Comparing Different Mathematical Definitions of 2D CFT

TL;DR

This paper compares various mathematical frameworks for 2D conformal field theories, establishing connections and equivalences among Wightman axioms, vertex algebras, and the Virasoro algebra representations.

Contribution

It provides a detailed comparison and proofs linking Wightman QFTs, vertex algebras, and unitary representations, clarifying their relationships in 2D CFTs.

Findings

Wightman CFTs give rise to conformal vertex algebras.

Finite generating fields imply unitarity of the associated vertex algebras.

Reversal of arguments shows equivalence between unitary vertex algebras and Wightman CFTs.

Abstract

We give introductions into the representation theory of the Virasoro algebra, Wightman axioms and vertex algebras in the first part. In the second part, we compare the above definitions. We give a proof of Luescher and Mack that a dilation invariant 2D QFT with an energy-momentum tensor gives rise to two commuting unitary representations of the Virasoro algebra. In the last chapter we compare Wightman QFTs to vertex algebras. We present Kac's proof that every Wightman Moebius CFT (a 2D Wightman QFT containing quasiprimary fields) gives rise to two commuting strongly-generated positive-energy Moebius conformal vertex algebras. If the number of generating fields of each conformal weight is finite, then these vertex algebras are unitary quasi-vertex operator algebras. As a corollary using Luescher--Mack's Theorem we obtain that a Wightman CFT (a Wightman Moebius CFT with an energy-momentum tensor) gives rise to a conformal vertex algebra which furthermore becomes a unitary vertex operator algebra, if the number of generating fields of each conformal weight is finite. We reverse Kac's arguments and get a converse proof that two unitary (quasi)-vertex operator algebras can be combined to give a Wightman (Moebius) CFT.