Emergent Spacetime from Modular Motives

Communications in Mathematical Physics (2011) DOI 10.1007/s00220-011-1179-z

Rolf Schimmrigk Emergent Spacetime from Modular Motives

Received: 28 April 2009 / Accepted: 4 October 2010 c Springer-Verlag 2011

Abstract The program of constructing spacetime geometry from string theoretic modular forms is extended to Calabi-Yau varieties of dimensions three and four, as well as higher rank motives. Modular forms on the worldsheet can be constructed from the geometry of spacetime by computing the L–functions associated to omega motives of Calabi-Yau varieties, generated by their holomorphic n−forms via Galois representations. The modular forms that emerge in this way are related to characters of the underlying rational conformal field theory. The converse problem of constructing space from string theory proceeds in the class of diagonal theories by de- termining the motives associated to modular forms in the category of pure motives with complex multiplication. The emerging picture suggests that the L–function can be viewed as defining a map between the geometric category of motives and the category of conformal field theories on the worldsheet.

Contents

1 Introduction ...... 2 2 Modularity ...... 6 3 Ω−Motives ...... 11 4 L–Functions for Calabi-Yau Threefolds and Fourfolds ...... 18 5 L–Functions Via Jacobi Sums ...... 20 6 Modularity Results for Rank two Motives in Dimension One and Two ...... 21 12 7 A Nonextremal K3 Surface X2 ⊂ P(2,3,3,4) ...... 23 6 8 Modular Motives of the Calabi-Yau Threefold X3 ⊂ P(1,1,1,1,2) ...... 26 12 9 The K3 Fibration Hypersurface X3 ⊂ P(2,2,2,3,3) ...... 28 6 5 10 A Modular Calabi-Yau Fourfold X4 ⊂ P ...... 31 11 Emergent Space from Characters and Modular Forms ...... 32 12 Further Considerations ...... 33 Indiana University South Bend, 1700 Mishawaka Ave., South Bend, IN 46634, USA. ne- [email protected] 2 R. Schimmrigk

1 Introduction

The present paper continues the program of applying methods from arithmetic geometry to the problem of understanding how spacetime emerges in string theory. The goal is to construct a direct relation between the physics on the worldsheet and the geometry of the extra dimensions. One way to formulate this question is by asking whether it is possible to explicitly determine the structure of the compact dimensions from the building blocks of the two-dimensional worldsheet theory. In this general, but vague, form the problem of constructing an emergent geometry in string theory could have been formulated more than thirty years ago. The reason that it was not can probably be traced to both the lack of a concrete framework, and the lack of useful tools. The framework of the heterotic string of the 1980s, in combination with the web of dualities between different string models discovered in the 1990s, motivates a more concrete version of this problem, which aims at the relation between Calabi-Yau varieties and worldsheet physics given by exactly solvable conformal field theories. Both Calabi-Yau varieties and conformal field theories define rich structures, raising a number of problems which have not been addressed in the past. The key ingredient of the program pursued here is the modular invariance of the theory. From a spacetime physics perspective it is initially somewhat surpris- ing that this feature of string theory should turn out to provide a useful tool for the understanding of its geometric consequences, because it is the modular invariance of the two-dimensional theory that would appear to be most difficult to explain from a geometric perspective. By now there exists a fair amount of evidence that shows that methods from arithmetic geometry provide promising tools for this problem, at least in lower dimensions. The purpose of the present paper is threefold. First, to describe the notion of the Ω−motive for arbitrary Calabi-Yau varieties (and more generally, Fano manifolds of special type), independent of any specific framework. Second, to extend the string modularity results obtained previously for the Ω-motive of diagonal Calabi-Yau manifolds in complex dimensions one and two to spaces of dimension three and four. Finally, to extend, in the process, previous results for rank two motives to higher rank motives. The basic problem of establishing modularity results for motivic L–functions in dimensions larger than one is made difficult by the fact that no generalization of the elliptic modularity theorem (1; 2; 3) is known, even conjecturally. This makes already the first step, the construction of modular forms from algebraic varieties, nontrivial. There exists, however, an extensive web of conjectures, the Langlands program, that suggests that in higher dimensions the Hasse-Weil L–functions of geometric structures have modular properties in a generalized sense. It is expected in particular that associated to each cohomology group is an automorphic representation, leading to an automorphic form. The class of automorphic L–functions contains a special type of object, the standard L–function (4), which generalizes the notion of a Hecke L–function. Modularity of Hecke L–functions is known by virtue of their analytic contin- uation and their functional equations (5; 6). Langlands’ vision thus extends results obtained by Artin and Hecke. While Artin considered representations ρ of the Galois group Gal(K/Q) of a number field K to define L–functions L(ρ,s), Hecke had previously introduced L–functions based on certain characters χ as- Emergent Spacetime from Modular Motives 3 sociated to number fields (called Großencharaktere¨ by Hecke, also called algebraic Hecke characters), whose structure was motivated by an attempt to establish modularity of the L–function. It turns out that these a priori different con- cepts lead to the same object in the sense that Artin’s L–functions and Hecke’s L–functions agree (7; 8). Langlands’ conjectures involve a generalization of the Artin-Hecke framework to GL(n). More precisely, the connection between geometry and arithmetic can be made because representations of the Galois group can be constructed by considering the `−adic cohomology as a representation space. This strategy has proven difficult to implement for higher n in general, and in particular in the context of obtaining a string theoretic interpretation of geometric modular forms beyond the case of elliptic curves and rigid Calabi-Yau varieties. For varieties of higher dimensions the results obtained so far indicate that it is more important to identify irreducible pieces of low rank in the cohomology groups, and to consider the L–functions associated to these subspaces. The dif- ficulty here is that at present there exists no general framework that provides guidance for the necessary decomposition of the full cohomology groups. Nev- ertheless, the Langlands program suggests that modularity, and more generally automorphy, are phenomena that transcend the framework of elliptic curves, and one can ask the question whether the methods described in (9; 10; 11) to establish modularity relations between elliptic curves and conformal field theories can be generalized to higher dimensional varieties. Results in this direction have been obtained for extremal K3 surfaces of Brieskorn-Pham type in ref. (12). In the present paper the string modularity results obtained previously for diagonal Calabi-Yau manifolds in complex dimensions one and two are extended to all higher dimensions that are of physical relevance. The idea is to consider particular subgroups of the intermediate cohomology group of a variety, defined by the representation of a Galois group associated to the manifold. For spaces of Calabi-Yau and special Fano type there exists at least one nontrivial orbit, defined by the holomorphic n−form in the case of Calabi-Yau spaces, and the corresponding cohomology group in the case of special Fano manifolds. This orbit will be called the omega motive MΩ (X), and its L–function will be noted by L(MΩ (X),s) = LΩ (X,s). The precise definition of these objects and their concrete implementation will be given in Sect. 2. The strategy developed here is completely general and can be applied to any Calabi-Yau variety, as well as Fano varieties of special type. In later sections this framework will be applied to Calabi-Yau varieties of Brieskorn-Pham type, the class of varieties for which Gepner (13) originally discovered a relation between the spectra of a certain type of conformal field theory and the cohomology of the manifolds. For such models the Ω−motive, as well as the other submotives, can be determined explicitly. A simplifying characteristic of extremal K3 surfaces of Brieskorn-Pham type analyzed in (12) is that their Ω−motive is of rank two. This is not the case in general, hence the problem arises of extending the string theoretic analysis from extremal K3 surfaces to K3 motives of higher rank. An example that leads to a motive of rank four is defined by the surface

n o 12 6 4 4 3 X2 = (z0 : z1 : z2 : z3) ∈ P(2,3,3,4) z0 + z1 + z2 + z3 = 0 . (1) 4 R. Schimmrigk

12 12 It is shown in Sect. 7 that the L–function LΩ (X2 ,s) of the Ω−motive of X2 is 4 6 obtained from those of the diagonal elliptic curves E ⊂ P(1,1,2) and E ⊂ P(1,2,3). It was shown in (11) that the L–functions of E4 and E6 have a string theoretic structure, i.e. the modular forms fE4 ∈ S2(Γ0(64)) and fE6 ∈ S2(Γ0(144)) are determined by the Hecke indeﬁnite theta series that appear in the conformal ﬁeld theory on the string worldsheet. The concrete relations are summarized in Theorem 6 12 in Sect. 6. This shows that LΩ (X2 ,s) is string theoretic in the same sense, i.e. its L–function is determined by modular forms that arise in the underlying string model on the worldsheet. More precisely, the following result will be shown.

12 Theorem 1 The L–series LΩ (X2 ,s) of the rank four Ω−motive of the K3 surface 12 X2 is given by the Rankin-Selberg product of the L–functions of the modular 4 6 forms fE4 ∈ S2(Γ0(64)) and fE6 ∈ S2(Γ0(144)) of the elliptic curves E and E ,

12 LΩ (X2 ,s) = L( fE4 ⊗ fE6 ,s). (2) The precise meaning of the tensor product of modular forms will become clear below. Previous work on the construction of space via modular forms arising in the conformal ﬁeld theory on the worldsheet was restricted to elliptic curves and K3 surfaces. To lift this restriction, consider ﬁrst the Calabi-Yau threefolds

6 6 6 6 6 3 X3 = {z0 + z1 + z2 + z3 + z4 = 0} ⊂ P(1,1,1,1,2), (3) 12 6 6 6 4 4 X3 = {z0 + z1 + z2 + z3 + z4 = 0} ⊂ P(2,2,2,3,3).

