ELLIPTIC GENUS OF CALABI–YAU MANIFOLDS

AND JACOBI AND SIEGEL MODULAR FORMS

V. Gritsenko^{1}

Abstract. In the paper we study two types of relations: a one is between the elliptic genus of Calabi–Yau manifolds and Jacobi modular forms, another one is between the second quantized elliptic genus, Siegel modular forms and Lorentzian Kac–Moody Lie algebras. We also determine the structure of the graded ring of the weak Jacobi forms with integral Fourier coefficients. It gives us a number of applications to the theory of elliptic genus and of the second quantized elliptic genus.

Introduction

For a compact complex manifold one can define its elliptic genus as a function in two complex variables. If the first Chern class c_{1}(M) of the complex manifold is equal to zero in H^{2}(M, ℝ), then the elliptic genus is an automorphic form in variables τ ∈ ℍ_{1} (ℍ_{1} is the upper-half plane) and z ∈ ℂ. More exactly, it is a Jacobi modular form with integral Fourier coefficients of weight 0 and index d/2, where d =dim_{ℂ}(M). The q^{0}-term of the Fourier expansion (q = e^{2πiτ}) of the elliptic genus is essentially equal to the Hirzebruch χ_{y}-genus. Thus we can analyze the arithmetic properties of the χ_{y}-genus of the Calabi– Yau manifolds and its special values such as signature (y = 1) and Euler number (y = −1) in terms of Jacobi modular forms. The famous Rokhlin theorem about divisibility by 16 of the signature of a compact, oriented, differentiable spin manifold of dimension 4 was one of the starting points of the theory of elliptic genera. Ochanine generalized this Rokhlin result to the manifolds of dim_{ℝ}M ≡ 4 mod 8. One can find an elegant proof the Ochanine’s theorem using modular forms in one variable with respect to Γ_{0}(2) in the lectures of Hirzebruch [HBJ, Chapter 8].

In this paper we study the ℤ-structure of the graded ring J_{0,∗}^{ℤ} = ⊕_{m≥1}J_{0,m}^{ℤ} of all weak Jacobi forms with integral Fourier coefficients (Theorem 1.9). We prove that this ring has four generators

J_{0,∗}^{ℤ} = ℤ[ϕ_{0,1}, ϕ_{0,2}, ϕ_{0,3}, ϕ_{0,4}]

where ϕ_{0,1}, …, ϕ_{0,4} are some fundamental Jacobi forms related to Calabi–Yau manifolds of dimension d = 2, 3, 4, 8. We consider some applications of this result to Calabi–Yau manifolds. Properties of the signature (i.e. the value of χ_{y}-genus at y = 1) modulo some powers of 2 are well known (see (2.8)). We analyze properties of the value of the elliptic genus and the Hirzebruch χ_{y}-genus at y = −1, y = ζ_{3} = e^{2πi/3} and y = i. For example, we prove that the Euler number of a Calabi-Yau manifold M_{d} of dimension d satisfies

1Supported by Max-Planck-Institut für Mathematik in Bonn

e(M_{d}) ≡ 0 mod 8 if d ≡ 2 mod 8

(see Proposition 2.4) and

χ_{y=ζ_{3}}(M_{d}) ≡ 0 mod 9 if d ≡ 2 mod 6

(see Proposition 2.5 for details).

The same Jacobi forms ϕ_{0,1}, …, ϕ_{0,4} are generating functions for the multiplicities of all positive roots of the four generalized Lorentzian Kac–Moody Lie algebras of Borcherds type constructed in [GN1–GN4]. In §3 we show that the second quantized elliptic genus (introduced by R. Dijkgraaf, E. Verlinde and H. Verlinde in [DVV]) of an arbitrary Calabi– Yau manifolds of dimension 4 and 6 can be written as a product of some powers of the denominator functions of the Lorentzian Kac–Moody algebras constructed in [GN4].

The results of this paper were presented at the seminar of Prof. Y. Manin and at the seminar ”Arithmetic Lunch” of MPI in Bonn in the spring semester of 1997. The author is grateful to Max-Planck-Institut für Mathematik for hospitality.

§1. Elliptic genus and weak Jacobi forms

Let M be a compact complex manifold of complex dimension d and T_{M} be the holomorphic tangent bundle of M. One defines the formal power series

𝔼_{q,y} = ⨂_{n≥0} ⋀_{−y^{−1}q^{n}}T_{M} ⊗ ⨂_{n≥1} ⋀_{−yq^{n}}T_{M}^{∗} ⊗ ⨂_{n≥1} S_{q^{n}}(T_{M} ⊕ T_{M}^{∗})

where ∧^{k} and S^{k} denote the k-th exterior product and k-th symmetric tensor product respectively and T_{M}^{∗} denotes the dual bundle. The elliptic genus of the complex manifold is the holomorphic Euler characteristic of this formal power series with vector bundle coefficients. This definition is standard in physical literature (see, for example, [W], [EOTY], [KYY], [AYS], [D]).

Definition. The elliptic genus of M is defined as follows

χ(M; τ, z) := y^{\frac{d}{2}} ∫_{M} ch(𝔼_{q,y}) td(M)  (1.1)

where td(M) is the Todd class of M. One applies the Chern character in ch(𝔼_{q,y}) to all coefficients of 𝔼_{q,y} and the integral ∫_{M} denotes the evaluation of the top degree differential form on the fundamental cycle of the manifold.

One can consider this definition as a limit case of the level N elliptic genus of Hirzebruch (see [H], [HBJ], [K], [Hö]). See also §3 for a physical definition. The coefficient f(m, l) of the elliptic genus

χ(M; τ, z) = ∑_{n≥0,l} f(n, l)q^{n}y^{l}

is equal to the index of the Dirac operator (see, for example, [HBJ, Appendix II]) twisted with the vector bundle E_{m,l− \frac{d}{2}} , where 𝔼_{q,y} = ⨁_{m, n} E_{m,n} · q^{m}y^{n}. In particular all coefficients of the elliptic genus are integral. According to the Riemann–Roch–Hirzebruch theorem one can see that the q^{0}-term of χ(M; τ, z) is essentially the Hirzebruch χ_{y}-genus of the manifold M:

χ(M; τ, z) = ∑_{p=0}^{d}(−1)^{p}χ_{p}(M) y^{\frac{d}{2} −p}+  (1.2)

q( ∑_{p=−1}^{d+1}(−1)^{p}y^{−p}(χ_{p}(M, T_{M}^{∗}) − χ_{p−1}(M, T_{M}^{∗}) + χ_{p}(M, T_{M}) − χ_{p+1}(M, T_{M}))) + …

where χ(M, E) = ∑_{q=0}^{d}(−1)^{q} dim H^{q}(M, E) and χ^{p}(M, E) = χ(M, ∧^{p}T_{M}^{∗} ⊗ E) or, for a Kähler manifold, χ_{p}(M) = ∑_{q}(−1)^{q}h^{p,q}(M).

It turns out that the elliptic genus of a Calabi–Yau d-fold is a weak Jacobi form of weight zero and index d/2. The Jacobi modular forms are modular forms with respect to the Jacobi group or in other words with respect to a parabolic subgroup of the integral symplectic group. In this paper we consider the Jacobi forms with respect to a maximal parabolic subgroup Γ_{∞} ⊂ Sp_{2}(ℤ), which consists of all element preserving an isotropic line. The Jacobi group Γ^{J} := Γ_{∞}/{±E_{4}} ≅ SL_{2}(ℤ) ⋉ H(ℤ). is the semi-direct product of SL_{2}(ℤ) and the integral Heisenberg group H(ℤ) which is the central extension of ℤ × ℤ. The binary character v_{H}([λ, μ; κ]) := (−1)^{λ+μ+λμ+κ} can be extended to a binary character v_{J} of the Jacobi group if we put v_{J}|_{SL_{2}(ℤ)} = 1. In this paper we use Jacobi forms of integral or half integral index.

Definition. A holomorphic function ϕ(τ, z) on ℍ_{1} × ℂ is called a weak Jacobi form of weight k ∈ ℤ/2 and index t ∈ ℤ/2 with respect to the Jacobi group Γ^{J} if the function ϕ̃(Z) := ϕ(τ, z) exp (2πitω) on the Siegel upper half-plane ℍ_{2} = \{Z = ( \begin{matrix}τ & z \\ z & ω \end{matrix} ) ∈ M_{4}(ℂ), Im(Z) > 0} is a Γ^{J}-modular form of weight k with character (or multiplier system) v_{J}^{2t}, i.e., if it satisfies the functional equations

ϕ(\frac{aτ + b}{cτ + d}, \frac{z}{cτ + d}) = (cτ + d)^{k}e^{2πitcz^{2}/(cτ+d)}ϕ(τ, z)  (( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ))  (1.3a)

and

ϕ(τ, z + λτ + μ) = (−1)^{2t(λ+μ)}e^{−2πit(λ^{2}τ+2λz)}ϕ(τ, z)  (λ, μ ∈ ℤ).  (1.3b)

and it has the Fourier expansion of the type

ϕ(τ, z) = ∑_{n≥0, l∈t+ℤ} f(n, l) exp (2πi(nτ + lz)).

We denote the space of all weak Jacobi forms of weight k and index t by J_{k,t}.

If the Fourier coefficients satisfy f(n, l) = 0 unless 4tn − l^{2} ≥ 0, then ϕ is holomorphic at infinity. We call such Jacobi forms holomorphic Jacobi forms. Weak Jacobi forms of integral index were firstly defined in [EZ]. The main advantage of this notion is that J_{k,t} is still finite dimensional and the graded ring of all weak Jacobi forms is finitely generated (over ℂ). In [EZ] only Jacobi forms of integral index were considered. In the next example we define the main week Jacobi forms of half-integral index (see Lemma 1.4 bellow).

