) R k i j l = j Γ k il = δ k l G i j + δ k i G l j C ilm C km j (2) where C ijk = Ω i j k Ω (Yukawa coupling or three-point function) and C jk ī = e 2K G j j G k kc ī j k, i = 1, 2, n = dimm.

11 Solving F g : BCOV holomorphic anomaly equations The genus g topological string partition function F g is a section of L 2 2g. For g = 1 case, it satises the holomorphic anomaly equation Bershadsky, Cecotti, Ooguri & Vafa (1993) ī j F 1 = 1 2 C jklc jk ī ( χ 24 1)G īj (3) where the quantity χ is the Euler characteristic of the mirror manifold Y (not of X ).

12 Solving F g : BCOV holomorphic anomaly equations For g 2 Bershadsky, Cecotti, Ooguri & Vafa (1993) īf g = 1 2 C jk ī g 1 +g 2 =g, g 1,g 2 1 D j F g 1 D k F g 2 + D j D k F g 1 (4) genus = g genus = g 1 genus = g 2 genus = g 1 ϕ ī D i = Cjk ī D k D j 1 2 Cjk ī where D i is the sum of the Chern connection associated to the Weil-Petersson metric G i j and the Kahler connection K i on the Hodge line bundle L. D k

13 Solving F g : polynomial recursion Using the special geometry relation and the holomorphic anomaly equation for genus 1, the equations could be solved recursively using integration by parts Bershadsky, Cecotti, Ooguri & Vafa (1993), Yamaguchi & Yau (2004), Alim & Länge (2007), for each genus g, the topological string partition function F g is found to take the form īf g = īp g, g 2 (5) where P g is a polynomial (coecients are holomorphic functions) of the propagators S ij, S i, S dened via ks ij = C ij k, ks i = G k ks ik, ks = G k ks k (6) and the Kahler connection K i. The coecients are kind of universal from recursion. These generators encode all of the anti-holomorphic dependence of F g.

14 Solving F g : differential ring of non-holomorphic generators Moreover, the ring generated by these non-holomorphic generators S ij, S i, S, K i is closed under the covariant derivative D i Alim & Länge (2007) D i S jk = δ j i S k + δ k i S j C imn S mj S nk + h jk i, D i S j = 2δ j i S C imns m S nj + h jk i K k + h j i, D i S = 1 2 C imns m S n hmn i K m K n + h j i K j + h i, D i K j = K i K j C ijk S k + C ijk S kl K l + h ij, (7) where h jk i, h j i, h i, h ij are holomorphic functions.

15 Solving F g : holomorphic ambiguities From īf g = īp g, g 2 (8) we get F g = P g (S ij, S i, S, K i ) + f g, g 2 (9) The function f g is purely holomorphic and is called the holomorphic ambiguity. It can not be determined by only looking at the holomorphic anomaly equations. Boundary conditions are needed to x it.

16 Solving F g : boundary condition at LCSL Mirror symmetry predicts that Bershadsky, Cecotti, Ooguri & Vafa (1993) FA 1 (X, t) := lim F 1 (X, t, t) LCSL F 1 GW (Y, t) = 1 24 h 1,1 (Y ) i=1 t i F g A (X, t) := lim F g (X, t, t) LCSL F g GW (Y, t) = ( 1)g χ 2 c 2 (TY )ω i + O(e 2πit i ) B 2g B 2g 2 2g(2g 2)(2g 2)! + O(e2πit i ), g 2 where t = (t 1, t 2, t n ) are the canonical coordinates on the moduli spaces and lim LCSL means the holomorphic limit at the large complex structure limit. These conditions are obtained from computing the constant map contribution to F g GW (Y, t).

17 Solving F g : boundary condition at conifold loci Moreover, at the conifold loci Bershadsky, Cecotti, Ooguri & Vafa (1993), Ghoshal & Vafa (1995), Huang & Klemm (2006) F 1 con(t i c) : = lim coni F 1 = 1 12 log i + regular terms F g con(t i c) : = lim coni F g = c g 1 B 2g 2g(2g 2)(t i c) 2g 2 + O((ti c) 0 ), g 2 where lim coni means the holomorphic limit at the conifold locus which is dened as the discriminant locus i = 0, i = 1, 2,, m. The quantity tc i is a suitably chosen coordinate normal to the conifold locus i = 0. This is called the gap condition.

