Advances in Theoretical and Mathematical Physics

Volume 20 (2016)

Number 6

Galois covers of $\mathcal{N}=2$ BPS spectra and quantum monodromy

Pages: 1227 – 1336



Sergio Cecotti (Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy)

Michele Del Zotto (Jefferson Physical Laboratory, Harvard University, Cambridge, Massachusetts, U.S.A.)


The BPS spectrum of many 4d $\mathcal{N} = 2$ theories may be seen as the (categorical) Galois cover of the BPS spectrum of a different 4d $\mathcal{N} = 2$ model. The Galois group $\mathbb{G}$ acts as a physical symmetry of the covering $\mathcal{N} = 2$ model. The simplest instance is $SU(2)$ SQCD with $N_f = 2$ quarks, whose BPS spectrum is a $\mathbb{Z}_2$-cover of the BPS spectrum of pure SYM. More generally, $\mathcal{N} = 2$ SYM with simply-laced gauge group $G$ admits $\mathbb{Z}_k$-covers for all $k \in \mathcal{N}$; e.g. the $\mathbb{Z}_2$-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 $\mathbb{G}$-symmetric points, in both finite and infinite chambers.

When the covering and quotient QFTs admit a geometric engineering, say for class $\mathcal{S}$ 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, $X(NM) \to X(M)$, or, more generally, by regular Grothendieck’s dessins d’enfants; the BPS spectra of the corresponding $\mathcal{N} = 2$ QFTs have magic properties.

The Galois covers allow to study effectively the action of the quantum (half) monodromy $\mathbb{K}(q)$ of 4d $\mathcal{N} = 2$ QFTs. We present several examples and applications of the spectral covering philosophy.

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