Example 1.1. Weak Jacobi forms ϕ_{0, \frac{3}{2}} and ϕ_{−1, \frac{1}{2}} . Let us define a Jacobi theta-series

ϑ(τ, z) = ∑_{n≡1 mod 2} (−1)^{\frac{n−1}{2}} exp (\frac{πi n^{2}}{4} τ + πi nz) = ∑_{m∈ℤ} (\frac{−4}{m}) q^{m^{2}/8} y^{m/2}
= −q^{1/8}y^{−1/2} ∏_{n≥1} (1 − q^{n−1}y)(1 − q^{n}y^{−1})(1 − q^{n})  (1.4)

where we use the formal variables q = e^{2πiτ}, y = e^{2πiz}. This is a holomorphic Jacobi form of weight 1/2 and index 1/2 with multiplier system v_{η}^{3} × v_{H} (v_{η} is the multipier system of the Dedekind eta-function η(τ)). One can check that

ϕ_{0, \frac{3}{2}} (τ, z) = \frac{ϑ(τ, 2z)}{ϑ(τ, z)} = y^{− \frac{1}{2}} ∏_{n≥1}(1 + q^{n−1}y)(1 + q^{n}y^{−1})(1 − q^{2n−1}y^{2})(1 − q^{2n−1}y^{−2}) ∈J_{0, \frac{3}{2}}

ϕ_{−1, \frac{1}{2}} (τ, z) = \frac{ϑ(τ, z)}{η(τ)^{3}} = −y^{−1/2} ∏_{n≥1} (1 − q^{n−1}y)(1 − q^{n}y^{−1})(1 − q^{n})^{−2} ∈ J_{−1, \frac{1}{2}}

are weak Jacobi forms with integral Fourier coefficients.

The next proposition is well-known in physical literature (see [KMY], [AYS]). See also [Hö] where another normalization for the elliptic genus function was used.

Proposition 1.2. If M_{d} is a compact complex manifold of dimension d with c_{1}(M) = 0 (over ℝ), then its elliptic genus χ(M_{d}; q, y) is a weak Jacobi form of weight 0 and index d/2.

Proof. For the convenience of the readers we give here a proof which is similar to the proof of the modular behavior of the level N elliptic genus (see [H] and [HBJ, Appendix III]). Let us represent χ(M; τ, z) in terms of the theta-series. Let c_{i}(T_{M}) ∈ H^{2i}(M, ℤ) be the Chern class of T_{M} and let c(T_{M}) and ch(T_{M}) be the total Chern class and the Chern character of M and x_{i} = 2πiξ_{i} (1 ≤ i ≤ d) be the formal Chern parameters of T_{M}:

c(T_{M}) = ∑_{i=0}^{r} c_{i}(T_{M}) = ∏_{i=1}^{r}(1 + x_{i}), ch(T_{M}) = ∑_{i=1}^{r} e^{x_{i}}.

We recall that

ch (⋀_{t}E) = ∏_{i=1}^{r}(1 + te^{x_{i}}), ch (S_{t}E) = ∏_{i=1}^{r} \frac{1}{1 − te^{x_{i}}} .

Then we have

ch (𝔼_{q,y}) td (T_{M}) = ∏_{i=1}^{d} ∏_{n=1}^{∞} \frac{(1 − q^{n−1}y^{−1}e^{−x_{i}})(1 − q^{n}ye^{x_{i}})}{(1 − q^{n−1}e^{−x_{i}})(1 − q^{n}ye^{x_{i}})}x_{i} .

Therefore

Φ(τ, z, ξ_{1}, …, ξ_{d}) = y^{d/2}ch (E_{q,y})td (T_{M}) = ∏_{i=1}^{d} \frac{ϑ(τ, z + ξ_{i})}{ϑ(τ, ξ_{i})}(2πiξ_{i})

is holomorphic function in τ, z and ξ_{i}. (We recall that ϑ(τ, z) has zero of order 1 along z = 0.) For arbitrary γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ) one has

\frac{ϑ(γ < τ >, \frac{z+ξ_{i}}{cτ+d})}{ϑ(γ < τ >, \frac{ξ_{i}}{cτ+d})} = e^{πic(z^{2}+2zξ_{i})/(cτ+d)} \frac{ϑ(τ, z + ξ_{i})}{ϑ(τ, ξ_{i})}.

Thus Φ(τ, z, ξ_{1}, …, ξ_{d}) transforms like a Jacobi form of weight d (due to d factors ξ_{i} after the quotient of the theta-series) and index \frac{d}{2} if c_{1}(M) = x_{1}+… x_{d} = 0 (over ℝ!). Therefore the coefficient of arbitrary monom ξ_{1}^{n_{1}} · . . . · ξ_{d}^{n_{d}} of total degree d in the Taylor expansion of Φ(τ, z, ξ_{1}, …, ξ_{d}) with respect to ξ_{i} is a weak Jacobi form of weight 0 and index \frac{d}{2}. It follows that after integration over M one gets a Jacobi form of weight zero and index d/2. The claim that the elliptic genus does not contain Fourier coefficients with negative powers of q follows directly from the definition (1.1).

Remark. The definition (1.1) has sense and Proposition 1.2 is still true for arbitrary compact manifold with a Spin^{c}-structure. Moreover one can give a similar definition for an arbitrary complex vector bundle E with c_{1}(E) = 0. A generalization of this staff for vector bundles with c_{1}(E) ≠ 0 see in [G2].

Example 1.3. Elliptic genus of Enriques and K3 surfaces. Let us consider the case of a complex surface M_{2} with c_{1}(M_{2}) = 0. Then

χ(M_{2}; τ, z) ∈ J_{0,1} = ℂ ϕ_{0,1}(τ, z)

where

ϕ_{0,1}(τ, z) = (y + 10 + y^{−1}) + q(10y^{−2} − 64y^{−1} + 108 − 64y + 10y^{2}) + q^{2}(…) (1.5)

is the unique, up to a constant, weak Jacobi form of weight 0 and index 1. The value of arbitrary weak Jacobi form of weight 0 at z = 0 is a holomorphic SL_{2}(ℤ)-modular form of weight 0, i.e. it is a constant. According (1.2)

χ(M_{d}; τ, 0) = ∑_{p=0}^{d}(−1)^{p}χ_{p}(M) = e(M_{d})  (1.6)

is the Euler number of M_{d}. Therefore we have

χ(Enriques; τ, z) = ϕ_{0,1}(τ, z), χ(K3; τ, z) = 2ϕ_{0,1}(τ, z)

for arbitrary Enriques and K3 surfaces.

The structure of the bigraded algebra over ℂ of all Jacobi forms of integral index was determined in [EZ]. We describe the structure of the graded ℤ-algebra of all Jacobi forms of weight zero with integral Fourier coefficients. Firstly we show how the weak Jacobi forms of integral and half-integral indices are related.

Lemma 1.4. Let m be integral, then we have for the weak Jacobi forms

J_{2k,m+ \frac{1}{2}} = ϕ_{0, \frac{3}{2}} · J_{2k,m−1}, J_{2k+1,m+ \frac{1}{2}} = ϕ_{−1, \frac{1}{2}} · J_{2k+2,m}

where ϕ_{0, \frac{3}{2}} and ϕ_{−1, \frac{1}{2}} are defined in Example 1.1.

Proof. From the functional equation (1.3b) one has

div(ϕ_{2k,m+ \frac{1}{2}} ) ⊃ \{z ≡ \frac{1}{2}, \frac{τ}{2} , \frac{τ + 1}{2} mod ℤτ + ℤ\}, div(ϕ_{2k+1,m+ \frac{1}{2}} ) ⊃ \{ℤτ + ℤ\}.

Thus any Jacobi form of half-integral index and even (resp. odd) weight is divisible by ϕ_{0, \frac{3}{2}} (resp. ϕ_{−1, \frac{1}{2}} ).

Example 1.5. Calabi–Yau 3-folds and 5-folds. Let c_{1}(M_{3}) = 0 and c_{1}(M_{5}) = 0. From Lemma 1.4 it follows that

J_{0, \frac{3}{2}} = ℂ ϕ_{0, \frac{3}{2}} , J_{0, \frac{4}{2}} = ℂ ϕ_{0, \frac{3}{3}} ϕ_{0,1}.

Thus

χ(M_{3}; q, y) = \frac{e(M_{3})}{2} [(y^{1/2} + y^{−1/2}) + q(−y^{5/2} + y^{1/2} + y^{−1/2} − y^{−5/2}) + q^{2}(…)],
χ(M_{5}; q, y) = \frac{e(M_{5})}{24}[(y^{± \frac{3}{2}} + 11y^{± \frac{1}{2}} ) + q(−y^{± \frac{7}{2}} + …) + q^{2}(−11y^{± \frac{9}{2}} + …)]

(y^{±l} means that we have two summands with y^{l} and y^{−l} respectively). As a simple corollary we have that the Euler characteristic of an arbitrary Calabi–Yau 5-fold is divisible by 24 and

χ_{1}(CY_{5}) = \frac{1}{24} e(CY_{5}), χ_{2}(CY_{5}) = \frac{11}{24} e(CY_{5}).

In particular, for the Hodge numbers of an arbitrary strict Calabi–Yau 5-fold we have

11(h^{1,1} − h^{1,4}) = h^{2,2} − h^{2,3} + 10(h^{2,1} − h^{3,1}).

To calculate the elliptic genus of a Calabi–Yau manifold of even dimension we introduce some other basic Jacobi forms. Let us denote by J_{0,m}^{ℤ} the ℤ-module of all weak Jacobi forms of weight 0 and index m with integral Fourier coefficients. We consider the graded ring

J_{0,∗}^{ℤ} = ⨁_{m∈ℤ_{≥0}} J_{0,m}^{ℤ}

of all Jacobi forms of integral index with integral Fourier coefficients and its ideal

J_{0,∗}^{ℤ}(q) = {ϕ ∈ J_{0,∗}^{ℤ} | ϕ(τ, z) = ∑_{n≥1, l∈ℤ} a(n, l)q^{n}y^{l}}

of the Jacobi forms without q^{0}-term.

Lemma 1.6. The ideal J_{0,∗}^{ℤ}(q) is principal. It is generated by a weak Jacobi form of weight 0 and index 6

ξ_{0,6}(τ, z) = Δ(τ)ϕ_{−1, \frac{1}{2}} (τ, z)^{12} = \frac{ϑ(τ, z)^{12}}{η(τ)^{12}} = q(y^{\frac{1}{2}} − y^{− \frac{1}{2}} )^{12} + q^{2}(…).

Proof. Let ψ ∈ J_{k,m} be an arbitrary Jacobi form. The product

Ψ(τ, z) = exp(−8π^{2}m G_{2}(τ)z^{2})ψ(τ, z)

transforms like a SL_{2}(ℤ)-modular form of weight k. We recall that G_{2}(τ) = −\frac{1}{24} + ∑_{n=1}^{∞} σ_{1}(n) q^{n} is a quasi-modular form of weight 2, i.e. it satisfies

G_{2}(γ < τ >) = (cτ + d)^{2}G_{2}(τ) − \frac{c(cτ + d)}{4πi}.

