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Extra info for Communications In Mathematical Physics - Volume 269
Example text
N on M, not necessarily closed. Then we should take the cup product of the pull-backs of the ωi ’s to Mg,n (M, β) under the evaluation maps, and take the push-forward of the resulting differential form to Mg,n . If the ωi ’s are smooth and have compact support, then one can show that the result is a differential form (not necessarily of top degree) on Mg,n . This is an example of a correlation function in our conformal field theory. But this is not the most general example. Other correlation functions correspond to other local observables, such as the vector fields on M realized as Lie derivatives acting on differential forms.
2) is satisfied by the following fields: ± (w, w) = exp ∓i w w0 ( p(z)dz + p(z)dz) , which are examples of the holomortex operators mentioned above. 1) with the term q 1/2 where (2) ± (2) + + (2) − , are the cohomological descendants (2) ± = ± (w, w)π(w)π(w)dwdw. The resulting deformed theory appears to be equivalent to the sigma model with the target P1 in the infinite volume limit (in the sense explained in Sect. 3). By construction, the part of a correlation function of this deformed theory that corresponds to degree n maps → P1 will appear with the overall factor q n .
Wn ∈ . The operator D(z, z) also has a transparent meaning from the point of view of the operator formalism. While operators of the form δ 2 (X )(z, z) are quite common, the operators δ 2 ( p)(z, z) may appear at first glance as somewhat more exotic. But the mystery disappears if one considers the corresponding state in the Hilbert space of the linear sigma model corresponding to a small circle around a point z ∈ . To simplify notation, set z = 0. Then this space contains the direct sum of the tensor products FN ⊗ F N , N ∈ Z, of the Fock representations FN , the Heisenberg algebra generated by the Fourier modes of the chiral fields X n z −n , X (z) = pn z −n−1 , p(z) = n∈Z n∈Z and their anti-holomorphic analogues F N .