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Quantum dilogarithm

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In mathematics, the quantum dilogarithm is a special function defined by the formula

ϕ ( x ) ≡ ( x ; q ) ∞ = ∏ n = 0 ∞ ( 1 − x q n ) , | q | < 1 {\displaystyle \phi (x)\equiv (x;q)_{\infty }=\prod _{n=0}^{\infty }(1-xq^{n}),\quad |q|<1}

It is the same as the q-exponential function e q ( x ) {\displaystyle e_{q}(x)} .

Let u , v {\displaystyle u,v} be "q-commuting variables", that is elements of a suitable noncommutative algebra satisfying Weyl's relation u v = q v u {\displaystyle uv=qvu} . Then, the quantum dilogarithm satisfies Schützenberger's identity

ϕ ( u ) ϕ ( v ) = ϕ ( u + v ) , {\displaystyle \phi (u)\phi (v)=\phi (u+v),}

Faddeev-Volkov's identity

ϕ ( v ) ϕ ( u ) = ϕ ( u + v − v u ) , {\displaystyle \phi (v)\phi (u)=\phi (u+v-vu),}

and Faddeev-Kashaev's identity

ϕ ( v ) ϕ ( u ) = ϕ ( u ) ϕ ( − v u ) ϕ ( v ) . {\displaystyle \phi (v)\phi (u)=\phi (u)\phi (-vu)\phi (v).}

The latter is known to be a quantum generalization of Rogers' five term dilogarithm identity.

Faddeev's quantum dilogarithm Φ b ( w ) {\displaystyle \Phi _{b}(w)} is defined by the following formula:

Φ b ( z ) = exp ⁡ ( 1 4 ∫ C e − 2 i z w sinh ⁡ ( w b ) sinh ⁡ ( w / b ) d w w ) , {\displaystyle \Phi _{b}(z)=\exp \left({\frac {1}{4}}\int _{C}{\frac {e^{-2izw}}{\sinh(wb)\sinh(w/b)}}{\frac {dw}{w}}\right),}

where the contour of integration C {\displaystyle C} goes along the real axis outside a small neighborhood of the origin and deviates into the upper half-plane near the origin. The same function can be described by the integral formula of Woronowicz:

Φ b ( x ) = exp ⁡ ( i 2 π ∫ R log ⁡ ( 1 + e t b 2 + 2 π b x ) 1 + e t d t ) . {\displaystyle \Phi _{b}(x)=\exp \left({\frac {i}{2\pi }}\int _{\mathbb {R} }{\frac {\log(1+e^{tb^{2}+2\pi bx})}{1+e^{t}}}\,dt\right).}

Ludvig Faddeev discovered the quantum pentagon identity:

Φ b ( p ^ ) Φ b ( q ^ ) = Φ b ( q ^ ) Φ b ( p ^ + q ^ ) Φ b ( p ^ ) , {\displaystyle \Phi _{b}({\hat {p}})\Phi _{b}({\hat {q}})=\Phi _{b}({\hat {q}})\Phi _{b}({\hat {p}}+{\hat {q}})\Phi _{b}({\hat {p}}),}

where p ^ {\displaystyle {\hat {p}}} and q ^ {\displaystyle {\hat {q}}} are self-adjoint (normalized) quantum mechanical momentum and position operators satisfying Heisenberg's commutation relation

[ p ^ , q ^ ] = 1 2 π i {\displaystyle ={\frac {1}{2\pi i}}}

and the inversion relation

Φ b ( x ) Φ b ( − x ) = Φ b ( 0 ) 2 e π i x 2 , Φ b ( 0 ) = e π i 24 ( b 2 + b − 2 ) . {\displaystyle \Phi _{b}(x)\Phi _{b}(-x)=\Phi _{b}(0)^{2}e^{\pi ix^{2}},\quad \Phi _{b}(0)=e^{{\frac {\pi i}{24}}\left(b^{2}+b^{-2}\right)}.}

The quantum dilogarithm finds applications in mathematical physics, quantum topology, cluster algebra theory.

The precise relationship between the q-exponential and Φ b {\displaystyle \Phi _{b}} is expressed by the equality

Φ b ( z ) = E e 2 π i b 2 ( − e π i b 2 + 2 π z b ) E e − 2 π i / b 2 ( − e − π i / b 2 + 2 π z / b ) , {\displaystyle \Phi _{b}(z)={\frac {E_{e^{2\pi ib^{2}}}(-e^{\pi ib^{2}+2\pi zb})}{E_{e^{-2\pi i/b^{2}}}(-e^{-\pi i/b^{2}+2\pi z/b})}},}

valid for Im ⁡ b 2 > 0 {\displaystyle \operatorname {Im} b^{2}>0} .

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