hep-th/9807087 YCTP-P17-98 Arithmetic and Attractors 8 9 9 1 l u Gregory Moore J 3 1 Department of Physics, Yale University 1 New Haven, CT 06511 v 7 [email protected] 8 0 7 0 8 We study relations between some topics in number theory and supersymmetric black holes. 9 These relations are based on the “attractor mechanism” of = 2 supergravity. In IIB / h N t string compactification this mechanism singles out certain “attractor varieties.” We show - p e that these attractor varieties are constructed from products of elliptic curves with complex h : multiplication for = 4,8 compactifications. The heterotic dual theories are related to v N i rational conformal field theories. In the case of = 4 theories U-duality inequivalent X N r backgrounds with the same horizon area are counted by the class number of a quadratic a imaginary field. The attractor varieties are defined over fields closely related to class fields of the quadratic imaginary field. We discuss some extensions to more general Calabi-Yau compactifications and explore further connections to arithmetic including connections to Kronecker’s Jugendtraum and the theory of modular heights. The paper also includes a short review of theattractormechanism. A much shorterversion of the paper summarizing the main points is the companion note entitled “Attractors and Arithmetic.” July 10, 1998 1. Introduction Many people learning modern string theory and supersymmetric gauge theory are struck by the fact that much of the necessary mathematical background is best learned from textbooks on number theory. For example, modular forms, congruence subgroups, and elliptic curves, are all mathematical objects of central concern both to string theorists and to number theorists. This suggests the obvious idea that there might be a deeper relationship between the two subjects. Such a relation, if truly valid, would clearly have a beneficial effect on both subjects. Of course, one should be cautious about speculations of this nature. For example, various partial differential equations occur in widely disparate fields of physics and en- gineering. The mere appearance of a technical tool in two disparate subjects does not necessarily imply a deeper unity (except insofar as the same equation appears). Indeed, upon closer examination, one is often disappointed to find that the precise questions of the number theorist and the string theorist generally seem to be orthogonal. One example will illustrate this orthogonality of the world-views of the string- and number- theorist. In string theory and supersymmetric gauge theory one often meets ellip- tic curves. These play a central role in conformal field theory, string perturbation theory, supersymmetric gauge theory, string duality, and F-theory. Yet, in all these applications, there has never been any compelling reason to restrict attention to elliptic curves defined over Q (or any other number field). On the other hand, it is the special properties of arithmetic elliptic curves which often take center stage in number theory. The point of this paper is that the “attractor mechanism” (explained below) used in the construction of supersymmetric black holes and black strings does provide a compelling reason to focus on certain varieties, and, at least in the examples where we can solve the equations exactly, these attractor varieties turn out to be arithmetic. We believe this observation opens up some opportunities for fruitful interactions between string theory and number theory, and the present paper is an attempt to explore some of those relations. The nonexpert reader should be warned that the author knows very little number theory. Here is an outline of the paper: A short introduction to this paper can be found in the separate text [1]. Section two contains a review of the attractor mechanism. It is primarily written for the mathematician who