Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Coset Constructions of Logarithmic (1,p)-Models

Thomas Creutzig, David Ridout, Simon Wood

TL;DR

The paper constructs coset realizations of the logarithmic (1,p) models M(p) and W(p) by tying Feigin–Semikhatov W^(2)_n algebras to a rank-two lattice screening framework. It shows that W(p) and M(p) can be obtained as commutants inside kernels of screenings, with a concrete B_p subalgebra whose OPEs align with W^(2)_{p−1} at level −(p−1)^2/p (proved for p≤5 and conjectured generally), enabling explicit coset decompositions for p=2,3. A key part of the approach is the identification of Howe pairs and the construction of surjective homomorphisms ω: W^(2)_{p−1} → B_p, which in turn yield realizations of M(p) as commutants and W(p) as cosets inside the screening kernel. The work provides detailed branching function calculations for p=2 and p=3, decomposing ŜL(2) modules into lattice and singlet/triplet characters and demonstrating how M(p) and W(p) characters arise from these decompositions. Overall, the paper deepens the structural understanding of (1,p) logarithmic models by connecting their representation theory to Feigin–Semikhatov algebras, coset constructions, and explicit branching data.

Abstract

One of the best understood families of logarithmic conformal field theories is that consisting of the (1,p) models (p = 2, 3, ...) of central charge c_{1,p} = 1 - 6 (p-1)^2 / p. This family includes the theories corresponding to the singlet algebras M(p) and the triplet algebras W(p), as well as the ubiquitous symplectic fermions theory. In this work, these algebras are realized through a coset construction. The W^(2)_n algebra of level k was introduced by Feigin and Semikhatov as a (conjectured) quantum hamiltonian reduction of affine sl(n)_k, generalising the Bershadsky-Polyakov algebra W^(2)_3. Inspired by work of Adamovic for p=3, vertex algebras B_p are constructed as subalgebras of the kernel of certain screening charges acting on a rank 2 lattice vertex algebra of indefinite signature. It is shown that for p <= 5, the algebra B_p is a homomorphic image of W^(2)_{p-1} at level -(p-1)^2 / p and that the known part of the operator product algebra of the latter algebra is consistent with this holding for p>5 as well. The triplet algebra W(p) is then realised as a coset inside the full kernel of the screening operator, while the singlet algebra M(p) is similarly realised inside B_p. As an application, and to illustrate these results, the coset character decompositions are explicitly worked out for p=2 and 3.

Coset Constructions of Logarithmic (1,p)-Models

TL;DR

The paper constructs coset realizations of the logarithmic (1,p) models M(p) and W(p) by tying Feigin–Semikhatov W^(2)_n algebras to a rank-two lattice screening framework. It shows that W(p) and M(p) can be obtained as commutants inside kernels of screenings, with a concrete B_p subalgebra whose OPEs align with W^(2)_{p−1} at level −(p−1)^2/p (proved for p≤5 and conjectured generally), enabling explicit coset decompositions for p=2,3. A key part of the approach is the identification of Howe pairs and the construction of surjective homomorphisms ω: W^(2)_{p−1} → B_p, which in turn yield realizations of M(p) as commutants and W(p) as cosets inside the screening kernel. The work provides detailed branching function calculations for p=2 and p=3, decomposing ŜL(2) modules into lattice and singlet/triplet characters and demonstrating how M(p) and W(p) characters arise from these decompositions. Overall, the paper deepens the structural understanding of (1,p) logarithmic models by connecting their representation theory to Feigin–Semikhatov algebras, coset constructions, and explicit branching data.

Abstract

One of the best understood families of logarithmic conformal field theories is that consisting of the (1,p) models (p = 2, 3, ...) of central charge c_{1,p} = 1 - 6 (p-1)^2 / p. This family includes the theories corresponding to the singlet algebras M(p) and the triplet algebras W(p), as well as the ubiquitous symplectic fermions theory. In this work, these algebras are realized through a coset construction. The W^(2)_n algebra of level k was introduced by Feigin and Semikhatov as a (conjectured) quantum hamiltonian reduction of affine sl(n)_k, generalising the Bershadsky-Polyakov algebra W^(2)_3. Inspired by work of Adamovic for p=3, vertex algebras B_p are constructed as subalgebras of the kernel of certain screening charges acting on a rank 2 lattice vertex algebra of indefinite signature. It is shown that for p <= 5, the algebra B_p is a homomorphic image of W^(2)_{p-1} at level -(p-1)^2 / p and that the known part of the operator product algebra of the latter algebra is consistent with this holding for p>5 as well. The triplet algebra W(p) is then realised as a coset inside the full kernel of the screening operator, while the singlet algebra M(p) is similarly realised inside B_p. As an application, and to illustrate these results, the coset character decompositions are explicitly worked out for p=2 and 3.

Paper Structure

This paper contains 18 sections, 16 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. $\mathcal{W}(p)$ and $\mathcal{M}(p)$ theories
  3. The free boson or Heisenberg vertex operator algebra
  4. The lattice vertex operator algebra $\mathcal{V}(p)$
  5. Screening operators and the singlet and triplet algebras
  6. Coset constructions for $\mathcal{M}(p)$ and $\mathcal{W}(p)$
  7. The Feigin-Semikhatov algebras $W_n^{(2)}$
  8. The triplet algebra as a coset
  9. A vertex algebra homomorphism
  10. The singlet as a commutant subalgebra
  11. Branching functions for $\mathcal{B}_2$ and $\mathcal{B}_3$
  12. $\widehat{\mathfrak{sl}} \left( 2 \right)$ at level $k$
  13. Branching functions for $\mathcal{B}_2$ and $\widehat{\mathfrak{sl}} \left( 2 \right)$ at level $k=-1/2$
  14. Branching functions and the singlet theory
  15. Branching functions and the triplet theory
  16. ...and 3 more sections

Key Result

Theorem 1

For $p=2,3,4,5$, the algebra $\mathcal{B}_p$ is a quotient of $W^{(2)}_{p-1}$ at level $-(p-1)^2/p$. In general, comparing operator product algebras is consistent with the conjecture that $\mathcal{B}_p$ is a quotient of $W^{(2)}_{p-1}$ at level $-(p-1)^2/p$ for all $p$,

Theorems & Definitions (29)

  • Theorem
  • Definition 1
  • Theorem
  • Definition 2
  • Conjecture 1
  • Proposition 1: Feigin-Semikhatov FS
  • Proposition 2: Feigin-Semikhatov FS
  • Proposition 3
  • proof
  • proof
  • ...and 19 more