18 Solving F g : traditional approach in fixing f g According to the above boundary conditions at the LCSL and conifold loci, one can try the following ansatz for f g Bershadsky, Cecotti, Ooguri & Vafa (1993), Katz, Klemm & Vafa (1999), Yamaguchi & Yau (2004) f g (z) = m i=1 A g i 2g 2 i (10) where A g i is a polynomial of z 1, z n of degree (2g 2)deg i Then one aims to solve for the coecients in A g i from the boundary conditions. For noncompact CY 3folds, a dimension counting Haghighat, Klemm & Rauch (2008) suggests that the numbers of unknowns is the same as the number of boundary conditions. So in principle, f g could be completely determined. For compact CY 3folds, further inputs, e.g., boundary conditions at the orbifold loci are needed Katz, Klemm & Vafa (1999), Huang & Klemm (2006).

19 Solving F g : difficulties - The boundary conditions are applied to the dierent holomorphic limits of the same function F g, namely, to F g A and F g con. But the relation between these two dierent functions is not quite clear.they are not related by a simple analytic continuation (recall that the denition of a holomorphic limit requires a choice of the base point). - The ansatz for the ambiguity f g is made based on regularity at the orbifold points (loci), which is not ensured for general CYs. - In practice, the generators S ij, S i, S, K i are computed as innite series in the complex coordinates on M, so the expression for F g is not compact. - Modularity is not manifest.

20 Solving F g : implement of modularity In our work, we could overcome these diculties by making use of the arithmetic properties of the moduli spaces, for certain noncompact CY 3folds X. We could solve F g in terms of modular forms according to the following procedure: - identify the moduli space M with a certain modular curve X Γ = H /Γ, Γ PSL(2, Z) - construct the ring of quasi modular forms and almost-holomorphic modular forms attached to X Γ, the latter turns out to be equivalent to the special geometry polynomial ring constructed out of purely geometric quantities (periods, connections, Yukawa couplings...)

21 Solving F g : implement of modularity Then we 1. express the quantities S ij, S i, S, K i and F g, g = 0, 1 in terms of the generators of the ring of almost-holomorphic modular forms 2. solve P g via polynomial recursion in terms of these generators, modularity then gives a very strong constraint and also a natural ansatz for f g in terms of modular forms, with under-determined coecients 3. explore the relation between F g A and F g con, realize the boundary conditions as certain regularity conditions imposed on the modular objects F g and F g A 4. solve the under-determined coecients in f g and thus express F g s (F g A s) as almost-holomorphic (quasi) modular forms

22 Local P 2 example: mirror CY 3 fold family The mirror CY family of Y = K P 2 (called local P 2 below) is a family of noncompact CY 3-folds π : X M = P 1 given by Chiang, Klemm, Yau & Zaslow (1999), Hori & Vafa (2000). More precisely, choose z as the parameter for the base M. For each z, the CY 3fold X z is itself a conic bration uv H(y i ; z) := uv (y 0 + y 1 + y 2 + y 3 ) = 0, (u, v) C 2 over the base C 2 parametrized by y i, i = 0, 1, 2, 3 with the following conditions: a. there is a C action: y i λy i, λ C, i = 0, 1, 2, 3; b. z = y 1y 2 y 3. y0 3 Straightforward computation shows that z = 0, 1/27, corresponds to the large complex structure limit, conifold point, orbifold of the CY 3fold family, respectively.

23 Local P 2 example: mirror curve family The degeneration locus of this conic bration is a curve E z (called the mirror curve) sitting inside X z : H(y i ; z) = y 0 + y 1 + y 2 + y 3 = 0, z = y 1y 2 y 3 y 3 0 This way, we get the mirror curve family π : E M. It is equivalent to with y 2 3 (y 0 + 1)y 3 = zy 3 0 (11) = (1 24z)3, j(z) = z z 3 (1 27z).

