1. Gravitational anomalies in curved space with non-abelian torsion S
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

12. 4． Tensorial form of anomalies

14. 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 even degree of the field strength.

15. Relation of the degree of F and the dimension of space
d = 4k (4-dim.)
d = 4k+2 (6-dim.)
Gravitational Anomalies
Odd
Even
Chiral U(1) Anomalies
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.