Thus we have Ψ(\frac{aτ+b}{cτ+d}, \frac{z}{cτ+d}) = (cτ + d)^{k}Ψ(τ, z). Therefore the coefficients f_{n}(τ) in the Taylor expansion

Ψ(τ, z) = ∑_{n∈ℤ} f_{n}(τ)z^{n}  (1.7)

are SL_{2}(ℤ)-modular forms of weight k + n. If ψ ∈ J_{0,m}^{ℤ}(q) has no q^{0}-term, then f_{2n}(τ) is a cusp form of weight 2n. Thus n ≥ 6 and ψ(τ, z) has zero of order at least 12 along z = 0. Therefore ψ(τ, z)/ξ_{0,6}(τ, z) is holomorphic on ℍ_{1} × ℂ.

Corollary 1.7. A weak Jacobi form of weight 0 is uniquely determined by its q^{0}-term if its index is less than 6 or equal to \frac{13}{2}.

Example 1.8. Calabi–Yau manifolds of dimension 4, 6, 8 and 10. The last corollary shows us that for these values of dimension of a Calabi–Yau manifold M_{d} one can define the elliptic genus of M_{d} if one knows its Hirzebruch χ_{y}-genus. In [GN1–GN4] we used the following important Jacobi forms of small index

ϕ_{0,2}(τ, z) = \frac{1}{2}η(τ)^{−4} ∑_{m ,n∈ℤ} (3m − n)(\frac{−4}{m})(\frac{12}{n})q^{\frac{3m^{2}+n^{2}}{24}} y^{\frac{m+n}{2}}
= (y + 4 + y^{−1}) + q(y^{±3} − 8y^{±2} − y^{±1} + 16) + q^{2}(…),  (1.8)
ϕ_{0,3}(τ, z) = ϕ_{0, \frac{3}{2}}^{2} (τ, z) = (y + 2 + y^{−1}) + q(−2y^{±3} − 2y^{±2} + 2y^{±1} + 4) + q^{2}(…),
ϕ_{0,4}(τ, z) = \frac{ϑ(τ, 3z)}{ϑ(τ, z)} = (y + 1 + y^{−1}) − q(y^{±4} + y^{±3} − y^{±1} − 2) + q^{2}(…).  (1.9)

Using these Jacobi forms together with ϕ_{0,1} (see (1.5)) one can construct a basic of the module of the weak Jacobi forms of weight 0 and index 2, 3, 4 and 5 and to write a formula for the elliptic genus of M_{d} in terms of the cohomological invariants χ_{p}(M_{d}):

ψ_{0,2}^{(1)} = ϕ_{0,2} = y + 4 + y^{−1} + q(…)
ψ_{0,2}^{(2)} = ϕ_{0,1}^{2} − 20ϕ_{0,2} = (y^{2} + 22 + y^{−2}) + q(…)
ψ_{0,3}^{(1)} = ϕ_{0,3} = ϕ_{0, \frac{3}{2}}^{2} = y + 2 + y^{−1} + q(…)
ψ_{0,3}^{(2)} = ϕ_{0,1}ϕ_{0,2} − 15ϕ_{0,3} = (y^{±2} − y^{±1} + 12) + q(…)  (1.10)
ψ_{0,3}^{(3)} = ϕ_{0,1}^{3} − 30ϕ_{0,1}ϕ_{0,2} + 117ϕ_{0,3} = ψ_{0,1}|T_{−}(3) − 3ϕ_{0,1} = (y^{±3} + 34) + q(…)
ψ_{0,4}^{(1)} = ϕ_{0,4} = y + 1 + y^{−1} + q(…)
ψ_{0,4}^{(2)} = ϕ_{0,1}ϕ_{0,3} − 12ϕ_{0,4} = (y^{±2} + 10) + q(…)  (1.11)
ψ_{0,4}^{(3)} = ϕ_{0,2}ψ_{0,2}^{(2)} − 4ψ_{0,4}^{(2)} − 24ϕ_{0,4} = (y^{±3} − y^{±1} + 24) + q(…)
ψ_{0,4}^{(4)} = ψ_{0,2}^{(2)}|T_{−}(2) − 2ϕ_{0,4}^{(2)} = (y^{±4} + 46) + q(…),

where T_{−}(m) is the standard Hecke operator (ϕ_{0,t}|T_{−}(m))(τ, z) = ∑_{n,l} f_{m}(n, l)q^{n}y^{l} with f_{m}(N, L) = m ∑_{a|(N,L,m)} a^{−1}f(\frac{Nm}{a^{2}}, \frac{L}{a}).

Fourier coefficient f(n, l) of a weak Jacobi form of weight 0 and index t depends only on the norm 4nt − l^{2} of its index and ±l mod 2t. For every Jacobi form written above its q^{0}-term contains all “orbits” of non-zero Fourier coefficients with negative norm 4nt − l^{2} of its index. This fact will be important in §3 bellow. For d = 10 the situation is a little bit more complicated. After some calculation we obtain the following Jacobi forms

ψ_{0,5}^{(1)} = 5y + 2 + 5y^{−1} + q(−y^{5} + …)
ψ_{0,5}^{(2)} = (y^{±2} + y^{±1} + 8) + q(y^{5} + …)
ψ_{0,5}^{(3)} = (y^{±3} + y^{±1} + 20) + q(…)  (1.12)
ψ_{0,5}^{(4)} = (y^{±4} − y^{±1} + 36) + q(−3y^{5} + …)
ψ_{0,5}^{(5)} = (y^{±5} + 58) + q(y^{±5} + …)

where we include in the formulae all “orbits” of non-zero Fourier coefficients with negative norm 20n − l^{2}. One knows (see [EZ]) that dim J_{0,m} = m for m = 1, 2, 3, 4 and 5. Thus the functions ψ_{0,m}^{(p)} constructed above (1 ≤ p ≤ m) form a basis of J_{0,m}^{ℤ}. Thus for d = 2, 4, 6, 8, 10 we can represent the elliptic genus of a compact complex manifold M_{d} with c_{1}(M_{d}) = 0 as a linear combination

χ(M_{d}; τ, z) = ∑_{p=0}^{\frac{d}{2} −2} χ_{p}(M_{d})ψ_{0, \frac{d}{2}}^{( \frac{d}{2} −p)}(τ, z) + cψ_{0, \frac{d}{2}}^{(1)} (τ, z)

with c ∈ ℤ. In particular, the coefficients χ_{p} of the Hirzebruch y-genus of the manifold M_{d} satisfy the following relations

\begin{aligned}χ_{2} & = 22χ_{0} − 4χ_{1} \\ χ_{3} & = 34χ_{0} − 14χ_{1} + 2χ_{2} \\ χ_{4} & = 46χ_{0} − 25χ_{1} + 10χ_{2} − χ_{3} \end{aligned}  and  e(M_{4}) ≡ 0 mod 6 ( for CY_{4}) and  e(M_{6}) ≡ 0 mod 4 ( for CY_{6}) and  e(M_{8}) ≡ 0 mod 3 ( for CY_{8}). (1.13)

For a Calabi–Yau 10-fold we have

χ_{5} = 58χ_{0} − 36χ_{1} + 20χ_{2} − 8χ_{3} + \frac{2}{5}(χ_{4} + χ_{3} − χ_{2} − χ_{1}).

We note that for dimension d ≥ 12 (d ≠ 13) an additional term cξ_{0,6} could appear in the formula for the elliptic genus. The odd dimensions d = 7, 9, 11 and 13 are related to the even dimensions considered above by the factor ϕ_{0,3/2}. For example,

χ(M_{7}; τ, z) = χ_{2}(M_{7})ϕ_{0, \frac{3}{2}} ϕ_{0,2} − χ_{1}(M_{7})ϕ_{0, \frac{3}{2}} ψ_{0,2}^{(2)}, e(M_{7}) = 12(χ_{2} − 4χ_{1}) ≡ 0 mod 12.

These examples show us that information about Jacobi forms provides some information about Calabi–Yau manifolds. In the next theorem we construct a basis of the module J_{0,m}^{ℤ} and we find generators of the graded ring J_{0,∗}.

Theorem 1.9. 1. Let m be a positive integer. The module

J_{0,m}^{ℤ}/J_{0,m}^{ℤ}(q) = ℤ [ψ_{0,m}^{(1)}, …, ψ_{0,m}^{(m)}]

is a free ℤ-module of rank m. Moreover there is a basis consists of ψ_{0,m}^{(n)} (1 ≤ n ≤ m) with the following q^{0}-term

[ψ_{0,m}^{(n)}]_{q^{0}} = y^{n} + ⋯ + y^{−n}  (3 ≤ n ≤ m)
[ψ_{0,m}^{(2)}]_{q^{0}} = y^{2} − 4y + 6 − 4y^{−1} + y^{−2}
[ψ_{0,m}^{(1)}]_{q^{0}} = \frac{1}{(12, m)} (my + (12 − 2m) + my^{−1})

where (12, m) is the greatest common divisor of 12 and m.

2. The graded ring of all weak Jacobi forms of weight 0 with integral coefficients is finitely generated

J_{0,∗}^{ℤ} = ⨁_{m} J_{0,m}^{ℤ} = ℤ [ϕ_{0,1}, ϕ_{0,2}, ϕ_{0,3}, ϕ_{0,4}]

where the corresponding Jacobi forms are defined in (1.8)–(1.9). ϕ_{0,1}, ϕ_{0,2}, ϕ_{0,3} are algebraicly independent and

4ϕ_{0,4} = ϕ_{0,1}ϕ_{0,3} − ϕ_{0,2}^{2}.

To prove Theorem 1.9 we need

Lemma 1.10. For arbitrary weak Jacobi form ϕ_{0,m} = ∑_{n,l} f(n, l)q^{n}y^{l} the following identities are valid

m ∑_{l} f(0, l) = 6 ∑_{l} l^{2}f(0, l), 24m ∑_{l} f(0, l) = ∑_{n}(m − 6n^{2})f(1, n).

Proof. We consider the Taylor expansion (1.7)

exp(−8π^{2}m G_{2}(τ)z^{2})ϕ_{0,m}(τ, z) = f_{0} + ∑_{n≥1} f_{n}(τ)z^{n}

where f_{n}(τ) is a SL_{2}(ℤ)-modular form of weight n. Thus f_{2}(τ) ≡ 0. The left hand side of the identities of the lemma are equal to the first two coefficients of f_{2}(τ). This proves the lemma. We remark that one can continue the list of similar identities calculating the second, third, … coefficients of f_{2}(τ).