24 Local P 2 example: arithmetic of moduli space Comparing it with the elliptic modular surface associated to Γ 0 (3), i.e., the Hesse family π Γ0 (3) : E Γ0 (3) X Γ0 (3) = H /Γ 0 (3) x x x 3 3 z 1 3 x1 x 2 x 3 = 0, j(z) = ( z) z(1 27z) 3 One can see that Aganagic, Bouchard & Klemm (2006), Haghihat, Klemm & Rauch (2008), [ASYZ] M = X 0 (3) := H /Γ 0 (3) In fact these two families are related by a 3isogeny, see e.g., Elliptic Curves (GTM) Husemöller. (2004). Taking the generator of the rational functional eld (Hauptmodul) of X 0 (3) to be α = 27z, then the points α = 0, 1, correspond to the large complex structure limit, conifold point, orbifold of the CY 3fold family respectively.

25 Local P 2 example: periods of elliptic curve family The Picard-Fuchs operator attached to the Hesse elliptic curve family is the hypergeometric operator Klemm, Lian, Roan & Yau (1994), Lian & Yau (1994) L elliptic = θ 2 α(θ )(θ ) (12) where θ = α α. A basis of the space of periods could be chosen to be ω 0 = 2 F 1 ( 1 3, 2 3, 1; α), ω 1 = i 2F 1 ( 1 3 3, 2, 1; 1 α) (13) 3 Then we get τ = ω 1 ω 0 = i 2F 1 ( 1, 2, 1; 1 α) F 1 ( 1, 2, 1; α) (14) 3 3 It follows then the points α = 0, 1, correspond to [τ] = [i ], [0], [exp 2πi/3], respectively.

26 Local P 2 example: periods of elliptic curve family Dene Maier (2006) then Moreover, one can show A = ω 0, B = (1 α) 1 3 A, C = α 1 3 A (15) A 3 = B 3 + C 3 Div A = 1 3 (α = ), Div B = 1 3 (α = 1), Div A = 1 3 (α = 0). (16)

27 Local P 2 example: modular forms It turns out that the A, B, C dened out of periods are modular forms (with multiplier systems) with respect to Γ 0 (3) (see e.g., Maier (2006), The of Modular Forms (2006) and references therein). Moreover, they have very nice θ or η expansions (q = exp 2πiτ) A(τ) = θ 2 (2τ)θ 2 (6τ) + θ 3 (2τ)θ 3 (6τ) = B(τ) = η(τ)3 η(3τ) = C(τ) = 3 η(3τ)3 η(τ) (m,n) Z 2 q m2 mn+n2 (m,n) Z 2 (exp( 2πi 3 ))m+n q m2 mn+n 2 = q 1 3 (m,n) Z 2 q m2 mn+n2+m+n = 1 2 (A(τ 3 ) A(τ))

28 Local P 2 example: quasi modular forms Now we consider the dierential ring structure. One can show that Dα = αβa 2 (17) where D = 1 2πi τ, β := 1 α. This is equivalent to A 2 = D log C 3 B (= 3E 2(3τ) E 2 (τ) 3 2 ) (18) which transforms as an honest modular form under Γ 0 (3). Now we dene the analogue of the quasi modular form E 2 by E = D log C 3 B 3 (= 3E 2(3τ) + E 2 (τ) ) (19) 4 It is easy to see that it transforms as a quasi modular form under Γ 0 (3). We denote its modular completion E + 3 πimτ ( ) by Ê.

29 Local P 2 example: ring of quasi modular forms It follows from the denitions and the Picard-Fuchs equation that the ring generated by A, B, C, E is closed upon taking derivative D = 1 2πi τ [ASYZ] DA = 1 6 A(E + C 3 B 3 ) A DB = 1 6 B(E A2 ) DC = 1 6 A(E + A2 ) (20) DE = 1 6 (E 2 A 4 ) This is the ring of quasi modular forms (with multiplier systems) for Γ 0 (3). The ring generated by Ê, A, B, C is then the ring of almost-holomorphic modular forms (with multiplier systems).