6 It is shown in Sect. 8 that the rank two Ω−motive of X3 leads to a string theoretic modular form of weight four, and that the remaining part of the interme- 12 diate cohomology leads to modular forms of weight two. The Ω−motive of X3 12 is of rank four and it is shown in Sect. 9 that its L–function LΩ (X3 ,s) is de- 4 termined by the weight two modular form fE4 of the elliptic curve E and the weight three cusp form f 6A ∈ S3(Γ1(27)) of the Ω−motive of the diagonal K3 X2 6A surface X2 ⊂ P(1,1,1,3), whose string theoretic interpretation was established in (12). More precisely, the following results will be shown.

6 Theorem 2 1) The inverse Mellin transform of the Ω−motivic LΩ (X3 ,s) of the 6 threefold X3 is a cusp form fΩ (q) ∈ S4(Γ0(108)). The L–function of the inter- 3 6 mediate cohomology group H (X3 ) decomposes into modular factors as

3 6 L(H (X3 ),s) = L( fΩ ,s) · ∏L( fi ⊗ χi,s), (4) i

where fi ∈ S2(Γ0(Ni)) with Ni = 27,144,432, and the χi are twist characters. 12 2) The Ω−motivic L–series of X3 is given by the Rankin-Selberg product of the 4 modular forms associated to E ⊂ P(1,1,2) and the extremal weighted Fermat 6A surface X2 ⊂ P(1,1,1,3),

12 LΩ (X3 ,s) = L( fE4 ⊗ f 6A ,s). (5) X2 Emergent Spacetime from Modular Motives 5

3 12 The L–series of the remaining part of the intermediate cohomology H (X3 ) decomposes into a product of factors that include L–series of modular forms of weight two and levels N = 27,64,144,432, possibly including a twist.

The results concerning the rank four motives in Theorems 1 and 2 can be viewed from a somewhat different perspective. A result shown by Ramakrishnan in ref. (14) implies that the Rankin-Selberg L–function associated to any pair of elliptic modular cusp forms comes from an automorphic form. Hence the fact that 12 12 the L–functions of the Ω−motives of both X2 and X3 are given in terms of the Rankin-Selberg convolution shows that these motives are automorphic in the sense that there exists an automorphic representation whose L–function agrees with that of the motive. In F-theory Calabi-Yau fourfolds are of interest, and as a ﬁnal example in this paper the modularity of the Ω−motive of the degree six fourfold ( ) 5 6 5 6 X4 = (z0 : ··· : z5) ∈ P ∑ zi = 0 (6) i=0 is shown in Sect. 10.

Theorem 3 The inverse Mellin transform of the L–function of the Ω−motive of 6 X4 is of the form

6 fΩ X4 ,q = f27(q) ⊗ χ3, (7) where f27(q) is a cusp Hecke eigenform of weight w = 5 and level N = 27, and χ3 is the Legendre character. There exists an Großencharacter¨ ψ27 with congruence 6 4 ideal m = (3) such that the motivic L–series is given by LΩ (X4 ,s) = L(ψ27,s) ⊗ χ3.

The basic question raised by these results is whether the Ω−motive is string automorphic in general. More generally one may view the L–function as a link between the geometry of spacetime and the physics of the worldsheet. One way to make this idea more explicit is by viewing L as a functor from the category of Fano varieties of special type (or rather their motives) to the category of superconformal field theories. The evidence obtained so far supports this perspective for a physical interpretation of L–functions. The notion e.g. of composing motives then translates into a corresponding composition of conformal field theories. A concrete example is the motivic tensor structure which maps into a tensor structure for conformal field theories. The basic tensor structure of motives is described in L–function terms by the Rankin-Selberg convolution L( f1 ⊗ f2,s) of the modular forms fi associated to the modular motives Mi, and also leads to the symmetric square of modular forms. Denote by Θχ = ∏i Θi ⊗ χ twisted products constructed from modular forms Θi on the string worldsheet. The motivic L–functions L(M(X),s) that emerge from Calabi-Yau varieties and special Fano varieties can be expressed in terms of string modular L–functions of the type L ,s, LSymr ,s, L 1 ⊗ 2 ,s , (8) Θχ Θχ Θχ1 Θχ2 6 R. Schimmrigk where L(Symr f ,s) describes the L–function associated to a symmetric tensor product of a modular form f . The emerging picture thus indicates that L–functions of automorphic forms facilitate a structural map between the space of Calabi-Yau varieties (and their generalizations) and the space of certain conformal field theories. The paper is organized as follows. Section 2 briefly introduces the necessary modular theoretic background. Section 3 describes the notion of a Grothendieck motive and introduces the general concept of Ω−motives for arbitrary Calabi-Yau manifolds, as well as for the class of Fano varieties of special type. This provides the framework for the relation between the geometry of spacetime and physics on the string worldsheet. Sections 4 and 5 describe the basic structure of the L– functions for Calabi-Yau surfaces and threefolds, derived from Artin’s zeta function. To make this paper more self-contained, Sect. 6 briefly reviews the results for modular motives of rank two that appear as building blocks for the examples of higher dimension. More details can be found in (11; 12). Sections 7 through 10 contain the string theoretic modularity analysis of the higher dimensional varieties and higher rank motives, leading to the results of Theorem 1, 2, 3. Section 11 shows how the converse problem can be approached in the context of modular motives, and Sect. 12 ends the paper with some final remarks.

2 Modularity

In order to make the paper more self-contained the following paragraphs briefly summarize the types of modular forms that eventually are reflected in the geometry of weighted hypersurfaces of Calabi-Yau and special Fano type. The affine Lie algebraic forms introduced by Kac and Peterson provide the structures on the worldsheet, while certain types of modular Hecke L–series will arise from the arithmetic of the geometry.

2.1 Modular forms from afﬁne Lie algebras

The simplest class of N = 2 supersymmetric exactly solvable theories is built in terms of the afﬁne SU(2) theory as a coset model

SU(2) ⊗ U(1) k 2 . (9) U(1)k+2,diag

Coset theories G/H lead to central charges of the form cG − cH , hence the supersymmetric afﬁne theory at level k still has central charge ck = 3k/(k + 2). The k k spectrum of anomalous dimensions ∆`,q,s and U(1)−charges Q`,q,s of the primary k ﬁelds Φ`,q,s at level k is given by

2 2 k `(` + 2) − q s ∆`,q,s = + , 4(k + 2) 8 (10) q s Qk = − + , `,q,s k + 2 2 Emergent Spacetime from Modular Motives 7 where ` ∈ {0,1,...,k}, ` + q + s ∈ 2Z, and |q − s| ≤ `. Associated to the primary ﬁelds are characters deﬁned as

c k −2πiu 2πiτ(L0− ) 2πiJ0 χ (τ,z,u) = e tr ` e 24 e `,q,s Hq,s k = ∑c`,q+4 j−s(τ)θ2q+(4 j−s)(k+2),2k(k+2)(τ,z,u), (11)

` where the trace is to be taken over a projection Hq,s to a deﬁnite fermion number (mod 2) of a highest weight representation of the (right-moving) N = 2 algebra with highest weight vector determined by the primary ﬁeld. The expression of the rhs in terms of the string functions (15), 8 R. Schimmrigk

Θ k (τ) ck (τ) = `,m , (12) `,m η3(τ)

k where η(τ) is the Dedekind eta function, and Θ`,m(τ) are the Hecke indeﬁnite modular forms

k 2πiτ((k+2)x2−ky2) Θ`,m(τ) = ∑ sign(x)e (13) −|x|

−2πimu 2πim`2τ+2πi`z θn,m(τ,z,u) = e ∑ e (14) n `∈Z+ 2m is useful because it follows from this representation that the modular behavior of the N = 2 characters decomposes into a product of the affine SU(2) structure in the ` index and into Θ-function behavior in the charge and sector index. It follows from the coset construction that the essential ingredient in the conformal field theory is the SU(2) affine theory. The issue of understanding the emergence of spacetime in string theory can now be reformulated in a more concrete way as the problem of relating string theoretic modular forms to geometric ones. It is apparent from the results of (9; 10; 11; 12) that more important than the string functions are the associated SU(2) k theta functions Θ`,m(τ). These indefinite Hecke forms are associated to quadratic number fields determined by the level of the affine theory. They are modular forms of weight 1 and cannot, therefore, be identified with geometric modular forms. However, products of these forms can lead to interesting motives, and it was shown in the above references that appropriate products lead to modular forms associated to Calabi-Yau varieties.

2.2 Modular forms from Großencharacters¨

The modularity of the L–series determined in this paper follows from the fact that they can be interpreted in terms of Hecke L–series associated to Großencharacters,¨ defined by Jacobi sums. A Großencharacter¨ can be associated to any number field. While some of the Jacobi sums encountered in this paper take values in higher degree cyclotomic fields, it will be sufficient for the modularity proofs that follow to focus on the special case of imaginary quadratic fields. Hecke’s modularity results (see e.g. (16; 17)) have been extended in work by Shimura (18) and Ribet (19), and introductions can be found e.g. in Miyake (20) and Iwaniec (21). In the brief summary of the necessary background that follows the notation of Ribet is adopted for the most√ part. Let K = Q( −D) be an imaginary quadratic field, σ : K −→ C an embedding, m ⊂ OK an integral ideal in the ring of algebraic integers, and w > 1 a positive integer. Denote further by mod×m the multiplicative congruence modulo m and by (z) the principal ideal of z ∈ K. Emergent Spacetime from Modular Motives 9

× Deﬁnition A Großencharacter¨ is a homomorphism ψ : Im(K) → C from the fractional ideals of K prime to the integral ideal m such that ψ((z)) = σ(z)w−1, for all z ∈ K× such that z ≡ 1(mod×m). The type of behavior of ψ on the principal ideals is called the inﬁnity type.

