Torus partition function of a free boson

For a noncompact free boson with target volume $V$, the torus partition function is $Z=V[2\pi \sqrt{\mathrm{Im}\tau }{]}^{-1}|\eta (\tau ){|}^{-2}$ in the normalization where the zero-mode Hamiltonian is $p^2/2$.

The Dedekind eta function isIt packages the oscillator contribution of a chiral free boson.

An eta-quotient is a finite product of integer powers of Dedekind eta functions at rescaled arguments:Negative exponents place factors in the denominator; its formal modular weight is $\frac{1}{2}{\sum }_{\delta }{r}_{\delta }$. A product of this form is not automatically a holomorphic modular form: its transformation law and behavior at cusps require additional conditions.

In string and conformal field theory calculations, inverse eta powers package oscillator-state counting. For example, $\eta (\tau {)}^{-24}=q^{-1}{\prod }_{n\ge 1}(1-q^n{)}^{-24}$ displays the vacuum offset and the degeneracies of 24 independent chiral bosonic oscillators. Checking its leading power and modular weight provides a useful consistency check before combining it with zero-mode or ghost contributions.