wishes to learn something about the subject. Section three illustrates the first connection between arithmetic and supersymmetric black holes. This connection is related to questions about the orbits of the arithmetic U-duality 1 groups. Sections 4, 5, and 6, discuss some exact attractor varieties in special compactifi- cations with a high degree of symmetry: compactifications of type II string theory with = 4,8 supersymmetry, and the FHSV model. Section 7 constitutes the second, and N main, connection to arithmetic. It attempts to explain why the attractor varieties in the = 4,8 examples are arithmetic. The essence of the matter is that the attractor varieties N are related to curves with “complex multiplication.” Section 8 discusses some extensions to larger classes of Calabi-Yau 3-folds and states the main conjecture of the paper. Sec- tion 9 examines what can be said about attractors near a point of maximal unipotent monodromy/large radius. Section 10 explains a relation to heterotic compactification on rational conformal field theories. Section 11 explores an arithmetic property of the K3 mirror map. Section 12 explains the relation of the conjectures of section 8 to Kronecker’s Jugendtraum and Hilbert’s twelfth problem. Section 13 mentions some more speculative ideas including a relation to the absolute Galois group and to heights of arithmetic vari- eties. In the conclusions we list our principle results and some of the main speculations. Three appendices briefly cover some background material and some technical proofs. In an effort to make this unreadable paper readable we have explicitly marked digres- sions, remarks, and examples. No harm is done if they are ignored. A list of some of our notation appears in appendix D. Finally, the references in this paper are incomplete. There is a surprisingly substantial literature on the relation of arithmetic and physics. The reader might wish to consult the proceedings of a Les Houches school [2][3] for an introduction to some aspects of the subject. 2. Review of the attractor mechanism TheattractormechanismisaninterestingphenomenondiscoveredbyFerrara,Kallosh, and Strominger in their work on dyonic black holes in supergravity [4]. The attractor equations [5][6] are central to the ideas of this paper. They were interpreted in terms of the minimization of the BPS mass by Ferrara, Gibbons and Kallosh in [7]. In this section we review some of this work. There is an extensive literature on the subject. Some recent reviews include [8][9][10], which the reader should consult for more complete references. 2 2.1. Electric-magnetic duality and the Gaillard-Zumino construction A common theme of modern string and gauge theory is the study of a families of abelian gauge theories with no natural electric/magnetic splitting of the fieldstrengths. We now review the standard formalism for describing such theories. Let M be a four-dimensional Minkowski signature spactetime. Let g = IRr be the Lie 4 algebra of the gauge group. Then the total 2-form fieldstrength (electric plus magnetic) F is valued in Ω2(M ;IR) V where V g g∗ is a real symplectic vector space with 4 ⊗ ≡ ⊕ symplectic product , : V V IR. h· ·i × → Inthephysical problemsthefamilyoftheoriesisoftenlabeledby acomplexsymplectic structure on V, i.e., a linear transformation : V V with 2 = 1 and v , v = 1 2 J → J − hJ · J · i v ,v . Given such a complex structure there are two natural constructions we can make. 1 2 h i First, we may define a symmetric bilinear form on V: (v ,v ) v , v = (v ,v ) (2.1) 1 2 J 1 2 2 1 J ≡ h J · i which will be used to write the Hamiltonian of the theory. Second, we can use to define the correct number of degrees of freedom of the theory. J Since 2 = 1 on Ω2(M ) the operator satisfies 2 = +1 and we can therefore ∗4 − 4 ∗T ≡ ∗4⊗J ∗T impose the all-important anti-self-duality constraint: = (2.2) T F −∗ F on real fieldstrengths . 