N=1 curves for trifundamentals

Abstract

We study the Coulomb phase of N=1 SU(2)^3 gauge theory coupled to one trifundamental field, and generalizations thereof. The moduli space of vacua is always one-dimensional with multiple unbroken U(1) fields. We find that the N=1 Seiberg-Witten curve which encodes the U(1) couplings is given by the double cover of a Riemann surface branched at the poles and the zeros of a meromorphic function.

Figures (6)

• Figure 1: The $U$-plane in the limit $\Lambda^4_2,\Lambda^4_3 \ll \Lambda^4_1$. There are four singular points in the $U$-plane where massless charged particles (monopole or dyon) appear. We denote two singular points close to $U\sim\Lambda_1^4$ as $a$ and $b$, and those close to $U\sim 0$ as $c$ and $d$.
• Figure 2: Cycles $\alpha_1,\alpha_2,\beta_1$ and $\beta_2$ in the curve (\ref{['eq:degreesixpolynomial']}). Wavy lines represent the branch cuts between the zero points of $f_6$.
• Figure 3: Left: the model studied in Intriligator:1994sm. Right: the model studied in section \ref{['sec:onetrifundamental']}. Circles represent $\mathrm{SU}(2)$ gauge groups, boxes represent global $\mathrm{SU}(2)$ symmetry groups, and trivalent vertices represent trifundamental fields. In this model, $g=0$ and $n=2$ (left) or $n=3$ (right).
• Figure 4: An example of a generalized model. Circles represent $\mathrm{SU}(2)$ gauge groups, boxes represent global $\mathrm{SU}(2)$ symmetry groups, and trivalent vertices represent trifundamental fields. In this example, $g=1$ and $n=7$.
• Figure 5: A genus-one graph can be obtained by gauging the diagonal subgroup of $\mathrm{SU}(2)_{n}$ and $\mathrm{SU}(2)_{2n+1}$ of this linear graph.