Anomaly cancellation in string theory

The anomaly cancellation in superstring theory is known to hold at leading order in the curvature for the gauge groups SO(32) and E-8 x E-8. The coefficients of the next-to-leading order terms may be evaluated, and a mechanism for cancellation is described, which would remain valid at higher genus when there exists a global splitting of the coordinates of supermoduli space. Since the spin-1 2 fields transform under the adjoint representation in these models, compactification of the heterotic string over G(2)/SU(3) or a fundamental 12-dimensional theory, from which superstrings are produced through an elliptic fibration, over G(2 )x SU(2) x U(1)/ SU(3) x U(1)' x U(1)'', provides another phenomenologically viable theory at lower energies compatible with the standard description of the elementary particle interactions. The 96 spin-1/2 fields now would transform under the fundamental representation of G(2) x SU(2) x U(1) and the spin-one gauge fields would belong to the adjoint representation. The sum of the anomaly polynomials for the particle content of the G(2) x SU(2) x U(1) model vanishes at n(1/2) similar or equal to 103.34271924. The contributions to the gravitational anomaly from the particles and antiparticles cancel by the CPT theorem and the duality transformations of polynomials of degree 6 in the curvature and the field strength. The existence of the interaction of a spin-2 charge, which is conserved only over a finite time interval, can be traced to nonlocal terms in the reduction of the string field theory to the gravitational sector. The source of the global gravitational anomaly cancellation in the modular form equations derived from an elliptic fibration of a 12-dimensional theory would restrict the compactifications and provide a method for preserving the absence of anomalies in four dimensions.