30 Local P 2 example: Fricke involution on modular curve There is a natural automorphism, called Fricke involution W N, of the family π Γ0 (3) : E Γ0 (3) X Γ0 (3). Here N = 3. In terms of coordinates, it is described by W N : τ 1, α β := 1 α (21) Nτ Using the interpretation of the modular curve as a moduli space X 0 (3) = {(E, C) C < E N = Z 2 N, C = N} then the Fircke involution has the following description W N : (E, C) (E/C, E N /C) The mirror curve family X is related to E Γ0 (3) by a 3isogeny and the Fricke involution is also an automorphism of X.

31 Local P 2 example: Fricke involution on modular curve Γ 0 (3) {( ) a b Γ 0 (N) = c 0 mod N} PSL(2, Z) c d Fricke involution W N : τ 1 Nτ For X 0 (N) = H /Γ 0 (N), N = 4, 3, 2, 1 which has three singular points, the Fricke involution exchanges the two distinguished cusps [i ] = [1/N] and [0] on the modular curve, and xes the last of the singularities on X 0 (N).

32 Local P 2 : Fricke involution on modular forms Under this transformation, one has Maier (2006), [ASYZ] N A(τ) τa(τ) i N B(τ) τc(τ) i N C(τ) τb(τ) i N Ê(τ, τ) ( τ) 2 Ê(τ, τ) i (22) This involution turns out to be a "duality" for the topological string partition functions F g, as I shall explain below.

33 Local P 2 example: Picard-Fuchs and periods of the CY 3-fold The Picard-Fuchs operator of the corresponding CY 3-fold X : uv H(y i, z) = 0 is Lerche, Mayr, & Warner (1996), Chiang, Klemm, Yau & Zaslow (1999), Haghighat, Klemm & Rauch (2008) L CY = L elliptic θ (23) The periods are given by X 0 = 1, t, t c = κ 1 F t, where t log α + near the LCSL α = 0, t c is the vanishing period at the conifold point α = 1, and κ is the classical triple intersection number of the mirror Y = K P 2 of X. Here we have chosen the normalization of t c so that Then θt = ω 0, θt c = ω 1 (24) τ = ω 1 ω 0 = θt c θt = κ 1 F tt (25)

34 Local P 2 example: holomorphic limit at the LCSL Recall the large complex structure limit is given by α = 0 or equivalently [τ] = [i ] on X 0 (3). In the following we shall compute the special geometry quantities (connections, Yukawa coupling ) in the holomorphic limit at this particular point. It is in this limit that F g becomes F g A (X ) and is mirror to the generating function F g GW (Y ) of the GW invariants on Y. Using the modularity. we shall see that the full non-holomorphic partition function F g could be recovered by its holomorphic limit F g A Huang & Klemm (2006), Aganagic, Bouchard & Klemm (2006)

35 Local P 2 example: special geometry quantities in the holomorphic limit Take the coordinate x = ln α near the LCSL. In this coordinate the holomorphic limit of the connections are The Yukawa coupling is lim K x = 0, lim Γ x xx = x log t x C xxx = κ β where κ is the classical triple intersection number, it is 1 in the 3 local P 2 case. Using the boundary conditions for F 1 at LCSL and at conifold, the holomorphic limit of the genus one partition function is solved to be where a = 1 12, b = 1 24 F 1 A = 1 2 log θt + log βa α b+ 1 2 c2 (TY )J = 2 24.

