Abstract
We construct two-dimensional ${\cal N} = (2, 2)$ supersymmetric gauge theories on a Euclidean spacetime lattice with matter in the two-index symmetric and anti-symmetric representations of SU($N_c$) color group. These lattice theories preserve a subset of the supercharges exact at finite lattice spacing. The method of topological twisting is used to construct such theories in the continuum and then the geometric discretization scheme is used to formulate them on the lattice. The lattice theories obtained this way are gauge-invariant, free from fermion doubling problem and exact supersymmetric at finite lattice spacing. We hope that these lattice constructions further motivate the nonperturbative explorations of models inspired by technicolor, orbifolding and orientifolding in string theories and the Corrigan-Ramond limit.
Highlights
With gauge group SU(Nc) × SU(Nf )
We construct the lattice theory with two-index matter in section 4 using the method of geometric discretization
Since there are two vector fields in the twisted theory, Am and Bm, and they both transform the same way under the twisted rotation group, it is natural to combine them to form a complexified gauge field, which we label as Am, and write down the twisted theory in a compact way
Summary
We can write down the gauge transformation rules for the lattice fields in the adjoint representation respecting the p-cell and orientation assignments on the lattice. These expressions reduce to the corresponding continuum results for the adjoint covariant derivative in the naive continuum limit They transform under gauge transformations like the corresponding lattice link field carrying the same indices. We need to define the action of the covariant difference operators on the lattice fields transforming in the two-index representations. We write down the following set of rules for the action of the covariant derivatives on fields in the two-index representations. For lattice variables in the complex conjugate representation Rwe have the following set of rules for the action of the covariant difference operator: Dm(+)ΦR (n) ≡ ΦR (n + νm) − ΦR (n)Um(n), D(m+)ΦR (n) ≡ ΦR (n + νm)U m(n) − ΦR (n), Dm(−)ΦR (n) ≡ Dm(+)ΦR (n − νm), D(m−)ΦR (n) ≡ D(m+)ΦR (n − νm). We note that the lattice action written above is Q-supersymmetric, gauge-invariant, local and free from the fermion doublers
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