On p. 13 of "Vertex Algebras for Beginners", 2nd edition, Kac writes:

"Under certain assumptions and with certain additional data one may reconstruct the whole QFT from these chiral algebras, but we shall not discuss this problem here".

However, he does not give any references to support this claim (at least not next to the claim) and I was not able to find any on my own.

Does anybody know of any references?

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    $\begingroup$ You can check that the correlation functions are just Laurent polynomials in the variables. Locality easily follows and fields associated with quasi primary vectors are covariant. Positivity of energy follows from the analytic properties. Wightman positivity is non-trivial and you need a unitary VOA. I don't know a good reference though $\endgroup$ Oct 3, 2015 at 1:31

2 Answers 2


You might find "An Introduction to Conformal Field Theory" by M. Gaberdiel (arXiv:hep-th/9910156v2) useful. He has a brief discussion of how in some cases chiral algebras can be assembled into Conformal Field Theories and has some further references. The language of this review is somewhere in-between the CFT language of physicists and the more formal VOA language of mathematicians.


I found Nikolov's paper "Vertex Algebras in Higher Dimensions and Globally Conformal Invariant Quantum Field Theory" arXiv:hep-th/0307235. The abstract:

We propose an extension of the definition of vertex algebras in arbitrary space-time dimensions together with their basic structure theory. An one-to-one correspondence between these vertex algebras and axiomatic quantum field theory (QFT) with global conformal invariance (GCI) is constructed.

Now he did not explicitly show how are these generalized vertex algebras related to the usual chiral vertex algebras. So in my master's thesis arXiv:1607.05078 I restricted myself to unitary (quasi-)vertex operator algebras and reversed Kac's proof.


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