By Paul Ginsparg
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Additional resources for Applied Conformal Field Theory
These very strong constraints were originally suggested to give a means of characterizing all conformally invariant systems in d dimensions (the procedure of solving the relations of ﬁg. 4 to ﬁnd conformal ﬁeld theories is known as ‘the conformal bootstrap’). This program however proved too diﬃcult to implement in practice. In two dimensions the problem becomes somewhat more tractable, since 43 one need only consider the primary ﬁelds, vastly reducing the number of independent quantities in the problem.
A linear combination of states that vanishes is known as a null state, and the representation of the Virasoro algebra with highest weight |h is constructed from the above Verma module by removing all null states (and their descendants). (It is useful at this point to contrast the situation in two dimensions with that of higher dimensions, where the conformal algebra is ﬁnite dimensional. The ﬁnite dimensional analog in two dimensions is the closed SL(2, C) subalgebra generated by L0,±1 , L0,±1 .
Thus we focus on eigenstates |ψ of L0 , say with L0 |ψ = hψ |ψ . Then since [L0 , Ln ] = −nLn , we have L0 Ln |ψ = (hψ − n)Ln |ψ and Ln lowers the eigenvalue of L0 for n > 0. But dilatation in z on the plane, generated by L0 + L0 , corresponds to translation in σ 0 on the cylinder, generated by the energy H. L0 + L0 should thus be bounded below in any sensible quantum ﬁeld theory. Since L0 and L0 reside in independent holomorphic and anti-holomorphic algebras, they must be separately bounded from below.