Physics Letters B 637 (2006) 97–101 www.elsevier.com/locate/physletb Integrality of instanton numbers and p-adic B-model Maxim Kontsevich a, Albert Schwarz b,∗, Vadim Vologodsky c a IHES, 91440 Bures-sur-Yvette, France b Department of Mathematics, University of California, Davis, Davis, CA 95616, USA c Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA Received 27 March 2006; accepted 7 April 2006 Available online 24 April 2006 Editor: L. Alvarez-Gaumé Abstract We study integrality of instanton numbers (genus zero Gopakumar–Vafa invariants) for quintic and other Calabi–Yau manifolds. We start with the analysis of the case when the moduli space of complex structures is one-dimensional; later we show that our methods can be used to prove integrality in general case. We give an expression of instanton numbers in terms of Frobenius map on p-adic cohomology; the proof of integrality is based on this expression. © 2006 Elsevier B.V. All rights reserved. 1. Introduction a particular case of GV-invariants. However, such a proof is unknown; moreover, there exists no rigorous definition of GV- The basic example of mirror symmetry was constructed invariants. in [1]. In this example one starts with holomorphic curves on The goal of present Letter is to prove the integrality of the quintic A given by the equation instanton numbers. However, we will be able to check only a weaker statement: the instanton numbers become integral af- 5 + 5 + 5 + 5 + 5 + = x1 x2 x3 x4 x5 ψx1x2x3x4x5 0 ter multiplication by some fixed number. We work in the frame- in projective space. (In other words, one considers A-model work of B-model definition. In Section 2 we consider the case on this quintic.) Mirror symmetry relates this A-model to when the moduli space of deformations of complex structure the B-model on B (on the quintic factorized with respect to on a Calabi–Yau threefold is one-dimensional. The proof can 3 the finite symmetry group (Z5) ). Instanton numbers are de- be generalized to the case when the moduli space is multidi- fined mathematically in terms of Gromov–Witten invariants, mensional (Section 3). The considerations of the Letter are not i.e., by means of integration over the moduli space of curves. rigorous. To make the Letter accessible to physicists we have The moduli space is an orbifold, therefore it is not clear that hidden mathematical difficulties in the exposition below. The this construction gives integer numbers. The mirror conjec- paper [5] will contain a rigorous mathematical proof of the re- ture proved by Givental [2] permits us to express the instanton sults of present Letter. numbers in terms of solutions of Picard–Fuchs equations on We will use freely the well-known mathematical results mirror quintic B; however, integrality is not clear from this about sigma-models on Calabi–Yau threefolds; see, for exam- expression. Gopakumar and Vafa [3] introduced BPS invari- ple, [6] or [7]. We will follow the notations of [7]. ants that are integer numbers by definition; it should be pos- The proof of integrality of instanton numbers is based on sible to prove that instanton numbers can be considered as an important statement that these numbers can be expressed in terms of arithmetic geometry. May be, the fact that physical quantities can be studied in terms of number theory is more * Corresponding author. E-mail addresses: [email protected] (M. Kontsevich), significant than the proof itself. [email protected] (A. Schwarz), [email protected] Instanton numbers we consider can be identified with (V. Vologodsky). genus 0 Gopakumar–Vafa (GV) invariants. GV-invariants can 0370-2693/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2006.04.012 98 M. Kontsevich et al. / Physics Letters B 637 (2006) 97–101 be expressed in terms of Gromov–Witten invariants, but their To prove the statement we notice that the following expres- integrality is not clear from this expression. One can give sion for mk in terms of nk can be derived from (1): a condition of integrality of GV-invariants in terms of Frobe- = 3 nius map generalizing Lemma 2 of present Letter. It seems mk nd d . (3) that GV-invariants also can be expressed in terms of p-adic d|k B-model. Let us suppose that k = pαr where r is not divisible by p. The relation between