Seiberg-Witten Geometry with Various Matter Contents

TL;DR

This work expands Seiberg-Witten geometry by deriving explicit ALE-fibration descriptions for 4D $N=2$ theories arising from breaking the $E_6$ theory: $SO(2N_c)$ with massive spinor and vector matter and $SU(N_c)$ with massive antisymmetric and fundamental matter (for appropriate $N_c$). The authors develop a general gauge-symmetry breaking framework within the SW setup, and apply it to breakings $E_6\to SO(10)$ and $E_6\to SU(6)$, producing detailed geometries that reduce correctly in scaling limits to the lower-rank theories, including massless limits that match known results. They validate these geometries by comparing with geometric engineering and brane dynamics and show that the same curves can be obtained from $N=1$ confining-phase superpotentials, reinforcing the consistency across methods. The results broaden the SW dictionary for diverse matter contents and gauge groups and set the stage for further extensions (e.g., $E_7$, $E_8$) and for extracting the Seiberg-Witten three-form and cycles in these ALE fibrations.

Abstract

We obtain the Seiberg-Witten geometry for four-dimensional N=2 gauge theory with gauge group SO(2N_c) (N_c \leq 5) with massive spinor and vector hypermultiplets by considering the gauge symmetry breaking in the N=2 $E_6$ theory with massive fundamental hypermultiplets. In a similar way the Seiberg-Witten geometry is determined for N=2 SU(N_c) (N_c \leq 6) gauge theory with massive antisymmetric and fundamental hypermultiplets. Whenever possible we compare our results expressed in the form of ALE fibrations with those obtained by geometric engineering and brane dynamics, and find a remarkable agreement. We also show that these results are reproduced by using N=1 confining phase superpotentials.

Figures (1)

• Figure 1: $E_6$ Dynkin diagram