Spontaneous Z2 Symmetry Breaking in the Orbifold Daughter of N=1 Super Yang-Mills Theory, Fractional Domain Walls and Vacuum Structure

TL;DR

This work analyzes the nonsupersymmetric ${Z}_2$ orbifold daughter of ${\cal N}=1$ SYM, arguing that spontaneous ${Z}_2$ breaking and a tachyon-like condensate occur, thereby signaling nonperturbative nonequivalence to the SUSY parent. By examining domain-wall dynamics in both four-dimensional and world-volume descriptions, the authors show that fractional electric and magnetic walls detach and interact in ways incompatible with planar equivalence, leading to a complex vacuum structure with two intertwined sectors. They connect these field-theoretic findings to type-0 string theory via brane realizations and discuss closed-string tachyon condensation, offering a unified view of nonperturbative dynamics in orbifold theories. The results challenge the applicability of nonperturbative planar equivalence for orbifold daughters and illuminate the role of twisted sectors and tachyons in the infrared behavior of nonsupersymmetric gauge theories.

Abstract

We discuss the fate of the Z2 symmetry and the vacuum structure in an SU(N)xSU(N) gauge theory with one bifundamental Dirac fermion. This theory can be obtained from SU(2N) supersymmetric Yang--Mills (SYM) theory by virtue of Z2 orbifolding. We analyze dynamics of domain walls and argue that the Z2 symmetry is spontaneously broken. Since unbroken Z2 is a necessary condition for nonperturbative planar equivalence we conclude that the orbifold daughter is nonperturbatively nonequivalent to its supersymmetric parent. En route, our investigation reveals the existence of fractional domain walls, similar to fractional D-branes of string theory on orbifolds. We conjecture on the fate of these domain walls in the true solution of the Z2-broken orbifold theory. We also comment on relation with nonsupersymmetric string theories and closed-string tachyon condensation.

Figures (7)

• Figure 1: The fermion loop expansion of the partition function at $N\to\infty$.
• Figure 2: The vacuum structure in the SU($2N$) SYM theory and its SU$(N)\times$SU($N$) orbifold daughter on the complex plane of the order parameter (the bifermion condensate $-\langle \lambda^{a}_{\alpha}\lambda^{a\,,\alpha} \rangle$ or $-\left\langle \bar{\Psi}\left( 1-\gamma_5\right)\Psi \right\rangle$, respectively).
• Figure 3: The vacuum structure of the SU(8)$\times$SU(8) orbifold theory.
• Figure 4: Topologically stable instanton-antiinstanton pair in the orbifold theory. Instanton belongs to the electric SU($N$) while antiinstanton to the magnetic SU($N$).
• Figure 5: The tachyon field potential. The ${Z}_2\ $ symmetry is dynamically broken in the true vacuum.