nLab Dedekind eta function

Idea

A modular form.

The Jacobi theta function for special values of its arguments…

Definition

$$\eta(\tau) = q^{1/24} \prod_{n=1}^\infty (1-q^n)$$

for $q \coloneqq e^{2\pi i \tau}$.

Properties

As functional determinant of Laplace operator on elliptic curve

For $\mathbb{C}/(\mathbb{Z}\oplus \tau \mathbb{Z})$ a complex torus (complex elliptic curve) equipped with its standard flat Riemannian metric, then the zeta function of the corresponding Laplace operator $\Delta$ is

$$\zeta_{\Delta} = (2\pi)^{-2 s} E(s) \coloneqq (2\pi)^{-2 s} \underset{(k,l)\in \mathbb{Z}\times\mathbb{Z}-(0,0)}{\sum} \frac{1}{{\vert k +\tau l\vert}^{2s}} \,.$$

The corresponding functional determinant is

$$\exp( E^\prime_{\Delta}(0) ) = (Im \tau)^2 {\vert \eta(\tau)\vert}^4 \,,$$

where $\eta$ is the Dedekind eta function.

(recalled e.g. in Todorov 03, page 3)

This kind of expression appears as the partition function of the bosonic string (e.g. section 6.4.2 in these lectures: pdf)

References

• Wikipedia, Dedekind eta function

• Michael Atiyah, The logarithm of the Dedekind $\eta$-function, Math. Ann. 278, 335-380 (1987) (pdf)

See also

• Andrey Todorov, The analogue of the Dedekind eta function for CY threefolds, 2003 pdf