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

• Published in: Progress of theoretical and experimental physics 2009-05, Vol.121 (5), p.983-1002
• Main Authors: Fukuda, Makoto, Yajima, Satoshi, Higashida, Yoji, Kubota, Shin-Ichiro, Tokuo, Shoshi, Kamo, Yuki
• Format: Article
• Language: English
• ISSN: 0033-068X; 2050-3911; 1347-4081
• DOI: 10.1143/PTP.121.983

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 艂) 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, Λ∼µν 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 6-dimensional 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.