The Hecke L–series of the character ψ is defined by 1 L(ψ,s) = . (15) ∏ ψ(p) p∈SpecOK 1 − Nps n The modularity of the corresponding q−series f (ψ,q) = ∑n anq associated to the L–series via the inverse Mellin transform is characterized by a Nebentypus character ε defined in terms of the Dirichlet character ϕD associated to K and a second character λ defined mod Nm by ψ((a)) λ(a) = , a ∈ Z. (16) σ(a)w−1

The Nebentypus character ε is given by the product ε = λϕD of these two characters. Modularity of the L–series follows from the results of Hecke and Shimura, adapted here following the formulation of Ribet (19). Denote by Nm the norm of the ideal m, and by DK the discriminant of K. Theorem 4 Let ψ be a Großencharacter¨ of the imaginary quadratic field K with w−1 infinity type σ . Define the coefficients cn as ∞ Na n ∑ ψ(a)q =: ∑ cnq . (17) (a,m)=1 n=1 aintegral

∞ n Then there exists a unique newform f (q)= ∑n=1 anq of weight w, level N = |DK|Nm, and character ε = λϕD such that

ap = cp ∀ p /| DNm. (18)

Of particular√ importance in this paper are Großencharacters¨ √ associated√ to the Gauss field Q( −1) and the Eisenstein field Q( −3). For Q( −1) consider prime ideals p = (zp) and define the character ψ32 by setting

ψ32(p) = zp, (19) where the generator zp is determined by the congruence relation

zp ≡ 1(mod (2 + 2i)) (20) for the congruence ideal√ m = (2 + 2i). For the ﬁeld Q( −3) two characters ψN, N =27,36 and their twists will appear. The congruence ideals here are given by

m27 = (3), (21) m36 = 2 + 4ξ3, 10 R. Schimmrigk

2πi/3 where ξ3 = e , leading to the characters

ψN(p) = zp, (22) where the generator is determined uniquely by the congruence relations

zp ≡ 1(mod mN). (23)

2.3 Rankin-Selberg products of modular forms

An important ingredient in the analysis of higher dimensional varieties is the fact that the L–series of motives carrying higher dimensional representations of the Galois group can sometimes be expressed in terms of L–functions of lower rank motives. This leads to Rankin-Selberg products of L–functions. This construction is quite general, but will be applied here only to products of L–series that are associated to Hecke eigenforms. If the L–series of two modular forms f ,g of arbitrary weight and arbitrary levels are given by

−s L( f ,s) = ∑ann , n −s (24) L(g,s) = ∑bnn , n it is natural to consider the naive Rankin-Selberg L–series associated to f ,g as

−s L( f × g,s) = ∑anbnn . n It turns out that a slight modification of this product has better properties, and is more appropriate for geometric constructions. If f ∈ Sw(Γ0(N)) and g ∈ Sv(Γ0(N)) are cusp forms with characters ε and λ, respectively, the modified Rankin-Selberg product is defined as

L( f ⊗ g,s) = LN(ελ,2s + 2 − (w + v))L( f × g,s), (25) where LN(χ,s) is the truncated Dirichlet L–series deﬁned by the condition that χ(n) = 0 if (n,N) > 1 (23). Hecke showed that such forms f ,g have Euler products given by

−s −s−1 L( f ,s) = ∏ 1 − αp p 1 − βp p , p (26) −s −s−1 L(g,s) = ∏ 1 − γp p 1 − δp p , p

w−1 v−1 where αp +βp = ap, γp +δp = bp and αpβp = p , γpδp = p . It can be shown that the modiﬁed Rankin-Selberg product has the Euler product L( f ⊗ g,s) −s −s −s −s−1 = ∏ 1 − αpγp p 1 − αpδp p 1 − βpγp p 1 − βpδp p . (27) p Emergent Spacetime from Modular Motives 11

The tensor notation is at this point formal, but it will become clear that this product is indicative of a representation theoretic tensor product. Furthermore, it also describes the L–series of the tensor product Mf ⊗Mg of motives Mf ,Mg associated to the modular forms f ,g via the constructions of Deligne (24), Jannsen (25) and Scholl (26),

L( f ⊗ g,s) = L(Mf ⊗ Mg,s). The motivic tensor product will be described below. The Rankin-Selberg products which will appear below involve modular forms of weight two and three, leading to rank four motives on Calabi-Yau varieties of dimension two and three.

2.4 Complex multiplication modular forms

A special class of modular forms that is relevant in this paper are forms which are sparse in the sense that a particular subset of the coefficients ap of their Fourier n expansion f (q) = ∑n anq vanish. A conceptual way to formulate this idea was introduced by Ribet (19). A modular form f (q) is called to be of complex multipli-√ cation (CM) type if there exists an imaginary quadratic number field K = Q( −d) such that the coefficients ap vanish for those rational primes p which are inert in K. It follows from this definition that any such form can be described by the inverse Mellin transform of an L–series associated to a Hecke Großencharacter,¨ which is the view originally adopted by Hecke, and also Shimura. Modular forms with complex multiplication are more transparent than general forms, in particular in the context of their associated geometry. This will become important further below in the construction of the Calabi-Yau motives from the conformal field theory on the worldsheet.

3 Ω−Motives

The results in the present and previous papers show that it is useful for modularity to consider the Galois orbit in the cohomology defined by the holomorphic n−form of a Calabi-Yau variety. This Galois orbit defines a geometric substruc- ture of the manifold, called here the Ω−motive, which provides an example of a Grothendieck motives in this manifolds. Similar orbits can be considered in the context of so-called special Fano varieties considered in (27; 28; 29), whose modularity properties were analyzed in the context of mirror pairs of rigid Calabi-Yau varieties in (30). The aim of the present section is to provide the geometric framework for the construction of the motivic structure from the conformal field theory. The circle of ideas that is concerned with the relations between characters, modular forms, and motives extends beyond the class of weighted Fermat varieties, and it is useful to formulate the constructions in its most general context. The outline of this section is to first describe the concept of Grothendieck motives, also called pure motives, then to define the notion of Ω−motives in complete generality, and finally to consider the Ω−motive in detail for weighted Fermat hypersurfaces, i.e. of Brieskorn-Pham type. 12 R. Schimmrigk

3.1 Grothendieck motives

The idea to construct varieties directly from the conformal field theory on the worldsheet without the intermediary of Landau-Ginzburg theories or sigma models becomes more complex as the number of dimensions increases. Mirror symmetry and other dualities show that it should not be expected that any particular model on the worldsheet should lead to a unique variety. Rather, one should view manifolds as objects which can be built from irreducible geometric structures. This physical expectation is compatible with Grothendieck’s notion of motives. The original idea for the existence of motives arose from a plethora of cohomology theories Grothendieck was led to during his pursuit of the Weil conjectures (31; 32; 33). There are several ways to think of motives as structures that support these various cohomology theories, such as Betti, de Rham, etale,´ crystalline cohomology groups, etc., and to view these cohomology groups as realization of motives. Grothendieck’s vision of motives as basic building blocks that support univer- sal structures is based on the notion of correspondences. This is an old concept that goes back to Klein and Hurwitz in the late 19th century. The idea is to define a relation between two varieties by considering an algebraic cycle class on their product. In order to do so an algebraic cycle is defined as a finite linear combination of irreducible subvarieties Vα ⊂ X of codimension r of a variety X. The set of all these algebraic cycles defines a group r Z = ∑nαVα Vα ⊂ X . (28) α This group is too large to be useful, hence one considers equivalence relations between its elements. There are a variety of such equivalence relations, resulting in quite different structures. The most common of these are rational, homological, and numerical equivalence. A description of these can be found in (34), but for the following it will not be important which of these is chosen. Given any of these equivalences one considers the group of equivalence classes Ar(X) = Zr(X)/ ∼ (29) of algebraic cycles to define the group of algebraic correspondences of degree r between manifolds X,Y of equal dimension d as Corrr(X,Y) = Ad+r(X ×Y). (30) Correspondences can be composed f · g, leading to the notion of a projector p such that p · p = p. The first step in the construction of Grothendieck motives is the definition of an effective motive, obtained by considering a pair defined by a variety and a projector M = (X, p), where p is a projector in the ring of algebraic correspondences of degree zero, p ∈ Corr0(X,X). Maps between such objects are of the form (35) 0 Hom((X, p),(Y,q)) = q ◦ Corr (X,Y)Q ◦ p. (31) This formulation of morphisms between effective motives is equivalent to the original view of Grothendieck described in (33). Emergent Spacetime from Modular Motives 13

It is important for physical applications of motives to enlarge the class of effective motives by introducing twists of effective motives by powers of the inverse Lefschetz motive L. This is an effective motive defined as L = (P1,1 − Z), where Z is the cycle class Z ∈ A1(P1 × P1) defined by the cycle P1 × pt. It is possible to tensor effective motives M by L and its inverse. Combining the notions of effective motives and the Lefschetz motive leads to the concept of a Grothendieck motive. Definition A Grothendieck motive is a triple M(m) = (X, p,m), where M = (X, p) is an effective motive, and m ∈ Z.M(m) = (M,m) is the m−fold Tate twist of M. If N(n) = (Y,q,n) is another motive morphisms are defined as Hom(M(m),N(n)) := p ◦ Corrn−m(X,Y) ◦ q. (32)

The tensor product of two Grothendieck motives Mi = (Xi, pi,mi),i = 1,2 is deﬁned as

M1 ⊗ M2 = (X1 × X2, p1 ⊕ p2,m1 + m2). (33) A discussion of the virtues and disadvantages of the various realizations in terms of speciﬁc equivalence relations can be found in (35; 36), building on earlier references, such as (31; 32; 33). A more detailed discussion of motives can be found in (37).

