Quantum Polylogarithms: Theory & Applications

• Quantum polylogarithms are deformations of classical polylogarithms using a parameter (ħ or q) that quantizes their algebraic and analytic frameworks.
• They satisfy holonomic difference and q-difference equations, uniting methodologies from analytic theory, geometric integrals, and Hopf algebra deformations.
• Their applications span Feynman integral evaluations, quantum group representations, and renormalization techniques, providing new insights in mathematical physics.

Quantum polylogarithms generalize classical polylogarithms and multiple polylogarithms by introducing a deformation parameter—traditionally denoted $\hbar$ or $q$—which controls the "quantization" of their algebraic and analytic structures. The theory encompasses diverse formulations: analytic through higher-order $q$-series, geometric via rational exponential integrals, and algebraic through deformations of shuffle and Hopf algebras. Quantum polylogarithms unify classical period computations in algebraic geometry, Feynman integral evaluations in quantum field theory, the representation theory of quantum groups, and the study of motivic Galois structures. Recent developments have clarified their role as deformations of periods of mixed Tate motives and have furnished explicit holonomic difference-differential systems, rich duality properties, and connections to vacuum diagrams, all extending deep structures underlying polylogarithmic functions.

1. Definitions and Deformations of Polylogarithms

Quantum polylogarithms originate as deformations of the classical multiple polylogarithms (MPLs). For complex parameters $\hbar$ (the quantum deformation parameter), the depth-$m$ quantum polylogarithm is defined by an $m$-fold rational-exponential integral: $\Li^{(\hbar)}_{s_1,\dots,s_m}(z_1,\dots,z_m) = \Fcal^\hbar_{\mathbf a,\mathbf b,\mathbf n}(\omega_1,\dots,\omega_m) = i^{|\mathbf n| - m} \int_{(+i0)^m} \prod_{k=1}^m \frac{e^{-i p_k \omega_k} \sinh^{a_k}(\pi p_k) \sinh^{b_k}(\pi \hbar p_k) } {(p_1 + \cdots + p_k)^{n_k} dp_k }$ where $z_k = e^{\omega_k}$, $\mathbf a,\mathbf b \in \mathbb Z_{\ge 0}^m$, $\mathbf n \in \mathbb Z^m$, and $(+i0)^m$ denotes contours just above the real axis. In the limit $\hbar \to 0$, these integrals recover the classical MPLs as leading asymptotic terms, demonstrating the deformation structure explicitly (Goncharov, 1 Jan 2026).

In the $q$-deformed setting, the $q$-multiple polylogarithms (q-MPLs) are given by

$\Li^q_{k_1,\dots,k_n}(z) := \sum_{m_1 > \cdots > m_n > 0} \frac{z^{m_1}}{[m_1]_q^{k_1} \cdots [m_n]_q^{k_n}}$

where $[m]_q = (1-q^m)/(1-q)$ for $|q|<1$ (Ebrahimi-Fard et al., 2015, Yamamoto, 2020). Several distinct $q$-models exist—e.g., the Bradley–Zhao and Schlesinger–Zudilin models—each tied to particular applications in enumerative geometry, quantum groups, and $q$-deformations of algebraic objects.

Quantum dilogarithm, the $m=1$ case, emerges as the Barnes integral

$\Fcal^\hbar_{1,1,1}(\omega) = \int_{+i0} \frac{e^{-i p \omega} }{ \sinh(\pi p) \sinh(\pi \hbar p) } \frac{dp}{p}$

central to representations of quantum enveloping algebras and statistical mechanical transfer matrices (Goncharov, 1 Jan 2026).

2. Analytic Structures: Difference Equations and Holonomic Systems

Quantum polylogarithms satisfy a rich system of holonomic difference equations. For the rational-exponential integrals,

$$\Delta^{(\omega)}_{i\pi} f(\omega) = f(\omega + i\pi) - f(\omega - i\pi)$$

which acts as a finite difference operator, and similarly for shifts by $i\pi\hbar$. The basic kernels evolve under these differences as

$$\Delta^{(\omega)}_{i\pi} K^\hbar_{a,b} = K^\hbar_{a-1, b}, \qquad \Delta^{(\omega)}_{i\pi\hbar} K^\hbar_{a, b} = K^\hbar_{a, b-1}$$

yielding, upon termwise integration, modular difference equations for the quantum polylogarithms: $\Delta^{(\omega_k)}_{i\pi} \Fcal^\hbar_{\mathbf a,\mathbf b,\mathbf n} = \Fcal^\hbar_{\mathbf a - \mathbf e_k,\, \mathbf b,\, \mathbf n}, \quad \Delta^{(\omega_k)}_{i\pi\hbar} \Fcal^\hbar_{\mathbf a,\, \mathbf b,\, \mathbf n} = \Fcal^\hbar_{\mathbf a,\, \mathbf b - \mathbf e_k,\, \mathbf n}$ These relations, together with a system of first-order differential equations,

$d\,\Fcal^\hbar_{\mathbf a,\mathbf b,\mathbf n} = \sum_{k=1}^m \Fcal^\hbar_{\mathbf a,\mathbf b,\mathbf n-\mathbf e_k}\, d(\omega_k - \omega_{k+1}),$

establish a flat, holonomic difference-differential framework (Goncharov, 1 Jan 2026).

In the $q$-analogue, $q$-polylogarithms admit natural $q$-difference equations in the argument $z$, connected to $q$-hypergeometric series and contiguous recurrence relations (Yamamoto, 2020).

