Theta Functions: A Lightning Review Johar M. Ashfaque
We review some of the properties of the theta functions. We begin y explaining Jacobi’s triple product formula; then we define theta functions in terms of series and infinite products. We also express the Dedekind η function in terms of the theta functions. We finally derive the conformal properties of the theta functions and of the Dedekind η function.
1
The Jacobi Triple Product
We consider the Jacobi’s triple product identity ∞ Y
(1 − q n )(1 + q n−1/2 t)(1 + q n−1/2 /t) =
n=1
X
qn
2
/2 n
t .
n∈Z
This identity is valid for |q| < 1 and t 6= 0 and can be demonstrated by combinatorial methods. Here we shall argue that this identity is correct by analogy with the fermion-antifermion system. Consider a set of fermion oscillators bn and their antifermion counterparts bn with the Hamiltonian X † H = E0 r(b†r br + br br ). r∈N+1/2
The fermion number operators is N=
X
†
(b†r br − br br ).
r∈N+1/2
Now we consider the grand partition function X Z(q, t) = e−β(E−µN ) ,
q = e−βE0 ,
t = eβµ .
states
We will evaluate this quantity in two different ways leading to the two sides of the following equation which is manifestly equivalent to the Jacobi’s triple product identity ∞ Y
(1 + q r t)(1 + q r /t) =
∞ Y
1 X n2 /2 n q t . 1 − qn n=1
(1)
n∈Z
r∈N+1/2
First, the grand partition function factorizes into a product of grand partition functions, each associated with a single fermion oscillator. This, of course, follows from the fact that H and N decouple into sums over different fermion modes. Since the grand partition functions for a fermion and antifermion modes labelled r are respectively (1 + q r t) and (1 + q r /t) (there are two occupation states), the complete grand partition function coincides with the left hand side of Eq. (1). Second, the grand partition function may be written as X Z(q, t) = tn Zn (q) n∈N
where Zn (q) is the ordinary partition function for a fixed fermion number n. We consider first Z0 , the partition function with no net fermion number. The lowest energy states are given in the following table. Energy 0 1
Degeneracy 1 1
States |0i † † b1/2 b1/2 |0i
2
2
b†3/2 b1/2 |0i, b†1/2 b3/2 |0i
3
3
b†5/2 b1/2 |0i, b†3/2 b3/2 |0i, b†1/2 b5/2 |0i
4
5
b†7/2 b1/2 |0i, b†5/2 b3/2 |0i, b†3/2 b5/2 |0i, b†1/2 b7/2 |0i, b†3/2 b3/2 b†1/2 b1/2 |0i
†
†
†
†
†
†
†
1
†
†
†
†
The number of creation operators in these states is always even and the sum of their indices is equal to the normalized energy level m = E/E0 . We notice so far that the degeneracy at level m is equal to the partition number p(m). Therefore Z0 is equal to ∞ X
Z0 =
p(m)q m =
m=0
∞ Y
1 . 1 − qn n=1
This confirms Eq. (1) as far as the t0 term is concerned. We now consider Zn . The lowest energy state with fermion number n is obtained by exciting the lowest n oscillators b†1/2 b†3/2 ...b†n−1/2 |0i
n X
E/E0 =
r=1
1 n− 2
=
n2 . 2
It turns out that the excitations on top of this ground state have exactly the same structure as the excitations of the n = 0 sector. Therefore Zn = q n
2
/2
∞ Y
1 1 − qn n=1
and Eq. (1).
2
Theta Functions
Jacobi’s theta functions are defined as follows θ1 (z|τ )
=
X
−i
(−1)r−1/2 y r q r
2
/2
r∈Z+1/2
θ2 (z|τ )
X
=
yr qr
2
/2
r∈Z+1/2
θ3 (z|τ )
=
X
yn qn
2
/2
n∈Z
θ4 (z|τ )
=
X
(−1)n y n q n
2
/2
n∈Z
where z is a complex variable and τ a complex parameter living on the upper half-plane. We have defined q = exp(2πiτ ) and y = exp(2πiz). Jacobi’s triple product allows us to write these functions in the form of infinite products θ1 (z|τ )
=
−iy 1/2 q 1/8
∞ Y
(1 − q n )
n=1
θ2 (z|τ )
=
y 1/2 q 1/8
∞ Y
θ3 (z|τ )
=
θ4 (z|τ )
=
(1 − q n )
(1 − q n )
(1 + yq n+1 )(1 + y −1 q n )
(1 + yq r )(1 + y −1 q r )
r∈N+1/2
∞ Y
∞ Y
n=1
∞ Y
n=0 ∞ Y
n=1
(1 − q n )
(1 − yq n+1 )(1 − y −1 q n )
n=0
n=1 ∞ Y
∞ Y
(1 − yq r )(1 − y −1 q r )
r∈N+1/2
For instance, the equivalence of the two expressions for θ1 is obtained by setting t = yq 1/2 in ∞ Y
(1 − q n )(1 + q n−1/2 t)(1 + q n−1/2 /t) =
n=1
X n∈Z
2
qn
2
/2 n
t .
