Spontaneous Supersymmetry Breaking, Localization and Nicolai Mapping in Matrix Models

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Supersymmetry (SUSY) is a symmetry between bosons and fermions. It leads to degeneracies of mass spectra between bosons and fermions. Although such degeneracies have not been observed yet, there is a possibility for SUSY being realized in nature as a spontaneously broken symmetry. From a theoretical viewpoint, SUSY provides a unified framework describing physics in high energy regime beyond the standard model (Sohnius, 1985). Spontaneous breaking of SUSY is one of the most interesting phenomena in quantum field theory. Since in general SUSY cannot be broken by radiative corrections at the perturbative level, its spontaneous breaking requires understanding of nonperturbative aspects of quantum field theory (Witten, 1981). In particular, recent developments in nonperturbative aspects of string theory heavily rely on the presence of SUSY. Thus, in order to deduce predictions to the real world from string theory, it is indispensable and definitely important to investigate a mechanism of spontaneous SUSY breaking in a nonperturbative framework of strings. Since one of the most promising approaches of nonperturbative formulations of string theory is provided by large-N matrix models (Banks et al., 1997; Dijkgraaf et al., 1997; Ishibashi et al., 1997), it will be desirable to understand how SUSY can be spontaneously broken in the large-N limit of simple matrix models as a first step. Analysis of SUSY breaking in simple matrix models would help us find a mechanism which is responsible for possible spontaneous SUSY breaking in nonperturbative string theory. For this purpose, it is desirable to treat systems in which spontaneous SUSY breaking takes place in the path-integral formalism, because matrix models are usually defined by the path integrals, namely integrals over matrix variables. In particular, IIB matrix model defined in zero dimension can be formulated only by the path-integral formalism (Ishibashi et al., 1997). Motivated by this, we discuss in the next section the path-integral formalism for (discretized) SUSY quantum mechanics, which includes cases that SUSY is spontaneously broken. Analogously to the situation of ordinary spontaneous symmetry breaking, we introduce an external field to choose one of degenerate broken vacua to detect spontaneous SUSY breaking. The external field plays the same role as a magnetic field in the Ising model introduced to detect the spontaneous magnetization. For the supersymmetric system, we deform the boundary condition for fermions from the periodic boundary condition (PBC) to a twisted boundary condition (TBC) with twist α, which can be regarded as such an external 17