3. Algebraic and Hopf-Algebraic Properties

Quantum polylogarithms generalize the shuffle algebra familiar from classical MPLs. For $q$-MPLs, the shuffle Hopf algebra structure is encoded in an algebra $H_{-1}$ generated by Lyndon words in an alphabet $L = \{p, d, y\}$ with specific shuffle product $\sh^{(-1)}$ and cocommutative coproduct $\Delta_{-1}$. The $q$-MPLs become algebra morphisms: $\Li^q_{u}(z) \Li^q_{v}(z) = \Li^q_{u \sh^{(-1)} v}(z)$ ensuring the compatibility of product and coproduct (Ebrahimi-Fard et al., 2015).

For generalised polylogarithms (GPLs), the algebra obeys the shuffle identity

$$G(\vec{a}; y) G(\vec{b}; y) = \sum_{\vec{c} \in \vec{a} \amalg \vec{b}} G(\vec{c}; y)$$

where the sum is over all interleavings preserving internal order (Naterop et al., 2019).

The Hopf algebraic perspective is crucial for renormalization procedures via the Connes–Kreimer Birkhoff decomposition, particularly when assigning finite values to polylogarithms at divergent arguments (Ebrahimi-Fard et al., 2015). The algebraic structure is also essential for the symbol calculus, which encodes the leading weight terms of single-valued quantum polylogarithms in Feynman diagram calculations (Drummond, 2012).

4. Single-Valued, Motivic, and Geometric Interpretations

Single-valued quantum polylogarithms arise naturally in the evaluation of Feynman integrals and are characterized by vanishing monodromy around real points, ensuring physical observables are real-valued after analytic continuation. For instance, Drummond constructs a class of single-valued polylogarithms $f_m(z, \bar z)$ for generalized ladder integrals, satisfying: $\Disc_z f_m - \Disc_{\bar{z}} f_m = 0$ and recursively solves inhomogeneous second-order PDEs with holomorphic-anti-holomorphic ambiguity fixed by single-valuedness (Drummond, 2012).

In the motivic context, classical MPLs are periods of variations of mixed Tate motives, and quantum polylogarithms with rational deformation parameters $\hbar$ remain algebraic-periods. At irrational $\hbar$, their integral expressions fall outside this category, being "rational exponential integrals"—integrals of differential forms built from both rational functions and exponentials thereof (Goncharov, 1 Jan 2026).

These motivic and single-valued aspects directly connect to the general principle that motivic periods should admit quantum deformations governed by analogous finite-difference systems.

5. Applications in Quantum Field Theory, Representation Theory, and Beyond

Quantum polylogarithms are indispensable in evaluating multi-loop Feynman integrals. The hierarchy of single-valued quantum polylogarithms constructed recursively provides explicit all-loop expressions for generalised ladder, wheel, and zigzag diagrams, with residues matching simple zeta values and confirming longstanding conjectures for the analytic forms of vacuum graphs (Drummond, 2012).

In representation theory and quantum geometry, quantum dilogarithms figure prominently in the study of cluster algebra quantization, the modular double of quantum groups, the bootstrap of correlation functions in Liouville theory, and the Baxter $Q$-operator in statistical mechanics (Goncharov, 1 Jan 2026).

Duality relations for one-variable $q$-polylogarithms have concrete realizations in terms of non-commutative algebraic involutions, with proofs leveraging hypergeometric connectors and contiguous relations, further suggesting connections to the $q$-deformation of associators and quantum group $R$-matrix monodromy (Yamamoto, 2020).

In numerical applications, the handyG library implements rapid and stable evaluation of generalised polylogarithms (weights $\leq 8$), using recursion, analytic continuation, and shuffle algebra to enable direct integration in Monte Carlo simulations of physically relevant cross-sections, greatly enhancing computational throughput compared to existing symbolic frameworks (Naterop et al., 2019).

6. Special Features, Limits, and Renormalization

Quantum polylogarithms display several notable limits and regularization behaviors:

• Classical limit ($\hbar \to 0$): The quantum polylogarithm $\Li^{(\hbar)}$ asymptotically produces a hierarchy of classical polylogarithms, with expansion coefficients given by higher weight MPLs (Goncharov, 1 Jan 2026).
• Rational deformation: For rational $\hbar$, distribution relations express every quantum polylogarithm as a finite $\mathbb Q$-linear combination of classical MPLs, confirming their status as motivic periods (Goncharov, 1 Jan 2026).
• Renormalization: The Connes–Kreimer algebraic Birkhoff decomposition assigns finite, shuffle-compatible values to $q$-MPLs at non-positive arguments, yielding results that match those of the classical theory in the $q \to 1$ limit (Ebrahimi-Fard et al., 2015).

A summary of main features is shown below:

Feature	Classical Polylogarithm	Quantum Polylogarithm ($\hbar, q$)
Functional Equations	Differential (e.g. dlog)	Difference, $q$-difference, holonomic
Algebraic Structure	Shuffle/Hopf algebra	Deformed shuffle, quantum Hopf algebra
Special Values	Periods of algebraic forms	Exponential/rational integrals, sometimes periods
Renormalization	Meromorphic continuation	Birkhoff decomposition for $q$-regulator

In total, quantum polylogarithms constitute a comprehensive generalization of classical period integrals, introducing modular difference equations, deep algebraic and motivic structures, and analytic features central to multiple areas of modern mathematics and theoretical physics. Their deformation theory is conjectured to be a broad organizing principle underlying quantization of periods for general mixed motives (Goncharov, 1 Jan 2026, Drummond, 2012, Ebrahimi-Fard et al., 2015, Yamamoto, 2020, Naterop et al., 2019).