Types of quantum field thories
Matching asymptotic expansions of certain functions like Airy functions and more generally of solutions to the wave equations (say in short wave approximation or solutions to non-Fuchsian meromorphic ODEs) comes with sharp changes (disconinuities) at certain phase angles (Stokes lines) described by certain factors, Stokes matrices.
G. G. Stokes has discovered these phenomena – now also known more generally as wall crossing phenomena – in the study of geometric optics, more specifically the study of Airy function which is an example of a solution to a linear meromorphic ordinary differential equation which has an irregular singular point, hence the equation is not Fuchsian. The Stokes phenomenon does not happen to Fuchsian equations. Their formal meromorphic solutions are automatically convergent. Exactly the presence of the irregular singularities makes the appearance of formal solutions with zero radius of convergence. Now look around the origin. One can try to prove that there are asymptotic expressions at best in some regions of argument. There are jumps at certain slopes. In fact there is Stokes sheaf and the first nonabelian cohomology of the Stokes sheaf measures the obstruction for a formal meromomorphic expansion to be build up of sectorial true meromorphic expansions.
Stokes factors (multipliers, matrices) appear also in the study of stability in the geometry of moduli spaces of sheaves. According to Toledano Laredo “Joyce’s wall-crossing formulae for invariants counting semistable objects in an abelian category A may be understood as Stokes phenomena for a connection on the Riemann sphere taking value in the Ringel-Hall Lie algebra of A.”.
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Marco Gualtieri, Songhao Li, Brent Pym, The Stokes groupoids, arxiv/1305.7288
We construct and describe a family of groupoids over complex curves which serve as the universal domains of definition for solutions to linear ordinary differential equations with singularities. As a consequence, we obtain a direct, functorial method for resumming formal solutions to such equations.
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