Using (1.2) we get

\frac{e(M)d}{12} = ∑_{p}(−1)^{p}χ^{p}(M)(\frac{d}{2} − p)^{2}  (1.14)

if c_{1}(M) = 0 over ℝ.

Remark. Identity (1.14) was rediscovered several times in the mathematical and physical literature. See, for example, [LW], [S] for a proof based on the Riemann–Roch–Hirzebruch formula and [AYS] where (1.14) is related to the sum rule for the charges in N = 2 superconformal field theory. Lemma 1.11 provides us with an automorphic proof of (1.14) (see [GN3, Lemma 2.2] for a more general statement).

Proof of Theorem 1.9. For a weak Jacobi form the polynom [ϕ]_{q^{0}} is of order not greater than m (see [EZ]). It can not be a constant according to the first identity of Lemma 1.10. Moreover if [ϕ]_{q^{0}} is “linear”, i.e. [ϕ]_{q^{0}} = ay + b + ay^{−1}, then a is divisible by \frac{m}{(m,12)}. It follows that m Jacobi forms of type dicribed in the first statement of the theorem, if they would exist, form a basis of the module J_{0,m}^{ℤ}/J_{0,m}^{ℤ}(q).

Let us construct a basis by induction. For m = 1, 2, 3, 4, 6, 8, 12 we set ψ_{0,m}^{(1)} = ϕ_{0,m} where ϕ_{0,1}, …, ϕ_{0,4} were defined above and

ϕ_{0,6}(τ, z) = ϕ_{0,2}ϕ_{0,4} − ϕ_{0,3}^{2} = (y + y^{−1}) + q(…),
ϕ_{0,8}(τ, z) = ϕ_{0,2}ϕ_{0,6} − ϕ_{0,4}^{2} = (2y − 1 + 2y^{−1}) + q(…),
ϕ_{0,12}(τ, z) = ϕ_{0,4}ϕ_{0,8} − 2ϕ_{0,6}^{2} = (y − 1 + y^{−1}) + q(…).

Now we can define the weak Jacobi forms ψ_{0,m}^{(1)} using the following procedure. (We omit the weight 0 bellow to simplify the notation). Let m ≥ 5. We set

ψ_{m,I} = ψ̃_{m−4}ϕ_{4} + ψ̃_{m−2}ϕ_{2} − 2ψ̃_{m−3}ϕ_{3} = my^{±1} + (12 − 2m) + q(…)

where by definition ψ̃_{m} := (12, m)ψ_{m}^{(1)}. We can take ψ_{m,I} as ψ_{m}^{(1)} if m is coprime with 12. If m ≡ 0 mod 2 and m ≥ 6 we can define a form ψ_{m,II} = \frac{1}{2}ψ_{m,I} with integral Fourier coefficients. If m ≡ 0 mod 3 and m ≥ 9 we define

ψ_{m,III} = \frac{2}{3}ψ̃_{m−3}ϕ_{3} + \frac{1}{3}ψ̃_{m−6}ϕ_{6} − ψ̃_{m−4}ϕ_{4} = \frac{m}{3}y^{±1} + (4 − \frac{2m}{3}) + q(…).

For m ≡ 0 mod 4 and m > 12 we take

ψ_{m,IV} = \frac{1}{4}[ψ̃_{m−12}ϕ_{12} + ψ̃_{m−4}ϕ_{4} − ψ̃_{m−8}ϕ_{8}] = \frac{m}{4}y^{±1} + (3 − \frac{m}{2}) + q(…).

In the remain cases m ≡ 0 mod 6 and m ≡ 0 mod 12 (m > 12) we put

ψ_{m,V I} = \frac{1}{2}ψ_{m,III} = \frac{m}{6}y^{±1} + (2 − \frac{m}{3}) + q(…),
ψ_{m,XII} = ψ_{m,III} − ψ_{m,IV} = \frac{m}{12}y^{±1} + \frac{6 − m}{6} + q(…).

Thus we finish the construction of functions ψ_{m}^{(1)} if we put ψ_{m}^{(1)} := ψ_{m,D} for (m, 12) = D. Next we construct ψ_{m}^{(2)}. We put

ψ_{0,m}^{(2)} = ψ̃_{0,m−3}ϕ_{0,3} − ψ̃_{0,m−4}ϕ_{0,4} − ψ̃_{0,m} (m ≥ 5)

and

ψ_{0,2}^{(2)} = ϕ_{0,1}^{2} − 24ϕ_{0,2}, ψ_{0,3}^{(2)} = ϕ_{0,1}ϕ_{0,2} − 18ϕ_{0,3}, ψ_{0,4}^{(2)} = ϕ_{0,1}ϕ_{0,3} − 16ϕ_{0,4}.

One can see that these forms have the q^{0}-term equal to y^{2} − 4y + 6 − 4y^{−1} + y^{−2}. For 3 ≤ n ≤ m − 2 one can use ϕ_{0,3} and the forms ϕ_{0,m−3}^{(n−1)} in order to construct ϕ_{0,m}^{(n)}. To finish the proof of the first statement of the theorem we put

ϕ_{0,m}^{(m−1)} = ϕ_{0,1}^{m−2} · ϕ_{0,2}, ϕ_{0,m}^{(m)} = ϕ_{0,1}^{m}.

We note that each function of the basis {ψ_{m}^{(n)}}_{n=1}^{m} of the module J_{0,m}^{ℤ}/J_{0,m}^{ℤ}(q) constructed above is a polynom in the basic Jacobi forms ϕ_{0,1}, …, ϕ_{0,4}. The torsion relation 4ϕ_{0,4} = ϕ_{0,1}ϕ_{0,3} − ϕ_{0,2}^{2} follows from Corollary 1.7. The form ξ_{0,6} generating the principle ideal J_{0,∗}^{ℤ}(q) is also a polynom in ϕ_{0,1}, …, ϕ_{0,4}:

ξ_{0,6} = −ϕ_{0,1}^{2}ϕ_{0,4} + 9ϕ_{0,1}ϕ_{0,2}ϕ_{0,3} − 8ϕ_{0,2}^{3} − 27ϕ_{0,3}^{2}.  (1.15)

(To prove the last formula one needs to check that the q-constant term of the write hand side is zero and to compair coefficients at the first power q.)

Let us prove that ϕ_{0,1}, ϕ_{0,2} and ϕ_{0,3} are algebraicly independent. For this end we consider its values at z = \frac{1}{2}. We have

ϕ_{0,2}(τ, \frac{1}{2}) ≡ 2, ϕ_{0,3}(τ, \frac{1}{2}) ≡ 0, ϕ_{0,4}(τ, \frac{1}{2}) ≡ −1.

(The two last identities follow from definition and the first one is a corollary of the torsion relation.) The restriction of

ϕ_{0,1}(τ, \frac{1}{2}) = α(τ) = 8 + 2^{8}q + 2^{11}q^{2} + 11 · 2^{10}q^{3} + 3 · 2^{14}q^{4} + 359 · 2^{9}q^{5} + … (1.16)

is a modular function with respect to Γ_{0}(2) with a character of order 2 (see §2). The square of this function is, up to factor 2^{12}, the “Hauptmodul” for the congruence subgroup group Γ_{0}(2). Since only one function obtained from the Jacobi forms of different indices is a non-constant function for z = 1/2, then they are algebraicly independent.

§2. Special values of the elliptic genus

In this section we analyze the value of the elliptic genus at the following special points z = 0 (Euler number), z = \frac{1}{2} (signature), z = \frac{τ+1}{2} (Â-genus) and z = \frac{1}{3}, \frac{1}{4}, \frac{1}{6}. For this end we have to study the restriction of the main generators of the graded ring of the integral week Jacobi forms. A special value of a Jacobi form is a modular form in τ. In the next lemma we give a little more precise statement than in [EZ, Theorem 1.3].

12 V. GRITSENKO

Lemma 2.1. Let ϕ ∈ J_{0,t} (t ∈ ℤ/2) and X = (λ, μ) ∈ ℚ^{2}. Then

ϕ|_{X}(τ, 0) = ϕ(τ, λτ + μ) exp(2πit(λ^{2}τ + λμ))

is an automorphic form of weight 0 with a character with respect to the subgroup

Γ_{X} = {M ∈ SL_{2}(ℤ) | XM − X ∈ ℤ^{2}}.

Let us consider some particular examples which we need in this section. Example 2.2. Let X = (0, \frac{1}{N}). Then

Γ_{X} = Γ_{0}^{(1)}(N) = \{M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ) | M ≡ ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ) mod N\}.

If t is integer, then the character of the function

ϕ|_{X}(τ, 0) = ϕ(τ, \frac{1}{N})

is given by v(M) = exp(−2πitc/N^{2}) and it has has order \frac{N}{(t,N)}.

It is easy to see that if ϕ ∈ J_{k,m}^{ℤ} with integral m, then the form ϕ(τ, \frac{1}{N}) still has integral Fourier coefficients if N = 1, …, 6. In particular, the value of ξ_{6}(τ, z) at these points is related to the “Hauptmodule” for the corresponding group Γ_{0}(N):

\begin{aligned}ξ_{6}(τ, \frac{1}{2}) & = 2^{12} \frac{Δ(2τ)}{Δ(τ)}, \\ ξ_{6}(τ, \frac{1}{3}) & = 3^{6} (\frac{Δ(3τ)}{Δ(τ)})^{1/2} , \end{aligned}  \begin{aligned}ξ_{6}(τ, \frac{1}{4}) & = 2^{6} (\frac{Δ(4τ)}{Δ(τ)})^{1/2} , \\ ξ_{6}(τ, \frac{1}{6}) & = (\frac{Δ(τ)Δ(6τ)}{Δ(2τ)Δ(3τ)})^{1/2} . \end{aligned}  (2.1)

Let us analyze the corresponding values of the four generators ϕ_{0,n} of the graded ring J_{0,∗}^{ℤ}. From the definition (see (1.8)–(1.9)) and the identity 4ϕ_{0,4} = ϕ_{0,1}ϕ_{0,3} − ϕ_{0,2}^{2} we obtain

