Surveys in Differential Geometry

Volume 18 (2013)

Classification of complete $N = 2$ supersymmetric theories in 4 dimensions

Pages: 19 – 101



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

Cumrun Vafa (Jefferson Physical Laboratory, Harvard University, Cambridge, Mass., U.S.A.)


We define the notion of a complete $N = 2$ supersymmetric theory in 4 dimensions as one which has a maximal allowed dimension for a UV complete moduli space for the coupling constants, masses and Coulomb branch parameters. We classify all such theories whose BPS spectrum can be obtained via a quiver diagram. This is done using the 4d/2d correspondence and by showing that such complete $N = 2$ theories map to quivers of finite mutation type. The list of such theories is given by the Gaiotto theories consisting of two 5-branes wrapping Riemann surfaces with punctures, as well as 11 additional exceptional cases, which we identify.

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