[Paper Review] Full vertex algebra and bootstrap -- consistency of four point functions in 2d CFT

Yuto Moriwaki|arXiv (Cornell University)|Jun 29, 2020

Algebraic structures and combinatorial models14 references4 citations

TL;DR

This paper introduces a mathematical framework called a full vertex algebra, a real analytic generalization of $η$-graded vertex algebras, to formalize the consistency of four-point correlation functions in 2D conformal field theory (CFT). By deriving the full vertex algebra structure from the bootstrap equation and conformal symmetry, the authors prove that the bootstrap condition ensures consistency of four-point functions under arbitrary operator product expansions, with an explicit construction of a deformable family of full vertex algebras from toroidal string compactifications.

ABSTRACT

In physics, it is believed that the consistency of two dimensional conformal field theory follows from the bootstrap equation. In this paper, we introduce the notion of a full vertex algebra by analyzing the bootstrap equation, which is a "real analytic" generalization of a $\mathbb{Z}$-graded vertex algebra. We also give a mathematical formulation of the consistency of four point correlation functions in two dimensional conformal field theory and prove it for a full vertex algebra with additional assumptions on the conformal symmetry. In particular, we show that the bootstrap equation together with the conformal symmetry implies the consistency of four point correlation functions. As an application, a deformable family of full vertex algebras parametrized by the Grassmanian is constructed, which appears in the toroidal compactification of string theory. This give us examples satisfying the above assumptions.

Motivation & Objective

• To establish a mathematical formulation of the consistency of four-point correlation functions in 2D conformal field theory.
• To define a full vertex algebra as a real analytic generalization of vertex algebras, derived directly from the bootstrap equation.
• To prove that the bootstrap equation, combined with conformal symmetry, implies the consistency of four-point functions under arbitrary operator product expansions.
• To construct a deformable family of full vertex algebras parametrized by the Grassmannian, arising from toroidal compactifications in string theory.

Proposed method

• Introduce a space $\mathrm{Cor}_4$ of real analytic functions to represent four-point correlation functions.
• Define parenthesized correlation functions $S_A$ associated with binary trees and operator product expansions, using formal variables $x_i = z_i - z_j$.
• Use tree-based recursion to assign variables and convergence domains to each parenthesization, ensuring analyticity in specific regions.
• Prove that the bootstrap equation—expressed as a monodromy-invariant condition—implies consistency of $S_A$ across different operator ordering choices.
• Construct a family of full vertex algebras from lattice vertex algebras via deformation, parameterized by the Grassmannian, satisfying the required symmetry and analyticity conditions.
• Verify the conjecture that $S_A$ corresponds to the expansion of a real analytic function $\phi(z_1,\dots,z_n)$ in the specified convergence regions, proven for $n \leq 4$ and $n=5$ with vacuum insertion.

Experimental results

Research questions

• RQ1 Can the consistency of four-point correlation functions in 2D CFT be rigorously formulated and proven using a real analytic algebraic structure?
• RQ2 Does the bootstrap equation, when combined with conformal symmetry, ensure that four-point functions are independent of operator ordering?
• RQ3 Can a full vertex algebra be defined independently of vertex algebras, with holomorphic and anti-holomorphic subalgebras naturally emerging from its axioms?
• RQ4 Does the full vertex algebra framework allow for the construction of deformable families of CFTs, such as those from toroidal compactifications?
• RQ5 Is the parenthesized correlation function $S_A$ for a given binary tree $A$ equivalent to the expansion of a globally defined real analytic function $\phi$ in a specified domain?

Key findings

• The bootstrap equation, when combined with conformal symmetry, implies the consistency of four-point correlation functions under arbitrary operator product expansions in the full vertex algebra framework.
• The full vertex algebra is a real analytic generalization of a $\mathbb{Z}$-graded vertex algebra, with holomorphic and anti-holomorphic subalgebras naturally arising from its axioms.
• For $n \leq 4$ and $n=5$ with vacuum insertion, the parenthesized correlation function $S_A$ corresponds to the expansion of a globally defined real analytic function $\phi(z_1,\dots,z_n)$ in a specified convergence domain.
• A deformable family of full vertex algebras is constructed from lattice vertex algebras, parametrized by the Grassmannian, modeling irrational CFTs from toroidal string compactifications.
• The change of variables $x_i = z_i - z_j$ is systematically derived from the tree structure, with convergence regions determined by hierarchical distance conditions $|x_k| > |x_i|$ for descendant nodes.
• The conjecture that $S_A$ is the expansion of a real analytic function $\phi$ is proven for $n \leq 4$ and $n=5$ with vacuum insertion, providing a key consistency check.