Wednesday, 16 December 2015

quantum field theory - Level quantization of 7d SO(N) Chern-Simons action


In 3d, one can write down the SO(N) Chern-Simons action to be S(A)=k192πMTr(AdA+23A3),

where A is an SO(N) connection. The level quantization can be derived as follows:


Let M be a bounding 4-manifold of M. We can always find such M since ΩSO3=0. Extend A to M and define S(A)=k192πMTr(FF),

where F is the curvature 2-form of A. We need exp(iSM(A)) to the independent of the choice of M, and the extension of A from M to M. Let M be another bounding manifold of M, then the difference of S is δS=k192πMˉMTr(FF),
where ˉM denotes the orientation reversal of M. δS can be rewritten as δS=kπ24p1(MˉM)=kπ8σ(MˉM),
where p1 is the first Pontryagin number, and σ is the signature of a 4-manifold. We also used the Hirzbruch signature theorem σ(X)=p1(X)/3 for 4-manifolds X. Since σ(X) is an integer, exp(iSM(A)) is well-defined for k equals multiples of 16.


One can use the above argument, together with the fact that Ωspin3=0 and the Rohlin theorem which implies that the signature of a closed spin 4-manifold is divisible by 16, to argue that for a spin 4-manifold, exp(iS) is well-defined for kZ.



I'm trying to derive the quantization condition of k using similar arguments as above, for 7d SO(N) Chern-Simons action (simply replace M by a 7-manifold, and A by 3-form ). The following facts may be helpful: ΩSO7=0, Ωspin7=0, σ(X)=(7p2(X)p21(X))/45

for 8-manifold X.




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