36 Local P 2 example: special geometry quantities in terms of modular forms Recall α = C 3 A 3, β = B3 A 3, using the η expansions C = 3 η(3τ)3 η(τ), B = η(τ)3 η(3τ) and the denition of A by A = ω 0 = θt, we then get C ttt = 1 (X 0 ) C xxx( x 2 t )3 = κ B = 1 η(3τ) η(τ) 9 F 1 A = 1 12 log B 3 C 3 = 1 2 log η(τ)η(3τ), DF 1 A = 1 12 E here as before D = 1 2πi τ. In particular, F 1 = 1 2 log Imτ Im3τη(τ)η(τ)η(3τ)η(3τ), DF 1 = 1 12 Ê

37 Local P 2 example: generators in terms of modular forms The generators S x, S are chosen so that: lim S x = lim S = 0, lim K x = 0 while lim S xx is solved according to the integrated special geometry relation In this case, it simplies to Γ k ij = δ k i K j + δ k j K i C ijm S mk + s m ij That is, lim Γ x xx = lim 2K x C xxx lim S xx + s x xx θ log A = κ β lim S xx + s x xx

38 Local P 2 example: generators in terms of modular forms Recall that Dα = αβa 2 with D = 1 2πi τ, one has θ = α τ = 2πiα α α D = 1 βa D 2 Then the above equation becomes 1 βa D log A = βa 2 6 (E + C 3 B 3 ) = κ A β lim S xx + sxx x A natural choice for s x xx is sxx x = C 3 B 3 = C 3 B 3 6βA 3 6B 3 so that lim S xx = E E 6κA 2 = 1 2 A 2

39 Local P 2 example: differential ring of generators Recall the dierential ring of generators D i S jk = δ j i S k + δ k i S j C imn S mj S nk + h jk i, D i S j = 2δ j i S C imns m S nj + h jk i K k + h j i, D i S = 1 2 C imns m S n hmn i K m K n + h j i K j + h i, D i K j = K i K j C ijk S k + C ijk S kl K l + h ij, (26) where h jk i, h j i, h i, h ij are holomorphic functions. According to our choices, the only nontrivial equation in the above is DS xx = C xxx S xx S xx + h x xx

40 Local P 2 example: differential ring of generators Taking the holomorphic limit, one then gets x lim S xx + 2 lim Γ xx lim S xx = C xxx lim S xx S xx + h xx x It follows that h xx x = 1 12 A 3 B 3 Note that the quantities s = sxx x = C 3 B 3, h = h xx 6B 3 x = 1 are 12 B 3 modular and thus honest holomorphic objects in the non-holomorphic completion (due to the fact that the non-holomorphic completion is the same as the completion to almost-holomorphic modular objects.) That is, these holomorphic ambiguities are really holomorphic as they should be. A 3

41 Local P 2 example: polynomial part P g Polynomial recursion gives P 2 = 5 24 C 2 S ChS 3 8 CS 2 s S 2 C (27) Straightforward computations show that its holomorphic limit at LCSL is P 2 A = E(6A4 9A 2 E + 5E 2 ) 1728B 6 (28) This implies in return that the non-holomorphic quantity P 2 is P 2 = Ê(6A4 9A 2 Ê + 5Ê 2 ) 1728B 6 (29) That is, the non-holomorphic quantity P 2 could be obtained from its holomorphic limit PA 2, thanks to (quasi) modularity.

42 Local P 2 example: holomorphic ambiguity in terms of modular forms By induction, one can prove that P g and F g are non-holomorphic completions of quasi modular functions (that is, modular weights are 0) for g 2. That is, they are almost-holomorphic modular functions. The above result for P 2 suggests the following ansatz for the holomorphic ambiguity f 2 f 2 = c 1A 6 + c 2 A 3 B 3 + c 3 B B 6 (30) (In fact, this form could be obtained by considering the singularities of F g on the deformation space M.)

43 Local P 2 example: boundary condition at LCSL Boundary condition at the LCSL given by α = 0 or equivalently t = i is F g A = ( 1)g χ 2 B 2g B 2g 2 2g(2g 2)(2g 2)! + O(e2πit ), g 2 It is easy to apply this boundary condition since the holomorphic limits of the quantities A, B, C, Ê based at the LCSL are very easy to compute from their expressions in terms of hypergeometric functions. For example, A(α) = 2 F 1 ( 1 3, 2 3, 1; α) = 1+2α 9 +10α α α α B(α) = (1 α) F 1 ( 1 3, 2 3, 1; α) = 1 α 9 5α α α

