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A quantized Riemann-Hilbert problem in Donaldson-Thomas theory. (English) Zbl 1494.14055

In physics and mathematics literature there has been recent interest in a class of Riemann-Hilbert problems that are naturally suggested by the form of the wall-crossing formula in Donaldson-Thomas (DT) theory. These problems involve piecewise holomorphic maps from the complex plane to the group of automorphisms of a Poisson algebraic torus, with discontinuities along a collection of rays prescribed by the DT invariants. The authors consider the special case of a refined BPS structure satisfying certain conditions. The basic example is the one arising from the refined DT theory of the \(A_1\) quiver. The article proposes an explicit solution to the corresponding quantum Riemann-Hilbert problem in terms of products of modified gamma functions. The solutions are also written in adjoint form using a modified version of the Barnes double gamma function; the latter arises in expressions for the partition functions of supersymmetric gauge theories.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
30E25 Boundary value problems in the complex plane
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
33E99 Other special functions
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