Ivancevic_Applied-Diff-Geom

1202 Applied Differential Geometry: A Modern Introductionthe path integral only depends on the topology of the surface, and hencenot on the size of the hemisphere [Vonk (2005)].Fig. 6.24 A graphical representation of the correlation function (6.311).We assume that all states that we are interested in are of the above form,which in particular means that we will integrate only over Riemann surfaceswithout boundary. Moreover, we assume these surfaces to be orientable.An important property of topological field theories in two dimensions isnow that its correlation functions factorize in the following way:〈O 1 · · · O n 〉 Σ = ∑ a,b〈O 1 · · · O i O a 〉 Σ1 η ab 〈O b O i+1 · · · O n 〉 Σ2 , (6.312)where the genus of Σ is the sum of the genera of Σ 1 and Σ 2 . This statementis explained in Figure 6.25. By using the topological invariance, we candeform a Riemann surface Σ with a set of operator insertions in such a waythat it develops a long tube. From general quantum field theory, we knowthat if we stretch this tube long enough, only the asymptotic states – thatis, the states in the physical part of the Hilbert space – will propagate. Butas we have just argued, instead of inserting these asymptotic states, we mayjust as well insert the corresponding operators at a finite distance. However,to conclude that this leads to (6.312), we have to show that this definitionof ‘physical states’ – being the ones that need to be inserted as asymptoticstates – agrees with our previous definition in terms of Q−cohomology. Letus argue that it does by first writing the factorization as〈O 1 · · · O n 〉 Σ = ∑ A,B〈O 1 · · · O i |O A 〉 Σ1 η AB 〈O B |O i+1 · · · O n 〉 Σ2 . (6.313)where the O A with capital index now correspond to a complete basis ofasymptotic states in the Hilbert space. The reader may be more used tothis type of expressions in the case where η AB = δ AB , but since we havenot shown that with our definitions 〈O A |O B 〉 = δ AB , we have to work withthis more general form of the identity operator, where η is a metric thatwe will determine in a moment.