3.2 Ω−motives for manifolds of Calabi-Yau and special Fano type

In the context of the emergent geometry problem via string theoretic modular forms it is necessary to consider L–functions associated to motives of low rank, not of the full cohomology groups of a variety. For higher genus curves and higher dimensional varieties the experimental evidence (9; 10; 11; 12) suggests that the relevant physical information is encoded in subspaces of the cohomology. A possible strategy therefore is to consider the factorization of L–functions and to ask whether modular forms arise from the emerging pieces, and if so, whether these modular forms admit a string theoretic Kac-Moody interpretation. This section formulates a general strategy, valid for any Calabi-Yau variety X of dimension dimCX = n, for decomposing the L–function of its intermediate cohomology Hn(X), into pieces which lead to L–functions with integral coefficients. These L–functions then have the potential, when modular, to admit a factorization into Kac-Moody theoretic modular forms along the lines discussed in (9; 10; 11) for elliptic curves and higher genus curves. The basic strategy outlined below is a generalization of the method described in (12), which was based on Jacobi sums associated to hypersurfaces embedded in weighted projective spaces. The idea is to consider an orbit in the cohomology which is generated by the n,0 holomorphic n−form Ω ∈ H (X) via the action of the Galois group Gal(KX /Q) of a number field KX that is determined by the arithmetic properties of the variety as dictated by the Weil conjectures (38), proven by Grothendieck (39) and Deligne (40). The resulting orbit of this action turns out to define a motive in the sense of Grothendieck, as will be shown further below. To see how the group structure appears in full generality it is necessary to briefly describe the arithmetic structure of arbitrary Calabi-Yau varieties. 14 R. Schimmrigk

For a general smooth algebraic variety X deﬁned over the rational integers the reduction mod q leads to the congruence zeta function of X/Fq, ! tr Z(X/ ,t) ≡ exp #(X/ r ) . (34) Fq ∑ Fq r r∈N

Here the sum is over all ﬁnite extensions Fqr of Fq of degree r = [Fqr : Fq]. By deﬁnition Z(X/Fq,t) ∈ 1 + Q[[t]], but the expansion can be shown to be integer valued by writing it as an Euler product. The main virtue of Z(X/Fq,t) is that the numbers Npr = #(X/Fpr ) show a simple behavior, as a result of which the zeta function can be shown to be a rational function.

1) The ﬁrst step is to consider the rational form of Artin’s congruence zeta function. This leads to a link between the purely arithmetic geometric objects Nr,p = #(X/Fpr ) for all primes p and the cohomology of the variety. This was ﬁrst shown for curves by F.K. Schmidt in the thirties in letters to Hasse (41; 42; 43). Further experience by Hasse, Weil, and others led to the conjecture that this phenomenon is more general, culminating in the cohomological part of the Weil conjecture. According to Weil (38) and Grothendieck (39) Z(X/Fp,t) is a rational function which can be written as

n 2 j−1 ∏ Pp (t) Z(X/ ,t) = j=1 , (35) Fp n 2 j ∏ j=0 Pp (t)

where

0 2n n Pp(t) = 1 −t, Pp (t) = 1 − p t, (36) and for 1 ≤ j ≤ 2n − 1,

j j deg Pp(t) = b (X), (37)

j th j j where b (X) denotes the j Betti number of the variety, b (X) = dim HdR(X). The rationality of the zeta function was first shown by Dwork (44) by adelic methods. j The resulting building blocks given by the polynomials Pp(t) associated to the full cohomology group H j(X) are not useful in the present context, leading to L–series whose Mellin transforms in general cannot directly be identified with string theoretic modular forms of the type considered in (9; 10; 11; 12; 30) and in the present paper. The idea instead is to decompose these objects further, which leads to the factorization of the polynomials i Pp(t). 2) The most difficult part of the Weil conjectures is concerned with the nature of the factorization of the polynomials

b j j j Pp(t) = ∏ 1 − γi (p)t . (38) i=1 Emergent Spacetime from Modular Motives 15

Experience with Jacobi and Gauss sums in the context of diagonal weighted j projective varieties indicates that the inverse eigenvalues γi (p) are algebraic integers that satisfy the Riemann hypothesis

j j/2 γi (p) = p , ∀i. (39)

This part of the Weil conjectures was finally proven by Deligne (40). j 3) The algebraic nature of the inverse roots of the polynomials Pp(t) can be used to define a number field KX associated to the intermediate cohomology of the variety of complex dimension n as

n n KX = Q({γi |i = 1,...,b }).

The field KX is separable and therefore one can consider orbits within the cohomology with respect to the embedding monomorphisms. For n−dimensional Calabi-Yau varieties one can, in particular, consider the orbits OΩ associated to the holomorphic n−forms Ω ∈ Hn,0(X), while for special Fano manifolds of charge Q one can consider Ω ∈ Hn−(Q−1),(Q−1)(X). The orbits of these forms generated by the embeddings of the field KX define a projection pΩ on the intermediate cohomology which can be used to define a motive.

Deﬁnition The Ω−motive of varieties of Calabi-Yau or special Fano type is de- ﬁned as the motive MΩ = (X, pΩ ,Q) which cohomologically is represented as the orbit of the class Ω ∈ Hn−(Q−1),(Q−1)(X) with respect to the Galois group Gal(KX /Q).

3.3 L–function of Ω−motives

Given the Ω−motive MΩ one can combine the local factors of the zeta functions, leading to the motivic L–function of the variety

. 1 L (X,s) = L(M ,s) = , Ω : Ω ∏ Ω −s (40) p Pp (p ) where PΩ (t) are the polynomials described by the orbit of Ω ∈ Hn−(Q−1),(Q−1)(X). p . The symbol = is the common notation to indicate that the bad primes are neglected in the discussion. Denoting the Weil number corresponding to Ω in (38) by γΩ , the polynomial can be expressed for the good primes p as

Ω Pp (t) = ∏ (1 − σ(γΩ )t). (41) σ∈Hom(K,C)

The σ−orbits defined via the embedding monomorphisms define traces of the number field KX , which implies that the corresponding Ω−motives have L– functions LΩ (X,s) with integral coefficients. 16 R. Schimmrigk

3.4 String theoretic modularity and automorphy

With the above structures in place we can ask in full generality for any Calabi-Yau variety the following

Questions. When is the L–function LΩ (X,s) of the Ω−motive MΩ of a Calabi- Yau variety modular? Further, if it is modular, can LΩ (X,s) be expressed in terms of string theoretic forms associated to the Kac-Moody algebras that arise in the conformal field theories on the worldsheet? Modularity of LΩ (X,s) here is understood to include the usual linear opera- tions on modular forms. More generally, this question can be raised for the class of motives associated to Fano varieties of special type. It has been shown in refs. (9; 10; 11; 12) that the answer to this question is affirmative at least sometimes in lower dimensions, and generalizations to higher dimensions will be established below. In such cases the L–function can be viewed as a map that takes motives and turns them into conformal field theoretic objects. This framework therefore leads to the following picture. Conjecture. The L–function defines a map from the category of Calabi-Yau motives (more general special Fano type motives) to the category of N = 2 supersymmetric conformal field theories. These ideas can be generalized to the context of automorphic motives via automorphic representations. The Langlands program leads to the expectation that every pure motive is automorphic, and at this level of speculation the question becomes whether the L–functions of the irreducible automorphic motives admit a string theoretic interpretation.

3.5 Grothendieck motives for Brieskorn-Pham varieties

For weighted projective hypersurfaces of Brieskorn-Pham type the motives that emerge can be described concretely in terms of the Jacobi sums that reﬂect the cohomology of these varieties. d Consider a diagonal hypersurface Xn of degree d and dimension n in a n+2 weighted projective space with weights (k0,...,kn+1) ∈ N . With di = d/ki one d can consider the group Gn = ∏i(µdi ), where µdi is the cyclic group with generator 2πi/di ξdi = e . This group acts on the projective space as

a0 an+1 gz = (ξ z ,...,ξ zn+ ) (42) d0 0 dn+1 1

n+2 for a vector a = (a0,...,an+1) ∈ Z . Motives of Fermat type can be deﬁned via projectors that are associated to the characters of the symmetry group. Denote the d d d dual group of Gn by Gbn and associate to a ∈ Gbn a projector pa as

1 −1 pa = d ∑ a(g) g, (43) |Gn| d g∈Gn Emergent Spacetime from Modular Motives 17 where the character deﬁned by a is given by

n+1 a(g) = ξ ai . (44) ∏ di i=0

Combining the projectors pa within a Gal(Q(µd)/Q)−orbit then leads to

pO := ∑ pa. (45) a∈O

The projectors pO can now be regarded as algebraic cycles on X × X with rational coefﬁcients by considering their associated graphs. By abuse of notation, effective motives of Fermat type can therefore be deﬁned as objects that are de- d termined by the Galois orbits O as (Xn , pO ). Including Tate twists then leads to Grothendieck motives,

d MO := (Xn , pO ,m). (46)

3.6 Ω−motives for CY and SF BP hypersurfaces

For varieties of Calabi-Yau and special Fano type there exists a particularly important cohomology group which can be parametrized explicitly. For Calabi-Yau manifolds of complex dimension n these forms take values in Hn,0(X), while for special Fano varieties of complex dimension n and charge Q one has more gen- n−(Q− ),(Q− ) erally Ω ∈ H 1 1 (X), where Q ∈ N. The Ω−motive associated to the Galois orbit of Ω of a variety of charge Q will be denoted by

Q d MΩ = (Xn , pΩ ,Q − 1). (47) 1 For Q = 1 one recovers the Calabi-Yau case, which will be denoted by MΩ = MΩ .

