Yuki Kamo

An Irreducible Form of Gamma Matrices for HMDS Coefficients of the Heat Kernel in Higher Dimensions

The heat kernel method is used to calculate 1-loop corrections of a fermion interacting with general background fields. To apply the Hadamard-Minakshisundaram-DeWitt-Seeley (HMDS) coefficients aq(x, x � ) of the heat kernel to calculate the corrections, it is meaningful to decompose the coefficients into tensorial components with irreducible matrices, which are the totally antisymmetric products of γ matrices. We present formulae for the tensorial forms of the γ-matrix-valued quantities X, ˜ Λμν and their product and covariant derivative in terms of the irreducible matrices in higher dimensions. The concrete forms of HMDS coefficients obtained by repeated application of the formulae simplifies the derivation of the loop corrections after the trace calculations, because each term in the coefficients contains one of the irreducible matrices and some of the terms are expressed by commutator and the anticommutator with respect to the generator of non-abelian gauge groups. The form of the third HMDS coefficient is useful for evaluating some of the fermionic anomalies in 6dimensional curved space. We show that the new formulae appear in the chiral U (1) anomaly when the vector and the third-order tensor gauge fields do not commute. Subject Index: 131, 133, 454

Analyses of Resonances in 4- and 5-Neutrino Oscillations in Matter

We analytically investigate the resonance conditions among n neutrinos in matter with the discriminant for the characteristic equation of the effective Hamiltonian. This discriminant is expressed in terms of the coefficients of the characteristic equation, without solving this equation. The graphical representation of the discriminant reveals the matter densities where resonance occurs. We apply the discriminant for n neutrino oscillation to the cases of 4and 5-neutrino oscillation.

Gravitational anomalies in higher dimensional Riemann-Cartan space

By applying the covariant Taylor expansion method of the heat kernel, the covariant Einstein anomalies associated with a Weyl fermion of spin in four-, six- and eight-dimensional Riemann–Cartan space are manifestly given. Many unknown terms with torsion tensors appear in these anomalies. The Lorentz anomaly is intimately related to the Einstein anomaly even in Riemann–Cartan space. The explicit form of the Lorentz anomaly corresponding to the Einstein anomaly is also obtained.

An irreducible form for the asymptotic expansion coefficients of the heat kernel of fermions in four-dimensional curved space

We consider the heat kernel for a massless fermion of spin 1/2 interacting with all types of non-Abelian boson fields, i.e. scalar, pseudo-scalar, vector, axial-vector and antisymmetric tensor fields, in a four-dimensional Riemannian space. The couplings of the fermion with the boson fields contain irreducible matrices of the product of the γ-matrices. In this model, the components of the first and second asymptotic expansion coefficients of the heat kernel with respect to the irreducible matrices are explicitly presented. The form of the second coefficients is useful for evaluation of some fermionic anomalies in four-dimensional curved space, and the concrete forms of the chiral U(1) and the trace anomalies are presented.