ϕ_{0,1}(τ, 0) = 12, ϕ_{0,2}(τ, 0) = 6, ϕ_{0,3}(τ, 0) = 4, ϕ_{0,4}(τ, 0) = 3  (2.2)

and

\begin{aligned}ϕ_{0,1}(τ, \frac{1}{2}) & = α(τ) \\ ϕ_{0,2}(τ, \frac{1}{2}) & = 2 \\ ϕ_{0,3}(τ, \frac{1}{2}) & = 0 \\ ϕ_{0,4}(τ, \frac{1}{2}) & = −1 \end{aligned}  \begin{aligned}ϕ_{0,1}(τ, \frac{1}{3}) & = β^{2}(τ) \\ ϕ_{0,2}(τ, \frac{1}{3}) & = β(τ) \\ ϕ_{0,3}(τ, \frac{1}{3}) & = 1 \\ ϕ_{0,4}(τ, \frac{1}{3}) & = 0 \end{aligned}  \begin{aligned}ϕ_{0,1}(τ, \frac{1}{4}) & = \frac{γ(τ)^{4} + 4}{γ(τ)} \\ ϕ_{0,2}(τ, \frac{1}{4}) & = 4γ^{2}(τ) \\ ϕ_{0,3}(τ, \frac{1}{4}) & = 2γ(τ) \\ ϕ_{0,4}(τ, \frac{1}{4}) & = 1. \end{aligned}  (2.3)

The automorphic functions α(τ), β(τ) and γ(τ) are automorphic forms of weight 0 with respect to the group Γ_{0}(2), Γ_{0}^{(1)}(3) and Γ_{0}^{(1)}(4) respectively. These functions have integral Fourier coefficients. The identity (1.15) gives us the following relations for the modular forms α, β and γ

2^{12} \frac{Δ(2τ)}{Δ(τ)} = α(τ)^{2} − 64, 3^{6} (\frac{Δ(3τ)}{Δ(τ)})^{1/2} = β(τ)^{3} − 27

2^{6} (\frac{Δ(4τ)}{Δ(τ)})^{1/2} = 4((\frac{γ(τ)}{2})^{2} − (\frac{2}{γ(τ)})^{2}).

It follows that

α(τ) − 8 ≡ 0 mod 2^{8},  β(τ) − 3 ≡ 0 mod 3^{3}  (2.4)

(compare with (1.16)). Using the definition of ϕ_{0,3} and γ(τ) and the relations between the Jacobi theta-series ϑ_{ab} of level 2 we have

γ(τ) = \frac{ϑ_{00}(2τ)}{ϑ_{01}(2τ)} = \frac{ϑ_{00}(2τ, 0)}{ϑ_{01}(2τ, 0)}.  (2.5)

Using Corollary 1.7 we check that

ϕ_{0,1}(τ, 2z) = ϕ_{0,2}^{2}(τ, z) − 8ϕ_{0,4}(τ, z).

Thus

α(τ) = 16γ(τ)^{4} − 8 = 16\frac{ϑ_{00}^{4}(2τ)}{ϑ_{01}^{4}(2τ)} − 8.  (2.6)

In particular all Fourier coefficients of γ(τ) and α(τ) are positive.

In connection with (2.5)–(2.6) we note that one can write the generators of the graded ring as symmetric polynomials in ξ_{ab}(τ, z) = ϑ_{ab}(τ, z)/ϑ_{ab}(τ, 0):

ϕ_{0,1} = 4(ξ_{00}^{2} + ξ_{10}^{2} + ξ_{01}^{2}), ϕ_{0, \frac{3}{2}} = 4ξ_{00}ξ_{10}ξ_{01}
ϕ_{0,2} = 2((ξ_{00}ξ_{10})^{2} + (ξ_{00}ξ_{01})^{2} + (ξ_{10}ξ_{01})^{2}).  (2.7)

To check these formulae one can use Corollary 1.7 and the fact that the generators of modular group transform ξ_{ab} to each other.

Example 2.3. Let X = (\frac{1}{N} , \frac{1}{N}). Then Γ_{X} contains the principle congruence subgroup Γ_{1}(N). In some cases Γ_{X} will be strictly larger. For example, if X_{2} = (\frac{1}{2}, \frac{1}{2}), then

ϕ|_{X_{2}}(τ, 0) = ϕ(τ, \frac{τ + 1}{2}) exp(\frac{πi}{2}(τ + 1))

is an automorphic form with respect of the so-called theta-group

Γ_{θ} = \{M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ) | M ≡ ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) or ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) mod 2\}.

The corresponding character is given by ϵ_{2}(M) = exp(2πim(d + b − a − c)/4) = ±1. This character is trivial if the index m of Jacobi form is even and it has order 2 if the index is odd. Let us consider Γ_{θ}-automorphic function

ϕ̂_{m}(τ) = q^{− \frac{m}{4}} ϕ_{0,m}(τ, −\frac{τ + 1}{2}).

We have

ϕ̂_{3} = 0, ϕ̂_{4} = −1, ϕ̂_{2} = −2, ξ̂_{6} = ϕ̂_{1}^{2} + 64 = (\frac{ϑ_{00}}{η} )^{12}

where

ϕ̂_{1}(τ) = 4\frac{ϑ_{10}^{4} − ϑ_{01}^{4}}{ϑ_{01}^{2}ϑ_{10}^{2}} = −q^{− \frac{1}{4}} + 20q^{\frac{1}{4}} + ⋯ ∈ 𝔐_{0}^{ℤ}(Γ_{θ}, ϵ_{2}).

Let us analyze some special values of the elliptic genus. As it easy follows from (1.2) we get Euler number and signature for z = 0 (d is arbitrary) and z = \frac{1}{2} (d is even)

χ(M_{d}, τ, 0) = e(M_{d}),
χ(M_{d}, τ, \frac{1}{2}) = σ_{M}(τ) = (−1)^{\frac{d}{2}} s(M_{d}) + q(…) ∈ 𝔐_{0}^{ℤ}(Γ_{0}(2), v_{2}), v_{2}(( \begin{matrix}a & b \\ c & d \end{matrix} )) = e^{πim \frac{c}{2}} .

In §1 we calculated Euler number e(M_{d}) for small d in terms of Jacobi forms. It gave us some divisibility of Euler number of Calai–Yau manifolds. We note that the quotient e(M)/24 appears in physics as obstruction to cancelling the tadpole (see [SVW] where it was proved that e(M_{4}) ≡ 0 mod 6).

Proposition 2.4. Let M_{d} be an almost complex manifold of complex dimension d such that c_{1}(M) = 0 in H^{2}(M, ℝ). Then

d · e(M_{d}) ≡ 0 mod 24.

If c_{1}(M) = 0 in H^{2}(M, ℤ), then we have a more strong congruence

e(M) ≡ 0 mod 8 if d ≡ 2 mod 8.

Proof. The first fact follows simply from (2.2) or from (1.14). If d ≡ 2 mod 8 one can write the elliptic genus as a polynom over ℤ in the generators ϕ_{0,∗}

e(M_{d}) ≡ P (ϕ_{0,1}, ϕ_{0,2}, ϕ_{0,3}, ϕ_{0,4})|_{z=0} ≡ c_{1,m}(ϕ_{0,1}|_{z=0})(ϕ_{0,4}|_{z=0})^{\frac{d−2}{8}} mod 8.

If one put z = −\frac{τ+1}{2}, i.e., y = −q^{1/2} (see Example 2.3), then one obtain that the series

𝔼_{q,−q^{1/2}} = ⨂_{n≥1} ⋀_{q^{n/2}}T_{M} ⊗ ⨂_{n≥1} ⋀_{q^{n/2}}T_{M}^{∗} ⊗ ⨂_{n≥1} S_{q^{n}}(T_{M} ⊕ T_{M}^{∗})

is ∗-symmetric. According to the Serre duality all Fourier coefficients of χ̂(M_{d}, τ) are even. The constant c_{1,m} from the last congruence is equal to the coefficient of χ̂(M_{d}, τ) at the minimal negative power of q. Therefore c_{1,m} is even and we obtain divisibility of e(M_{8m+2}) by 8.

We note that divisibility of d · e(M) by 3 was proved by F. Hirzebruch in 1960. For a hyper-Kähler compact manifold the claim of the proposition above was proved by S. Salamon in [S]. After my talk on the elliptic genus at a seminar of MPI in Bonn in April 1997 Professor F. Hirzebruch informed me that the result of Proposition 2.4 was known for him (non-published). Using some natural examples he also proved that this property of divisibility of the Euler number modulo 24 is strict (see [H2]).

Formulae (2.3) provide us with a formula for the signature χ(M_{d}; τ, \frac{1}{2}) as a polynom in α(τ). As a corollary of (2.3) and Theorem 1.9 we have that for an arbitrary Jacobi form of integral index

\begin{aligned}ϕ_{0,4m}(τ, \frac{1}{2}) & = c + 2^{13}q(…) \\ ϕ_{0,4m+2}(τ, \frac{1}{2}) & = 2c + 2^{12}q(…) \end{aligned}  \begin{aligned}ϕ_{0,4m+1}(τ, \frac{1}{2}) & = 8c + 2^{8}q(…) \\ ϕ_{0,4m+3}(τ, \frac{1}{2}) & = 16c + 2^{9}q(…). \end{aligned}  (2.8)

Similar to the proof of Proposition 2.4 we obtain a better congruence for the signature of a manifold with dim≡ 2 mod 8 and c_{1}(M) = 0:

χ(M_{8m+2}; τ, z) = 16c + 2^{9}q(…).  (2.9)

This gives us another proof of the Oshanine’s theorem in this particular case.

This is interesting that the values of the Hirzebruch y-genus at y = e^{2πi/3} and y = i also have some properties of divisibility. For z = \frac{1}{3} (resp. z = \frac{1}{4}) we can write ϕ_{0,m}(τ, \frac{1}{3}) (resp. ϕ_{0,m}(τ, \frac{1}{4})) as a polynom in β(τ) = 3 + 27(q + …) (resp. in γ(τ)^{±1}). This gives us the following results

ϕ_{0,3m}(τ, \frac{1}{3}) = c + 3^{6}q(…), ϕ_{0,3m+1}(τ, \frac{1}{3}) = 9c + 3^{4}q(…)

ϕ_{0,3m+2}(τ, \frac{1}{3}) = 3c + 3^{3}q(…).  (2.10)

Thus we have

Proposition 2.5. If c_{1}(M) = 0 (over ℝ), then

χ(M_{6m}; τ, \frac{1}{3}) = c_{1} mod 3^{6}, χ(M_{6m+2}; τ, \frac{1}{3}) = 9c_{2} mod 3^{4},

χ(M_{6m+4}; τ, \frac{1}{3}) = 3c_{3} mod 3^{3}.

