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Motivation An omalies and the heat kernel Tensorial form of anomalies Discussion

Gravitational anomalies in curved space with non-abelian torsion S. Yajima , M. Fukuda, M. Oka, S. Yamashita, T. Yamamoto Dept. of Physics, Kumamoto Univ. (Japan). Motivation An omalies and the heat kernel Tensorial form of anomalies Discussion. １． Motivation.

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Motivation An omalies and the heat kernel Tensorial form of anomalies Discussion

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  1. Gravitational anomalies in curved space with non-abelian torsion S. Yajima ,M. Fukuda, M. Oka, S. Yamashita, T. Yamamoto Dept. of Physics, Kumamoto Univ. (Japan) Motivation Anomalies and the heat kernel Tensorial form of anomalies Discussion

  2. １．Motivation • In the supergravity action, the 4-Fermi interaction term of the Rarita-Schwinger field “　　　” appears, • By a suitable gauge fixing of the , the kinetic part of the action is treated as a fermion of spin ½, regarding “　 ” as an index of the internal space of SO(２n).

  3. In 1-loop fermion diagrams, 4-Fermi interactions are regarded as some 2-Fermi interactions due to the Fierz transformation. • The fermionic (gravitino) part of the action fixed the gauge

  4. We assume that has only the vector gauge and the third order antisymmetric torsion tensor fields, which do not commute (non-abelian), and that is a massless Weyl fermion of spin ½：

  5. 2． Anomalies and the heat kernel • Some kinds of fermionic anomalies in even 2n dimensions are expressed by the heat kernel for fermions of spin ½.

  6. Einstein anomaly: • Lorentz anomaly: • Chiral U(1) anomaly:

  7. Anomalies and the heat kernel

  8. Ansatz of the heat kernel

  9. Anomalies and • Properties of anomalies

  10. The covariant Taylor Expansion of non-local quantity

  11. 4． Tensorial form of anomalies

  12. 5. Discussion • In 4 (and 6) dimensions, the Einstein, the Lorentz and the chiral U(1) anomalies are evaluated in massless Weyl fermion of spin ½ interacting with only the vector gauge and the axial vector (or the torsion tensor) fields, which do not commute. • The degree of the strength of the vector gauge field in some anomalies depends on the dimension of space. If these boson fields commute, then the anomalies have terms containing either odd or evendegree of the field strength.

  13. Relation of the degree of F and the dimension of space • If these boson fields do not commute, the dimensional dependence of the field strength does not hold. • The new terms in the anomalies are expressed by the commutator of the torsion tensor and the field strength.

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