Exceptional structure of the dilute A${}_3$ model: E${}_8$ and E${}_7$ Rogers--Ramanujan identities

Abstract

The dilute A${}_3$ lattice model in regime 2 is in the universality class of the Ising model in a magnetic field. Here we establish directly the existence of an E${}_8$ structure in the dilute A${}_3$ model in this regime by expressing the 1-dimensional configuration sums in terms of fermionic sums which explicitly involve the E${}_8$ root system. In the thermodynamic limit, these polynomial identities yield a proof of the E${}_8$ Rogers--Ramanujan identity recently conjectured by Kedem {\em et al}. The polynomial identities also apply to regime 3, which is obtained by transforming the modular parameter by $q\to 1/q$. In this case we find an A$_1\times\mbox{E}_7$ structure and prove a Rogers--Ramanujan identity of A$_1\times\mbox{E}_7$ type. Finally, in the critical $q\to 1$ limit, we give some intriguing expressions for the number of $L$-step paths on the A${}_3$ Dynkin diagram with tadpoles in terms of the E${}_8$ Cartan matrix. All our findings confirm the E${}_8$ and E${}_7$ structure of the dilute A${}_3$ model found recently by means of the thermodynamic Bethe Ansatz.

9 pages, 1 postscript figure

References in corpus (5)

• Fermionic Sum Representations for Conformal Field Theory Characters
• Fermionic Quasi-Particle Representations for Characters of $(G^{(1)}{)}_1\times (G^{(1)}{)}_1ø(G^{(1)}{)}_2$
• Fermionic Character Sums and the Corner Transfer Matrix
• Order Parameters of the Dilute A Models
• Lattice Ising model in a field: E${}_8$ scattering theory

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