Integrality of Instanton Numbers and P-Adic B-Model

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. − ∗ = 3 Y ϕ Y δ ψ. We considered quintic as a Calabi–Yau complex threefold. M Z The above statements show that it is natural to apply p-adic However, it is possible to consider it and over or over Z methods attempting to prove integrality of instanton numbers. the ring p of integer p-adic numbers and to study its coho- Z Recall that B-model is formulated in terms of Hodge fil- mology over p. It is well known that the Hodge filtration and tration on the middle-dimensional cohomology of Calabi–Yau weight filtration on cohomology can be defined also in this case manifold; one should consider deformation of complex struc- [10–12]. It is natural to assume that in p-adic setting all of the ture of Calabi–Yau manifold and corresponding variations of statements (1)–(4) remain valid. This can be proven under cer- Hodge filtration. Analogous problems can be considered in tain conditions, however, the proof is not simple. A rigorous p-adic setting. proof for general Calabi–Yau threefolds is given in [5].(For If we work with mirror quintic B (and more general if the quintic one can derive these statements from known integrality moduli space M of complex structures on a Calabi–Yau three- of mirror map [13] and integrality of one of periods.) Notice, fold is one-dimensional) then one can find a local coordinate that the Yukawa coupling in p-adic situation remains the same, q in a neighborhood of maximally unipotent boundary point but the coefficients mk are considered as p-adic integers. 0 1 It is important to emphasize that in our consideration we of M (canonical coordinate) and a basis e (q), e (q), e1(q), should assume that the manifold at hand remains nonsingular e0(q) in three-dimensional cohomology that satisfy the follow- ing conditions: after reduction with respect to prime number p. This require- ment can be violated for finite number of primes. (For example, 0 1 0 the mirror quintic B becomes singular after reduction mod 5.) (1) The basis e , e , e1, e0 is symplectic: e0,e =−1, 1 In the p-adic theory there exists an additional symmetry: the e1,e =1, all other inner products vanish. (2) Gauss–Manin connection acts in the following way: so-called Frobenius map. (In the situation we need the Frobe- nius map was analyzed in [10,12].1) Namely, the map q → qp 0 ∇δe = 0, (8) of the moduli space of quintics into itself can be lifted to a ho- 1 0 momorphism Fr of cohomology groups of corresponding quin- ∇δe = e , (9) tics. Here q is considered as a formal parameter or as a p-adic ∇ e = Y(q)e1, (10) δ 1 integer; cohomology are taken with coefficients in Zp. We will ∇δe0 = e1. (11) express instanton numbers in terms of this map, assuming that p>3. d ∇ Here δ stands for logarithmic derivative q dq and δ for corre- Notice first of all that the Frobenius map Fr preserves the sponding Gauss–Manin covariant derivative weight filtration Wk. It does not preserve the Hodge filtra- (3) tion F p, but it has the following property: 0 ∈ F 0 ∩ W e 0, Fr F s ⊂ psF 0. (12) e1 ∈ F 1 ∩ W , 2 Notice that to prove (12) one should assume that p>3(the 2 e1 ∈ F ∩ W4, proof is based on the inequality s

 =±1. In what follows we assume that  = 1; the modifica- Conversely, one can express the Frobenius map in terms of in- tions necessary in the case  =−1 are obvious. stanton numbers using Lemma 3 and (20)–(24). (Notice, that Taking into account (12) and (13) we can write we are talking about Frobenius map in canonical coordinates.) One should notice that we assumed that diagonal entries of the 0 = 0 Fr e e , (15) matrix of the Frobenius map are positive; the assumption that 1 1 0 Fr e = pe + pm12e , (16) they are negative leads to the change of sign of all entries of this 2 2 1 2 0 matrix. One should mention also that our considerations do not Fr e1 = p e1 + p m23e + p m13e , (17) fix the value of m at the point q = 0; it seems, however, that = 3 + 3 + 3 1 + 3 0 14 Fr e0 p e0 p m34e1 p m24e p m14e , (18) one can prove that this value is equal to zero. where mij ∈ Zp[q] are q-series with integer p-adic coefficients. (The powers of p in RHS come from (12).) Using (13) we can 3. Integrality of instanton numbers: general case obtain also that In the considerations of Section 2 we restricted ourselves − + = − + + = m34 m12 0, m23m34 m24 m13 0. to the case of quintic or, more generally, to the case when the Applying Fr to (9) and using (14) we obtain moduli space of complex structures is one-dimensional. Let us analyze the case when the dimension of moduli 0 + 0 = 0 pe pδm12e pe . (19) space M of complex structures is equal to r>1. This means that m is a constant. One can prove [5] that Under certain conditions one can find a basis in three- 12 ∈ 3,3 m = 0. We see that dimensional cohomology consisting of vectors e0 I , 12 2,2 1 r 1,1 0 0,0 p,p e1,...,er ∈ I , e ,...,e ∈ I , e ∈ I where I stands p m34 = m12 = 0,m24 + m13 = 0. (20) for F ∩ W2p. In appropriate coordinate system q1,...,qr on M (in canonical coordinates) Gauss–Manin connection acts Similarly, from (10) we obtain   in the following way: Y(q)− Y qp = δm ,m+ δm = 0 (21) 23 23 13 ∇ 0 = δi e 0, hence ∇ k = =   δi e δike0,k1,...,r, p − = 2  Y q Y(q) δ m13. (22) ∇ = k = δi ej Yij k (q)e ,j 1,...,r, From (11) we see that k ∇ = δi eo ei. m34Y + δm24 = m23,δm34 = 0, Here ∇δ denotes the covariant derivative that corresponds to m24 + δm14 = m13. (23) j the logarithmic derivative δj = qj ∂/∂qj . We know that m34 = 0, hence m23 = δm24. Using the last equa- We are working in the neighborhood of maximally unipo- tion in (23) and (20) we conclude that tent boundary point q = 0. We assume that the B-model at hand can be obtained as a mirror of A-model. Then one can say that 2m = δm . (24) 13 14 the so-called integrality conjecture of [6] (see also [7], Sec- Combining this equation with (22) we obtain tion 5.2.2) is satisfied; this is sufficient to derive the above rep- resentation for Gauss–Manin connection. As in the case r = 1 Lemma 3. the conditions we imposed leave some freedom in the choice of   canonical coordinates and of the basis. We will require that the p 1 3 Y q − Y(q)= δ m14. (25) vectors e0(0), e1(0),...,er (0) constitute a Z-basis of W . 2 2 The Yukawa couplings Yij k (q) can be considered as power Theorem. Instanton numbers are p-adic integers if p>3. series with respect to canonical coordinates q1,...,qr ; these series have integral coefficients. (The integrality of these coef- This statement follows immediately from Lemmas 2 and 3. ficients will not be used in the proof; it can be derived from the Eq. (25) together with (5) leads to the representation of in- integrality of Frobenius map and the formula (27).) stanton numbers in terms of Frobenius map: Again one can prove [5] that the Gauss–Manin connection    has the same form in p-adic situation; the Yukawa couplings r d3 Q [ ] n a = μ M a , (26) should be considered as elements of p q1,...,qr in this case. p r d p d r3 d|r The Frobenius map has the form where we assume that r is not divisible by p and use the Fr e0 = e0, notation M for the coefficients of power series expansion k k = k + k 0 − 1 Fr e pe p(m12) e , of 2 m14: 2 2 k 2 0  Fr ej = p ej + p (m23)jke + p (m13)j e , 1 k − m14 = Mkq . = 3 + 3 j + 3 k + 3 0 2 Fr e0 p e0 p (m34) ej p (m24)ke p m14e , M. Kontsevich et al. / Physics Letters B 637 (2006) 97–101 101

