GENERALIZED ELLIPTIC GENERA

V. GRITSENKO (Lille/POMI)

In this talk we present a new link between the theory of automorphic forms and geometry. Let M be an (almost) complex compact manifold M of (complex) dimension d and E be a holomorphic vector bundle over M. We set q = exp(2πiτ) and y = exp(2πiz) (τ ∈ ℍ_{1}, z ∈ ℂ). One can define a formal series E_{q,y} ∈ K(M)[[q, y^{±1}]] (see [W], [H1])

E_{q,y} = ⨂_{n=0}^{∞} ⋀_{−y^{−1}q^{n}}E^{∗} ⊗ ⨂_{n=1}^{∞} ⋀_{−yq^{n}}E ⊗ ⨂_{n=1}^{∞} S_{q^{n}}T_{M}^{∗} ⊗ ⨂_{n=1}^{∞} S_{q^{n}}T_{M}

where E^{∗} is dual to E, T_{M} denotes the holomorphic tangent bundle of M, ⋀_{x}E = ∑_{k≥0} (∧^{k}E)x^{k} and S_{x}E = ∑_{k≥0}(S^{k}E)x^{k}. We propose the following

Definition. Modified Witten genus (MWG) of E → M is a function

χ(M, E; τ, z) = q^{(r−d)/12}y^{r/2} ∫_{M} exp(\frac{1}{2}(c_{1}(E) − c_{1}(T_{M})))·

exp((p_{1}(E) − p_{1}(T_{M})) · G_{2}(τ)) exp(−\frac{c_{1}(E)}{2πi} \frac{ϑ_{z}}{ϑ} (τ, z)) ch(E_{q,y}) td(T_{M})

where c_{1}(E) and p_{1}(E) are the first Chern and Pontryagin class of E, td is the Todd class, ch(E_{q,y}) is the Chern character applied to each coefficient of the formal power series and ∫_{M} denotes the evaluation of the top degree differential form on the fundamental cycle of the manifold. In the definition we use Jacobi theta-series of level two

ϑ(τ, z) = −q^{1/8}y^{−1/2} ∏_{n≥1} (1 − q^{n−1}y)(1 − q^{n}y^{−1})(1 − q^{n}), ϑ_{z}(τ, z) = \frac{∂ϑ}{∂z}(τ, z).

G_{2}(τ) = −\frac{1}{24} + ∑_{n=1}^{∞} σ_{1}(n) q^{n} is a quasi-modular Eisenstein series of weight 2.

Witten genus. Let E = 0 be the trivial vector bundle of rank 0. Let us assume that M admits a spin structure and p_{1}(M) = 0. In this case

q^{d/12}χ(M, 0; τ, z) = q^{−d/12} ∫_{M} ch(⨂_{n=1}^{∞} S_{q^{n}}(T_{M} ⊕ T_{M}^{∗}))Â(M) = \frac{Witten genus (M, τ)}{η(τ)^{2d}}.

It is known that the Witten genus is a SL_{2}(ℤ)-modular form of weight 2k.

Elliptic genus of a Calabi–Yau manifold. This case is of some interest in physics. Let M be an arbitrary complex compact manifold and E = T_{M}. The holomorphic Euler characteristic of E_{q,y}

ϕ_{ell}(M; τ, z) = y^{d/2} ∫_{M} ch(E_{q,y})td(T_{M})

is called (in physics) elliptic genus of M. One can see that the elliptic genus of M (as a function in τ, z) is equal to the zeroth term in a sum of d + 1 summands of the modified Witten genus for E = T_{M}. These summands correspond to the powers of the first Chern class of M

χ(M, T_{M}; τ, z) = ϕ_{ell} (M; τ, z) + ∑_{n=1}^{d}(∫_{M} c_{1}(M)^{n}(…)).