We finish with some relations for z = \frac{1}{4}:

χ(M_{8m+2}; τ, \frac{1}{4}) = 4c + 2^{4}q(…), ϕ_{0,4m+2}(τ, \frac{1}{4}) = 4c + 2^{5}q(…)

ϕ_{0,4m+3}(τ, \frac{1}{4}) = 2c + 2^{8}q(…).  (2.11)

§3. Second quantized elliptic genus (SQEG) of Calabi–Yau manifolds

3.1. SQEG of Calabi–Yau manifolds and Siegel modular forms. The notion of elliptic genus of N = 2 super-symmetric theories was introduced more than 10 years ago (see, for example, [W1], [W2], [EOTY], [AYS], [KYY]). In physics the elliptic genus of a Calabi–Yau manifold M_{d} is defined as the genus one partition function of the supersymmetric sigma model whose target space is M_{d}. By definition this is the trace of an operator over the Ramond-Ramond sector of the sigma model

χ(M_{d}; τ, z) = Tr_{H} (−1)^{F} q^{L_{0}− \frac{c}{24}} \overline{q}^{\overline{L}_{0}− \frac{c}{24}} y^{F_{L}}

where q^{L_{0}− \frac{c}{24}} is the evolution operator, F_{L} is the fermion number and L_{0} (\overline{L_{0}}) is Virasoro generator of of left (right) movers, (−1)^{F} is the fermion parity operator in N = 2 supersymmetric theory and c is the central charge.

We can consider n-fold symmetric product of the manifold M, i.e., the orbifold space S^{n}M = M^{n}/S_{n}, where S_{n} is the symmetric group of n elements. This is a singular manifold but one can define the orbifold elliptic genus of S^{n}M (see for details the talk of R. Dijkgraaf at ICM-1998 in Berlin [D]). It gives us a string version of the elliptic genus of the symmetric product of a Calabi–Yau manifold. One can compare this construction with the definition of the string Euler number of symmetric products (see [HH]). Using some arguments from the conformal field theory on orbifolds it was proved in [DVV] and [DMVV] that the string elliptic genus of the second quantization ∪_{n≥1}S^{n}M of a Calabi– Yau manifold M coincides with the second quantized elliptic genus of the given manifold:

∑_{n=0}^{∞} p^{n}χ_{orb}(S^{n}M; q, y) = ∏_{m≥0, l, n>0} \frac{1}{(1 − q^{m}y^{l}p^{n})^{f(mn,l)}}  (3.1)

where

χ(M, τ, z) = ∑_{m≥0, l∈ℤ (or ℤ/2)} f(m, l)q^{m}y^{l}

is the elliptic genus of M.

For a K3 surface, the product in the left hand side of (3.1) is essentially the power −2 of the infinite product expansion of the product of all even theta-constants (see [GN1])

Δ_{5}(Z)^{2} = 2^{−12} ∏_{^{t}ab=0 mod 2} Θ_{a,b}(Z)^{2} = (qpr) ∏_{(m,l,n)>0}(1 − q^{m}y^{l}p^{n})^{f(mn,l)}

where

Z = ( \begin{matrix}τ & z \\ z & ω \end{matrix} ) ∈ ℍ_{2}, q = e^{2πiτ} , y = e^{2πir}, p = e^{2πiω}

belongs to the Siegel upper half-plane, Θ_{a,b}(Z) are the even Siegel theta-constants of level 2 and f(m, l) is the Fourier coefficient of 2ϕ_{0,1}(τ, z) (see Example 1.3). The modular form Δ_{5}(Z) is the first cusp form with respect to the Siegel modular group Sp_{4}(ℤ) (with a nontrivial binary character). In [DVV] (3.1) was conjectured as a generalization of a formula for the orbifold Euler numbers

∑_{n≥1} e_{orb}(S^{n}M) p^{n} = ∏_{n>0}(1 − p^{n})^{−24} = ∏_{m≥0, l, n>0}(1 − q^{m}y^{l}p^{n})^{−f(mn,l)}|_{y=1}.

In the case of an algebraic K3 surface e_{orb}(S^{n}M) is the topological Euler characteristic of the Hilbert scheme M^{(n)} of zero dimension subschemas of length n (see [Gö]).

Following [DVV, §4] we call the product

Ψ(M; Z) = ∏_{m≥0, l, n>0}(1 − q^{m}y^{l}p^{n})^{−f(mn,l)},  (3.2)

where f(m, l) are coefficients of the elliptic genus of M, the second-quantized elliptic genus (SQEG) of the manifold M.

In this section we prove that the SQEG multiplied by a factor (Hodge anomaly) depending only on invariants χ_{p}(M) of the Calabi–Yau d-fold

E(CY_{d}; Z) = Hodge anomaly(CY_{d}; Z) · Ψ(CY_{d}; Z)

is a Siegel automorphic form with respect to the paramodular group of polarization (1, d) (resp. (1, 2d)) for even (resp. odd) dimension d. We also calculate E(CY_{d}; Z) for small d in terms of the basic Siegel modular forms for the corresponding paramodular groups. For small dimensions d = 2, 3, 4, 6 the function E(CY_{d}; Z) is product of some powers of the denominator function of generalized Lorentzian Kac–Moody Lie algebras constructed in [GN1–GN4].

We considered in [GN1]–[GN4] the following infinite product for an arbitrary nearly holomorphic Jacobi form ϕ_{0,t}(τ, z) = ∑_{n,l} c(n, l)q^{n}y^{l} of weight 0 and integral index t

Exp-Lift(ϕ_{0,t})(Z) = q^{A}y^{B}s^{C} ∏_{\begin{matrix}n,l,m∈ℤ \\ (n,l,m)>0 \end{matrix}}(1 − q^{n}y^{l}s^{tm})^{c(nm,l)},  (3.3)

where

A = \frac{1}{24} ∑_{l} f(0, l), B = \frac{1}{2} ∑_{l>0} lf(0, l), C = \frac{1}{4} ∑_{l} l^{2}f(0, l)

and (n, l, m) > 0 means that if m > 0, then l and n are arbitrary integers, if m = 0, then n > 0 and l ∈ ℤ or l < 0 if n = m = 0.

In order to define the corresponding product for Calabi–Yau manifolds of odd dimension it is useful to modify the definition of SQEG. We remark that for arbitrary ϕ_{0, \frac{d}{2}} (τ, z) ∈ J_{0, \frac{d}{2}} the Jacobi form ϕ_{0, \frac{d}{2}} (τ, 2z) ∈ J_{0,2d} has even index. To see this one can apply symplectic transformation Λ_{2} = diag(1, 2, 1, 2^{−1}) to the corresponding Γ^{J}-modular form:

(F |Λ_{2}) ( \begin{matrix}τ & z \\ z & ω \end{matrix} ) = F ( \begin{matrix}τ & 2z \\ 2z & 4ω \end{matrix} ) .

The next theorem is a particular case of Theorem 2.1 from [GN4].

Theorem 3.1. Let M = M_{d} be a compact complex manifold of dimension d with trivial c_{1}(M),

χ(M; τ, z) = ∑_{m≥0, l∈ℤ (or ℤ/2)} f(m, l)q^{m}y^{l}

be its elliptic genus and SQEG(M; Z) (Z ∈ ℍ_{2}) be its second quantized elliptic genus (3.3). We define a factor

H(M; Z) = \{ \begin{matrix}η(τ)^{− \frac{1}{2} (e−3χ′_{d_{0}} )} ∏_{p=1}^{d_{0}}(ϑ(τ, pz) e^{πip^{2}ω})^{−χ′_{d_{0}−p}} & if d = 2d_{0} \\ η(τ)^{− \frac{1}{2} e} ∏_{p=1}^{d_{0}}(ϑ(τ, \frac{2p−1}{2}z) e^{\frac{1}{4} πi(2p−1)^{2}ω})^{−χ′_{d_{0}−p+1}} & if d = 2d_{0} + 1 \end{matrix}

where e = e(M) is Euler number of M and χ′_{p} = (−1)^{p}χ_{p}(M) (see (1.2)). Then the product

E(M; Z) = Ψ(M; Z) · SQEG(M; Z)  (d = 2d_{0})
E^{(2)}(M; Z) = (E|Λ_{2})(M; Z)  (d = 2d_{0} + 1)

determines a Siegel automorphic form of weight −\frac{1}{2}χ′_{d_{0}}(M) if d is even and of of weight 0 if d is odd with a character or a multiplier system of order 24/(24, e) with respect to a double extension of the paramodular group Γ_{d}^{+} (resp. Γ_{2d}^{+}), if d is even (resp. d is odd). The divisor of E(M; Z) (resp. of E^{(2)}(M; Z)) on A_{d}^{+} = Γ_{d} \ ℍ_{2} (resp. A_{2d}^{+}) is the union of a finite number of Humbert modular surfaces H_{D}(b) of discriminant D = b^{2} −4ad (resp. D = 4b^{2} − 8ad) with multiplicities

m_{D,b} = − ∑_{n>0} f(n^{2}a, nb).

Remark. We call the factor H(M; Z) defined above Hodge anomaly of SQEG. The divisor of ϑ(τ, pz) is pz ∈ τℤ + ℤ. Thus the zeros and poles of H(M; Z) (but not of E(M; Z)!) are completely defined by χ_{p}(M).

We would like to consider applications of this result to Calabi–Yau manifolds of dimension 3, 5 and 2, 4, …, 10. Before doing this we recall the definitions of the notions we used above. The paramodular group Γ_{t} is isomorphic to the integral symplectic group of the skew-symmetric form with elementary divisors (1, t). It can be realized as a subgroup of Sp_{4}(ℚ)

Γ_{t} := \{( \begin{matrix}∗ & t∗ & ∗ & ∗ \\ ∗ & ∗ & ∗ & t^{−1}∗ \\ ∗ & t∗ & ∗ & ∗ \\ t∗ & t∗ & t∗ & ∗ \end{matrix} ) ∈ Sp_{4}(ℚ) | all ∗ are integral \} .

If t ≠ 1, the group Γ_{t} is not a maximal discrete subgroup and it has normal extensions (see, for example, [GH1]). By definition,

Γ_{t}^{+} = Γ_{t} ∪ Γ_{t}V_{t}, V_{t} = \frac{1}{\sqrt{t}} ( \begin{matrix}0 & t & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & t & 0 \end{matrix} ) .

