Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1203Now, we can write the Hilbert space as a direct sum, H = H 0 ⊕ H 1 ,where H 0 consists of states |ψ〉 for which Q|ψ〉 = 0 and H 1 is its orthogonalcomplement. SinceQ (O 1 . . . O i |0〉) = 0, (6.314)the states in H 1 are in particular orthogonal to states of the formO 1 . . . O i |0〉, and hence the states in H 1 do not contribute to the sum in(6.313). Moreover, changing O A to O A + {Q, Λ} does not change the resultin (6.313), so we only need to sum over a basis of H 0 /I({Q, ·}), which isexactly the space H phys of ‘topologically physical’ states.Fig. 6.25 A correlation function on a Riemann surface factorizes into correlation functionson two Riemann surfaces of lower genus (see text for explanation).Finally, let us determine the metric η ab . We can deduce its form byfactorizing the 2–point functionin the above way, resulting inC ab = 〈O a O b 〉 (6.315)C ab = C ac η cd C db . (6.316)In other words, we find that the metric η ab is the matrix inverse of the2–point function C ab , which for this reason we will write as η ab from nowon.