Quantum Riemann–Hilbert Problem

• Quantum Riemann–Hilbert Problem is a quantum extension of the classical problem, incorporating non-commutative structures such as quantum tori and operator algebras.
• It leverages quantum dilogarithms and refined Donaldson–Thomas theory to address non-perturbative corrections and wall-crossing phenomena in gauge and string theory.
• The framework enables explicit solutions using advanced special functions and categorical enhancements to capture quantum monodromy and integrable system dynamics.

The Quantum Riemann–Hilbert Problem (QRHP) generalizes the classical Riemann–Hilbert problem by encoding quantum, non-commutative, or q-deformed structures in monodromy data, jump conditions, or moduli spaces. Across quantum field theory, integrable systems, Donaldson–Thomas theory, random matrix theory, and representation theory, QRHPs serve as a framework for formulating non-perturbative, wall-crossing, and operator-valued problems that require quantum deformations of traditional analytic and algebraic methods.

1. Classical versus Quantum Riemann–Hilbert Problems

The classical Riemann–Hilbert problem concerns the construction of analytic (typically matrix-valued) functions with prescribed jumps across contours in the complex plane, often related to monodromy data from linear differential equations. In QRHP, the setting is “quantized”—the jump data, solution spaces, or the target of the sought-for map are replaced by operator algebras, quantum tori, or categories reflecting non-commutative or q-deformed structures.

For example, in refined Donaldson–Thomas theory, classical wall-crossing automorphisms (built from the commutative dilogarithm) are replaced in the quantum version by automorphisms constructed from quantum dilogarithms acting on quantum torus algebras (Barbieri et al., 2019, Chuang, 2022, Alexandrov et al., 22 Jul 2025). The solution space becomes the automorphism group of a non-commutative algebra, and the jump and connection data are accordingly upgraded.

A direct comparison appears in the resolved conifold case:

	Classical (commutative)	Quantum (non-commutative)
Jump	$S(\ell)(x) = \exp(\operatorname{Li}_2(x))$	$$S_q(\ell)(x) = \prod_{k \ge 0} (1 - q^{k+1/2} x)^{(-1)^{…}}$$
Solution	Functions on a torus	Automorphisms of a quantum torus algebra
Wall-crossing	Classical dilogarithm	Quantum dilogarithm

This quantum uplift is crucial for capturing non-perturbative and refined structures in gauge theory and string theory compactifications (Chuang, 2022, Alexandrov et al., 22 Jul 2025).

2. Algebraic and Non-Commutative Structure

Quantum Riemann–Hilbert problems are formulated on spaces with quantum group, quantum torus, or Moyal star-algebra structures. For refined BPS invariants, the key quantum deformation is encoded in a non-commutative product controlled by a fugacity parameter $y$:

$$x_\gamma \star x_{\gamma'} = y^{\langle \gamma, \gamma' \rangle} x_{\gamma + \gamma'}$$

where $\star$ indicates the Moyal star product, and $\langle \cdot, \cdot \rangle$ is a symplectic (or skew-symmetric) pairing (Alexandrov et al., 22 Jul 2025).

Jump conditions across BPS rays or Stokes rays are expressed via quantum Kontsevich–Soibelman (K–S) symplectomorphisms:

$X_\gamma^- = U_{\ell, \gamma} \star X_\gamma^+$

where $U_{\ell, \gamma}$ is an automorphism built from quantum dilogarithms and refined indices, and all products and exponentials are defined with respect to the non-commutative algebra structure.

In the language of TBA (Thermodynamic Bethe Ansatz), the classical product in integral equations is replaced by the star product, producing quantum-deformed integral or difference systems whose solutions encapsulate wall-crossing and quantum corrections (Alexandrov et al., 22 Jul 2025):

$X_0 = X_0 \star [ 1 + {}_1 K_{01} X_1 ]$

This non-commutativity is reflected in gauge/string non-perturbative physics, where the QRHP encodes quantum (refined) corrections to moduli space geometries.

3. Explicit Solutions and Special Functions

Explicit solutions of QRHPs in refined DT theory are constructed using quantum dilogarithms and higher Barnes or multiple sine functions (Barbieri et al., 2019, Chuang, 2022). For instance, for the doubled $A$ quiver:

$$\Psi^\pm(t)(y_{\alpha^\vee}) = \Lambda^\pm(z, 1 \mp \theta + \tau/(2\pi i t)) \cdot y_{\alpha^\vee}$$

where $\Lambda$ is a modified Barnes double gamma function, and the solution is piecewise holomorphic except at the prescribed "active rays". The jumps across these rays respect non-trivial commutation relations that encode refined Donaldson–Thomas/BPS invariants.

