12.4. Bosonic partition function at the self-dual point Consider the expression (12.2.56) of the partition function of a bosonic field with a compactification radius R equal to the self-dual value R =

√

2 Z(

√

2) = 1

|η(τ )|2 

n,m∈Z q14

(n+m)2 ¯ q14

(n−m)2 .

Prove that this expression can be written as Z(

√

2) = |C0|2 +|C1|2 where C0(τ ) = 1

η



k∈Z qm2 = θ3(2τ)

η(τ )

C1(τ ) = 1

η



k∈Z q(m+1/2)2 = θ2(2τ)

η(τ )

.