ANALYTIC PROPERTIES OF MIRROR MAPS

Duke Math

Description

Niveau: Supérieur, Doctorat, Bac+8
ANALYTIC PROPERTIES OF MIRROR MAPS C. KRATTENTHALER† AND T. RIVOAL†† Abstract. We consider a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin [Math. Notes 71 (2002), 604–616]. This family in- cludes many of the known one-variable mirror maps as special cases, in particular many of modular origin and the celebrated example of Candelas, de la Ossa, Green and Parkes [Nucl. Phys. B359 (1991), 21–74] associated to the quintic hypersurface in P4(C). In [Duke Math. J. 151 (2010), 175–218], we proved that all coefficients in the Taylor expan- sions at 0 of these canonical coordinates (and, hence, of the corresponding mirror maps) are integers. Here we prove that all coefficients in the Taylor expansions at 0 of these canonical coordinates are positive. Furthermore, we provide several results pertaining to the behaviour of the canonical coordinates and mirror maps as complex functions. In particular, we address analytic continuation, points of singularity, and radius of conver- gence of these functions. We present several very precise conjectures on the radius of convergence of the mirror maps and the sign pattern of the coefficients in their Taylor expansions at 0. 1. Introduction In the focus of this article there is a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin [39] (to be defined below).

• variable mirror


• generalised hypergeometric


• fuchsian differential


• mirror maps


• theorem provides precise


• multi-variable mirror


• nk ≥


• hypergeometric differential operator

Sujets

• Taylor

Informations

• Publié par: mijec
• Nombre de lectures: 11
• Langue: English

Extrait

ANALYTIC PROPERTIES OF MIRROR MAPS

C. KRATTENTHALER † AND T. RIVOAL ††

Abstract.We consider a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin [Math. Notes71(2002), 604–616]. This family in- cludes many of the known one-variable mirror maps as special cases, in particular many of modular origin and the celebrated example of Candelas, de la Ossa, Green and Parkes [Nucl. Phys.B359(1991), 21–74] associated to the quintic hypersurface inP 4 (C). In [Duke Math. J.151that all coefficients in the Taylor expan-(2010), 175–218], we proved sions at 0 of these canonical coordinates (and, hence, of the corresponding mirror maps) are integers. Here we prove that all coefficients in the Taylor expansions at 0 of these canonical coordinates are positive. Furthermore, we provide several results pertaining to the behaviour of the canonical coordinates and mirror maps as complex functions. In particular, we address analytic continuation, points of singularity, and radius of conver- gence of these functions. We present several very precise conjectures on the radius of convergence of the mirror maps and the sign pattern of the coefficients in their Taylor expansions at 0.

1.Introduction

In the focus of this article there is a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin [39] (to be defined below). This family contains many of the known one-variable mirror maps as special cases, including many of modular origin and the celebrated example of Candelas, de la Ossa, Green and Parkes [10] associated to the quintic hypersurface inP 4 (C the). Forgeometricsignificance of these maps see [10, 26, 27, 36]. Thenumber-theoreticproperties of the coefficients in the Taylor expansions at 0 of these canonical coordinates and mirror maps have been investigated recently in [20] (cf. [13] for a far-reaching generalisation). Our aim here is to provide an as detailed as possible analysis of theanalyticproperties of canonical coordinates and mirror maps. Apart from the intrinsic interest in this kind of investigation, one motivation comes from the hope of finding more applications of the Diophantine method of✭uniformisation ade´liquesimultan´ee✮atlbtynoe´2[,]onofAndr.nsioludom-notautisra

Date: February 26, 2011. 2000Mathematics Subject Classification.Primary 30B10; Secondary 11F11 14J32 30B40 33C20. Key words and phrases.mirror maps, canonical coordinates, analytic continuation, singular expansion, generalised hypergeometric functions, modular forms. † Research partially supported by the Austrian Science Foundation FWF, grants Z130-N13 and S9607- N13, the latter in the framework of the National Research Network “Analytic Combinatorics and Proba- bilistic Number Theory”. †† Research partially supported by the project Q-DIFF, ANR-10-JCJC-0105, of the french “Agence Na- tionale de la Recherche”. 1