44 Local P 2 example: gap condition at the conifold point The expression t c (β) is clear near the conifold point α = 1 or equivalently β = 0 since t c is chosen to be the vanishing period of the Picard-Fuchs equation of the CY 3fold at the this point. From the expansion of the period t c (β) = β + 11β β β β solved as the vanishing period of L CY = L elliptic θ, we can invert the series to get β = β(t c ): β(t c ) = t c 11t2 c t3 c t4 c t5 c

45 Local P 2 example: gap condition at the conifold point To apply the gap condition F g con(t c ) = c g 1 B 2g 2g(2g 2)(t c ) 2g 2 + O(t0 c ), g 2, one needs to evaluate the holomorphic limit F g con from F g = P g + f g. In the holomorphic limit based at the conifold, the holomorphic limit of f g is itself, while the holomorphic limit P g con of P g is dierent from (the analytic continuation of) P g A since a dierent base point is taken. Then one needs to compute the β or t c expansion of F g con.

46 Local P 2 example: series expansion in dual coordinates We have obtained the α, ᾱ expansions of F g (α, ᾱ), we could try to do analytic continuation to get (F g (α, ᾱ))(β, β). But it is extremally complicated to do this directly.

47 Local P 2 example: series expansion in dual coordinates Instead of doing analytic continuation to get (F g (α, ᾱ))(β, β), we make use of the Fricke involution as follows 1 A(α) = N i τ A(β) B(α) = (1 α) A(α) = β 3 N i τ A(β) = 1 N i τ C(β) C(α) = α A(α) = N i τ B(β) 1 Ê(α, ᾱ) = ( N i τ )2 Ê(β, β) here we treat A, B, C, E as functions A( ), B( ),. More precisely, A( ) = 2 F 1 ( 1, 2, 1; ), 3 3

48 Local P 2 example: Fricke involution At rst glance, it seems that we only used the denition of τ in terms of A(α), A(β) to get the series expansion in the dual β coordinate for all of the generators. However, what is really working is the Fricke involution: W N : τ 1 Nτ α β 2F 1 ( 1 3, 2 3, 1; α) 2F 1 ( 1 3, 2, 1; β) 3 with A(τ) A(W N τ) F g (τ, τ) F g WN (W N τ, W N τ) N i τ = A(W Nτ) A(τ) = 2 F 1 ( 1, 2, 1; β) 3 3 2F 1 ( 1, 2, 1; α) 3 3

49 Local P 2 example: Fricke involution Since F g has weight 0, the N i τ factors are cancelled out. From P 2 (α, ᾱ) = Ê(α, ᾱ)(6a(α)4 9A(α) 2 Ê(α, ᾱ) + 4Ê(α, ᾱ) 2 ) 1728B 6 (α) f 2 (α) = c 1A 6 (α) + c 2 A 3 (α)b 3 (α) + c 3 B 6 (α) 1728B 6 (α) we then get P 2 (α(β, β), ᾱ(β, β)) = ( Ê(β, β))(6a(β) 4 9A(β) 2 ( Ê(β, β)) + 4( Ê(β, β) 2 )) 1728C 6 (β) f 2 (α(β)) = c 1A 6 (β) + c 2 A 3 (β)c 3 (β) + c 3 C 6 (β) 1728C 6 (β)

50 Local P 2 example: solving the unknowns Therefore, we can easily get the β expansion of F 2 con(β) = lim(p 2 (α(β, β), ᾱ(β, β)) + f 2 (α(β)) From the expansion = ( E(β)(6A(β)4 9A(β) 2 ( E(β) + 4( E(β) 2 )) 1728C 6 (β) + c 1A 6 (β) + c 2 A 3 (β)c 3 (β) + c 3 C 6 (β) 1728C 6 (β) β(t c ) = t c 11t2 c t3 c t4 c t5 c we can express F 2 con(β) in terms of t c series. We can then make use of the gap conation to get linear equations satised by c 1, c 2, c 3.

