Abstract
We examine the possibility of dynamical supersymmetry breaking in two-dimensional $\mathcal{N} = (2, 2)$ supersymmetric Yang-Mills theory. The theory is discretized on a Euclidean spacetime lattice using a supersymmetric lattice action. We compute the vacuum energy of the theory at finite temperature and take the zero temperature limit. Supersymmetry will be spontaneously broken in this theory if the measured ground state energy is non-zero. By performing simulations on a range of lattices up to $96 \times 96$ we are able to perform a careful extrapolation to the continuum limit for a wide range of temperatures. Subsequent extrapolations to the zero temperature limit yield an upper bound on the ground state energy density. We find the energy density to be statistically consistent with zero in agreement with the absence of dynamical supersymmetry breaking in this theory.
Highlights
The investigations of supersymmetric gauge theories on a spacetime lattice are important for understanding the nonperturbative structure of such theories and in particular they can address the question of whether dynamical supersymmetry (SUSY) breaking takes place in such theories
We examine the possibility of dynamical supersymmetry breaking in two-dimensional N 1⁄4 ð2; 2Þ supersymmetric Yang-Mills theory
We find the energy density to be statistically consistent with zero in agreement with the absence of dynamical supersymmetry breaking in this theory
Summary
It is the simplest two-dimensional supersymmetric theory that can be studied on the lattice This theory is a interesting theory in the continuum because of its exotic phases as discussed by Witten in Ref. The two-dimensional N 1⁄4 ð2; 2Þ SYM theory is the simplest supersymmetric gauge theory which admits topological twisting [36] and satisfies the requirements for a supersymmetric lattice construction following the prescription given in Refs. The continuum covariant derivatives are replaced by covariant difference operators and they act on the twisted fields in the following way: Dða−ÞfaðnÞ 1⁄4 faðnÞUaðnÞ − Uaðn − μaÞfaðn − μaÞ; DðaþÞfbðnÞ 1⁄4 UaðnÞfbðn þ μaÞ − fbðnÞUaðn þ μbÞ: The lattice field strength is given by F abðnÞ 1⁄4 DðaþÞUbðnÞ, and is antisymmetric It transforms like a lattice 2-form and yields a gauge-invariant loop on the lattice when contracted with χabðnÞ. The term involving the covariant backward difference operator, Dða−ÞUaðnÞ, transforms as a 0-form or site field and can be contracted with the site field ηðnÞ to yield a gauge-invariant expression
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