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

${\displaystyle \Phi ^{\hbar }(z):=\exp \left(-{\frac {1}{4}}\int _{C}{\frac {e^{-ixz}}{\sinh(\pi x)\sinh(\pi \hbar x)}}{\frac {dx}{x}}\right)}$,

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

${\displaystyle 2\pi i\hbar d\log \Phi ^{\hbar }(z)=\phi ^{\hbar }(z)dz}$(Here denotes the quantum logarithm .) ${\displaystyle \phi ^{\hbar }(z)}$

${\displaystyle \lim _{x\rightarrow -\infty }\Phi ^{\hbar }(x+iy)=1\ \ \forall \ y\in \mathbb {R} }$

${\displaystyle \lim _{h\rightarrow 0}{\frac {\Phi ^{\hbar }(z)}{\exp {\frac {-Li_{2}(-e^{z})}{2\pi \hbar }}}}=1}$(Here denotes the classic dilogarithm .) ${\displaystyle Li_{2}}$

${\displaystyle \Phi ^{\hbar }(z)\Phi ^{\hbar }(-z)=\exp \left({\frac {z^{2}}{4\pi i\hbar }}\right)e^{-{\frac {\pi i}{12}}(\hbar +\hbar ^{-1})}}$

${\displaystyle {\overline {\Phi ^{\hbar }(z)}}=(\Phi ^{\hbar }({\overline {z}}))^{-1}}$, in particular ${\displaystyle \mid \Phi ^{\hbar }(z)\mid =1\ \ \forall \ z\in \mathbb {R} }$

${\displaystyle \Phi ^{\hbar }(z)=\Phi ^{\frac {1}{\hbar }}\left({\frac {z}{\hbar }}\right)}$

${\displaystyle \Phi ^{\hbar }(z+2\pi i\hbar )=\Phi ^{\hbar }(z)(1+e^{\pi ih+z})}$

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

${\displaystyle \Phi ^{1}(z)=e^{({\frac {\pi ^{2}}{6}}-Li_{2}(1-e^{z}))/2\pi i}}$

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

${\displaystyle \Pi _{l=-r}^{r}\Pi _{m=-s}^{s}\Phi ^{\hbar }\left(z+{\frac {2\pi i}{2r+1}}l+{\frac {2\pi i\hbar }{2s+1}}m\right)=\Phi ^{{\frac {2r+1}{2s+1}}\hbar }((2r+1)z)}$

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

literature

• VV Fock, AB Goncharov: The quantum dilogarithm and representations of quantum cluster varieties. Invent. Math. 175 (2009), no. 2, 223-286. (Chapter 4.2)