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Galois covers of N=2 BPS spectra and quantum monodromy

The BPS spectrum of many 4d N=2 theories may be seen as the (categorical) Galois cover of the BPS spectrum of a different 4d N=2 model. The Galois group G acts as a physical symmetry of the covering N=2 model. The simplest instance is SU(2) SQCD with N_f=2 quarks, whose BPS spectrum is a Z_2-cover of the BPS spectrum of pure SYM. More generally, N=2 SYM with simply--laced gauge group admits Z_k-covers for all k; e.g. the Z_2-cover of SO(8) SYM is SO(8) SYM coupled to two copies of the E_6 Minahan-Nemeshanski SCFT. Galois covers simplify considerably the computation of the BPS spectrum at G-symmetric points, in both finite and infinite chambers. When the covering and quotient QFTs admit a geometric engineering, say for class S models, the categorical spectral cover may be realized as a covering map in the geometry. A particularly nice instance is when the spectral Galois cover is induced by a modular cover of principal modular curves, X(NM)-> X(M), or, more generally, by regular Grothendieck's dessins d'enfants; the BPS spectra of the corresponding N=2 QFTs have magic properties. The Galois covers allow to study effectively the action of the quantum (half)monodromy of 4d N=2 QFTs. We present several examples and applications of the spectral covering philosophy.