1 The equation of motion and Bianchi identity of the electromag- F netic theory are combined in the single equation: d = 0 (2.3) F ~ ~ If we choose a space/time splitting M = M IR, the electric and magnetic fields E,B 4 3 × ∈ Ω3(M ) g are not functionally independent because of (2.2). Moreover, the formula for 3 ⊗ the energy density in terms of the spatial components ~ is simply: F = (~, ~) (2.4) J H F F Equations (2.2)(2.3)(2.4) constitute a manifestly dual formulation of the abelian theory. It is impossible to write a local Lorentz invariant and symplectically invariant action. 1 We choose the ( ) sign to agree with several standard conventions below. − 3 In order to connect to more standard treatments of the subject (see, e.g. [8][11]) we proceed as follows. If we complexify we can simultaneously diagonalize the operators 4 ∗ and : J Ω2(M ;C) = Ω2,+ Ω2,− 4 ⊕ = iΠ+ iΠ− 4 ∗ − ⊕ (2.5) V C = V1,0 V0,1 ⊗ ⊕ = +iΠ1,0 iΠ0,1 J ⊕− Here Π are projection operators and we use the notation ξ− = 1(ξ i ξ) for any two- 2 − ∗4 form ξ. Note that ξ− = +iξ−. If we now choose a symplectic (Darboux) basis αˆ ,βˆI, 4 I ∗ I = 1,...,r for V with α ,βˆJ = δ J then we may always choose a basis f for V0,1 h I i I { I} with f = αˆ +τ βˆJ. Let f¯ be the complex conjugate basis. By symplectic invariance of I I IJ I it follows that the period matrix τ = τ is symmetric. Equivalently, the symplectic IJ JI J form , is of type (1,1) with respect to the complex structure . h· ·i J The components of the total fieldstrength are, by definition, = FIαˆ G βˆI . (2.6) I I F − On the other hand, by the self-duality constraint (2.2) we have = FI,+f +FI,−f¯ (2.7) I I F and combining these we arrive at G− = τ¯ FJ,− I − IJ (2.8) G = (Imτ ) FJ (Reτ )FJ I IJ 4 IJ ∗ − and hence we recognize (2.3) as the standard Bianchi identities and equation of motion. The energy is, after a short calculation: = Imτ E~I E~J +B~I B~J (2.9) IJ H · · (cid:0) (cid:1) where E~I,B~I are the standard spatial components of FI. Thus, physically, we require Imτ > 0, that is, τ where is the Siegel upper half plane for r r matrices. IJ r r ∈ H H × Hence, if Λ is the integral span of αˆ ,βˆI for a symplectic basis then V/Λ is a principally I polarized abelian variety. 4 As a final remark one might attempt to form a local, Lorentz invariant, and symplectic invariant action using ( , ) . A short calculation reveals this to be zero. (Although M4 F F J the symmetric form (2R.1) is positive definite, Ω2(M ) has nilpotents.) If one chooses a 4 symplectic basis then ( , ) is naturally written as a sum of two cancelling terms, M4 F F J either one of which proRvides an action: G FI = +2 Im τ¯ FI,− FJ,− (2.10) I 4 IJ ∧∗ ∧ ZM4 ZM4 (cid:18) (cid:19) 2.2. Low energy supergravity for Calabi-Yau compactification of IIB supergravity In order to explain the attractor phenomenon we will focus attention on the compact- ification of IIB string theory on Calabi-Yau 3-folds. In this section we review a few of the relevant details of the resulting low-energy d = 4, = 2 supergravity needed to describe N the dyonic black holes. In the present discussion a key role is played by the abelian gauge fields in the theory. Gauge fields in the four-dimensional theory arise from the 5-form fieldstrength G of IIB supergravity. The equations of motion and Bianchi identity follow from the anti-self- duality constraint: G = G. Let X be a Calabi-Yau3-fold, that is, a compact, complex 10 −∗ 3-fold with Ricci flat K¨ahler metric. If b (X) = 0 this is the only source of abelian gauge 1 fields and consequently the total fieldstrength is Ω2(M ) H3(X;IR). (2.11) 4 F ∈ ⊗ We are now exactly in the general setup of the previous section since V = H3(X;IR) has symplectic form: γˆ ,γˆ γˆ γˆ (2.12) 1 2 1 2 h i ≡ ∧ ZX and a metric on X defines a complex structure = : H3 H3. The selfduality X J ∗ → constraint (2.2) is just that