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Wall crossing invariants from spectral networks. (English) Zbl 1394.81149

On about 50 pages plus appendices and references a new approach towards BPS monodromies in four-dimensional \(\mathcal{N} = 2\) supersymmetric theories of “class \(\mathcal{S}\)” is outlined, where unfortunately “class \(\mathcal{S}\)” is never explained on any of these pages. In a similar manner, the non-expert reader suffers from missing explanations that have been left out in order not to make the paper even longer.
The new construction is supposed to be manifestly invariant under wall crossing, where it is developed at loci of Coulomb branches on which the BPS spectrum is actually ill-defined that are walls of marginal stability.
The first section gives an introduction into the problem and wall crossing, which is reviewed in more detail somewhat self-contained in the second section. The spectrum changes discontinuously at the wall of marginal stability leading to the (dis)appearance of bound states which is shown in some of the figures. In the third section, BPS monodromy from marginal stability is derived which is at the special locus of the Coulomb branch and the BPS states are sitting on the critical locus \(\mathcal{B}_c\). Finally, classical and quantum monodromy is derived from critical graphs. Section four gives a vast list of examples, starting from the simplest case of Argyres-Douglas \(AD_3\) theory which has a one-dimensional Coulomb branch and is divided into two chambers by a wall of marginal stability. Similarly, \(SU(2)\) SYM theory is shown which has the same property. Eventually, there are generalisations of the previous two examples and furthermore \(T_2\) and \(T_3\) theory, where again, unfortunately, the reader has to be familiar with the classification. In the last section, future directions are outlined in some bullet points.
The appendices contain necessary notational conventions and formal variables; furthermore some technical developments and \(T_3\) generating functions are listed.

MSC:

81T13 Yang-Mills and other gauge theories in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
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