3.7 Lower weight motives from higher dimensional varieties

The Ω−motive is not the only motive of Calabi-Yau and special Fano varieties that can lead to interesting modular forms. Instead, one can construct Grothendieck motives MO (X) associated to Galois orbits O that do not arise from the Ω−form, but come from motives represented by subgroups of the remainder of the intermediate cohomology group. The associated L–series L(MO (X),s) may, or may not lead to interesting modular forms. In the case they do in the examples below, the resulting modular forms can be described as determined by Tate twists of forms

L(MO (X),s) = L( fO ,s − 1), where fO (q) is a modular associated to the motive MO (X). In general one would expect that these lower weight motives lead to Tate twists of automorphic forms of lower dimensional varieties that are embedded in the higher-dimensional manifolds. 18 R. Schimmrigk

4 L–Functions for Calabi-Yau Threefolds and Fourfolds

In this paper the focus is on Calabi-Yau manifolds in complex dimensions two, three and four. String theoretic examples of rank two and dimension one and two are considered in (9; 10; 11; 12). Combining these examples with the varieties discussed in the present paper covers all physically interesting dimensions in string Emergent Spacetime from Modular Motives 19 theory, M-theory, and F-theory. Calabi-Yau threefolds with ﬁnite fundamental group lead to zeta functions of the form 3 Pp(t) Z(X/Fp,t) = 2 4 3 (48) (1 −t)Pp(t)Pp(t)(1 − p t) with 3 (2,1) deg Pp(t) = 2 + 2h , 2 (1,1) deg Pp(t) = h . This follows from the fact that for Calabi-Yau threefolds without a torus factor we have b1 = 0. For Calabi-Yau threefolds with h1,1 = 1 the zeta function reduces to P3(t) Z (X/ ,t) = p , (49) Fp (1 −t)(1 − pt)(1 − p2t)(1 − p3t)

3 and therefore becomes particularly simple. The coefﬁcients βi (p) of the polynomial

b3(X) 3 3 i Pp(t) = ∑ βi (p)t (50) j=0 are related to the cardinalities of the variety via the expansion 1 Z(X/ ,t) = 1 + N t + (N2 + N )t2 Fp 1,p 2 1,p 2,p 1 1 1 3 3 4 + N ,p + N ,pN ,p + N t + O(t ) (51) 3 3 2 1 2 6 1,p as

3 2 3 β1 (p) = N1,p − (1 + p + p + p ), 1 β 3(p) = (N2 + N ) − N (1 + p + p2 + p3) 2 2 1,p 2,p 1,p +(1 + p + p2 + p3)2 + (1 + p + p2) + p2(1 + p) + p4(1 + p + p2), (52) etc. This procedure is useful because the knowledge of a ﬁnite number of terms in the L–function determines it uniquely. In dimension four the cohomology of Calabi-Yau varieties is more compli- cated, leading to zeta functions

3 5 Pp(t)Pp(t) Z(X4/Fp,t) = 2 4 6 4 (53) (1 −t)Pp(t)Pp(t)Pp(t)(1 − p t) for spaces without torus factors, but the procedure is the same as above. For smooth hypersurfaces the cohomology groups except of degree given by the dimension 20 R. Schimmrigk are either trivial or inherited from the ambient space, leading to the intermediate L–function 1 L(X,s) = ∏ 4 −s (54) p Pp(p ) as the only nontrivial factor.

5 L–Functions Via Jacobi Sums

For the class of hypersurfaces of Brieskorn-Pham type it is possible to gain insight into the precise structure of the L–function by using a result of Weil (38) which expresses the cardinalities of the variety in terms of Jacobi sums of finite fields. In this context there the L–function of the Ω−motive of such weighted Fermat hypersurfaces can be made explicit. For any degree vector d = (d0,...,dn+1) and for any prime p define the numbers ei = (di, p − 1) and the set ( )

p,d n+2 An = (α0,...,αn+1) ∈ Q 0 < αi < 1,eiαi ≡0(mod 1),∑αi ≡0(mod 1) . i (55)

Theorem 5 The number of solutions of the smooth projective variety ( ) n+1 n+1 di Xn = (z0 : ··· : zn+1) ∈ P ∑ bizi = 0 (56) i=0 over the ﬁnite ﬁeld Fp is given by 2 n ¯ Np(Xn) = 1 + p + p + ··· + p + ∑ jp(α)∏ χαi (bi), (57) d α∈An where 1 j (α) = χ (u )··· χ (u ). (58) p p − 1 ∑ α0 0 αn+1 n+1 ui∈Fp u0+···+un+1=0

With these Jacobi sums jq(α) one deﬁnes the polynomials

!1/ f n n/2 |n| n ¯ i f Pp(t) = (1 − p t) ∏ 1 − (−1) jp f (α)∏ χαi (b )t (59) d i α∈An and the associated L–function

(n) 1 L (X,s) = ∏ n −s . (60) p Pp(p ) Emergent Spacetime from Modular Motives 21

Here |n| = 1 if n is even and |n| = 0 if n is odd. A slight modiﬁcation of this result is useful even in the case of smooth weighted projective varieties because it can be used to compute the factor of the zeta function coming from the invariant part of the cohomology, when viewing these spaces as quotient varieties of projective spaces. The Jacobi-sum formulation allows to write the L–function of the Ω−motive M of weighted Fermat hypersurfaces in a more explicit way. Deﬁne the vector Ω k0 kn+1 αΩ = d ,..., d corresponding to the holomorphic s−form, and denote its d Galois orbit by OΩ ⊂ An . Then −1/ f n − f s LΩ (X,s) = ∏ ∏ 1 − (−1) jp f (α)p . (61) p α∈OΩ

6 Modularity Results for Rank two Motives in Dimension One and Two

In the examples of higher rank motives considered in Theorems 1 and 2 the geometry of the varieties can be constructed from that of lower-dimensional manifolds. This leads to an induced string theoretic modular structure for the higher rank motive by using known modularity results for the building blocks involved. To make this paper more self-contained this section briefly summarizes the necessary ingredients obtained in (11) for the diagonal weighted elliptic curves and in (12) for the diagonal weighted extremal K3 surfaces. More details can be found in these references. The simplest possible framework in which the problem of an emergent spacetime can be raised is for compactifications on tori, in particular Brieskorn-Pham curves Ed of degree d embedded in the weighted projective plane. The ellip- d tic curves E are defined over the rational number Q, and therefore modular in lieu of the Shimura-Taniyama conjecture, proven in complete generality in ref. (3), based on Wiles’ paradigmatic results (1; 2). This theorem says that any elliptic curve over the rational numbers is modular in the sense that the inverse Mellin transform of the Hasse-Weil L–function is a modular form of weight two for some congruence subgroup Γ0(N). This raises the question whether the modular forms derived from these Brieskorn-Pham curves are related in some way to the characters of the conjectured underlying conformal field theory models. The conformal field theory on the string worldsheet is fairly involved, and a priori there are a number of different modular forms that could play a role in the geometric construction of the varieties. The first string theoretic modularity result showed that the modular form associated to the cubic Fermat curve E3 ⊂ P2 factors into a product of SU(2)−modular forms that arise from the characters of the underlying world sheet (9). More precisely, the worldsheet modular forms that encode the structure of the compact spacetime geometry are Hecke indefinite theta series associated to Kac-Moody theoretic string functions introduced by Kac and Peterson. For the remaining two elliptic weighted Fermat curves this relation requires a modification involving a twist character that is physically motivated by the number field generated by the quantum dimensions of the string model (10; 11). The modular forms that emerge from the weighted Fermat curves provide a 22 R. Schimmrigk string theoretic interpretation of the Hasse-Weil L–function in terms of the exactly solvable Gepner models at central charge c = 3. Ref. (11) furthermore identifies the criteria that lead to the derivation of these elliptic curves from the conformal field theory itself, with no a priori input from the geometry. The class of elliptic Brieskorn-Pham curves is given by

3 2 3 3 3 E = (z0 : z1 : z2) ∈ P | z0 + z1 + z2 = 0 , 4 4 4 2 E = (z0 : z1 : z2) ∈ P(1,1,2) | z0 + z1 + z2 = 0 , (62) 6 6 3 2 E = (z0 : z1 : z2) ∈ P(1,2,3) | z0 + z1 + z2 = 0 . The modular forms associated to these curves are cusp forms of weight two with respect to congruence groups of level N, Γ0(N) ⊂ SL(2,Z), deﬁned by a b a b ∗ ∗ Γ0(N) = ∈ SL2(Z) ≡ (modN) . (63) c d c d 0 ∗

k Recalling the string theoretic Hecke indeﬁnite theta series Θ`,m(τ) deﬁned in k Sect. 2 in terms of the Kac-Peterson string functions c`,m(τ), the geometric modular forms decompose as

f (Ed,q) = Θ ki (qai ) ⊗ χ , (64) ∏ `i,mi d i

d where ai ∈ N, and χd(·) = · is the Legendre symbol. More precisely, given the worldsheet theta series

1 2 Θ1,1(τ) = η (τ), 2 (65) Θ1,1(τ) = η(τ)η(2τ), the elliptic modular forms factorize in the following way (11).

