Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1207The L m are called the Virasoro generators, and it is a well–known resultfrom conformal field theory that in the quantum theory their commutationrelations are[L m , L n ] = (m − n)L m+n + c12 m(m2 − 1)δ m+n .The number c depends on the details of the theory under consideration,and it is called the central charge. When this central charge is nonzero, oneruns into a technical problem. The reason for this is that the equation ofmotion for the metric field readsδSδh αβ = T αβ = 0.In conformal field theory, one imposes this equation as a constraint in thequantum theory. That is, one requires that for physical states |ψ〉,L m |ψ〉 = 0(for all m ∈ Z).However, this is clearly incompatible with the above commutation relationunless c = 0. In string theory, this value for c can be achieved by takingthe target space of the theory to be 10D. If c ≠ 0 the quantum theory isproblematic to define, and we speak of a ‘conformal anomaly’ [Vonk (2005)].The whole above story repeats itself for ¯T (¯z) and its modes ¯L m . Atthis point there is a crucial difference between open and closed strings.On an open string, left–moving and right–moving vibrations are relatedin such a way that they combine into standing waves. In our complexnotation, ‘left–moving’ translates into ‘z−dependent’ (i.e. holomorphic),and ‘right–moving’ into ‘¯z−dependent’ (i.e. anti–holomorphic). Thus, on anopen string all holomorphic quantities are related to their anti–holomorphiccounterparts. In particular, T (z) and ¯T (¯z), and their modes L m and ¯L m ,turn out to be complex conjugates. There is therefore only one independentalgebra of Virasoro generators L m .On a closed string on the other hand, which is the situation we have beenstudying so far, left– and right–moving waves are completely independent.This means that all holomorphic and anti–holomorphic quantities, and inparticular T (z) and ¯T (¯z), are independent. One therefore has two sets ofVirasoro generators, L m and ¯L m .Let us now analyze the problem of central charge in the twisted theories.To twist the theory, we have used the U(1)−symmetries. Any globalU(1)−symmetry of our theory has a conserved current J α . The fact thatit is conserved again means that J z ≡ J(z) is holomorphic and J¯z ≡ ¯J(¯z)