j j k where (m34) + (m12) = 0, (m23)jk(m34) + (m13)j + It is clear from this formula that instanton numbers are (m24)j + 0. p-adic integers for p>3. (We use the fact that matrix elements We can repeat the considerations of the case r = 1 to obtain of Frobenius map, in particular, m14 are power series having p-adic integers as coefficients.) Notice, that the appearance of (m )j = (m )j = 0, 34  12 negative power of d in (30) does not contradict p-adic integral- p Yij k q − Yij k (q) = δi(m23)jk = δiδj (m13)k, ity, because d is not divisible by p.

2(m13)j = δj m14. Acknowledgements We come to a conclusion that     1 We are indebted to A. Goncharov, S. Katz and A. Ogus for Y qp − Y (q) = δ δ δ m . (27) ij k ij k i j k 2 14 useful discussions. We acknowledge the stimulating atmosphere of IHES where This equation permits us to prove integrality of instanton num- this project was started. bers in the case at hand. The second author was partially supported by NSF grant Recall that the Yukawa couplings can be represented in the DMS-0505735. form The third author was partially supported by NSF grant DMS-  qs 0401164. Y (q) = const + n s s s ij k s 1 − qs i j k s References  qs = const + n s sisj sk. (28) d3 d s,d|s [1] P. Candelas, X. de la Ossa, P. Green, L. Parkes, Nucl. Phys. B 359 (1991) 21. = s = s1 ··· sr [2] A. Givental, alg-geom/9701016. Here s is a multiindex: s (s1,...,sr ) and q q1 qr .In the second sum d runs over all positive integers dividing the [3] R. Gopakumar, C. Vafa, hep-th/9809187; integer vector s. R. Gopakumar, C. Vafa, hep-th/9812127. [4] V. Vologodsky, On the canonical coordinates, in preparation. The numbers ns can be identified with instanton numbers of [5] M. Kontsevich, A. Schwarz, V. Vologodsky, On integrality of instanton the mirror A-model. The formula (28) remains correct in p-adic numbers, in preparation. setting if we consider ns as p-adic numbers. [6] D. Morisson, alg-geom/9202004; Comparing the above formula with (27) we see that D. Morisson, alg-geom/9304007; D. Morisson, alg-geom/9609021.  n pas [7] D. Cox, S. Katz, Mirror Symmetry and Algebraic Geometry, American d = M a , (29) Mathematical Society, Providence, 1999. d3 p s d|s [8] P. Candelas, X. de la Ossa, F. Rodriguez-Villegas, hep-th/0012233; P. Candelas, X. de la Ossa, F. Rodriguez-Villegas, hep-th/0402133. − 1 [9] D. Wan, alg-geom/0411464. where Mr are coefficients of the power expansion of 2 m14 and the vector s in (29) is not divisible by p. Using Moebius [10] B. Mazur, Ann. Math. 98 (1973) 58. inversion formula we obtain from (29) an expression of instan- [11] J. Steenbrink, Invent. Math. 31 (1976). [12] G. Faltings, Crystalline cohomology and p-adic Galois representations, ton numbers ns in terms of Mr ; this expression has the form in: Algebraic Analysis, Geometry and Number Theory, 1989.  = −3 [13] B. Lian, S.T. Yau, hep-th/9507151. npar d μ(d)M par , (30) [14] A. Schwarz, I. Shapiro, Supergeometry and arithmetic geometry, in prepa- d d|r ration. [15] J. Stienstra, alg-geom/0212061; where r is an integer vector that is not divisible by p and the J. Stienstra, alg-geom/0212067. sum runs over all positive divisors of r.