Theorem 6 The inverse Mellin transforms f (Ed,q) of the Hasse-Weil L–functions d d d LHW(E ,s) of the curves E ,i = 3,4,6 are modular forms f (E ,q) ∈ S2(Γ0(N)), with N = 27,64,144 respectively. These cusp forms factor as

3 1 3 1 9 f (E ,q) = Θ1,1(q )Θ1,1(q ), 4 2 4 2 f (E ,q) = Θ1,1(q ) ⊗ χ2, (66) 6 1 6 2 f (E ,q) = Θ1,1(q ) ⊗ χ3. Consider next the class of extremal K3 surfaces that can be constructed as weighted Fermat varieties n o 4 3 4 4 4 4 X2 = (z0 : ··· : z3) ∈ P z0 + z1 + z2 + z3 = 0 , n o 6A 6 6 6 2 X2 = (z0 : ··· : z3) ∈ P(1,1,1,3) z0 + z1 + z2 + z3 = 0 , (67) n o 6B 6 6 3 3 X2 = (z0 : ··· : z3) ∈ P(1,1,2,2) z0 + z1 + z2 + z3 = 0 . Emergent Spacetime from Modular Motives 23

For the remainder of this section let X2 be any of these three varieties. All three K3 surfaces X2 lead to an Ω−motive that is modular and admits a string theo- 2 retic interpretation. Denote by M(X) ⊂ H (X2) the cohomological realization of a motive M, with MΩ (X) the motive associated to the holomorphic 2-form, and let LΩ (X,s) = L(MΩ (X),s) be the associated L–series, with fΩ (X,q) denoting the inverse Mellin transform of LΩ (X,s). The main result of (12) shows that the modular forms determined by the Ω−motives of these extremal K3 surfaces are determined by the string theoretic modular forms that enter in Theorem 6.

2 Theorem 7 Let MΩ ⊂ H (X2) be the irreducible representation of Gal(Q(µd)/Q) 2,0 associated to the holomorphic 2−form Ω ∈ H (X2) for any of the K3 surfaces X2 in (67). Then the q−series fΩ (X2,q) of the L–function LΩ (X2,s) are modular forms given, respectively, by

4 6 4 fΩ X2 ,q = η (q ), 6A 3 2 3 2 9 fΩ X2 ,q = ϑ q η q η (q ), (68) 6B 3 2 3 6 fΩ X2 ,q = η q η (q ) ⊗ χ3.

These functions are cusp forms of weight three with respect to Γ0(N) with levels 4 6A 16,27 and 48, respectively. For X2 and X2 the L–functions can be written as

4 2 LΩ X2 ,s = L ψ64,s , (69) 6A 2 LΩ X2 ,s = L ψ27,s , (70) where ψd with d = 27,64 are Großencharacters¨ associated to cusp forms fd(q) of weight two and levels 64 and 27, respectively, given by the elliptic forms

4 2 4 2 8 f4 τ = f E ,q = η q η q ⊗ χ2, (71) 3 2 3 2 9 f6A τ = f E ,q = η q η q .

6B 6B 2 For X2 the L–series is given by LΩ (X2 ,s) = L(ψ144 ⊗ χ3,s), leading to the cusp form of level 144,

6 4 6 f6B(τ) = f (E ,q) = η (q ) ⊗ χ3. (72) Several of these results will enter in the discussion below of higher dimensional varieties.

12 7 A Nonextremal K3 Surface X2 ⊂ P(2,3,3,4) String theoretic modularity of the class of extremal K3 surfaces of Brieskorn- Pham type has been established in (12). Extremal K3 surfaces are special in the sense that their Picard number is maximal, i.e. ρ = 20, and the resulting motive has rank two. In this section the results of (12) are extended to a nonextremal K3 surface with a rank four Ω−motive. Consider the Brieskorn-Pham hypersurface (1) of degree twelve in weighted projective space P(2,3,3,4). The Galois group of 24 R. Schimmrigk

12 4 6 Table 1 Coefﬁcient comparison of the surface X2 and the curves E− and E p f 13 25 37 49 61 73 4 ap f (E ) −6 −1 2 −7 10 −6 6 bp f (E ) 2 −5 −10 9 14 −10 12 cp f X2 −12 30 −20 −14 140 60 the cyclotomic ﬁeld Q(µ12) has order four, hence the Ω−motive has rank four. The four Jacobi sums which parametrize this motive are given by

jp(σαΩ ), σ ∈ Gal(Q(µ12)/Q), (73) i.e. jp(α) with 1 1 1 1 5 1 1 2 α ∈ , , , , , , , , (74) 6 4 4 3 6 4 4 3 and their conjugates α¯ = 1−α, where 1 denotes the unit vector. The computation of these sums leads to the L–function of the Ω−motive . 12 30 20 14 140 60 L (X12,s) = 1 − + − − + + + ··· . (75) Ω 2 13s 25s 37s 49s 61s 73s Insight into the structure of this L–function can be obtained by noting that the 12 surface X2 can be constructed as a quotient of a product of elliptic curves

E1 × E2/ι, where ι is an involution on the product. More precisely, the elliptic curves are given by

4 2 4 4 E− = x0 − (x1 + x2 = 0) ⊂ P(2,1,1), (76)

6 and the degree six curve E ⊂ P(1,2,3). The twist map (48; 49) (see also (50; 51))

Φ : P(2,1,1) × P(3,1,2) −→ P(3,3,2,4) (77) is deﬁned by

1/3 2/3 1/2 1/2 ((x0,x1,x2),(y0,y1,y2)) 7→ y0 x1,y0 x2,x0 y1,x0 y2 . (78)

4 6 12 This maps the product E− × E and leads to the K3 surface of degree twelve X2 . 4 4 4 −s 6 The coefﬁcients of L(E−,s) = L(E ,s) = ∑n an(E )n and L(E ,s) = 6 −s 4 6 ∑n bn(E )n of the Hasse-Weil L–functions of the elliptic curves E and E , respectively, can be obtained by expanding the results of Theorem 6. Multiplying 4 6 the coefﬁcients ap(E ) and bp(E ) leads to

4 6 12 ap(E )bp(E ) = cp(X2 ). (79) For low primes the results are collected in Table 1. Emergent Spacetime from Modular Motives 25

The inverse Mellin transform of the L–function of the two building blocks E4 6 12 and E of the surface X2 are given in terms of string theoretic theta functions as described in Theorem 6. The modular forms of E4 and E6 are both of complex multiplication type, leading to an interpretation of the L–series in terms of Hecke’s Großencharacters.¨ More precisely, they are described by twists by Legendre sym- bols χn,n = 2,3 of the characters ψ32 and ψ36, as described in (11),

4 L(E ,s) = L(ψ32 ⊗ χ2,s), (80) 6 L(E ,s) = L(ψ36 ⊗ χ3,s).

√ Here the characters ψ32√ and ψ36 are associated to the Gauss field Q( −1) and the Eisenstein field Q( −3) respectively, as described in 2. 12 The factorization of the coefficients cp(X2 ) suggests that the rank four 12 4 6 Ω−motive MΩ (X2 ) is the tensor product of the elliptic motives of E and E , but a priori leaves open the precise nature of the L–function product. This can be determined by combining the iterative reduction of Jacobi sums introduced by Weil (52) in his discussion of cyclotomic fields with the results of the elliptic curves E4 and E6 given in Sect. 6. Applying Weil’s reduction to the Jacobi sums associated 12 to the Ω−motive of the surface X2 shows that they can be expressed for the relevant primes p = 1 + 12n, n ∈ N, completely in terms of the Jacobi sums of E4 6 12 and E . Furthermore, the resulting local factors of the L–function LΩ (X2 ,s) are precisely those of the modified Rankin-Selberg product considered in Sect. 2, applied to the modular forms f64 ∈ S2(Γ0(64)) and f144 ∈ S2(Γ0(144)) of the curves E4 and E6 respectively,

12 LΩ X2 ,s = L Mf64 ⊗ Mf144 ,s ,

where M are the elliptic motives of f . fNi Ni 26 R. Schimmrigk

The string theoretic interpretation of the Hasse-Weil L–series of E4 and E6 described in Theorem 6 therefore induces a string interpretation of the modular 12 blocks of the K3 surface X2 . The CM property of these forms will allow a sys- tematic discussion in Sect. 11 of the reverse construction of emergent space from the modular forms of the worldsheet ﬁeld theory.

6 8 Modular Motives of the Calabi-Yau Threefold X3 ⊂ P(1,1,1,1,2) In this section string theoretic modularity is proven for the Ω−motive of the diag- 6 onal weighted Calabi-Yau threefold X3 , extending the lower-dimensional results of (11; 12).

