Conformal Field Theories, Representations and Lattice Constructions

Abstract

An account is given of the structure and representations of chiral bosonic meromorphic conformal field theories (CFT's), and, in particular, the conditions under which such a CFT may be extended by a representation to form a new theory. This general approach is illustrated by considering the untwisted and $Z_2$-twisted theories, $H(\Lambda )$ and $\tilde{H}(\Lambda )$ respectively, which may be constructed from a suitable even Euclidean lattice $\Lambda$. Similarly, one may construct lattices ${\Lambda }_C$ and ${\tilde{\Lambda }}_C$ by analogous constructions from a doubly-even binary code $C$. In the case when $C$ is self-dual, the corresponding lattices are also. Similarly, $H(\Lambda )$ and $\tilde{H}(\Lambda )$ are self-dual if and only if $\Lambda$ is. We show that $H({\Lambda }_C)$ has a natural ``triality'' structure, which induces an isomorphism $H({\tilde{\Lambda }}_C)\equiv \tilde{H}({\Lambda }_C)$ and also a triality structure on $\tilde{H}({\tilde{\Lambda }}_C)$. For $C$ the Golay code, ${\tilde{\Lambda }}_C$ is the Leech lattice, and the triality on $\tilde{H}({\tilde{\Lambda }}_C)$ is the symmetry which extends the natural action of (an extension of) Conway's group on this theory to the Monster, so setting triality and Frenkel, Lepowsky and Meurman's construction of the natural Monster module in a more general context. The results also serve to shed some light on the classification of self-dual CFT's. We find that of the 48 theories $H(\Lambda )$ and $\tilde{H}(\Lambda )$ with central charge 24 that there are 39 distinct ones, and further that all 9 coincidences are accounted for by the isomorphism detailed above, induced by the existence of a doubly-even self-dual binary code.

65 pages

References in corpus (1)

• Continuous Symmetries of Lattice Conformal Field Theories and their $Z_2$-Orbifolds

Cited by in corpus (44)

• An Introduction to Conformal Field Theory
• Constraints on extremal self-dual CFTs
• Duality Defects in $E_8$
• Genus Two Partition Functions and Renyi Entropies of Large c CFTs
• Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I
• Quadratic spaces and holomorphic framed vertex operator algebras of central charge 24
• Non-rational Narain CFTs from codes over $F_4$
• Orbifold Constructions and the Classification of Self-Dual c=24 Conformal Field Theories
• Narain CFTs and error-correcting codes on finite fields
• Conformal Field Theories with Sporadic Group Symmetry
• Fermionic CFTs from classical codes over finite fields
• Narain CFTs and Quantum Codes at Higher Genus
• Dimension Formulae and Generalised Deep Holes of the Leech Lattice Vertex Operator Algebra
• Classical Codes and Chiral CFTs at Higher Genus
• Systematic Orbifold Constructions of Schellekens' Vertex Operator Algebras from Niemeier Lattices
• Holographic description of Narain CFTs and their code-based ensembles
• Relation between spectra of Narain CFTs and properties of associated boolean functions
• Monstrous branes
• Representation theory in chiral conformal field theory: from fields to observables
• Dimension Formulae in Genus Zero and Uniqueness of Vertex Operator Algebras
• The Genus Two Partition Function for Free Bosonic and Lattice Vertex Operator Algebras
• Third and Higher Order NFPA Twisted Constructions of Conformal Field Theories from Lattices
• Vertex Ring-Indexed Lie Algebras
• Global Symmetries, Code Ensembles, and Sums Over Geometries
• Reverse orbifold construction and uniqueness of holomorphic vertex operator algebras
• Lectures on conformal field theory and Kac-Moody algebras
• Rational Generalised Moonshine from Abelian Orbifoldings of the Moonshine Module
• New meromorphic CFTs from cosets
• General Genus Zhu Recursion for Vertex Operator Algebras
• The automorphism group of the Z_2-orbifold of the Barnes-Wall lattice vertex operator algebra of central charge 32
• Non-Chiral Vertex Operator Algebra Associated To Lorentzian Lattices And Narain CFTs
• On the Uniqueness of the Twisted Representation in the Z_2 Orbifold Construction of a Conformal Field Theory from a Lattice
• Transformations among large c conformal field theories
• Fermionic CFTs from topological boundaries in abelian Chern-Simons theories
• Bimodule and twisted representation of vertex operator algebras
• Bootstrapping Chiral CFTs at Genus Two
• Lattices, Vertex Algebras and Modular Categories
• Intertwiners in Orbifold Conformal Field Theories
• Continuous Symmetries of Lattice Conformal Field Theories and their $Z_2$-Orbifolds
• Beauty And The Beast Part 2: Apprehending The Missing Supercurrent
• Reduction cohomology of Riemann surfaces
• Automorphism groups of the holomorphic vertex operator algebras associated with Niemeier lattices and the $-1$-isometries
• Cohomology of Jacobi forms
• Optimal Narain CFTs from Codes