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Dedekind eta function

Contents

Idea

A modular form.

The Jacobi theta function for special values of its arguments…

Definition

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

for qe 2πiτq \coloneqq e^{2\pi i \tau}.

Properties

As functional determinant of Laplace operator on elliptic curve

For /(τ)\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 Δ\Delta is

ζ Δ=(2π) 2sE(s)(2π) 2s(k,l)×(0,0)1|k+τl| 2s. \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 Δ (0))=(Imτ) 2|η(τ)| 4, \exp( E^\prime_{\Delta}(0) ) = (Im \tau)^2 {\vert \eta(\tau)\vert}^4 \,,

where η\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

See also

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

Last revised on July 18, 2015 at 04:19:51. See the history of this page for a list of all contributions to it.