QPFT operator algebras and commutative exterior differential calculus (Q1309102)

Let \(H\) be a linear space. A quantum field theory (QFT) operator algebra is defined as a linear mapping \(l_ u\), analytically depending on \(u\in \mathbb{C}\), relating to each vector \(\varphi\in H\) an operator \(l_ u(\varphi)\) acting in \(H\) and the so called duality relation \[ l_ u (\varphi) l_ v(\psi)= l_ v (l_{u-v} (\varphi)\psi) \] is required. Apparently in the rest of the paper it is assumed that a singularity at the origin \(u=0\) can occur. Because of this and in virtue of the duality relation, the product \(l_ u(\varphi) l_ v(\psi)\) is, in general, singular for \(u=v\). This is why one introduces a ``renormalization'' which is the operator \(\varphi(f):= \text{res}_{u=0} \{f(u) l_ u(\varphi) u^{-1}du\}\) depending on a function \(f(u)\). A quantum projective field theory operator algebra is defined as a QFT operator algebra acting in a direct sum or direct integral of Verma modules \(V_ \alpha\) over the Lie algebra \({\mathfrak {sl}} (2,\mathbb{C})\) and obeying some additional conditions. The paper presents a theorem according to which the result of renormalization of the pointwise product of operator fields in the algebra \(\text{Vert } ({\mathfrak {sl}} (2,\mathbb{C}))\) of vertex operators can be identified with the algebra of the so called commutative exterior differential calculus. The commutative exterior calculus was introduced by the author in [J. Math. Phys. 33, 3112-3116 (1992; Zbl 0768.17012)] and it is generated by \(x\), \(d\), \(xi\) and \(dx^ \lambda\), \(\lambda=1,2,\dots\), with explicitly prescribed commutation relations.