complex geometry

# Contents

## Idea

Jacobi forms are power series of two variables which in one variable behave like a modular form and in the other have an “elliptic” nature. They arise naturally as the characteristic series of the elliptic genus/Witten genus (Zagier 86, pages 8-9).

## Definition

For $k, n \in \mathbb{Z}$, a Jacobi form of weight $k$ and index $n$ is a function of the form

$\phi \;\colon\; H \times \mathbb{C} \longrightarrow \mathbb{C}$

hence from the product of the upper half plane with the full complex plane which transforms under

$\left( \array{ a & b \\ c & d } \right) \in SL_2(\mathbb{Z})$

as

$\phi \left( \frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d} \right) = (c \tau+ d)^k \exp(2 \pi i n c z^2 / (c \tau + d)) \phi(\tau, z) \,.$

## Examples

### Jacobi theta-functions

The most important examples are the Jacobi theta-functions. The four Jacobi $\theta$-functions are (with $q = e^{2\pi i \tau}$)

$\theta(z,\tau) \coloneqq 2 q^{1/8} sin(\pi z) \prod_{j = 1}^{\infty} \left( \left( 1 - q^{j} \right) \left( 1 - e^{2\pi i z} q^{j} \right) \left( 1 - e^{-2 \pi i z} q^{j} \right) \right)$
$\theta_1(z,\tau) \coloneqq 2 q^{1/8} cos(\pi z) \prod_{j = 1}^{\infty} \left( \left( 1 - q^{j} \right) \left( 1 + e^{2\pi i z} q^{j} \right) \left( 1 + e^{-2 \pi i z} q^{j} \right) \right)$
$\theta_2(z,\tau) \coloneqq \;\;\;\;\;\;\;\;\; \prod_{j = 1}^{\infty} \left( \left( 1 - q^{j} \right) \left( 1 - e^{2\pi i z} q^{j - 1/2} \right) \left( 1 - e^{-2 \pi i z} q^{j - 1/2} \right) \right)$
$\theta_3(z,\tau) \coloneqq \;\;\;\;\;\;\;\;\; \prod_{j = 1}^{\infty} \left( \left( 1 - q^{j} \right) \left( 1 + e^{2\pi i z} q^{j - 1/2} \right) \left( 1 + e^{-2 \pi i z} q^{j - 1/2} \right) \right)$

See for instance (KL 95, section 2.4, Chen-Han-Zhang 10, section 2) for a review in the context of elliptic genera.

As part of this, the Kac-Weyl character of an integral highest-weight loop group representation is a Jacobi form (KL 95, section 2.2).

The Jacobi identity (see at Jacobi triple product) asserts that these are related by

$\theta'(0,\tau) \coloneqq \frac{\partial}{\partial z}\theta(0,\tau) = \pi \theta_1(0,\tau) \theta_2(0,\tau) \theta_3(0,\tau) \,.$

### Weierstrass function

(…)

The original canonical account is

• Martin Eichler, Don Zagier, The theory of Jacobi forms, Progress in Mathematics 55, Boston, MA: Birkhäuser Boston (1985), ISBN 978-0-8176-3180-2, MR 781735

Discussion of Jacobi forms as coefficients of the elliptic genus/Witten genus includes

• Don Zagier, pages 8,9 of Note on the Landweber-Stong elliptic genus 1986 (pdf)

• Kefeng Liu, On modular invariance and rigidity theorems, J. Differential Geom. Volume 41, Number 2 (1995), 247-514 (EUCLID, pdf)

• Matthew Ando, Christopher French, Nora Ganter, The Jacobi orientation and the two-variable elliptic genus, Algebraic and Geometric Topology 8 (2008) p. 493-539 (pdf)

• Qingtao Chen, Fei Han, Weiping Zhang, Generalized Witten Genus and Vanishing Theorems, Journal of Differential Geometry 88.1 (2011): 1-39. (arXiv:1003.2325)