Theta function

In mathematics, theta functions are special functions of several complex variables. They are important in several areas, including the theories of abelian varieties and moduli spaces, and of quadratic forms. They have also been applied to soliton theory. When generalized to a Grassman algebra, they also appear in quantum field theory, specifically string theory and D-branes.

The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions (sometimes called quasi-periodicity, though this is not related to the use of that term for dynamical systems). In the abstract theory this is shown to come from a line bundle condition of descent.

 Contents

Jacobi theta function

The Jacobi theta function is a function defined for two complex variables z and τ, where z can be any complex number and τ is confined to the upper half plane, which means it has positive imaginary part. It is given by the formula

[itex]\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp (\pi i n^2 \tau + 2 \pi i n z)[itex]

If τ is fixed, this becomes a Fourier series for a periodic entire function of z with period 1; in this case, the theta function satisfies the identity

[itex]\vartheta(z+1; \tau) = \vartheta(z; \tau).[itex]

The function also behaves very regularly with respect to addition by τ and satisfies the functional equation

[itex]\vartheta(z+a+b\tau;\tau) = \exp(-\pi i b^2 \tau -2 \pi i b z)\vartheta(z;\tau)[itex]

where a and b are integers.

Auxiliary functions

It is convenient to define three auxiliary theta functions, which we may write

[itex]\vartheta_{01} (z;\tau) = \vartheta(z+1/2;\tau)[itex]
[itex]\vartheta_{10}(z;\tau) = \exp(\pi i \tau/4 + \pi i z)\vartheta(z+\tau/2;\tau)[itex]
[itex]\vartheta_{11}(z;\tau) = \exp((\pi i \tau/4 + \pi i (z+1/2))\vartheta(z+(\tau+1)/2;\tau)[itex]

This notation follows Riemann and David Mumford; Jacobi's original formulation was in terms of the nome [itex]q = \exp(\pi \tau)[itex] rather than τ, and theta there is called [itex]\theta_3[itex], with [itex]\vartheta_{01}[itex] termed [itex]\theta_0[itex], [itex]\vartheta_{10}[itex] named [itex]\theta_2[itex], and [itex]\vartheta_{11}[itex] called [itex]-\theta_1[itex].

If we set z = 0 in the above theta functions, we obtain four functions of τ only, defined on the upper half plane (sometimes called theta constants.) These can be used to define a variety of modular forms, and to parametrize certain curves; in particular the Jacobi identity is

[itex]\vartheta(0;\tau)^4 = \vartheta_{01}(0;\tau)^4 + \vartheta_{10}(0;\tau)^4[itex]

which is the Fermat curve of degree four.

Jacobi identities

Jacobi's identities describe how theta functions transform under the modular group. Let

[itex]\alpha = (-i \tau)^{1/2} \exp\left({i \tau z^2 \over \pi}\right)[itex]

Then

[itex]\vartheta_1 (z; -1/\tau) = -i \alpha \vartheta_1 (\tau z; \tau)[itex]
[itex]\vartheta_2 (z; -1/\tau) = \alpha \vartheta_4 (\tau z; \tau)[itex]
[itex]\vartheta_3 (z; -1/\tau) = \alpha \vartheta_3 (\tau z; \tau)[itex]
[itex]\vartheta_4 (z; -1/\tau) = \alpha \vartheta_2 (\tau z; \tau)[itex]

Product representations

The Jacobi theta function can be expressed as a product, through the Jacobi triple product theorem:

[itex]\vartheta(z; \tau) = \prod_{m=1}^\infty

\left( 1-\exp 2i\pi \tau m \right) \left( 1+\exp i\pi \left[(2m-1)\tau +2z \right]\right) \left( 1+\exp i\pi \left[(2m-1)\tau -2z \right]\right) [itex]

The auxiliary functions have the expressions:

[itex]\vartheta_1 (z;q) = 2 q^{1/4} \sin z \prod_{n=1}^\infty (1 - q^{2n}) (1 - 2 q^{2n} \cos 2 z + q^{4n})[itex]
[itex]\vartheta_2 (z;q) = 2 q^{1/4} \cos z \prod_{n=1}^\infty (1 - q^{2n}) (1 + 2 q^{2n} \cos 2 z + q^{4n})[itex]
[itex]\vartheta_3 (z;q) = \prod_{n=1}^\infty (1 - q^{2n}) (1 + 2 q^{2n-1} \cos 2 z + q^{4n-2})[itex]
[itex]\vartheta_4 (z;q) = \prod_{n=1}^\infty (1 - q^{2n}) (1 - 2 q^{2n-1} \cos 2 z + q^{4n-2})[itex]

Integral representations

The Jacobi theta functions have the following integral representations:

[itex]\vartheta_1 (z; \tau) = -e^{iz + i \pi \tau / 4}

\int_{i - \infty}^{i + \infty} {e^{i \pi \tau u^2} \cos (2 u z + \pi \tau u) \over \sin (\pi u)} du[itex]

[itex]\vartheta_2 (z; \tau) = -i e^{iz + i \pi \tau / 4}

\int_{i - \infty}^{i + \infty} {e^{i \pi \tau u^2} \cos (2 u z + \pi u + \pi \tau u) \over \sin (\pi u)} du[itex]

[itex]\vartheta_3 (z; \tau) = -i

\int_{i - \infty}^{i + \infty} {e^{i \pi \tau u^2} \cos (2 u z + \pi u) \over \sin (\pi u)} du[itex]

[itex]\vartheta_4 (z; \tau) = -i

\int_{i - \infty}^{i + \infty} {e^{i \pi \tau u^2} \cos (2 u z) \over \sin (\pi u)} du[itex]

Relation to the Riemann zeta function

Note that

[itex]\vartheta(0;-1/\tau)=(-i\tau)^{1/2} \vartheta(0;\tau)[itex]

This relation was used by Riemann to prove the functional equation for Riemann's zeta function, by means of the integral

[itex]\Gamma\left(\frac{s}{2}\right) \pi^{-s/2} \zeta(s) =

\frac{1}{2}\int_0^\infty\left[\vartheta(0;it)-1\right] t^{s/2}\frac{dt}{t}[itex] which can be shown to be invariant under substitution of s by 1 − s. The corresponding integral for z not zero is given in the article on the Hurwitz zeta function.

Relation to the Weierstrass elliptic function

The theta function was used by Jacobi to construct (in a form adapted to easy calculation) his elliptic functions as the quotients of the above four theta functions, and could have been used by him to construct Weierstrass's elliptic functions also, since

[itex]\wp(z;\tau) = -(\log \vartheta_{11}(z;\tau))'' + c[itex]

where the second derivative is with respect to z and the constant c is defined so that the Laurent expansion of [itex]\wp(z)[itex] at z = 0 has zero constant term.

Some relations to modular forms

Let η be the Dedekind eta function. Then

[itex]\vartheta(0;\tau)=\frac{\eta^2\left(\frac{\tau+1}{2}\right)}{\eta(\tau+1)}[itex].

As solution to heat equation

The Jacobi theta function is the unique solution to the one-dimensional heat equation with periodic boundary conditions at time zero. This is most easily seen by taking z = x to be real, and taking τ = it with t real and positive. Then we can write

[itex]\vartheta (x,it)=1+2\sum_{n=1}^\infty \exp(-\pi n^2 t) \cos(2\pi nx)[itex]

which solves the heat equation

[itex]\frac{\partial}{\partial t} \vartheta(x,it)=\frac{1}{4\pi} \frac{\partial^2}{\partial x^2} \vartheta(x,it).[itex]

That this solution is unique can be seen by noting that at t = 0, the theta function becomes the Dirac comb:

[itex]\lim_{t\rightarrow 0} \vartheta(x,it)=\sum_{n=-\infty}^\infty \delta(x-n)[itex]

where δ is the Dirac delta function. Thus, general solution can be specified by convolving the (periodic) boundary condition at t = 0 with the theta function.

Relation to the Heisenberg group

The Jacobi theta function can be thought of as belonging to a representation of the Heisenberg group in quantum mechanics, sometimes called the theta representation. This can be seen by explicitly constructing the group. Let f(z) be a holomorphic function, let a and b be real numbers, and fix a value of τ. Then define the operators Sa and Tb such that

[itex](S_a f)(z) = f(z+a)[itex]

and

[itex](T_b f)(z) = \exp (i\pi b^2 \tau +2\pi ibz) f(z+b\tau)[itex]

Note that

[itex]S_{a_1} (S_{a_2} f) = (S_{a_1} \circ S_{a_2}) f = S_{a_1+a_2} f[itex]

and

[itex]T_{b_1} (T_{b_2} f) = (T_{b_1} \circ T_{b_2}) f = T_{b_1+b_2} f,[itex]

but S and T do not commute:

[itex]S_a \circ T_b = \exp (2\pi iab) \; T_b \circ S_a.[itex]

Thus we see that S and T together with a unitary phase form a nilpotent Lie group, the (continuous real) Heisenberg group, parametrizable as [itex]H=U(1)\times\mathbb{R}\times\mathbb{R}[itex] where U(1) is the unitary group. A general group element [itex]U(\lambda,a,b)\in H[itex] then acts on a holomorphic function f(z) as

[itex]U(\lambda,a,b)\;f(z)=\lambda (S_a \circ T_b f)(z) =

\lambda \exp (i\pi b^2 \tau +2\pi ibz) f(z+a+b\tau)[itex]

where [itex]\lambda \in U(1)[itex]. Note that U(1) = Z(H) is the center of H, the commutator subgroup [HH].