6 8.1 The modular Ω−motive of X3

6 Consider the Calabi-Yau variety X3 defined as the double cover branched over the degree six Fermat surface in projective threespace P3. This manifold can be viewed as a smooth degree six hypersurface of Brieskorn-Pham type in the weighted projective fourspace P(1,1,1,1,2) as in (3). The Galois group of Q(µ6) has − M (X6) order two, hence the Ω motive Ω 3 has rank two. The Jacobi sums√ associated to this motive take values in the imaginary quadratic field K = Q( −3), and therefore define a Großencharacter¨ of the type described in Sect. 2. The results 6 of Hecke and Shimura thus imply that the L–series of the motive MΩ of X3 is modular. Computing sufficiently many coefficients of this L–function . 17 89 107 125 308 433 L (X6,s) = 1 + + + − + − + ··· , (81) Ω 3 7s 13s 19s 25s 31s 37s therefore determines the resulting modular form uniquely. It follows from (81) that this motivic L–function is the Mellin transform of a modular form of weight 4 and level N = 108, fΩ ∈ S4(Γ0(108)). This form cannot be written as a product or a quotient of Dedekind eta- functions (53; 54), but the fact that it admits complex multiplication by the imagi- 6 nary quadratic field K implies that LΩ (X3 ,s) can be written as the Hecke L–series of a twisted Großencharacter¨ associated to the field K. Such characters were considered in (11; 12) in the context of the elliptic Brieskorn-Pham curve E3 ⊂ P2 and 6 3 E ⊂ P(1,2,3), as well as modular K3 surfaces. The Hasse-Weil L–series L(E ,s) is a Hecke series for the character ψ27 considered in 2, which can be written as (9),

3 1 3 1 9 L(E ,s) = L(ψ27,s) = L Θ1,1(q )Θ1,1(q ),s . (82)

The character ψ27 turns out to be the fundamental building block of the L–series 6 of the Ω−motive of X3 . For higher dimensional varieties it is possible to generalize such relations by considering powers of the Großencharacter¨ in order to obtain higher weight mod- 3 ular forms, as described above in Hecke’s theorem. In the present case ψ27 leads to a modular form which can be written in terms of the Dedekind eta function 8 3 η(q) as η (q ) ∈ S4(Γ0(9)), which also appears as a motivic form in a number of Emergent Spacetime from Modular Motives 27

6 6 geometries different from X3 . In order to obtain the motivic L–series LΩ (X3 ,s) computed above it is necessary to introduce a twist character. This can be chosen to be the cubic residue power symbol, denoted here by (3) 2 χ2 (p) := , (83) p 3 where p|p and the congruence ideal is chosen to be m = (3). With this character 6 the Hecke interpretation of the motivic L–function of X3 takes the form 6 3 (3) 2 LΩ (X3 ,s) = L(ψ27 ⊗ (χ2 ) ,s). (84) 6 The inverse Mellin transform fΩ (X3 ,q) of this L–series 6 . 7 13 19 25 31 37 fΩ (X3 ,q) = q + 17q + 89q + 107q − 125q + 308q − 433q + ··· (85) is therefore of complex multiplication type in the sense of Ribet (19), and the string theoretic nature of the character ψ27 implies the same for the weight four 6 modular form of X3 .

6 8.2 Lower weight modular motives of X3

6 The degree six Calabi-Yau hypersurface X3 provides an example of the phenomenon noted in Sect. 3 that the intermediate cohomology can lead to modu- 6 lar motives beyond the Ω−motive. For X3 the motives are of rank two, given by the Galois group Gal(Q(µ6)/Q) with certain multiplicities and twists. Mod- ulo these twists and multiplicities the group H2,1(X) ⊕ H1,2(X) leads to three different types of modular motives of weight two and rank two, denoted in the following by MA ∈ {MI,MII,MIII}. The L–series L(MA,s) that result from these A motives have coefﬁcients ap that are all divisible by the prime p. By introducing A A the twisted coefﬁcients ap = ap/p, these L–series lead to modular forms of weight two fA ∈ S2(Γ0(NA)), where the level NA is determined by the motive MA, 6 L MA(X3 ),s = L( fA,s − 1).

The modular forms are of levels NA = 27,144,432, the ﬁrst two given by the curves E3,E6 described in Theorem 6, 3 fI(q) = f (E ,q), (86) 6 fII(q) = f (E ,q), while the level NIII = 432 form is given by . 7 13 19 31 fIII(q) = q − 5q − 7q + q + 4q + ··· , 3 6 It follows that the L–series L(H (X3 ),s) of the intermediate cohomology group decomposes into modular pieces in the sense that each factor arises from a modular form

3 6 ai L(H (X3 ),s) = L( fΩ ,s)∏L( fi ⊗ χi,s − 1) , i 28 R. Schimmrigk

6 Table 2 The Jacobi sums of X3 that lead to non-isogenic modular motives Type Jacobi sum Level N 1 1 1 1 2 I jp 3 , 3 , 3 , 3 , 3 27 1 1 1 5 1 II jp 6 , 6 , 2 , 6 , 3 144 1 1 2 2 1 III jp 6 , 6 , 3 , 3 , 3 432 where ai ∈ N, fΩ ∈ S4(Γ0(108)) is as determined above, the fi(q) are modular forms of weight two and levels Ni = 27,144,432, and χi is a Legendre character (which can be trivial). The Jacobi sums corresponding to the motives MI,MII,MII are listed in Table 2, together with the level NA of the corresponding modular form fA ∈ S2(Γ0(NA)). 6 The weight two modular forms fA that emerge from X3 have a natural ge- 6 ometric interpretation. The threefold X3 contains divisors given by degree six 6 2 6 Fermat curves C ⊂ P and C ⊂ P(1,1,2), obtained from the original hypersurface via intersections with coordinate hyperplanes. These curves are of genus ten and can be shown to decompose into ten elliptic factors of three different types 3 6 EI = E ,EII = E , and EIII an elliptic curve of conductor 432. Hence its L–function factors as 6 6 3 L(C ,s) = L(EI,s)L(EII,s) L(EIII,s) , 6 taking into account their multiplicities. The curve C is of genus four and leads to the same modular forms, with different multiplicities. These results establish and 6 make precise the structure of X3 described in Theorem 2.

12 9 The K3 Fibration Hypersurface X3 ⊂ P(2,2,2,3,3)

12 The diagonal hypersurface X3 given in Eq. (3) is the second manifold that extends previous results to three dimensions. It differs in character from the degree six hypersurface considered in the previous section because its motive is of higher rank.

12 9.1 The Ω−motive of X3

12 The Galois group of X3 is of order four, leading to an Ω−motive of rank four. The 1 1 1 1 1 1 1 1 3 3 Jacobi sums that parametrize this motive are given by jp 6 , 6 , 6 , 4 , 4 , jp 6 , 6 , 6 , 4 , 4 and their complex conjugates. The computation of these sums for low p f leads to the L–series . 6 150 94 497 1210 582 L X12,s = 1 + − + − − + + ··· . (87) Ω 3 13s 25s 37s 49s 61s 73s 12 The structure of the Ω−motivic L–series of X3 can be understood by noting 6A that the threefold is a K3 fibration with typical fiber X2 given in (67). The inter- 12 pretation of LΩ (X3 ,s) in terms of the fibration is also useful because it makes the Emergent Spacetime from Modular Motives 29 complex multiplication structure of the associated modular form transparent. The threefold can be constructed explicitly as the quotient of a product of a torus E and a K3 surface

E × K3/ι, (88)

where ι is an involution acting on the product. More precisely, the elliptic curve 4 12 is the weighted quartic curve E− considered in the discussion of X2 , and the K3 6A surface is X2 . Applying the twist map of (48; 49) gives the map

Φ : P(2,1,1) × P(3,1,1,1) −→ P(3,3,2,2,2) (89)

deﬁned by

1/3 1/3 1/2 1/2 1/2 ((x0,x1,x2),(y0,y1,y2,y3)) 7→ y0 x1,y0 x2,x0 y1,x0 y2,x0 y3 . (90)

4 6A 12 This maps the product E− × X2 to threefold X3 . 30 R. Schimmrigk

12 Table 3 Coefﬁcient comparison for the threefold X3 p f 13 25 37 49 61 73 4 ap f (E ) −6 −1 2 −7 10 −6 6A bp f (X2 ) −1 25 47 120 −121 −97 12 cp f (X3 ) 6 −150 94 −994 −1210 582

12 The fibration structure suggests that the L–function of the threefold X3 can be understood in terms of those of its building blocks. The L–function of the K3 fiber of this threefold was determined in Theorem 7 to be given by the Mellin 6A transform of the cusp form fΩ (X2 ,q) of weight w = 3 and level N = 27, and 4 the L–function of the quartic curve is given by f (E ,q) ∈ S2(Γ0(64)) according to Theorem 6 (11). 12 The geometric structure of X3 suggests that a comparison of the coefficients 12 4 of the L–function of the Ω−motive of X3 with those of its building blocks E 6A and X2 should lead to a composite structure. Table 3 illustrates that this is indeed 4 4 the case. The coefficients ap(E ) arise from the L–series of the quartic E , while 6A the surface expansion bp(X2 ) is that of (68) in Theorem 7, which leads to the expansion

6A . 7 13 19 25 31 37 f (X2 ,q) = q − 13q − q + 11q + 25q − 46q + 47q −22q43 + 120q49 − 121q61 − 109q67 − 97q73 + ··· . (91)

4 6A It follows that the products ap(E )bp(X2 ) agree with the expansion coefﬁcients 12 12 cp(X3 ) of the threefold X3 . 12 The factorization of the Ω−motivic L–series LΩ (X3 ,s) also shows that its building blocks have complex multiplication. For the curve E4 this was already 12 6A discussed above in the context of the K3 surface X2 (11). For the K3 surface X2 it was shown in (12) that the modular form of Theorem 7 is the Mellin transform√ 2 of the Hecke L–series of a Großencharacter¨ ψ27 of the Eisenstein ﬁeld Q( −3),

