Seiberg-Witten geometries revisited

Abstract

We provide a uniform solution to 4d \( \mathcal{N} = 2 \) gauge theory with a single gauge group G = A, D, E when the one-loop contribution to the beta function from any irreducible component R of the hypermultiplets is less than or equal to half of that of the adjoint representation. The solution is given by a non-compact Calabi-Yau geometry, whose defining equation is built from explicitly known polynomials W G and X R , associated respectively to the gauge group G and each irreducible component R. We provide many pieces of supporting evidence, for example by analyzing the system from the point of view of the 6d \( \mathcal{N} = \left( {2,0} \right) \) theory compactified on a sphere.