For the resolved conifold, solutions are written in terms of multiple sine functions with unequal parameters, providing necessary analytic continuation and controlling the q-dependence as parameters move in the space of stability conditions (Chuang, 2022).

4. QRHPs and Quantum Integrable Systems

QRHPs emerge naturally in the quantization of integrable models, both via Lax pair formalisms and isomonodromy deformations. In the context of AdS${}_5$/CFT${}_4$ integrability, the spectral problem is reformulated as a non-linear matrix RH problem for a set of Q-functions—the quantum spectral curve (Gromov et al., 2013). Similarly, q-deformed problems (arising in quantum groups, random partition measures, and q-orthogonal polynomials) are treated via q--difference Riemann–Hilbert problems, whose compatibility conditions yield quantum (q--)Painlevé equations (Joshi et al., 2021, Kimura, 2 Mar 2025, Joshi et al., 2019, Joshi et al., 2022).

This integrable systems perspective reveals a universality: determinantal structures, isomonodromic deformations, and Painlevé connections persist and are governed by quantum or q-deformed RHPs in the operator-valued or non-commutative setting.

5. Geometric, Categorical, and Sheaf-Theoretic Extensions

Quantum Riemann–Hilbert problems have deep geometric and categorical significance. The extension of the Riemann–Hilbert correspondence to quantum and irregular settings requires enhancement of categories (e.g., replacing perverse sheaves by perverse enhanced subanalytic sheaves) to capture Stokes data, quantum monodromy, and non-perturbative features (D'Agnolo et al., 8 Aug 2024, Chekhov et al., 2017). Decorated character varieties, their quantum deformations, and moduli of flat connections (with irregular or higher order poles) provide the non-commutative phase space for quantum monodromy data (Chekhov et al., 2017).

Categorial enhancements, such as Tannakian or microlocal/ind-sheaf frameworks, bridge the description of quantum differential or difference equations, monodromy/stokes data, and the interpretation of non-commutative jump conditions.

6. Applications and Impact

Gauge and String Theory

QRHP formulations provide systematic frameworks for computing non-perturbative corrections to moduli spaces in compactified string/gauge theories, especially via wall-crossing and refined BPS indices (Alexandrov et al., 22 Jul 2025). The QRHP encodes modular, duality, and wall-crossing properties, allowing the reconstruction of twistor space data and generating functions (τ-functions) for supersymmetric effective actions (Alexandrov et al., 22 Jul 2025, Chuang, 2022).

Donaldson–Thomas Theory and Non-Perturbative Topological Strings

By solving the quantum RH problem for DT invariants (e.g., on the resolved conifold), one obtains a possible non-perturbative definition of refined DT theory and the refined topological string partition function, capturing both perturbative and exponentially suppressed contributions through explicit quantum special functions (Chuang, 2022).

Random Partitions, q-Deformation, and Quantum Probability

Quantum Riemann–Hilbert problems are essential in analyzing determinantal processes for q-deformed random partitions, providing integrable structures, Lax representations, and encoding dynamics in q-Painlevé equations (Kimura, 2 Mar 2025). The RHP setup organizes the spectral data and establishes universality, gap probabilities, and asymptotic kernel behavior in quantum random matrix ensembles.

7. Quantum Stokes Phenomena and Future Directions

QRHPs naturally encode Stokes phenomena in quantum settings, where non-perturbative and asymptotic effects (resurgence, exact WKB) become integral to moduli and category-theoretic descriptions. Enhanced sheaf-theoretic frameworks, as developed for the irregular Riemann–Hilbert correspondence, suggest parallel extensions to quantum D-modules, likely to be central in future developments (D'Agnolo et al., 8 Aug 2024).

Outstanding challenges include:

• Extending QRHP theory to general quantum/categorical settings and higher dimensions.
• Developing a full quantum Tannakian or ind-sheaf theory capturing quantum monodromy and quantum Stokes data.
• Understanding modular and duality properties of the quantum generating functions and their role in wall-crossing and non-perturbative supersymmetric dynamics.

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In summary, the Quantum Riemann–Hilbert Problem unifies non-commutative, operator-algebraic, and categorical structures in quantum integrable models, refined Donaldson–Thomas theory, quantized moduli spaces, random partitions, and beyond. By extending the analytic and homological tools of the classical Riemann–Hilbert paradigm to these quantum domains, QRHPs provide the natural language for non-perturbative phenomena, wall-crossing, and quantum geometry, with explicit realizations across gauge theory, string theory, algebraic geometry, and mathematical physics.