Supersymmetry Breakings and Fermat's Last Theorem

A mechanism of supersymmetry breaking in two or four-dimensions is given, in which the breaking is related to the Fermat's last theorem. It is shown that supersymmetry is exact at some irrational number points in parameter space, while it is broken at all rational number points except for the origin. Accordingly, supersymmetry is exact {\it almost everywhere}, as well as broken {\it almost everywhere} on the real axis in the parameter space at the same time. This is the first explicit mechanism of supersymmetry breaking with an arbitrarily small change of parameters around any exact supersymmetric model, which is possibly useful for realistically small non-perturbative supersymmetry breakings in superstring model building. As a byproduct, we also give a convenient superpotential for supersymmetry breaking only for irrational number parameters. Our superpotential can be added as a ``hidden'' sector to other useful supersymmetric models.Comment: 12 pages (LATEX), UMDEPP 94--96 (Feb. 1994

Lagrangian and Covariant Field Equations for Supersymmetric Yang-Mills Theory in 12D

We present a lagrangian formulation for recently-proposed supersymmetric Yang-Mills theory in twelve dimensions. The field content of our multiplet has an additional auxiliary vector field in the adjoint representation. The usual Yang-Mills field strength is modified by a Chern-Simons form containing this auxiliary vector field. This formulation needs no constraint imposed on the component field from outside, and a constraint on the Yang-Mills field is generated as the field equation of the auxiliary vector field. The invariance check of the action is also performed without any reference to constraints by hand. Even though the total lagrangian takes a simple form, it has several highly non-trivial extra symmetries. We couple this twelve-dimensional supersymmetric Yang-Mills background to Green-Schwarz superstring, and confirm fermionic kappa-invariance. As another improvement of this theory, we present a set of fully Lorentz-covariant and supercovariant field equations with no use of null-vectors. This system has an additional scalar field, whose gradient plays a role of the null-vector. This system exhibits spontaneous breaking of the original Lorentz symmetry SO(10,2) for twelve-dimensions down to SO(9,1) for ten-dimensions.Comment: 14 pages, latex, no figure

N=2 Chiral Supergravity in (10 + 2)-Dimensions As Consistent Background for Super (2 + 2)-Brane

We present a theory of N=2 chiral supergravity in (10+2)-dimensions. This formulation is similar to N=1 supergravity presented recently using null-vectors in 12D. In order to see the consistency of this theory, we perform a simple dimensional reduction to ten-dimensions, reproducing the type IIB chiral supergravity. We also show that our supergravity can be consistent background for super (2+2)-brane theory, satisfying fermionic invariance of the total action. Such supergravity theory without manifest Lorentz invariance had been predicted by the recent F-theory in twelve-dimensions.Comment: 14 pages, LATEX, with minor corrections in typos and expression

N=1 Supersymmetric Non-Abelian Compensator Mechanism for Extra Vector Multiplet

We present a variant formulation of N=1 supersymmetric compensator mechanism for an arbitrary non-Abelian group in four dimensions. This formulation resembles our previous variant supersymmetric compensator mechanism in 4D. Our field content consists of the three multiplets: (i) A Non-Abelian Yang-Mills multiplet (A_\mu^I, \lambda^I, C_{\mu\nu\rho}^I), (ii) a tensor multiplet (B_{\mu\nu}^I, \chi^I, \varphi^I) and an extra vector multiplet (K_\mu^I, \rho^I, C_{\mu\nu\rho}^I) with the index I for the adjoint representation of a non-Abelian gauge group. The C_{\mu\nu\rho}^I is originally an auxiliary field dual to the conventional auxiliary field D^I for the extra vector multiplet. The vector K_\mu^I and the tensor C_{\mu\nu\rho}^I get massive, after absorbing respectively the scalar \varphi^I and the tensor B_{\mu\nu}^I. The superpartner fermion \rho^I acquires a Dirac mass shared with \chi^I. We fix all non-trivial cubic interactions in the total lagrangian, all quadratic terms in supersymmetry transformations, and all quadratic interactions in field equations. The action invariance and the super-covariance of all field equations are confirmed up to the corresponding orders.Comment: 11 pages, no figure