New bosonization and conformal field theory over \({\mathbb{Z}}\) (Q1119982)

In conformal field theory one is faced to the problem of calculating multiloop amplitudes whose divergences correspond to deformations of Riemann surfaces. One is led to construct an infinite dimensional universal Graßmann manifold (over \({\mathbb{C}})\) in which all Riemann surfaces of genus g are treated as single points. This universal Graßmannian can be embedded into projective fermion Fock space \({\mathbb{P}}({\mathcal F})\). One can also define the boson Fock space \({\mathcal H}\) and the bosonization operator \(B: {\mathcal F}\to {\mathcal H}.\) In the underlying paper this formalism is developed over \({\mathbb{Z}}\) (or any commutative ring A). The construction of the universal Graßmann manifold UGM over \({\mathbb{Z}}\) (or A) and of fermion and boson Fock spaces over \({\mathbb{Z}}\) (or A) is straightforward. As to the bosonization operator B one has to introduce new variables which can be considered as the coordinates of the universal Witt scheme \(W_{\infty}\). B extends to an A-linear isomorphism \(\tilde B: {\mathcal F}(A\to {\mathcal H}(A).\) Instead of the complex Riemann surfaces one considers proper smooth morphisms of schemes \(\pi: C\to Spec(A)\) with generic fiber a non-singular algebraic curve of genus g. Denoting by \(\pi: {\mathcal C}_{g,\ell}\to {\mathcal M}_{g,\ell}\) the universal family over the moduli space of curves of genus g with level \(\ell\) structure, a principal \({\mathcal G}_ n\)-bundle \(f_ n: {\mathcal C}^{(n)}_{g,\ell}\to {\mathcal C}_{g,\ell}\) with \({\mathcal G}_ n(A)=Aut_ A(A[[\zeta]]/(\zeta^{n+1}))\) can be constructed for any \(n\geq 2\). The A-points of \({\mathcal C}^{(n)}_{g,\ell}\) can be described explicitly in terms of the morphisms \(\pi: C\to Spec(A)\) with a section. For \(n=1\) there is a somewhat different statement. A point \(X\in \hat {\mathcal C}_{g,\ell}(A):=\varprojlim_ n{\mathcal C}^{(n)}_{g,\ell}(A) \) is shown to define a point of UGM(A).