Define the subgroup [itex]\Gamma\subset H[itex] as

[itex]\Gamma = \{ U(1,a,b) \in H : a,b \in \mathbb{Z} \}.[itex]

Then we see that the Jacobi theta function is an entire function of z that is invariant under Γ, and it can be shown that the Jacobi theta is the unique such function.

The above theta representation of the Heisenberg group can be related to the canonical Weyl representation of the Heisenberg group as follows. Fix a value for τ and define a norm on entire functions of the complex plane as

[itex]\Vert f \Vert ^2 = \int_{\mathbb{C}}

\exp \left( \frac {-2\pi y^2} {\Im \tau} \right) |f(x+iy)|^2 \ dx \ dy [itex]

Let [itex]\mathcal{J}[itex] be the set of entire functions f with finite norm. Note that [itex]\mathcal{J}[itex] is a Hilbert space, and that [itex]U(\lambda,a,b)[itex] is unitary on [itex]\mathcal{J}[itex], and that [itex]\mathcal{J}[itex] is irreducible under this action. Then [itex]\mathcal{J}[itex] and L2(R) are isomorphic as H-modules, where H acts on L2(R) as

[itex]U(\lambda,a,b)\;\psi(x)=\lambda \exp (2\pi ibx) \psi(x+a)[itex]

for [itex]x\in\mathbb{R}[itex] and [itex]\psi\in L^2(\mathbb{R})[itex].

See also the Stone-von Neumann theorem for additional development of these ideas; note that in relation to that article, [itex]S=\exp iP/\hbar[itex] and [itex]T=\exp i2\pi Q[itex].

Generalizations

If F is a quadratic form in n variables, then the theta function associated with F is

[itex]\theta_F (z)= \sum_{m\in Z^n} \exp(2\pi izF(m))[itex]

with the sum extending over the lattice of integers Zn. This theta function is a modular form of weight n/2 (on an appropriately defined subgroup) of the modular group. In the Fourier expansion,

[itex]\theta_F (z) = \sum_{k=0}^\infty R_F(k) \exp(2\pi ikz)[itex],

the numbers RF(k) are called the representation numbers of the form.

Riemann theta function

Let

[itex]\mathbb{H}_n=\{F\in M(n,\mathbb{C}) \; \mathrm{s.t.}\, F=F^T \;\textrm{and}\; \mbox{Im} F >0 \}[itex]

be set of symmetric square matrices whose imaginary part is positive definite. Hn is called the Siegel upper half space and is the multi-dimensional analog of the upper half-plane. The n-dimensional analogue of the modular group is the symplectic group Sp(2n,Z); note that for n = 1, Sp(2,Z) = SL(2,Z). The n-dimensional analog of the congruence subgroups is played by [itex]\textrm{Ker} \{\textrm{Sp}(2n,\mathbb{Z})\rightarrow \textrm{Sp}(2n,\mathbb{Z}/k\mathbb{Z}) \}[itex].

Then, given [itex]\tau\in \mathbb{H}_n[itex], the Riemann theta function is defined as

[itex]\theta (z,\tau)=\sum_{m\in Z^n} \exp\left(2\pi i

\left(\frac{1}{2} m^T \tau m +m^T z \right)\right). [itex]

Here, [itex]z\in \mathbb{C}^n[itex] is an n-dimensional complex vector, and the superscript T denotes the transpose. The Jacobi theta function is then a special case, with n=1 and [itex]\tau \in \mathbb{H}[itex] where [itex]\mathbb{H}[itex] is the upper half-plane.

The Riemann theta converges absolutely and uniformly on compact subsets of [itex]\mathbb{C}^n\times \mathbb{H}_n.[itex]

The functional equation is

[itex]\theta (z+a+\tau b, \tau) = \exp 2\pi i

\left(-b^Tz-\frac{1}{2}b^T\tau b\right) \theta (z,\tau)[itex]

which holds for all vectors [itex]a,b \in \mathbb{Z}^n[itex], and for all [itex]z \in \mathbb{C}^n[itex] and [itex]\tau \in \mathbb{H}_n[itex].

References

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 486-61272-4 . (See section 16.27ff.)
• Naum Illyich Akhiezer, Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island ISBN 0-8218-4532-2
• Hershel M. Farkas and Irwin Kra, Riemann Surfaces (1980), Springer-Verlag, New York. ISBN 0-387-90465-4 (See Chapter 6 for treatment of the Riemann theta)
• David Mumford, Tata Lectures on Theta I (1983), Birkhauser, Boston ISBN 3-7643-3109-7
• James Pierpont Functions of a Complex Variable, Dover
• Harry E. Rauch and Hershel M. Farkas, Theta Functions with Applications to Riemann Surfaces, (1974) Williams & Wilkins Co. Baltimore ISBN 683-07196-3.

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