Thus, χ(M, T_{M}; τ, z) = ϕ_{ell} (M; τ, z) if c_{1}(T_{M}) = 0. For E = T_{M} and c_{1}(T_{M}) ≠ 0 the integral in the definition of MWG contains the only correction term

exp(−\frac{c_{1}(T_{M})}{2πi} \frac{ϑ_{z}}{ϑ} (τ, z)).

It is known that the elliptic genus of a Calabi–Yau manifold is a weak Jacobi form of weight 0 and index d/2 (see [EOTY], [Hö], [KYY], [L]). If c_{1}(T_{M}) ≠ 0, then the elliptic genus of M is not a modular form in τ and z. We add the three correction factors in the definition in order to obtain a function with a good behavior with respect to the modular transformations in τ and z. In general the elliptic genus is not an automorphic form in two variables but the modified Witten genus is.

Theorem 1. Let χ(M, E; τ, z) be the modified Witten genus. Then the product

χ(M, E; τ, z) (\frac{ϑ(τ, z)}{η(τ)})^{d−r}

is a weak Jacobi form of weight 0 and index d/2.

We recall the definition of Jacobi forms. A holomorphic function ϕ(τ, z) on ℍ_{1} × ℂ is called a nearly-holomorphic Jacobi form of weight k and index m (m ∈ ℤ/2) if it satisfies the functional equations

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

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

The form ϕ(τ, z) has the Fourier expansion of the type ϕ(τ, z) = ∑_{n, l} f(n, l) q^{n}y^{l}. If a(n, l) ≠ 0 only for n ≥ 0, then ϕ is called weak. We denote the space of all week Jacobi forms of weight k and index m by J_{k,t}. In particular, ϑ(τ, z) ∈ J_{\frac{1}{2} , \frac{1}{2}} (v_{η}^{3}) (v_{η} is the multiplier system of the Dedekind η-function). The next lemma is very useful.

Lemma. The following formula is valid

exp(−4π^{2}G_{2}(τ)ξ^{2} − \frac{ϑ_{z}}{ϑ} (τ, z)ξ)ϑ(τ, z + ξ) = ϑ(τ, z) exp(− ∑_{n≥2} ℘^{(n−2)}(τ, z)\frac{ξ^{n}}{n!} )

where ℘(τ, z) = z^{−2} + ∑_{ω∈ℤτ+ℤ}^{ω≠0}((z + ω)^{−2} − ω^{−2}) ∈ J_{2,0}^{mer} is the Weierstrass ℘-function

and ℘^{(n)}(τ, z) = \frac{∂^{n}}{∂z^{n}} ℘(τ, z) ∈ J_{n+2,0}^{mer}.

(See [G1], [G2] for more details.)

Let c(E) = ∑_{i=0}^{r} c_{i}(E) = ∏_{i=1}^{r}(1+x_{i}) be the total Chern class of E where x_{i} (1 ≤ i ≤ r) are the formal Chern roots of E. We denote by x′_{j} (1 ≤ j ≤ d) the Chern roots of T_{M}. Using the formula of the lemma we can split the Chern roots {x_{i}} and {x′_{j}} under the integral in MWG. More exactly, we have

χ(M, E; τ, z) = \frac{ϑ^{r}}{η^{r+2d}} ∫_{M} P (E; τ, z) · W (M; τ).  (S)

The first factor depends only on the vector bundle E

P (E; τ, z) = exp(− ∑_{n≥2} \frac{℘^{(n−2)}(τ, z)}{(2πi)^{n}n!}(∑_{i=1}^{r} x_{i}^{n})).

The second factor is the Witten factor

W (M; τ) = exp(2 ∑_{k≥2} \frac{G_{2k}(τ)}{(2k)!}(∑_{j=1}^{d} x′_{j}^{2k}))

which determines the Witten genus of the manifold M. Theorem 1 is a direct corollary of the representation (S) above. (See [G2] for more details.)

