Assuming the Lorentz and CPT invariances we show that neutron-antineutron oscillation implies breaking of CP along with baryon number violation -- i.e. two of Sakharov conditions for baryogenesis. The oscillation is produced by the unique operator in
the effective Hamiltonian. This operator mixing neutron and antineutron preserves charge conjugation C and breaks P and T. External magnetic field always leads to suppression of oscillations. Its presence does not lead to any new operator mixing neutron and antineutron.
We study N=(0,2) deformed (2,2) two-dimensional sigma models. Such heterotic models were discovered previously on the world sheet of non-Abelian strings supported by certain four-dimensional N=1 theories. We study geometric aspects and holomorphic pr
operties of these models, and derive a number of exact expressions for the beta functions in terms of the anomalous dimensions analogous to the NSVZ beta function in four-dimensional Yang-Mills. Instanton calculus provides a straightforward method for the derivation. The anomalous dimensions are calculated up to two loops implying that one of the beta functions is explicitly known up to three loops. The fixed point in the ratio of the couplings found previously at one loop is not shifted at two loops. We also consider the N=(0,2) supercurrent supermultiplet (the so-called hypercurrent) and its anomalies, as well as the Konishi anomaly. This gives us another method for finding exact $beta$ functions. We prove that despite the chiral nature of the models under consideration quantum loops preserve isometries of the target space.
The chiral symmetry of QCD shows up in the linear Weyl--Wigner mode at short Euclidean distances or at high temperatures. On the other hand, low-lying hadronic states exhibit the nonlinear Nambu--Goldstone mode. An interesting question was raised as
to whether the linear realization of the chiral symmetry is asymptotically restored for highly excited states. We address it in a number of ways. On the phenomenological side we argue that to the extent the meson Regge trajectories are observed to be linear and equidistant, the Weyl--Wigner mode is not realized. This picture is supported by quasiclassical arguments implying that the quark spin interactions in high excitations are weak, the trajectories are linear, and there is no chiral symmetry restoration. Then we use the string/gauge duality. In the top-down Sakai--Sugimoto construction the nonlinear realization of the chiral symmetry is built in. In the bottom-up AdS/QCD construction by Erlich et al., and Karch et al. the situation is more ambiguous. However, in this approach linearity and equidistance of the Regge trajectories can be naturally implemented, with the chiral symmetry in the Nambu--Goldstone mode. Asymptotic chiral symmetry restoration might be possible if a nonlinearity (convergence) of the Regge trajectories in an intermediate window of $n,J$, beyond the explored domain, takes place. This would signal the failure of the quasiclassical picture.
