Abstract
Twisted and orbifold formulations of lattice ${\cal N}=4$ super Yang-Mills theory which possess an exact supersymmetry require a $U(N)=SU(N)\otimes U(1)$ gauge group. In the naive continuum limit, the $U(1)$ modes trivially decouple and play no role in the theory. However, at non-zero lattice spacing they couple to the $SU(N)$ modes and can drive instabilities in the lattice theory. For example, it is well known that the lattice $U(1)$ theory undergoes a phase transition at strong coupling to a chirally broken phase. An improved action that suppresses the fluctuations in the $U(1)$ sector was proposed in arXiv:1505.03135 . Here, we explore a more aggressive approach to the problem by adding a term to the action which can entirely suppress the $U(1)$ mode. The penalty is that the new term breaks the $\mathcal{Q}$-exact lattice supersymmetry. However, we argue that the term is $1/N^2$ suppressed and the existence of a supersymmetric fixed point in the planar limit ensures that any SUSY-violating terms induced in the action possess couplings that also vanish in this limit. We present numerical results on supersymmetric Ward identities consistent with this conclusion.
Highlights
In recent years a great deal of effort has been devoted to the construction and numerical studies of lattice formulations of N 1⁄4 4 super Yang-Mills theory which retain one exact supersymmetry at nonzero lattice spacing—see the reviews [1,2,3] and references therein
We have shown that simulations of lattice N 1⁄4 4 super Yang-Mills targeting the SUðNÞ rather than the UðNÞ theory are possible at moderately strong coupling λlat ≤ 4
This is a stronger coupling than has been achieved with the improved action described in [21], where only λlat ≤ 3 was possible
Summary
In recent years a great deal of effort has been devoted to the construction and numerical studies of lattice formulations of N 1⁄4 4 super Yang-Mills theory which retain one exact supersymmetry at nonzero lattice spacing—see the reviews [1,2,3] and references therein. In this approach the link fields appearing in the lattice theory take their values in the algebra of the group, denoted by glðN; CÞ.1 This is readily apparent from the (twisted) scalar supersymmetry (SUSY) transformation. We have attempted to address this problem in a different way by adding to the lattice action a term which explicitly suppresses the Uð1Þ sector for each link field (we call this the detlink term) We argue that this term is 1=N2 suppressed and the exact scalar SUSY Q should be recovered in the large N limit. To maintain lattice gauge invariance, for this hybrid action we only truncate the bosonic sector down to SUðNÞ This construction restores Q supersymmetry in the limit N → ∞. We show detailed numerical results in four dimensions consistent with the claimed 1=N2 suppression
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