Quantum logarithm

The quantum logarithm is a function of mathematical physics .

definition

Be it . The quantum logarithm ${\displaystyle \hbar >0}$

${\displaystyle \phi ^{\hbar }\colon \mathbb {C} \to \mathbb {C} }$

is defined by

${\displaystyle \phi ^{\hbar }(z):=-2\pi \hbar \int _{C}{\frac {e^{-ixz}}{(e^{\pi x}-e^{-\pi x})(e^{\pi \hbar x}-e^{-\pi \hbar x})}}dx}$,

where is a curve running along the real axis from to and revolving around the zero point from above, for example . ${\displaystyle C\subset \mathbb {C} }$ ${\displaystyle -\infty }$ ${\displaystyle \infty }$ ${\displaystyle C=\left[-\infty ,-1\right]\cup \left\{e^{it}\colon \pi \geq t\geq 0\right\}\cup \left[1,\infty \right]}$

(For every curve with these properties, integration of this integrand over the curve gives the same value.)

properties

${\displaystyle \lim _{\hbar \rightarrow 0}\phi ^{\hbar }(z)=\log(e^{z}+1)}$

${\displaystyle \phi ^{\hbar }(z)-\phi ^{\hbar }(-z)=z}$

${\displaystyle {\overline {\phi ^{\hbar }(z)}}=\phi ^{\hbar }({\overline {z}})}$

${\displaystyle {\frac {1}{\hbar }}\phi ^{\hbar }(z)=\phi ^{\frac {1}{\hbar }}({\frac {z}{\hbar }})}$

${\displaystyle \phi ^{\hbar }(z+i\pi \hbar )-\phi ^{\hbar }(z-i\pi \hbar )={\frac {2\pi i\hbar }{e^{-z}+1}}}$

${\displaystyle \phi ^{\hbar }(z+i\pi )-\phi ^{\hbar }(z-i\pi )={\frac {2\pi i}{e^{-{\frac {z}{\hbar }}}+1}}}$

${\displaystyle \phi ^{1}(z)={\frac {z}{1-e^{-z}}}}$

${\displaystyle \phi ^{\hbar }(z)=\phi ^{\hbar +1}(z+\pi i)+\phi ^{\frac {\hbar }{\hbar +1}}(z-\pi \hbar i)=\phi ^{\hbar +1}(z-\pi i)+\phi ^{\frac {\hbar }{\hbar +1}}(z+\pi \hbar i)}$

${\displaystyle \sum _{l=-r}^{r}\sum _{m=-s}^{s}\phi ^{\hbar }(z+{\frac {2\pi i}{2r+1}}l+{\frac {2\pi i\hbar }{2s+1}}m)=\phi ^{{\frac {2r+1}{2s+1}}\hbar }((2r+1)z)}$

The 1-form is meromorphic , it has simple poles with a residual in the points with . ${\displaystyle \phi ^{\hbar }(z)dz}$ ${\displaystyle \pm 2\pi i\hbar }$ ${\displaystyle (2n-1)\hbar \pi i\pm (2m-1)\pi i}$ ${\displaystyle n,m\in \mathbb {N} }$

literature

• VV Fock, AB Goncharov: The quantum dilogarithm and representations of quantum cluster varieties. Invent. Math. 175 (2009), no. 2, 223-286. (Chapter 4.1)