The double quotient

A_{t} ⟶^{2:1} A_{t}^{+} = Γ_{t}^{+} \ ℍ_{2}

of A_{t} can be interpreted as a moduli space of lattice-polarized K3 surfaces for arbitrary t or as the moduli space of Kummer surfaces of (1, p)-polarized Abelian surfaces for a prime t = p (see [GH1, Theorem 1.5]). Any Humbert surface in A_{t}^{+} of discriminant D can be represented in the form

H_{D}^{+}(b) = π_{t}^{+} ( ⋃_{g∈Γ_{t}^{+}} g^{∗}({Z ∈ ℍ_{2} | aτ + bz + tω = 0}))

where a, b ∈ ℤ, D = b^{2} − 4ta, 0 ≤ b < 2t and π_{t}^{+} : ℍ_{2} → A_{t}^{+} is the natural projection. According (3.3) the infinite product E(M; Z) can be written as follows

E(M_{d}; Z) = \{ \begin{matrix}Exp-Lift(−χ(M_{d}; τ, z)) & if d is even \\ Exp-Lift(−χ(M; τ, 2z)) & if d is odd. \end{matrix}

We note that for odd d the product E(M; Z) (without Λ_{2}-modification) is an automorphic function with respect to the group Λ_{2}Γ_{2d}Λ_{2}^{−1} conjugated to the paramodular group.

3.2. SQEG of Calabi–Yau manifolds of even dimension and Lorentzian Kac– Moody algebras.

1. The case of CY_{2}. One of the starting points for considerations in [DVV] was the infinite product expansion formula for the modular form Δ_{5}(Z). According to Example 1.3 we have for the SQEG of surfaces with trivial c_{1}

E(Enriques surface; Z) = Exp-Lift(−ϕ_{0,1})(Z) = Δ_{5}(Z)^{−1},
E(K3; Z) = Exp-Lift(−2ϕ_{0,1})(Z) = Δ_{5}(Z)^{−2}.

In [GN1] we proved that the modular form Δ_{5}(Z) defines an automorphic correction of the Kac–Moody algebra of hyperbolic type with the Cartan matrix

A_{1,II} = ( \begin{matrix}2 & −2 & −2 \\ −2 & 2 & −2 \\ −2 & −2 & 2 \end{matrix} ) .

It means that the Fourier expansion of the modular form Δ_{5}(Z) determines a generalized Kac–Moody Lie super-algebra with a system of the real simple roots of type A_{1,II}. We would like to note that the Lorentzian Kac–Moody algebras with Cartan matrix of types A_{1,0} and A_{1,I} constructed in [GN2] are related to SQEG of some vector bundles of rank 2 over a manifold of dimension 14 (see [G2]).

2. The case of CY_{4}. The basic Jacobi modular forms for this dimension are the Jacobi forms ϕ_{0,2} and ψ_{0,2}^{(2)} (see (1.8) and (1.10)). They correspond to the following cusp forms for the paramodular group Γ_{2} (see [GN1] and [GN4]):

Δ_{2}(Z) = Exp-Lift(ϕ_{0,2}(τ, z)) = Lift(η(τ)^{3}ϑ(τ, z))

= ∑_{N≥1} ∑_{\begin{matrix}n, m>0, l∈ℤ \\ n, m≡1 mod 4 \\ 2nm−l^{2}=N^{2} \end{matrix}}N(\frac{−4}{Nl}) ∑_{a | (n,l,m)}(\frac{−4}{a}) q^{n/4}y^{l/2}s^{m/2} ∈ 𝔐_{2}^{cusp}(Γ_{2}, v_{η}^{6} × v_{H})

and

Δ_{11}(Z) = Lift(η(τ)^{21}ϑ(τ, 2z)) = Exp-Lift(ψ_{0,2}^{(2)}(τ, z)) ∈ 𝔑_{11}(Γ_{2}).

For an arbitrary Calabi–Yau 4-fold M_{4} we have the following formula for its SQEG

E(M_{4}; Z) = Δ_{11}(Z)^{−χ_{0}(M)}Δ_{2}(Z)^{χ_{1}(M)}.  (3.4)

Its divisor is equal to (χ_{1}(M) − χ_{0}(M))H_{1} − χ_{0}H_{4}. We note that Δ_{2}(Z)^{4} is the first Γ_{2}-cusp form with trivial character and Δ_{11}(Z) is the first cusp form of odd weight with respect to Γ_{2}. Thus, if the Euler number of M_{4} is divisible by 24, then the automorphic form E(M_{4}; Z) has trivial character. The Fourier expansion of the cusp forms Δ_{2}(Z), Δ_{11}(Z) and \frac{Δ_{11}(Z)}{Δ_{2}(Z)} coincide with the Weyl–Kac–Borcherds denominator formula of generalized Kac–Moody super-algebras with generalized Cartan matrix A_{1,II}, A_{2,II} and A_{2,0} respectively:

A_{2,II} = ( \begin{matrix}2 & −2 & −6 & −2 \\ −2 & 2 & −2 & −6 \\ −6 & −2 & 2 & −2 \\ −2 & −6 & −2 & 2 \end{matrix} ) , A_{2,0} = ( \begin{matrix}2 & −2 & −2 \\ −2 & 2 & 0 \\ −2 & 0 & 2 \end{matrix} ) , A_{2,I} = ( \begin{matrix}2 & −2 & −4 & 0 \\ −2 & 2 & 0 & −4 \\ −4 & 0 & 2 & −2 \\ 0 & −4 & −2 & 2 \end{matrix} )

(see [GN1]–[GN4]). Thus, the formula (3.4) gives us three particular cases of Calabi–Yau 4-folds of Kac–Moody type when the second quantized elliptic genus is a power of the denominator function of the corresponding Lorentzian Kac–Moody algebra:

E(M_{4}; Z) = Δ_{11}(Z)^{−χ_{0}}  if χ_{1} = 0
E(M_{4}; Z) = (\frac{Δ_{11}(Z)}{Δ_{2}(Z)})^{−χ_{0}}  if χ_{0}(M) = −χ_{1}(M)
E(M_{4}; Z) = Δ_{2}(Z)^{χ_{1}}  if χ_{0}(M) = 0.

At this point we would like to discuss some relations of Hecke type between the basic Siegel modular forms occurred in SQEG-construction. It would be interesting to find a geometric or a physical interpretation of such relations. We recall that the Hecke operators T_{−}(m) on the space of Jacobi forms of weight zero play the main role in the construction of the Borcherds products (3.3) (see [B2], [GN1], [GN3]) and in the proof of the identity (3.1) for the second quantized elliptic genus (see [DMVV]). We start with a relation

ψ_{0,2}^{(2)} = ϕ_{0,1}|T_{−}(2) − 2ϕ_{0,2}.

Thus, according to [GN4, Theorem 3.3] we can represent Δ_{11}(Z) in the following form

Δ_{11}(Z) = Δ_{5}(Z)Δ_{5}(( \begin{matrix}τ & 2z \\ 2z & 4ω \end{matrix} ))Δ_{5}(( \begin{matrix}τ & z \\ z & ω+ \frac{1}{2} \end{matrix} ))Δ_{2}(Z)^{−2}.

Moreover we can write Δ_{11}^{2} only in terms of the cusp form Δ_{2} since

2ψ_{0,2}^{(2)} = ϕ_{0,2}|T_{0}(2),

where T_{0}(2) is another Hecke operator which does not change the index of Jacobi forms. If f(n, l) = g(8n−l^{2}) is the Fourier coefficient of ϕ_{0,2} (for a prime index t Fourier coefficients depend only on the norm 4tn − l^{2}), then the Fourier coefficient f_{2}(n, l) = g_{2}(8n − l^{2}) of ϕ_{0,2}|T_{0}(2) is given by the formula

g_{2}(N) = 8g(4N) + 2(\frac{−N}{2} )g(N) + g(\frac{N}{4} ).

According [GN2, Theorem A.7], Δ_{11}(Z)^{2} = [Δ_{2}(Z)]_{T (2)}/Δ_{2}(Z)^{4}, where

[Δ_{2}(Z)]_{T (2)} = ∏_{a,b,c mod 2} Δ_{2}(\frac{z_{1}+a}{2}, \frac{z_{2}+b}{2}, \frac{z_{3}+c}{2}) ∏_{a mod 2} Δ_{2}(\frac{z_{1}+a}{2}, z_{2}, 2z_{3}) Δ_{2}(2z_{1}, z_{2}, \frac{z_{3}+a}{2})

× Δ_{2}(2z_{1}, 2z_{2}, 2z_{3}) ∏_{b mod 2} Δ_{2}(2z_{1}, −z_{1} + z_{2}, \frac{z_{1}−2z_{2}+z_{3}+b}{2}).

3. The case of CY_{6}. In this case the elliptic genus of a Calabi–Yau 6-fold is a sum of three basic Jacobi forms ψ_{0,3}^{(i)} (see (1.11)). Thus there are three basic Siegel modular forms

Δ_{1}(Z) = Exp-Lift(ψ_{0,3}^{(1)}) = Lift(η(τ)ϑ(τ, z))

= ∑_{M≥1} ∑_{\begin{matrix}m>0, l∈ℤ \\ n, m≡1 mod 6 \\ 4nm−3l^{2}=M^{2} \end{matrix}}(\frac{−4}{l})(\frac{12}{M} ) ∑_{a|(n,l,m)}(\frac{6}{a})q^{n/6}y^{l/2}s^{m/2},

D_{6}(Z) = Exp-Lift(ψ_{0,3}^{(2)}) = Lift(η(τ)^{12}ϕ_{0, \frac{3}{2}} ), Δ_{17}(Z) = Exp-Lift(ψ_{0,3}^{(3)})

with divisors H_{1}, H_{4} and H_{1} + H_{9} respectively. (These Humbert modular surfaces have only one component in A_{6}^{+}). The Fourier expansion of the cusp forms Δ_{1}, D_{6} and Δ_{7}(Z) = Δ_{1}(Z)D_{6}(Z) determine three generalized Kac–Moody Lie super-algebras with the following Cartan matrix of the real simple roots:

A_{3,II} = ( \begin{matrix}2 & −2 & −10 & −14 & −10 & −2 \\ −2 & 2 & −2 & −10 & −14 & −10 \\ −10 & −2 & 2 & −2 & −10 & −14 \\ −14 & −10 & −2 & 2 & −2 & −10 \\ −10 & −14 & −10 & −2 & 2 & −2 \\ −2 & −10 & −14 & −10 & −2 & 2 \end{matrix} ) , A_{3,I} = ( \begin{matrix}2 & −2 & −5 & −1 \\ −2 & 2 & −1 & −5 \\ −5 & −1 & 2 & −2 \\ −1 & −5 & −2 & 2 \end{matrix} ) , A_{3,0} = ( \begin{matrix}2 & −2 & −2 \\ −2 & 2 & −1 \\ −2 & −1 & 2 \end{matrix} ) .