2

Another connection between number theory and mirror maps can be found in the study ofthearithmeticnatureofvaluesoftheRiemannzetafunctionatintegers.Ape´ry’sproof of the irrationality ofζ(3) (cf. [35]) was recast in terms of modular forms by Beukers [7]. The search for an extension of Beukers’ ideas toζ(n),n≥4, led Almkvist and Zudilin [1] to study systematically mirror maps associated to Fuchsian differential equations, not necessarily of hypergeometric type. In [21], the authors showed that, in fact, many of the examples of Beukers and of Almkvist and Zudilin can be obtained from suitable specialisa- tions of hypergeometric multi-variable mirror maps. It would therefore be of great interest to extend the investigations undertaken in this paper addressing the analytic behaviour of the family of mirror maps to be introduced below to the family of multi-variable mirror maps in [21]. Let us now introduce this family of canonical coordinates and corresponding mirror maps. For a given integerN≥1, letr 1  r 2      r d denote the integers in{12     N}which are coprime toN. It is well-known thatd=ϕ(N), Euler’s totient function, which is given by ϕ(N) =N Q p|N  1− p 1  SetC N :=N ϕ(N) Q p|N p ϕ(N)(p−1) which is an integer becausep−1 dividesϕ(N) for any primepdividingNLet us also define the Pochhammer symbol (α) m for complex numbersαand non-negative integersmby (α) m :=α(α+ 1)  (α+m−1) ifm≥1, and (α) 0 [39, Lemma 1]) that, for any integer It can be proved (see := 1.m≥0, (N) B N (m) :=C N ϕ m Y (r j N m ) ! m (1.1) j=1 is an integer. Let us consider the hypergeometric differential operatorLdefined by L:=  z d d z  ϕ(N 1 )++ϕ(N k ) −C N z k j Y =1i ϕ = ( Y N 1 j )  z d dz + N r j ji  (1.2) Here,C N =C N 1 C N 2   C N k and ther ij ∈ {12     N j }form the residue classes modulo N j which are coprime toN j . Unlessk= 1 andN= (2), the differential equationLy= 0 is of order≥ Set can construct two solutions as follows.2. WeH(x m) := P m n= − 0 1 x+ 1 n and ϕ(N) H N (m) := X H(r j N m)−ϕ(N)H(1 m) j=1 Then,F N (z) andG N (z)+log(z)F N (z) are twoC-linearly independent solutions toLy= 0, where F N (z) := m ∞ X =0  j= k Y 1 B N j (m)  z m and G N (z) := m ∞ X =1  j= k X 1 H N j (m)  j Y = k 1 B N j (m)  z m (1.3)

3

and where log(z simplicity, we write For) denotes the principal branch of the logarithm. B N (m) := Q j k =1 B N j (m) andH N (m) := P j k =1 H N j (m). SinceB 1 (m) = 1 andH 1 (m) = 0 for allm≥0, the seriesF N (z) andG N (z) do not change if one omits or adds components of 1 from/toN. We may therefore assume without loss of generality thatN j ≥2 for allj, which we shall do throughout the paper. The power seriesF N (z) andG N (z) have radius of convergence 1C N  prove in. We Section 3 that the functionsF N (z) andG N (z) + log(z)F N (z) can be analytically continued toC[1C N +∞) andC  (−∞0]∪[1C N +∞)  , respectively. Given the notation above, we define thecanonical coordinateq N (z) as the exponential of the quotient of the above two solutions, that is, by

q N (z) :=zexp(G N (z)F N (z))(1.4) Its compositional inverse, which we denote byz N (q), is called (the corresponding)mirror map. Whenk= 1 andN= (2), we haveF (2) (z) = (1−4z) −12 , which satisfies the differential equation (1−4z)y ′ −2y= 0The functionG (2) (z) + log(z)F (2) (z) defined formally by the above formula is not solution of that differential equation, but it turns out that all theorems stated below are still true in this case because q (2) (z) = (1− √ 1−4z) 2 (4z)(1.5) However, certain proofs do not work for this case, and we will say when. The special caseN= (5) has been of particular interest since it produces the earlier mentioned example of Candelas et al. [10].

