Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 741antisymmetric tensor with ɛ 12 = 1, then(ds) 2 = − i 2 ɛ da α dā βαβ dudū. (4.231)du dūThis is manifestly invariant under linear transformations that preserve ɛ andcommute with complex conjugation (the latter condition ensures that a αand ā α transform the same way). These transformations make the groupSL(2, R) (or equivalently Sp(2, R)). Also, (4.231) is obviously invariantunder adding a constant to a D or a. So if we arrange (a D , a) as a columnvector v, the symmetries that preserve the general structure are: v →Mv + c, where M is a 2 × 2 matrix in SL(2, R), and c is a constant vector.In general, this group of transformations „ « can be thought of as the group of1 03 × 3 matrices of the form , acting on the three objects (1, ac MD , a).Generalization to Dimension Greater than OneNow, let us briefly discuss the generalization to other gauge groups. If thegauge group G has rank r, then M has complex dimension r. Locally, from(4.217), it follows that the metric is( ∂(ds) 2 2 )F= Im∂a i ∂a j da i dā j ,with distinguished local coordinates a i and a holomorphic function F. Weagain reformulate this by introducingThen we can writea D,j = ∂F∂a j . (4.232)(ds) 2 = Im ∑ ida D,i dā i .To formulate this invariantly, we introduce a complex space X ∼ = C 2r withcoordinates a i , a D,j . We endow X with the symplectic formω = i 2∑ (da i ∧ dā D,i − da D,i ∧ dā i)of type (1, 1) and also with the holomorphic 2–formiω h = ∑ ida i ∧ da D,i .