We note that the definition of MWG, Theorem 1 and its proof are valid for an arbitrary compact Spin^{c}-manifold M and a Hermitian C^{∞}-vector bundle E over M. Representation (S) is new also in the case of the elliptic genus of Calabi–Yai manifolds. It shows us that ϕ_{ell}(M, τ, z) is a twisted version of the Witten genus. In order to define a generalization of the elliptic genus of Oshanine–Landweber as a function in τ and z one should use some other vector bundle instead of E_{q,y} above.

Examples for dimension d = 2, 3, 4 and d = 6, r = 2. For d = 2 and d = 3 we consider an arbitrary E over M. It follows from (S) that

χ(M_{2}, E_{r}; τ, z) = \frac{1}{2(2πi)^{2}} (∑_{i=1}^{r} x_{i}^{2})[M]\frac{ϑ^{r}}{η^{r+4}} ℘ = \frac{1}{24}(c_{1}(E)^{2} − 2c_{2}(E)) ϕ_{0,1}(\frac{ϑ}{η})^{r−2}

where ϕ_{0,1}(τ, z) = (y + 10 + y^{−1}) + q(…) is the unique, up to a constant, weak Jacobi form of weight 0 and index 1. Let ϕ_{0, \frac{3}{2}} (τ, z) = ϑ(τ, 2z)/ϑ(τ, z). For d = 3 we have

χ(M_{3}, E_{r}; τ, z) = \frac{1}{6}(c_{1}(E)^{3} − 3c_{1}(E)c_{2}(E) + 3c_{3}(E))ϕ_{0, \frac{3}{2}} (τ, z)(\frac{ϑ}{η})^{r−3}.

Let us consider the case d = 4, r ≥ 4 with p_{1}(E_{4}) = p_{1}(M_{4}) (c_{1}(E_{4}) is arbitrary). We have the following formula for MWG

χ(M_{4}, E_{r}; τ, z) = [Â(M_{4})ψ_{0,2}^{(2)}(τ, z) + (rÂ(M_{4}) − Â(M_{4}, E_{r}))ϕ_{0,2}(τ, z)](\frac{ϑ(τ, z)}{η(τ)})^{r−4},

where ϕ_{0,2}(τ, z) = y + 4 + y^{−1} + q(…) is a Jacobi form related to a generalized Lorentzian Kac–Moody Lie algebra (see [GN1]) and ψ_{0,2}^{(2)} = ϕ_{0,1}^{2} − 24ϕ_{0,2}.

Let us consider the case d = 6, r = 2, p_{1}(E) = p_{1}(T_{M}). We assume also that MWG is holomorphic. (In the case under consideration this condition is equivalent to the relation < c_{1}(E)^{4}c_{2}(E) − \frac{1}{6}c_{1}(E)^{6} > [M] = 0). Then

χ(M_{6}, E_{2}; τ, z) = [Â(M)E_{6}(τ)ϕ_{−2,1} + (2Â(M) − Â(M, E))E_{4,1}(τ, z)]η(τ)^{−8},

where ϕ_{−2,1} = ϑ^{2}/η^{6}, E_{6}(τ) is the Eisenstein series of weight 6 and E_{4,1}(τ, z) is the Eisenstein–Jacobi series of weight 4 and index 1. (E_{4,1} is equal to the Jacobi theta-series of the lattice 𝔼_{8}.)

One can obtain many others exact formulae. The fact that MWG of a vector bundle is a Jacobi modular form has some number of applications in the theory of vector bundles (relations between Â-genera and between other cohomological invariants, some obstructions if rank<dim, arithmetic of special values). As one can see from the definition, the coefficients of MWG are rational with bounded denominators. In the next theorem we determine ℤ-structure of the graded ring J_{0,∗}^{ℤ} of all weak Jacobi forms of weight 0 with integral Fourier coefficients. We denote by J_{0,m}^{ℤ}(q) the subspace of integral Jacobi form with q^{0}-term equal to 0.

