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Jacobi form
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Contents
Context
Complex geometry
Elliptic cohomology
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 ∈ ℤ k, n \in \mathbb{Z} , a Jacobi form of weight k k and index n n is a function of the form
ϕ : H × ℂ ⟶ ℂ
\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 ( ℤ )
\left(
\array{
a & b
\\
c & d
}
\right)
\in
SL_2(\mathbb{Z})
as
ϕ ( a τ + b c τ + d , z c τ + d ) = ( c τ + d ) k exp ( 2 π i n c z 2 / ( c τ + 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 2 π i τ q = e^{2\pi i \tau} )
θ ( z , τ ) ≔ 2 q 1 / 8 sin ( π z ) ∏ j = 1 ∞ ( ( 1 − q j ) ( 1 − e 2 π i z q j ) ( 1 − e − 2 π i z q 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 ) ≔ 2 q 1 / 8 cos ( π z ) ∏ j = 1 ∞ ( ( 1 − q j ) ( 1 + e 2 π i z q j ) ( 1 + e − 2 π i z q 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 ∞ ( ( 1 − q j ) ( 1 − e 2 π i z q j − 1 / 2 ) ( 1 − e − 2 π i z q j − 1 / 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 ∞ ( ( 1 − q j ) ( 1 + e 2 π i z q j − 1 / 2 ) ( 1 + e − 2 π i z q j − 1 / 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 ) ≔ ∂ ∂ z θ ( 0 , τ ) = π θ 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 February 15, 2025 at 04:22:08.
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