By shifting their arguments, theta functions may all be related to each other from their definitions, it is a simple matter to check that 1 = θ3 (z + |τ ) 2
θ4 (z|τ )
1 = −ieiπz q 1/8 θ4 (z + τ |τ ) 2 1 θ2 (z|τ ) = θ1 (z + |τ ) 2 θ1 (z|τ )
Theta functions are used to define doubly periodic functions on the complex plane. One sees that they are not periodic under z → z + 1 or z → z + τ but obey the simple relations θ1 (z + 1|τ )
=
−θ1 (z|τ )
θ2 (z + 1|τ )
=
−θ2 (z|τ )
θ3 (z + 1|τ )
=
θ3 (z|τ )
θ4 (z + 1|τ )
=
θ1 (z + τ |τ )
=
θ2 (z + τ |τ )
=
θ3 (z + τ |τ )
=
θ4 (z + τ |τ )
=
θ4 (z|τ ) 1 − θ1 (z|τ ) yq 1 θ2 (z|τ ) yq 1 θ3 (z|τ ) yq 1/2 1 − 1/2 θ4 (z|τ ) yq
It follows that doubly periodic functions may be easily constructed out of ratios or logarithmic derivatives of theta functions. The best known example is the Weierstrass function 2
℘(z|τ ) = − ∂ 2 ln θ1 (z|τ ) − 2η1 ∂z
where the constant η1 depends only on τ η1 = −
1 ∂z3 θ1 (0|τ ) . 6 ∂z θ1 (0|τ )
We shall also use the theta functions at z = 0 θi (τ ) ≡ θi (0|τ ) for i = 2, 3, 4. One can easily check that θ1 (0|τ ) = 0. Their explicit expressions, in terms of sums and products are θ2 (τ )
X
=
q (n+1/2)
2
/2
= 2q 1/8
θ3 (τ ) θ4 (τ )
=
=
qn
2
/2
=
∞ Y
n∈Z
n=1
X
2
(−1)n q n
/2
(1 − q n )(1 + q n−1/2 )2
=
∞ Y
(1 − q n )(1 − q n−1/2 )2
n=1
n∈Z
3
(1 − q n )(1 + q n )2
n=1
n∈Z
X
∞ Y
Dedekind’s η Function
Dedekind’s η function is defined as η(τ ) = q 1/24 ψ(q) = q 1/24
∞ Y
(1 − q n )
n=1
3
where ψ(q) is the Euler function. This function is related to the theta functions as follows 1 θ2 (τ )θ3 (τ )θ4 (τ ). 2
η 3 (τ ) =
This identity is an immediate consequence of the infinite product expressions for the theta functions at z = 0; we simply need to show that the function f (q) =
∞ Y
(1 + q n )(1 + q n−1/2 )(1 − q n−1/2 )
n=1
is equal to unity. But we may write f (q) =
∞ Y
(1 + q n )(1 − q 2n−1 ).
n=1
4
Modular Transformations of Theta Functions
We are interested in the behaviour of the theta functions θi (τ ) under the modular transformation τ → − τ1 . For this we need the following formula called the Poisson resummation formula 2 X 1 X π b exp(−πan2 + bn) = √ exp − k+ . a 2πi a n∈Z
k∈Z
This formula is easily demonstrated by using the identity X X δ(x − n) = e2πikx n∈Z
k∈Z
and by integrating it over exp(−πax2 + bx). We consider now the infinite series expression for θ3 (τ ). Applying the Poisson resummation formula with a = −iτ and b = 0 we immediately find √ θ3 (−1/τ ) = −iτ θ3 (τ ). If we set a = −iτ and b = −iπ, we obtain the modular transformation of θ2 √ θ2 (−1/τ ) = −iτ θ4 (τ ). Applying the modular transformation a second time, we find √ θ4 (−1/τ ) = −iτ θ2 (τ ). These simple transformation properties, as well as the relation 1 θ2 (τ )θ3 (τ )θ4 (τ ) 2
η 3 (τ ) = gives us directly the modular transformation
η(−1/τ ) =
√
−iτ η(τ ).
The modular properties under the shift τ → τ + 1 are easily derived from θ2 (τ )
=
X
q (n+1/2)
2
/2
= 2q 1/8
θ3 (τ ) θ4 (τ )
=
=
(1 − q n )(1 + q n )2
n=1
n∈Z
X
∞ Y
qn
2
/2
=
∞ Y
n∈Z
n=1
X
2
(−1)n q n
/2
(1 − q n )(1 + q n−1/2 )2
=
∞ Y
(1 − q n )(1 − q n−1/2 )2
n=1
n∈Z
4
The infinite product expansion for θ2 implies that θ2 (τ + 1) = eiπ/4 θ2 (τ ). On the other hand, the infinite series expressions for θ3 and θ4 yield X 2 2 θ3 (τ + 1) = q n /2 eiπn n∈Z
=
X
2
qn
/2 iπn2
e
(−1)n
n∈Z
= θ4 (τ ) Likewise, we find that θ4 (τ + 1) = θ3 (τ ).
5
Doubling Identities
The Jacobi theta functions satisfy the following doubling identities r θ3 (τ )2 − θ4 (τ )2 θ2 (2τ ) = 2 r 2 θ3 (τ ) + θ4 (τ )2 θ3 (2τ ) = 2 p θ4 (2τ ) = θ3 (τ )θ4 (τ )
whereas θ1 satisfies ∂z θ1 (2τ ) =
1 θ2 (τ )∂z θ1 (τ ) p . 2 θ3 (τ )θ4 (τ )
5