In [20], we proved that, for any positive integersN 1  N 2      N k , the canonical coordinate q N (z first result says that these coefficients are, in) has integral Taylor coefficients. Our fact, positive. Its proof is given in Section 2. An essential ingredient there is a classical result of Kaluza [19] on the sign of coefficients in certain power series expansions (see Lemma 2.2).

Theorem 1.1.For all integersN 1  N 2      N k ≥2, all Taylor coefficients ofq N (z)at0 are positive, except the constant coefficient.

A problem that suggests itself at this point is to find a combinatorial interpretation for the Taylor coefficients ofq N (z) or ofz N (q) (even if the latter may have negative coefficients, see Conjecture 1.8 below). Some progress in this direction can be found in [23]. The next theorem provides precise information on the radius of convergence and the asymptotic behaviour of the Taylor coefficients of the canonical coordinateq N (z) as a power series inz. Here, and in the sequel, givenN= (N 1  N 2      N k ), we employ the notation k Φ N := X ϕ(N j )(1.6) j=1

Theorem 1.2.For all integersN 1  N 2      N k ≥2, the following assertions hold:

4

(i)The radius of convergence of the Taylor series ofq N (z)is equal to1C N and the Taylor series converges for anyzsuch that|z|= 1C N . (ii)The functionq N (z)has a singularity atz= 1C N . (iii)For anyzsuch that|z| ≤1C N , we have|q N (z)| ≤1 (iv)IfΦ N = 1, then them-th Taylor coefficient ofq (2) (z)is equal to them-th Catalan number m 1 1+  2 m m  ,m≥1, and, hence, asm→ ∞, it is equal to 4 m m 32 (1 +o(1)) √  π (v)IfΦ N = 2, then, asmtends to∞, them-th Taylor coefficient ofq N (z)is equal to C m const× N mlog 2 (m 1(+ ) o(1)) where throughout the symbol “const” stands for a non-zero constant. (vi)IfΦ N ≥3, then, asmtends to∞, them-th Taylor coefficient ofq N (z)is equal to C m const× m Φ N N 2 (1 +o(1)) In (iii), the inequality is always strict, except atz= 1C N when the seriesF N (1C N ) diverges to +∞, which happens in five cases:k= 1 N 1 = 234 or 6, andk= 2,N 1 = N 2 = 2. Theorem 1.2 is proved in Section 5. There, the analytic continuation ofF N (z) and G N (zimportant role, as well as the fine behaviour), which is discussed in Section 3, plays an of these functions near the point 1C N , which is discussed in detail in Section 4. By (1.4), the canonical coordinateq N (z) is only defined forzin the disk of convergence of the seriesG N (z) andF N (z) involved in its definition. The knowledge of the analytic continuation ofF N (z) andG N (z2[ay]9ofemolP´atthorhebmnideiwitno,3oc)fromSec on zeroes of hypergeometric functions, allows us to show thatq N (z) can be continued to a function of the entire complex plane except for a branch cut. The corresponding proof is the subject of Section 6.

Theorem 1.3.For all integersN 1  N 2      N k ≥2, the following assertions hold: (i)The power seriesq N (z)can be continued to an analytic function onC[1C N +∞). (ii)The point1C N is a branch point. (iii)We have z→∞ xp  −πcot(πM  limq N (z) =−e N )(1.7) whereM N = max(N 1      N k ), and where the limit has to be performed along a path that avoids the cut[1C N +∞). The monodromy ofq N (z) atz= 1C N follows from applying the well-known monodromy theory of solutions to (generalised) hypergeometric equations (cf. [8]) to the seriesF N (z) andG N (z) + log(z)F N (z).