inherited from G. The principally polarized variety V/Λ is known as the Weil Jacobian. In the supergravity literature the period matrix is denoted as N = τ . IJ IJ − 5 2.2.1.Hodge structures We next want to decompose the gauge fields (2.11) into those coming from the dif- ferent supersymmetric multiplets namely the graviphoton in the gravity multiplet and the remaining vectors in the vectormultiplets. In order to do this we need to use a little of the theory of variation of Hodge structures, see, e.g., [12][13][14][15]. Consider the universal family π : . The fiber at s is X , the Calabi-Yau with complex structure s s X → M ∈ M (we fix a Kahler class and use Yau’s theorem). The complex structure at s determines a Hodge decomposition: H3(X ;C) = H3,0 H2,1 H1,2 H0,3 (2.13) s s ⊕ s ⊕ s ⊕ s in terms of which the Weil complex structure = is diagonal: X J ∗ = = iΠ3,0 iΠ2,1 iΠ1,2 +iΠ0,3 (2.14) X J ∗ − ⊕ ⊕− ⊕ Here Πp,q(v) is the component of v in Hp,q. We also use the notation vp,q Πp,q(v). ≡ Accordingly, the anti-self-duality constraint is solved by: = (Π2,1 Π0,3)( −)+(Π1,2 Π3,0)( +) (2.15) F ⊕ F ⊕ F Now choose a neighborhood and a holomorphic family Ω3,0(s) of nonwhere U ⊂ M zero holomorphic (3,0) forms. 2 Using Ω and Kodaira-Spencer theory we have the iso- morphism: T1,0 = H2,1(X ), and moreover, if we choose local holomorphic coordinates s M ∼ s zi, i = 1,...,h2,1(X), we have a basis for H2,1: ∂ Ω,Ω¯ χ eK/2Π2,1(∂ Ω) = eK/2 ∂ Ω h i iΩ . (2.16) i ≡ i i − Ω,Ω¯ (cid:18) h i (cid:19) Here K is a K¨ahler potential for the Weil-Peterson-Zamolodchikov (WPZ) metric: e−K = i Ω,Ω¯ . (2.17) h i A short calculation shows that g = i χ ,χ¯ . i¯j i ¯j − h i 2 In a more precise description we consider the Hodge line bundle over : = Rπ∗ωX/M M L where ωX/M is the relative dualizing sheaf. The fiber at s is H3,0(Xs). We choose a local holomorphic section of . L 6 2.2.2.Supersymmetry transformations The massless multiplets are the gravity multiplet, vectormultiplet, and hypermulti- plet. We generally follow the notation and conventions of [11] and denote the gravity multiplet as (g ,ψ ,A0), where the subscript A is an sl(2) R-symmetry index and A0 µν µA µ µ is the graviphoton. The vectormultiplets are denoted (zi,λAi,Ai), i = 1,...,n . The µ V vectormultiplet scalars are coordinates for a nonlinear sigma model with target the moduli of complex structures on a polarized 3-fold X: z : M (X). (2.18) 4 → M The kinetic energy follows from the WPZ metric. Variations δzi of the vectormultiplet scalars are related to tangents to , and are M also related by = 2 supersymmetry to the fieldstrengths of the vectormultiplets. Hence N we define the vectormultiplet fieldstrengths by: 1 Gi,− χ Π2,1( −) (2.19) i ⊗ ≡ −2 F The supersymmetry transformations for the associated gauginos must contain two terms corresponding to raising or lowering helicity: δλAi = i∂/ziǫA +Gi,−γµνεABǫ (2.20) µν B Here ǫA,ǫ are supersymmetry parameters of opposite chirality. εAB = iσ2 is a numerical B matrix. Since b (X) = 2+2h2,1(X) there is one remaining gauge field. This gauge field is the 3 graviphoton, whose fieldstrength is defined by the projection of − onto H0,3: F T− eK/2 Ω, − (2.21) ≡ h F i The corresponding susy transformation law is: δψ = ǫ +T−γνε ǫB µA Dµ A µν AB (2.22) 1 i ǫ = ∂ ωabγ + Q ǫ Dµ A µ − 4 µ ab 2 µ A (cid:0) (cid:1) where the covariant derivative is the standard spinor and Ka¨hler covariant derivative. Neglecting hypermultiplets, the bosonic part of the action is accordingly: 1 1 I = eR+ z 2 Im[τ¯ FI,−FJ,−] (2.23) boson IJ −2 k ∇ k −8π ZM4 where τ is the period matrix of the Weil Jacobian. IJ 7 2.3. Charge lattices Supersymmetric black holes and strings in D = 4,5,6-dimensional compactifications of string/M/F theory are charged under certain one-form or two-form gauge fields. The set of charges are valued in a lattice Λ. In the examples related to the attractor phenomenon