nLab
Jacobi form

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,nk, n \in \mathbb{Z}, a Jacobi form of weight kk and index nn is a function of the form

ϕ:H× \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

(a b c d)SL 2() \left( \array{ a & b \\ c & d } \right) \in SL_2(\mathbb{Z})

as

ϕ(aτ+bcτ+d,zcτ+d)=(cτ+d) kexp(2πincz 2/(cτ+d))ϕ(τ,z). \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 θ\theta-functions are (with q=e 2πiτq = e^{2\pi i \tau})

θ(z,τ)2q 1/8sin(πz) j=1 ((1q j)(1e 2πizq j)(1e 2πizq j)) \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)
θ 1(z,τ)2q 1/8cos(πz) j=1 ((1q j)(1+e 2πizq j)(1+e 2πizq j)) \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)
θ 2(z,τ) j=1 ((1q j)(1e 2πizq j1/2)(1e 2πizq j1/2)) \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)
θ 3(z,τ) j=1 ((1q j)(1+e 2πizq j1/2)(1+e 2πizq j1/2)) \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

θ(0,τ)zθ(0,τ)=πθ 1(0,τ)θ 2(0,τ)θ 3(0,τ). \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

(…)

References

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)

See also

Last revised on September 10, 2014 at 19:52:28. See the history of this page for a list of all contributions to it.