Twisted 3D $N=4$ Supersymmetric YM on deformed ${\mathbb{A}}_3^{\ast }$ Lattice

Abstract

We study a class of twisted 3D $N=4$ supersymmetric Yang-Mills (SYM) theory on particular 3-dimensional lattice denoted as ${\mathcal{L}}_{3D}^{s{u}_3\times u_1}$ and given by non trivial fibration ${\mathcal{L}}_{1D}^{u_1}\times {\mathcal{L}}_{2D}^{s{u}_3}$ with base ${\mathcal{L}}_{2D}^{s{u}_3}={\mathbb{A}}_2^{\ast }$, the weight lattice of $SU(3)$. We first, develop the twisted 3D $N=4$ SYM in continuum by using superspace method where the scalar supercharge $Q$ is manifestly exhibited. Then, we show how to engineer the 3D lattice ${\mathcal{L}}_{3D}^{s{u}_3\times u_1}$ that host this theory. After that we build the lattice action ${\mathcal{S}}_{latt}$ invariant under the 3 following: (i) $U(N)$ gauge invariance, (ii) BRST symmetry, (iii) the hidden $SU(3)\times U(1)$ symmetry of ${\mathcal{L}}_{3D}^{s{u}_3\times u_1}$. Other features such as reduction to twisted 2D supersymmetry with 8 supercharges living on ${\mathcal{L}}_{2D}^{s{u}_2\times u_1}$, the extension to twisted maximal 5D SYM with 16 supercharges on lattice ${\mathcal{L}}_{5D}^{s{u}_4\times u_1}$ as well as the relation with known results are also given.

Comments: LaTex, 81 pages, 4 figures