topological sigma-models and num- Then ber theory was anticipated long ago. The existence of such − = α+1 3 a relation is strongly supported by the fact that Picard–Fuchs mkp mk npα+1s p s . (4) equations that play important role in B-model appear also in s|r Dwork’s theory of zeta-functions of manifolds over finite fields. (We are summing over all divisors of kp that are not divisors Calabi–Yau manifolds over finite fields and arithmetic ana- of k, i.e., over all pα+1s, where s|r.) log of mirror conjecture where considered in very interesting 3(α+1) We see immediately that mkp − mk is divisible by p . papers by Candelas, de la Ossa and Rodriguez-Villegas [8],see To derive integrality of nk from this property one can use the also [9]. We go in different direction: our main goal is to obtain Moebius inversion formula the information about sigma-models over complex numbers us- 3 ing methods of number theory. nkk = μ(d)m k , (5) d Notice that p-adic methods were used in [13] to prove inte- d|k grality of mirror map for quintic and in [4] to prove integrality where μ(d) stands for Moebius function. Recall, that Moebius in general case. The idea to use the Frobenius map on p-adic co- function can be defined by means of the following properties homology to prove some integrality statements related to mirror μ(ab) = μ(a)· μ(b) if a and b are relatively prime, μ(p) =−1 symmetry appeared in [15]. if p is a prime number, μ(pα) = 0ifα>1. Again we rep- In arithmetic geometry one can consider Hodge structure on resent k as pαr where p does not divide r and a divisor d cohomology; this means that one can define the main notions of k as pβ s where s|r and β α. Taking into account that of B-model theory in p-adic framework [11,12]. Our computa- μ(pβ s) = 0ifβ>1 we obtain tions are based on the fact that in the situation we consider one can obtain the information about the conventional sigma-model 3 nkk = μ(s)m k + μ(sp)m k s sp from analysis of its p-adic analog. This is a nontrivial mathe- s|r matical fact; however, in this Letter we will skip the justification = μ(s)(m k − m k ). (6) of this statement referring to [5]. s sp s|r 2. Integrality of instanton numbers: the simplest case It follows from our assumption that the left-hand side of (5) is 3α divisible by p , hence nk does not contain p in the denomina- Instead of working with A-model we consider mirror B- tor (in other words, nk can be considered as an integer p-adic model. number). In the above calculation we assumed that α 1; the Our starting point is the well-known formula relating the case α = 0 is trivial. If the condition (2) is satisfied for every Yukawa coupling Y in canonical coordinates (normalized prime p we obtain that the numbers nk are integers. Yukawa coupling) to instanton numbers nk: ∞ d Lemma 2. The numbers nk defined in terms of Y(q)by the for- 3 q Y(q)= const + nd d . mula (1) are integers if and only if for every prime p there exists 1 − qd k d=1 such a series ψ(q)= skq having p-adic integer coefficients (This formula is valid in the case when the moduli space that of complex structures is one-dimensional; for the quintic Y(q)− Y(qp) = δ3ψ(q). (7) const = 5.) d Here δ stands for the logarithmic derivative q dq . Lemma 1. Let us assume that ∞ d ∞ It is easy to check that this lemma is a reformulation 3 q k nd d = mkq . (1) of Lemma 1.Thekth coefficient of the decomposition of 1 − qd p d=1 k=1 Y(q)− Y(q ) into q-series is equal to If the numbers n are integers then for every prime number p k m − m k = m α − m α−1 3(α+1) k p s p s the difference mkp − mk is divisible by p where α is p α defined as the number of factors equal to p in the prime de- if k = p s and α 1, and to mk if k is not divisible by p.From 3 3 composition of k. Conversely, if the other side, the coefficients of δ ψ(q) are equal to k sk. + Notice, that Lemma 2 can be formulated in terms of the p3(α 1)m − m (2) kp k Frobenius map ϕ. This map transforms q into qp; correspond- ∗ for every prime p and every k, the numbers nk are integers. ing map ϕ on functions of variable q transforms f(q) into M. Kontsevich et al. / Physics Letters B 637 (2006) 97–101 99 f(qp). The formula (7) can be rewritten in the form All of the statements above are well known; see, for exam- ple, [7], Chapters 5 and 6.