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Reconstruction of vertex algebras in even higher dimensions

Bojko N. Bakalov, Nikolay M. Nikolov

Abstract

Vertex algebras in higher dimensions correspond to models of quantum field theory with global conformal invariance. Any vertex algebra in dimension D admits a restriction to a vertex algebra in any lower dimension and, in particular, to dimension one. In the case when D is even, we find natural conditions under which the converse passage is possible. These conditions include a unitary action of the conformal Lie algebra with a positive energy, which is given by local endomorphisms and obeys certain integrability properties.

Reconstruction of vertex algebras in even higher dimensions

Abstract

Vertex algebras in higher dimensions correspond to models of quantum field theory with global conformal invariance. Any vertex algebra in dimension D admits a restriction to a vertex algebra in any lower dimension and, in particular, to dimension one. In the case when D is even, we find natural conditions under which the converse passage is possible. These conditions include a unitary action of the conformal Lie algebra with a positive energy, which is given by local endomorphisms and obeys certain integrability properties.
Paper Structure (18 sections, 19 theorems, 113 equations)

This paper contains 18 sections, 19 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Vertex Algebras in Dimension $1$ and $D$
  3. Vertex algebras in dimension $1$
  4. The definition of vertex algebras in dimension $D$
  5. Restriction to lower dimensions
  6. Conformal vertex algebras
  7. Local Endomorphisms of Vertex Algebras
  8. Local endomorphisms
  9. Local action of $\mathfrak{c}_D$
  10. The Reconstruction Theorem
  11. Formulation of the theorem
  12. Reconstructing the state-field correspondence
  13. Parity property
  14. The case $D=2$
  15. Pole bounds
  16. ...and 3 more sections

Key Result

Theorem 3.1

For a vertex algebra $V$ and a linear operator $X$ of $V$, the following two conditions are equivalent$:$ In either of these two cases, one has $X(\mathrm z)=e^{-\mathop{\mathrm{ad}}\nolimits(\mathrm z\cdot\mathrm T)} X$ and

Theorems & Definitions (47)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1
  • Remark 3.1
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Proposition 3.3
  • ...and 37 more