Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1209new Lorentz rotation generators,M A = M − F V = M − 1 2 (F L + F R )M B = M − F A = M − 1 2 (F L − F R ).A well–known result from string theory (see [Vonk (2005)]) is that thegenerator of Lorentz rotations is M = 2πi(L 0 − ¯L 0 ). Therefore, we findthat the twisting procedure in this new language amounts toA : L 0,A = L 0 − 1 2 J 0, ¯L0,A = ¯L 0 + 1 2 ¯J 0 ,B : L 0,B = L 0 − 1 2 J 0, ¯L0,B = ¯L 0 − 1 2 ¯J 0 .Let us now focus on the left–moving sector; we see that for both twistingsthe new Lorentz rotation generator is the difference of L 0 and 1 2 J 0. Thenew Lorentz generator should also correspond to a conserved 2–tensor, andfrom (6.321) and (6.322) there is a very natural way to get such a current:which clearly satisfies ¯∂ ˜T = 0 and˜T (z) = T (z) + 1 ∂J(z), (6.323)2˜L m = L m − 1 2 (m + 1)J m, (6.324)so in particular we find that ˜L 0 can serve as L 0,A or L 0,B . We should applythe same procedure (with a minus sign in the A−model case) in the right–moving sector. Equations (6.323) and (6.324) tell us how to implementthe twisting procedure not only on the conserved charges, but on the wholeN = 2 superconformal algebra – or at least on the part consisting of the J−and L−modes, but a further investigation shows that this is the only partthat changes. We have motivated, but not rigorously derived (6.323); for acomplete justification the reader is referred to the original papers [Lercheet. al. (1989)] and [Cecotti and Vafa (1991)].Now, we come to the crucial point. The algebra that the new modes˜L m satisfy can be directly calculated from (6.323), and we find[˜L m , ˜L n ] = (m − n)˜L m+n .That is, there is no central charge left. This means that we do not haveany restriction on the dimension of the theory, and topological strings willactually be well–defined in target spaces of any dimension.