6A 2 L X2 ,s = L ψ27,s , (92) where the character ψ27 has been deﬁned in Sect. 2. The motivic L–series of both 12 building blocks of X3 therefore are of complex multiplication type. The proof that the precise relation between the Ω−motivic L–function 12 LΩ (X3 ,s) and its building blocks is again given by the Rankin-Selberg convo- 12 lution proceeds in the same way in the case of the K3 surface X2 ⊂ P(2,3,3,4) discussed in Sect. 7. By using Weil’s dimensional reduction of the Jacobi sums it becomes clear that the only nontrivial ingredients at q = 1 + 12n are the lower- 4 6A dimensional Jacobi sums of E and X2 , and these combine in the local factors of 12 the L–function of X3 in precisely the way required by the Rankin-Selberg product. Combining this result with the theorem of Ramakrishnan (14) shows that the 12 Ω−motive of X3 is automorphic as well. Emergent Spacetime from Modular Motives 31

12 9.2 Lower weight modular forms of X3

6 2,1 1,2 Similar to the degree six threefold X3 the cohomology H ⊕ H of the degree 12 twelve hypersurface X3 leads to modular forms of weight two. There are again Jacobi sums that lead to precisely the same modular forms fA(q),A = I,II,III of 6 weight two and levels NA = 27,144,432, possibly including a twist, as for X3 . 12 6 2 This is expected because X3 contains the plane Fermat curve C ⊂ P already 6 encountered in X3 . There is a further L–series of a rank two motive that is deter- 4 mined by the quartic elliptic weighted Fermat curve E ⊂ P(1,1,2) considered in Theorem 6, 12 4 LIV X3 ,s = L E ,s − 1 .

6 5 10 A Modular Calabi-Yau Fourfold X4 ⊂ P Calabi-Yau varieties of complex dimension four are useful in the context of F- theory in four dimensions and M-theory in three dimensions. In this section it is shown that the Ω−motive of the fourfold of degree six in projective fivespace P5 defined by the Brieskorn-Pham hypersurface (6) is modular. The Galois orbit is of length two, and the motivic L–function is described 1 1 by computing the Jacobi sums jp(αΩ ) with αΩ = 6 ,..., 6 . The resulting L– function is given by . 71 337 601 625 194 529 L X6,s = 1 − − + + − − + ··· . (93) Ω 4 7s 13s 19s 25s 31s 37s 6 The associated q−expansion fΩ (X4 ,q) differs from that of a newform of weight five and level 27, 7 13 19 31 37 f27(q) = q + 71q − 337q − 601q + 194q − 529q + ··· , (94) only in signs. These signs can be adjusted to the quadratic character χ3 defined by the Legendre symbol 3 χ (p) = , (95) 3 p leading to 6 fΩ X4 ,q = f27(q) ⊗ χ3. (96) This modular form can also be described as a Hecke L√–series associated to the character ψ27 associated to the Eisenstein field K = Q( −3) and defined in Sect. 2 in (21). More precisely, the twisted Hecke L–series agrees with that of the 6 Ω−motive of X4 , 6 4 LΩ X4 ,s = L(ψ27,s) ⊗ χ3. (97) Hence the motivic L–function is again a purely algebraic object and its fundamental structure is determined by the L–series of a Hecke indefinite theta series, as noted in (82). 32 R. Schimmrigk

11 Emergent Space from Characters and Modular Forms

In the discussion so far the goal was to formulate a general framework of Ω−motives of varieties of Calabi-Yau type, and more generally, of special Fano type, in the context of Grothendieck’s framework of motives, and to test the conjecture that these motives are string modular in the sense that it is possible to identify modular forms on the worldsheet whose Mellin transform agrees with the L–function of the resulting motives. The problem of constructing spacetime geometry from fundamental string input involves the inverse problem of this strategy. The aim of the present section is to address this “emergent space” problem. As already mentioned earlier, the idea here must be to obtain a construction of the motivic pieces of the compact varieties from the basic conformal field theoretic modular forms. There are several ways to think about these objects and the following remarks describe how these various constructions, which a priori are independent, fit together in the context of the Ω−motive of Brieskorn-Pham type varieties. The main simplifying observation in the present context of weighted Fermat hypersurfaces is that all the motivic L–functions LΩ (X,s) that have been encountered so far in this program (9; 10; 11; 12; 55; 56) lead to modular forms which are of complex multiplication type (19) (see (22) for geometric aspects of CM). The structure of such forms has been described in detail in (11; 12; 57). The important point in the present context is that modular Ω−motives of CM type are algebraic objects whose L–functions are given by the Hecke L–series of a Großencharacter¨ (possibly including a twist). This construction of Großencharacters¨ from geometry can be inverted, and it is known how to construct motives directly from the characters. It is in particular possible to construct a Grothendieck motive of the form Mχ =(Aχ , pχ ), where Aχ is an abelian variety associated to the character χ by the theorem of Casselman (58), where pχ is a projector associated to χ. This leads to an apparent problem of riches, because given a modular form Sw(Γ0(N)) it is possible to construct a Grothendieck motive Mf =(Xf , p f ) by considering the cohomology of an associated Kuga-Sato variety Xf , as shown by Deligne (24), Jannsen (25), and Scholl (26). Combining the abelian motives and the Kuga-Sato motives with those of the Ω−motives thus leads to three a priori different motivic constructions associated to the Großencharacters¨ encountered here and in the earlier papers. It turns out that the motives Mχ , Mf and MΩ all are isomorphic because they arise from the same CM modular form. This follows because motives associated to CM modular forms have CM, and one can generalize Faltings result that L– series characterize abelian varieties up to isogeny to CM motives, as shown e.g. by Anderson (59). In the case of varieties of Brieskorn-Pham type it is possible to make the relation between abelian varieties and Ω−motives more concrete by noting that the cohomology of Fermat varieties has a well-known inductive structure, which was first noted by Shioda-Katsura (60) in the context of Fermat varieties (see also Deligne (61)). This inductive structure allows to reduce the cohomology of higher- dimensional varieties in terms of the cohomology of algebraic curves (modulo Tate twists). Hence the basic building blocks are abelian varieties derived from the Jacobians of these curves. It follows from results of Gross and Rohrlich (62) Emergent Spacetime from Modular Motives 33 that these Jacobians factor into simple abelian varieties and that these abelian factors have complex multiplication. The final step in the construction is provided by the fact that the L–function of the abelian variety attached to χ by Casselman’s result is given by the conjugates of the L–function of the Großencharacter.¨

12 Further Considerations

The goal of the program continued in this paper is to investigate the relation between the geometry of spacetime and the physics of the worldsheet by analyzing in some depth the connection between the modular symmetry encoded in exact models on the worldsheet and the modular symmetries that emerge from the nontrivial arithmetic structure of spacetime. The techniques introduced for this purpose provide a stronger, and more precise, alternative to the framework of Landau- Ginzburg theories and σ−models. The latter in particular presupposes the concept of an ambient space in which the string propagates, a notion that should emerge as a derived concept in a fundamental theory. The focus of the results obtained in previous work and the present paper has been on the class of diagonal models, related to Gepner’s construction. It would be interesting to extend these considerations to the more general class of hypersurfaces that are related to the Kazama-Suzuki models (63). Of particular interest in that class are certain ‘irreducible’ models which are not tensor products, hence a single conformal field theoretic quotient suffices to saturate the necessary central charge. Such models exist for both K3 surfaces and Calabi-Yau threefolds, and establishing modularity in the sense described here would be a starting point for the exploration of modular points in the moduli space of nondiagonal varieties. Results in this direction would illuminate relations between different conformal field theories. A second open problem is the analysis of families of varieties with respect to their modular properties. First steps in this direction have been taken in refs. (64; 65; 66), where the zeta functions for particular one-parameter families of Calabi-Yau threefolds are computed. It is of interest to understand how the modular behavior of these families is related to deformations along marginal directions of the associated conformal field theory. Deforming away from the weighted Fer- mat point in Brieskorn-Pham type families will in general change the structure of the motive, but the reciprocity conjecture of Langlands implies that automorphy is expected to occur for different fibers in such families. Of particular interest is the modular behavior of singular fibers in families of Calabi-Yau varieties. It is known from work of the 1980s that the moduli spaces of different Calabi-Yau spaces are connected via singular varieties (67; 68). This raises the question what the precise structure is of the conjectured family of modular and automorphic forms at such singular fibers in a given family. The results of such a modularity analysis should prove useful for a deeper understanding of the conformal field theoretic behavior of phase transitions between Calabi-Yau varieties, a problem that has proven challenging in the past. Progress in this direction will be described in (69).

Acknowedgements. It is a pleasure to thank Monika Lynker for conversations, and Rob Myers for raising the point of string dualities in the context of the motivic picture. Part of this work 34 R. Schimmrigk was completed while the author was supported as a Scholar at the Kavli Institute for Theoretical Physics in Santa Barbara. The work at the KITP was supported in part by the NSF under Grant No. PHY05-51164. The author is also grateful to the National Science Foundation for support provided under Grant No. PHY09-69875. This work was furthermore supported in part by Fac- ulty Research Grants from Indiana University South Bend. Thanks are due to the referee, whose comments and questions led to an improved presentation of this paper.

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