By Gnaedig P., Honyek G., Riley K.F.
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Extra resources for 300 puzzling physics problems
8 For simplicity we work in a unitary trivialization of the bundle V. 176 on Sat Jan 31 06:40:45 GMT 2015. 4 Gauge dualities The non–scalar fields live in some vector bundle V → M. We wish to study in more detail the geometry of these bundles in some relevant case, related to a class of dualities which generalize the usual electromagnetic duality. Again we do not assume supersymmetry. However, historically, the following results were first obtained in the context of supergravity (SUGRA), as they are needed in order to construct the Lagrangian of the maximal extended N = 8 supergravity in 4D (see Cremmer and Julia  and also [110, 150]).
176 on Sat Jan 31 06:40:45 GMT 2015. 7, we can easily answer this basic question. 91) so, in view of Eq. 75), it can be compensated by a change in the vielbein E of the form E → E S−1 . 93) that is, explicitly, E (si (φ)) = h(φ i) E (φ i ) S−1 where h(φ i ) ∈ K (2n). 95) Then the transformation F± → S F± , is a symmetry of both the scalars’ and the forms’ kinetic terms. In fact, it is an invariance of the full Lagrangian provided the expression on the RHS of Eq. 75), “other fields,” transforms covariantly under K (2n); that is, if the other fields live in the (pull–back of) right homogeneous bundles and the remaining couplings of the scalars preserve the isometry.
7 Let F a = dAa be the m–form field–strengths of a D = 2m field theory whose Lagrangian L is quadratic in the F a . 89) and [E ] has a representative as in Eq. 79) with M= Re(iN ) (N − N t )/2 Q= (Im(iN ))1/2 D = 4k, ( − (N D = 4k + 2. 90) C ONVERSELY, any such map μ specifies a unique Lagrangian coupling. All bundles on M which describe the couplings of the F a to other fields (φ, χ , ψμ , . . ) are the pull–back by μ of homogeneous bundles over the coset space in Eq. 88). 16 14 We call it μ since in the Maxwell theory it corresponds to the magnetic susceptibility.
300 puzzling physics problems by Gnaedig P., Honyek G., Riley K.F.