Theorem 2. 1. J_{0,m}^{ℤ}(q) = ξ_{0,6}J_{0,m−6}^{ℤ}, where ξ_{0,6}(τ, z) = (ϑ(τ, z)/η(τ))^{12}.

2. 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 we can chose a basis with the following q^{0}-terms

[ψ_{0,m}^{(n)}(τ, z)]_{q^{0}} = y^{n} + n^{2}y + (2n^{2} − 2) + n^{2}y^{−1} + y^{−n}  (2 ≤ n ≤ m),
[ψ_{0,m}^{(1)}]_{q^{0}} = \frac{1}{(12, m)} (my + (12 − 2t) + my^{−1}),

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

3. 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 ϕ_{0,1}, ϕ_{0,2}, ϕ_{0,3} = ϕ_{0, \frac{3}{2}}^{2} (defined in examples above) are algebraicly independent and 4ϕ_{0,4} = ϕ_{0,1}ϕ_{0,3} − ϕ_{0,2}^{2}.

The proof of the theorem see in [G1]. We remark that ξ_{0,6} is MWG of a rank 12 vector bundle over a point. One can consider the generators ϕ_{0,n} as analytic analogies of the Tate’s parameters b_{2}, b_{4}, b_{6}, b_{8} of the elliptic curve (see [T]). Then ξ_{0,6} is the discriminant of the curve.

One of the direct corollaries of the last theorem is the following property of Euler number of a Calabi–Yau manifold. Let us consider the case

d = r, p_{1}(E) = p_{1}(T_{M}), c_{1}(E) = 0 (over ℤ).

We consider spin bundle Δ^{+}(E) − Δ^{−}(E) such that ch(Δ^{+}(E) − Δ^{−}(E)) = ∏_{i=1}^{r}(e^{x_{i}/2} − e^{−x_{i}/2}). Then

Â(M, Δ^{+}(E) − Δ^{−}(E)) ≡ \{ \begin{matrix}0 mod 8 & d ≡ 1, 2, 5 mod 8 \\ 0 mod 4 & d ≡ 6 , 7 mod 8 \\ 0 mod 2 & d ≡ 3, 4 mod 8 \\ 0 mod 3 & d ≢ 0 mod 3. \end{matrix}

The same relations are true, in particular, for the Euler number e of a Calabi–Yau manifold. For example, e(CY_{5}) ≡ 0 mod 24 and e(CY_{10}) ≡ 0 mod 24. The divisibily of the Euler number by 3 was proved by Hirzebruch in 1960. After my talk at MPI on this subject Prof. Hirzebruch informed me that the properties of e(CY_{d}) modulo 8 were known for him (non-published, see [H2]). We note also that e(CY_{d})/24 is a special obstruction in the string theory. (See [SVW], where e(CY_{4}) ≡ 0 mod 6 was proved.)

We finish the talk with some possible strange “periodicity” for MWG. If p_{1}(E) = p_{1}(M) and c_{1}(E) = 0, then MWG is the partition function for (0, 2)-symmetric non-linear sigmamodel (see [KM]). Let us assume that χ(M_{1}, E_{2}, τ, z) ≡ χ(M_{1}, E_{2}, τ, z). Then Theorem 1 implies that r_{1} = r_{2} and d_{1} ≡ d_{2} mod 12.

Question. To construct a vector bundle E of rank 2 over a complex manifold M of dim = 14 such that p_{1}(E) = p_{1}(M), c_{1}(E) = 0, Â(M) = 0, Â(M, E) = −2 and Â(M, E ⊗ T_{M}) = −68. (The last condition is equivalent to Â(M, S^{2}E) + 2Â(M, T_{M}) = 0.) If such a vector bundle exist then its Second Quantized Elliptic Genus is equal, up to a simple factor, to the automorphic correction of the simplest hyperbolic Kac–Moody algebra (see [D], [G1], [GN2]).

References

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Département de Mathématique Université Lille I

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