Ivancevic_Applied-Diff-Geom

1134 Applied Differential Geometry: A Modern Introductionis an observable. However, unfortunately, as ˜W is cohomologically trivialbecause δ B ˜W = 0 but d˜W ≠ δB ˜W ′ for some ˜W ′ . Accordingly, ˜W does notgive any new topological invariant [Ohta (1998)].In topological Bogomol’nyi theory, there is a sequence of observablesassociated with a magnetic charge. For the Abelian case, it is given by∫W = F ∧ db. (6.223)YAs is pointed out for the case of Bogomol’nyi monopoles [Birmingham et. al.(1989)], we can not get the observables related with this magnetic chargeby the action of δ B as well, but we can construct those observables byanti–BRST variation δ B which maps (m, n)−form to (m, n − 1)−form. δ Bcan be obtained by a discrete symmetry which is realized as ‘time reversalsymmetry’ in 4D. In our 3D theory, the discrete symmetry is given byφ −→ −λ, λ −→ −φ, N −→ i √ 2µ, µ −→ i √2N,ψ i −→ χ i√ , χ i −→ √ 2ψ i , η −→ √ 2ξ, ξ −→ −√ η (6.224)2 2with b −→ −b, (6.225)where (6.225) represents an additional symmetry [Birmingham et. al.(1989)]. Note that we must also change N and µ (and their conjugates).The positive chirality condition for M should be used in order to check theinvariance of the action. In this way, we can get anti–BRST transformationrule by substituting (6.224) and (6.225) into (6.213) and then we can getthe observables associated with the magnetic charge by using the action ofthis anti–BRST variation [Birmingham et. al. (1989)].The topological observables available in this theory are the same withthose of topological Bogomol’nyi monopoles.Finally, let us briefly comment on our three dimensional theory. Firstnote that Lagrangian L and Hamiltonian H in dimensional reduction canbe considered as equivalent. This is because the relation between them isdefined byH = p ˙q − L,where q is any field, the overdot means time derivative and p is a canonicalconjugate momentum of q, and the dimensional reduction requires the timeindependence of all fields, thus H = −L in this sense. Though we haveconstructed the three dimensional action directly from the 3D monopole