the relevant charge lattices are given in Table 1. Td, = 32 K3 Td, = 16 X, = 8 N × N N II5,5 II21,5 H1,1(B;ZZ) II6,6 II21,5 H (X;ZZ) 2 II6,6 II21,5 H (X;ZZ) 4 ZZ28 ZZ28 II22,6 II22,6 H (X;ZZ),Hev(X˜;ZZ) el ⊕ mag el ⊕ mag 3 Table 1. In the first line we have listed charge lattices of 6D strings in various compactifications of type IIB string theory and F-theory. Here B is the base of an elliptic fibration π : X B. In the second and third lines we have listed the charge lattices of 5D bl→ack holes and strings, respectively. In the final line we have listed the electric/magnetic charge lattices of 4D supersymmetric black holes. A full explanation of all entries of this table is beyond the scope of this short review. We content ourselves with some explanation of the entry on the lower right corner, namely IIB on X. We have shown that the Lie algebra is such that g g∗ = H3(X;IR). To specify ∼ ⊕ the physical theory we must specify the corresponding Lie group G = IRrmod2πL where L IRr is a rank r “lattice.” 3 The electric charge lattice is the lattice of unitary irreps of ⊂ G and is just L∗ while the magnetic charge lattice is the lattice of chern classes of gauge bundles on a sphere at infinity and is just L. Together we form the symplectic rank 2r charge lattice Λ = L L∗. In this form it has a symplectic splitting. ⊕ Let us determine Λ for the compactification of IIB theory on X. The quantization of the abelian charges is justified by the existence of D-branes. For example, suppose a D3-brane wraps a real 3-cycle Σ in nine-dimensional space M in IIB theory. Suppose 3 9 Σ is a linking 5-cycle in M . Then: 5 9 G (2.24) 2π ZΣ5 3 See the remark below on the use of the word “lattice.” 8 measures the “number of enclosed D3-branes.” If there are no fractional D3-branes (2.24) is necessarily integral. In the compactification of IIB on M = M X described above 9 4 × supersymmetric configurations arise when a D3-brane wraps a supersymmetric 3-cycle Σ X [16]. If M is an asymptotically Minkowskian spacetime and we consider the 3 4 ⊂ compactification M X then we can choose the linking 5-cycle to be Σ = S2 Σ and 4 × 5 ∞ × 3 it follows that G = [Σ ],c ( ) (2.25) 3 1 2π h F i ZS∞2 ×Σ3 ZS∞2 where [Σ ] is the Poincar´e dual to the homologdy class [Σ ] and c ( ) = 1 [ ] is the 3 3 1 F 2π F Chern class of the G-bundle over S2 defining a topological sector of configuration space. ∞ d Thus, the magnetic charges are quantized and hence so are the electric charges. If the fundamental D3 brane has charge 1 then: Λ = H3(X;ZZ) (2.26) in IIB theory with the natural symplectic structure. Remark. In6Dthe charge latticeΛ ofstringsis, ongeneral principles a lattice inthesense that there is an integral symmetric bilinear form: ( , ) : Λ Λ ZZ [17]. In 4D the general · · × → principles only guarantee the existence of a nondegenerate integral symplectic structure on the electric/magnetic charge lattice. In the above discussion the word “lattice” means a rank r ZZ-submodule of IRr. In fact, in the theories under discussion L turns out to have an integral quadratic form and hence is a lattice in the usual sense. This does not follow from any general physical principles (as far as the author is aware) and is a deep consequence of mirror symmetry. Choosing a mirror map between a point of maximal unipotent monodromy and a large radius limit we have an isomorphism [18][19][20]: µ : Hodd(X;ZZ) Heven(X˜;ZZ) (2.27) → where X˜ is the mirror CY manifold. Hence the charge lattices Λ have a natural antisym- metric symplectic structure and a natural symmetric quadratic form. To choose a simple example, K3 T2 is self-mirror. By the Kunneth theorem: × H3(K3 T2;ZZ) = H2(K3;ZZ) H1(T2;ZZ) (2.28) ∼ × ⊗ and, if we make a choice of a,b cycles on T2, we can identify H3(K3 T2;ZZ) = II19,3 II19,3 (2.29) ∼ × ⊕ where II19,3 is the even unimodular lattice of signature (( 1)19,(+1)3). − 9