A quartet of fermionic expressions for $M(k,2k\pm 1)$ Virasoro characters via half-lattice paths

Abstract

We derive new fermionic expressions for the characters of the Virasoro minimal models $M(k,2k\pm 1)$ by analysing the recently introduced half-lattice paths. These fermionic expressions display a quasiparticle formulation characteristic of the ${\varphi }_{2,1}$ and ${\varphi }_{1,5}$ integrable perturbations. We find that they arise by imposing a simple restriction on the RSOS quasiparticle states of the unitary models $M(p,p+1)$. In fact, four fermionic expressions are obtained for each generating function of half-lattice paths of finite length $L$, and these lead to four distinct expressions for most characters ${\chi }_{r,s}^{k,2k\pm 1}$. These are direct analogues of Melzer's expressions for $M(p,p+1)$, and their proof entails revisiting, reworking and refining a proof of Melzer's identities which used combinatorial transforms on lattice paths. We also derive a bosonic version of the generating functions of length $L$ half-lattice paths, this expression being notable in that it involves $q$-trinomial coefficients. Taking the $L\to \infty$ limit shows that the generating functions for infinite length half-lattice paths are indeed the Virasoro characters ${\chi }_{r,s}^{k,2k\pm 1}$.

arXiv comment29 pages. v2: minor improvements, references added

References in corpus (21)

• Fermionic Sum Representations for Conformal Field Theory Characters
• Fermionic counting of RSOS-states and Virasoro character formulas for the unitary minimal series M(ν,ν+1). Exact results
• Fermionic Character Sums and the Corner Transfer Matrix
• Rogers-Schur-Ramanujan type identities for the $M(p,p^{\prime })$ minimal models of conformal field theory
• Continued Fractions and Fermionic Representations for Characters of M(p,p') minimal models
• Fermionic solution of the Andrews-Baxter-Forrester model II: proof of Melzer's polynomial identities
• Non compact conformal field theory and the a_2^{(2)} (Izergin-Korepin) model in regime III
• Renormalization group trajectories from resonance factorized S-matrices
• Exact resonance A-D-E S-matrices and their renormalization group trajectories
• New families of flows between two-dimensional conformal field theories
• Fermionic expressions for minimal model Virasoro characters
• Graded Parafermions
• Integrable perturbations of CFT with complex parameter: the $M_{3/5}$ model and its generalizations
• Polynomial Fermionic Forms for the Branching Functions of the Rational Coset Conformal Field Theories $\widehat{su}(2{)}_M\times \widehat{su}(2{)}_N/\widehat{su}(2{)}_{M+N}$}
• The perturbations ${\varphi }_{2,1}$ and ${\varphi }_{1,5}$ of the minimal models $M(p,p^{\prime })$ and the trinomial analogue of Bailey's lemma
• $q$-Trinomial identities
• RG Flows of Non-Unitary Minimal CFTs
• New path description for the M(k+1,2k+3) models and the dual Z_k graded parafermions
• Particles in RSOS paths
• A bijection between paths for the M(p,2p+1) minimal model Virasoro characters
• Half-lattice paths and Virasoro characters

Cited by in corpus (2)

• Fermionic fractional quantum Hall states: A modern approach to systems with bulk-edge correspondence
• One-Dimensional Sums and Finitized Characters of $2\times 2$ Fused RSOS Models