It follows that SQEG of an arbitrary Calabi–Yau 6-fold can be represented as

E(M_{6}, Z) = Δ_{17}(Z)^{−χ_{0}}Δ_{7}(Z)^{χ_{1}}Δ_{1}(Z)^{−χ_{2}}.

Thus there are three types of manifolds of Kac–Moody type: (χ_{0} = 0, χ_{1} = 0) or χ_{2} = −χ_{1} or χ_{2} = 0. Similar to Δ_{11} the cusp form Δ_{17} can be represented using a Hecke type product of Δ_{5} (see [GN4, Theorem 3.3]):

Δ_{17}(Z) = [Δ_{5}(( \begin{matrix}τ & 3z \\ 3z & 9ω \end{matrix} )) ∏_{b mod 3} Δ_{5}(( \begin{matrix}τ & z \\ z & ω+ \frac{b}{3} \end{matrix} ))]Δ_{1}(Z)^{−3}.

4. The case of CY_{8} and CY_{10}. In the case d = 8 we have four basic weak Jacobi forms of index 4 (see (1.9)). The exponential lifting of ϕ_{0,4}(τ, z) is the “most odd” even Siegel theta-constant

Exp-Lift(ϕ_{0,4})(Z) = Δ_{1/2}(Z) = \frac{1}{2} ∑_{n, m∈ℤ} (\frac{−4}{n})(\frac{−4}{m})q^{n^{2}/8}y^{nm/4}s^{m^{2}/8}

which is the denominator function of a generalized Lorentzian Kac–Moody super-algebra with a real root system of parabolic type (see [GN2–GN3]). For arbitrary Calabi–Yau 8-fold we have (see (1.11)):

E(CY_{8}; Z) = Δ_{1/2}(Z)^{χ_{1}−χ_{3}} · Δ_{5}(( \begin{matrix}τ & 2z \\ 2z & 4ω \end{matrix} ))^{−χ_{2}} · Δ_{12}(Z)^{χ_{1}} · Δ_{23}(Z)^{−χ_{0}},

where

Δ_{12}(Z) = Exp-Lift(ψ_{0,4}^{(3)}) ∈ 𝔐_{12}(Γ_{4}^{+}), Δ_{23}(Z) = Exp-Lift(ψ_{0,4}^{(4)}) ∈ 𝔐_{23}(Γ_{4}^{+})

are Γ_{4}^{+}-modular forms with trivial character. We remark that all forms above have multiplicities one along divisors! From the formulae above it follows that for d = 2, 4, 6, 8 the divisor of E(M_{d}; Z) is determined by the divisor of Hodge anomaly of M_{d}. For d = 10 this is not the case. From (1.12) we see that for d = 10 one new divisor appears. This is the Humbert surface H_{5} = π_{5}^{+}{τ + 5z + 5ω = 0} ⊂ A_{5}^{+}. The second quantized elliptic genus E(M_{11}; Z) is anti-invariant with respect to the involution defined by this rational quadratic divisor.

We remark also that for d < 12 (or d = 13) we have

E(CY_{d}; Z) = E(CY_{d}^{mir}; Z)^{−1}  (if d is odd)
E(CY_{d}; Z) = E(CY_{d}^{mir}; Z)  (if d is even)

where CY_{d}^{mir} is the mirror partner of Calabi–Yau d-fold.

3.3. SQEG of Calabi–Yau 3-folds and 5-folds. The elliptic genus of a Calabi–Yau 3-folds M_{3} is defined uniquely up to a constant (see Example 1.5):

χ(M_{3}, τ, z) = \frac{1}{2}e(M_{3})ϕ_{0, \frac{3}{2}} (τ, z).

According (3.3) and Theorem 3.1 the product

Φ_{3}(Z) = Exp-Lift(ϕ_{0, \frac{3}{2}} (τ, 2z)) =

q^{\frac{1}{12}} y^{\frac{1}{2}} s^{\frac{1}{2}} ∏_{\begin{matrix}n,l,m∈ℤ \\ (n,l,m)>0 \end{matrix}}(1 − q^{n}y^{l}s^{tm})^{c(nm,l)} ∈ 𝔐_{0}(Γ_{6}^{+}, v_{12}) (3.5)

is an automorphic function of weight 0 with respect to Γ_{6}^{+} with a character of order 12. (This character is induced by the square v_{η}^{2} of the multiplier system of the Dedekind eta-function η(τ) and by the binary character v_{H} of the Heisenberg group defined in §1.) Moreover

div_{A_{6}^{+}}(Φ_{3}(Z)) = H_{1}(0) − H_{1}(5) = π_{6}^{+}({Z ∈ ℍ_{2} | z = 0}) − π_{6}^{+}({Z ∈ ℍ_{2} | τ + 5z + 6ω = 0}).

The divisor H_{1}(5) appears because the Fourier expansion of ϕ_{0, \frac{3}{2}} (τ, 2z) contains the term −qy^{5}. We remark that Φ_{3}(Z) is anti-invariant with respect to the reflection in the vectors which define the Humbert surfaces H_{1}(0) and H_{1}(5). The character of the automorphic function Φ_{3}(Z) has the maximal possible order 12 since Γ_{6}^{+}/(Γ_{6}^{+})^{com} = ℤ/2 × ℤ/12 (see [GH2, Theorem 2.1]). One can say that Φ_{3}(Z) is a Siegel modular function with the simplest possible divisor. Such a function should have a zero and pole inside ℍ_{2}. The simplest divisor is a rational quadratic divisor (a Humbert surface) with discriminant one. The case t = 6 is the first case when there are two divisors of discriminant one. Φ_{3}(Z) has zero and poles of order one along these surfaces. Thus we can consider Φ_{3}(Z) as the best analogue of the modular invariant j(τ) in the case of Siegel modular functions!

It is interesting that Φ_{3}(Z) is related to Siegel modular threefolds of geometric genus 1 and 2. More exactly, the square of Φ_{3}(Z) can be written as quotient of two Γ_{6}^{+}-cusp forms constructed in [GN2, Example 4.6]

Φ_{3}(Z)^{2} = \frac{Lift(η(τ)^{3}ϑ(τ, z)^{2}ϑ(τ, 2z))}{Lift(η(τ)^{5}ϑ(τ, 2z))} = \frac{Exp-Lift(3ϕ_{0,3}^{2} − 2ϕ_{0,2}ϕ_{0,4})}{Exp-Lift(5ϕ_{0,3}^{2} − 4ϕ_{0,2}ϕ_{0,4})}.

The modular form in the numerator F_{1}(Z) = Lift(η(τ)^{3}ϑ(τ, z)^{2}ϑ(τ, 2z)) ∈ 𝔑_{3}(Γ_{6}, v_{2}) is cusp form of weight 3 with divisor 3H_{1}(1) + 2H_{1}(5) + H_{4}. Let us introduce the subgroup Γ_{6}(v_{2}) = Ker(v_{2}) ⊂ Γ_{6} of index 2 (v_{2} is character of order 2) and the covering

A_{6}(v_{2}) = Γ_{6}(v_{2}) \ ℍ_{2} ⟶^{2:1} A_{6}.

It is known that h^{3,0}(A_{6}(v_{2})) = 1 and the cusp form

F_{1}(Z)dτ ∧ dz ∧ dω ∈ H^{3,0}(A_{6}^{o}(v_{2}), ℂ)

defines the unique canonical differential form on a smooth model of A_{6}(v_{2}) (see [GH2, Theorem 3.1]). The form in the numerator F_{2}(Z) = Lift(η(τ)^{5}ϑ(τ, 2z)) ∈ 𝔑_{3}(Γ_{6}, v_{3}) is a cusp form of weight 3 with a character of order 3. The Siegel threefold

A_{6}(v_{3}) = Γ_{6}(v_{3}) \ ℍ_{2} ⟶^{3:1} A_{6}

has h^{3,0} = 2 and F_{2}(Z) defines one of the canonical differential forms on it.

Let us consider an arbitrary Calabi–Yau 3-fold M_{3} with Hodge numbers h^{1,1} and h^{2,1}. Its modified SQEG is the modular function

E^{(2)}(M_{3}; Z) = Φ_{3}(Z)^{h^{2,1}−h^{1,1}}.

In particular Φ_{3}(Z) is the modified SQEG of a Calabi–Yau 3-fold with Euler number −2. The first divisor (h^{2,1} − h^{2,2})H_{1}(0) of E^{(2)}(M_{3}; Z) comes from the Hodge anomaly H(M_{3}; Z), the second (h^{2,2} − h^{2,1})H_{1}(5) is the additional divisor which is, in some sense, “mirror symmetric” to the first one, because if M_{3}^{mir} is a mirror partner of M_{3}, then

E^{(2)}(M_{3}^{mir}; Z) = E^{(2)}(M_{3}; Z)^{−1}.

It turns out (see [GN5]) that the functions Φ_{3}(Z)^{±1} determine Lorentzian Kac–Moody super-algebras with an infinite system of real simple roots of the third possible type. This is the so-called hyperbolic type when the infinite system of real simple roots has a “limit” line (see [N]). The Lorentzian Kac–Moody algebras related to SQEG(CY_{d}) (d = 2, 4, 6, 8) have elliptic type (the system of real simple roots is finite) or parabolic type (the system of real simple roots has geometry similar to the real simple roots of the fake monster Lie algebra constructed by Borcherds in [B1]).

In the case d = 5 the elliptic genus is again defined uniquely by the Euler number. The basic Jacobi form is ϕ_{0, \frac{5}{2}} = ϕ_{0, \frac{3}{2}} ϕ_{0,1} (see Example 1.5). We have that

Φ_{5}(Z) = Exp-Lift(ϕ_{0, \frac{5}{2}} (τ, 2z)) ∈ 𝔐_{0}(Γ_{10}^{+})

is an automorphic function of weight 0 with respect to Γ_{10}^{+} with trivial character. The automorphic function Φ_{5}(Z) has the divisor consisting of four irreducible components

div_{A_{10}^{+}}(Φ(Z)) = H_{9}(3) − H_{9}(7) + 12H_{1}(1) − 12H_{1}(9).

For arbitrary Calabi–Yau fivefold M_{5} with Euler number χ(M_{5}) we have

E^{(2)}(M_{5}; Z) = Φ_{5}(Z)^{−e(M_{5})/24}.

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St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, 191011 St. Petersburg, Russia

Mathématiques-bât. M2, Université Lille I

F-59655 Villeneuve d’Ascq Cedex, France

   E-mail address: gritsenk@agat.univ-lille1.fr