51 Local P 2 example: predicting GW (GV) invariants It turns out that By using the mirror map c 1 = 8 5, c 2 = 2 5, c 3 = 8 3χ 10 (31) α = 27q 162q 2 243q q q 5 +, q = exp 2πit we then get the q = exp 2πit expansion of F 2 GW (Y, t) = E(6A4 9A 2 E + 5E 2 ) 1728B A A3 B χ 10 B B 6 This gives exactly the generating function of genus 2 GW (GV) invariants listed in Katz, Klemm, & Vafa (1999). For example, the rst few GV invariants n g=2 d, d = 1, 2, are 0, 0, 0, 102, 5430, , , , ,

52 Other examples: local dpn, n = 5, 6, 7, 8 One can easily work out the higher genus cases for local P 2 using the same approach. For the local del Pezzo geometries K dpn, n = 5, 6, 7, 8 with corresponding modular groups being Γ 0 (N) with N = 4, 3, 2, 1 respectively, the same procedure we outlined above constructs the ring of quasi modular forms from the periods, and solves F g. The F g s also predict the correct GV invariant. This approach making use of modularity works for noncompact CY 3-fold families whose mirror curves are of genus one, and whose base M could be identied with some modular curves.

53 Local CY examples: differential ring of special geometry generators vs differential ring of almost-holomorphic modular forms In these examples, rst we constructed the dierential ring of almost-holomorphic modular forms, then we expressed the dierential ring of generators S ij, S i, S, K i in terms of these almost-holomorphic modular forms. After that we did the polynomial recursion. We could have started from the dierential ring of generators S ij, S i, S, K i constructed from special geometry without knowing their relations to the generators A, B, C, E, provided that we know the correct notion of τ and their gradings as "modular weights" which tell how they transform.

54 Local CY examples: special geometry polynomial ring This ring of generators S ij, S i, S, K i constructed from special geometry (called the special geometry polynomial ring below) is as follows (for K P 2, κ = 1 3 ): lim S tt (= t E), θt(= A), Cttt = κ 2 τ (= κ 1 B 3 ) DS tt = S tt S tt 1 12κ (θt)4 Dθt = Cttt 1 θ 2 t = S tt θt + Cttt 1 sx xx DC 1 ttt = 3C 1 ttt S tt + C 1 ttt (θt) 2 ( x log C 1 xxx + 3s x xx) where sxx x = 1 α 6 β 1, 3 x log Cxxx 1 = α β. To make the ring closed, we add the holomorphic quantity x log C xxx = α β = C 3. Now its B 3 derivative lies in the ring of the above generators: D x log C xxx = ( x log C xxx )D log x log C xxx = ( x log C xxx )(θt) 2

55 Local CY examples: special geometry polynomial ring vs ring of quasi modular forms The holomorphic limit of the special geometry polynomial ring is essentially equivalent to the dierential ring of quasi modular forms [ASYZ] DA = 1 2r A(E + C r B r A r A 2 ) DB = 1 2r B(E A2 ) DC = 1 2r C(E + A2 ) (32) DE = 1 2r (E 2 A 4 ) where D = 1 2πi τ and r = 2, 3, 4, 6 for N = 4, 3, 2, 1 respectively.

56 Special geometry polynomial ring This seems to suggest that the special geometry polynomial ring constructed using connections, Yukawa couplings, etc. is a natural candidate of the ring of almost-holomorphic modular forms, even for the cases in which the arithmetic properties of the moduli space is unknown. This leads us to dene the following "special" special geometry polynomial ring on the moduli space (dimm=1)[asyz] K 0 = κ C 1 ttt (θt) 3, G 1 = θt, K 2 = κ C 1 ttt K t T 2 = S tt, T 4 = C 1 ttt S t, T 6 = C 2 ttt S 0, (33) with θ = z d dz, z is the algebraic coordinate. We furthermore need a generator C 0 = θ log z 3 C zzz for the coecients from their derivatives.

