A Note on the M5 Brane Anomaly

The problem of the M5 brane anomaly cancellation is addressed. We refor-mulate FHMM construction [1] making explicit the relation with the M5 brane SUGRA solution. We suggest another solution to the magnetic coupling equation which doesn't need anomalous SO(5) variation of the 3-form potential and coincides with the SUGRA solution outside smoothed out core of the magnetic source. Chern-Simons term evaluated on this solution generates the same anomaly inflow as achieved by FHMM. The potential anomaly of the normal bundle in the presence of the M5 brane has three contributions. The first one comes from a chiral theory of zero-modes on the brane world volume, second comes from the Chern-Simons coupling G 4 ∧ I 7 , where G 4 is the 4-form field strength and I 7 is a gravitational Chern-Simons. The problem of finding the third contribution that cancels the previous two is addressed in [1] in the following way. Instead of considering 11-d SUGRA in the background of the 11-d M5 brane solution the other theory is studied. It's 11-d SUGRA with a 4-form field-strength satisfying modified Bianchi identity corresponding to the (singular) magnetic coupling to a 6-dimensional submanifold (which represents the 5-brane in this picture). To have a well-defined delta-function in the magnetic coupling equation the source is smoothed out and the 4-form G satisfies dG = dρ(r) ∧ e 4 /2 , (1) where ρ is a bump function. It equals to −1 on the brane (r = 0) and to 0 far away from the brane, e 4 is an angular form on the SO(5) normal bundle. It is closed (de 4 = 0) and gauge invariant under SO(5) transformations of the normal bundle. Locally e 4 = de 3. The tubular neighborhood of the brane is removed and the resulting effective action in the bulk is