Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 745Coupling to GravityNow, we would like to compare the structure we have found to the ‘specialgeometry’ that appears if the chiral multiplet is coupled to N = 2supergravity. In N = 2 supergravity, the general Kähler metric for a systemof r chiral superfields is described locally by a holomorphic functionG 0 (a 1 , . . . , a r ) of r complex variables a i . The Kähler potential is(K grav = − ln2i(G 0 − Ḡ0) + i 2∑(iā i ∂G ) )0 ∂Ḡ0− ai∂ai ∂ā i . (4.235)In global supersymmetry we had a local holomorphic function F withK = −i ∑(ā i ∂F )∂ ¯F− ai2 ∂ai ∂ā i . (4.236)iOne would expect that there is some limit in which gravitational effects aresmall and (4.235) would reduce to (4.236). How does this occur?It suffices to setG 0 = −i M P l 2+ F, (4.237)4with M P l the Planck mass. Then if M P l is much larger than all relevantparameters, we getK grav = − ln M P l 2 +KM P l2 + O(M P l −4 ).The constant term -ln M 2 P l does not contribute to the Kähler metric, soup to a normalization factor of 1/MP 2 l, the Kähler metric with supergravityreduces to that of global N = 2 supersymmetry as M P l → ∞ keepingeverything else fixed.More fundamentally, we would like to compare the allowed monodromygroups. In supergravity, the global structure is exhibited as follows. Oneintroduces an additional variable a 0 and sets G = (a 0 ) 2 G 0 . One also introducesa D,j = ∂G/∂a j for j = 0, . . . , r. Then one finds that the specialKähler structure of (4.235) allows Sp(2r + 2, R) transformations acting on(a Di , a j ). 28 Now, in decoupling gravity, we consider G to be of the specialform in (4.237). In that case, a D,0 = −i M P l2 . The other ai , a D,j are independentof M P l . To preserve this situation in which M P l appears only in28 In the gauge fields this is reduced to Sp(2r + 2, Z). The symplectic form preservedby Sp(2r + 2, R) becomes the usual one P i dai ∧ da D,i .