57 Special geometry polynomial ring This dierential ring, which has a nice grading called the weight, is given by τ K0 = 2K 0 K2 K 2 0 G 2 1 ( h z zzz + 3(s z zz + 1)), τ G1 = 2G 1 K2 κg 1 T2 + K 0G 3 1 (s z zz + 1), τ K2 = 3K 2 2 3κK 2 T2 κ 2 T 4 + K 2 0 G 4 1 k zz K 0 G 2 1 K 2 hz zzz, τ T2 = 2K 2 T2 κt κT κ K 2 0 G 4 1 h z zz, τ T4 = 4K 2T4 3κT 2 T4 + 2κT 6 K 0 G 2 1 T 4 hz zzz 1 κ K 2 0 G 4 1 T 2kzz (34) + 1 κ 2 K 3 0 G 6 1 h zz, τ T6 = 6K 2 T6 6κT 2 T6 + κ 2 T κ K 2 0 G 4 1 T 4 kzz + 1 κ 3 K 4 0 G 8 1 h z 2 K 0 G 2 1 T 6 hz zzz, where τ = κ 1 F tt and the sub-indices are the weights [ASYZ]. For each g 2, the normalized topological string partition function (X 0 ) 2 2g F g is a rational function of weight 0 in these generators.

58 Candidate of the ring of almost-holomorphic modular forms As we have discussed, for local P 2 and local del Pezzo cases, this ring, in the holomorphic limit, is essentially equivalent to the ring of quasi modular forms. For a general CY 3fold X (compact or noncompact), we could use this ring, as a guidance for the study of modular forms.

59 Candidate of the ring of almost-holomorphic modular forms For example, for the mirror quintic family, one gets τ C0 = C 0 (1 + C 0) K 0 G 2 1, τ K0 = 2K 0 K2 C 0K 2 0 G 2 1, τ G1 = 2G 1 K2 5G 1 T2 3 5 K0G 3 1, τ K2 = 3K K 2 T2 25T K 2 0 G 4 1 ( ) C0 2 K 0 G 1 K 2, τ T2 = 2K 2 T2 5T T (1 + C0) K 2 0 G 4 1, ( ) 9 τ T4 = 4K 2T4 15T 2 T4 + 10T C0 2 K 0 G 1 T K 2 0 G 4 1 T (1 + C0)K 3 0 G 6 1, τ T6 = 6K 2 T6 30T 2 T T K 2 0 G 4 1 T (1 + C0)K 4 0 G 8 1 ( ) C0 K 0 G 2 1 T 6. (35)

60 Candidate of the ring of almost-holomorphic modular forms Since we know how to take the holomorphic limit in this special geometry polynomial ring, we can get the candidate for the ring of quasi modular forms from it. But if the arithmetic properties of the moduli space is not clear, we cann't really say that the special polynomial ring is the ring of almost-holomorphic modular forms. Also if there is no suitable description of the modular group, it is not clear whether there is an analogue of Fricke involution as an duality of the topological string partition functions.

61 Conclusions In summary, we have - computed F g s in terms of modular forms for certain CY manifolds whose moduli spaces have nice arithmetic descriptions: M = H /Γ 0 (N) - found a duality acting on topological string partition functions for these examples: F g con = F g A W N - constructed the special geometry polynomial ring which has a nice grading. The normalized topological string partition functions are rational functions of degree zero in these generators. This ring is a natural candidate for the ring of almost-holomorphic modular forms.

62 Discussions and future directions There are some interesting questions - interpretation of τ coordinate in generality, e.g. for compact CY, multi-moduli cases? - enumerative meaning of the q τ = exp 2πiτ expansions? (example: quintic, where q t = exp(2πit) is used in GW generating functions) q t = q τ 575q 2 τ q 3 τ q 4 τ enumerative meaning of holomorphic limits of F g at some other points on the moduli space. - interpretation of gap condition in mathematics - Fricke involution on the level of moduli space of CYs - How exactly is the special geometry polynomial ring related to the ring of quasi modular forms for general CYs, e.g, mirror quintic?

63 Thanks to... - collaborators Murad Alim, Emanuel Scheidegger and Shing-Tung Yau - You for your attention