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Gravitational anomalies in curved space with non-abelian torsion S

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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

11

12 4. Tensorial form of anomalies

13

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.


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