Charged free fermions, vertex operators and the classical theory of conjugate nets

1998/03/20 by Adam Doliwa, Manuel Manas, Manuel Mañas +6 · 5 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Bilinear interpolation #Diagonal #Discrete mathematics #Fermion #Geometry #Integrable system #Laplace transform #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Physics #Pure mathematics #Quantum mechanics #Vertex (graph theory) #hep-th #math.DG #nlin.SI #solv-int

paper · pdf · doi:10.1088/0305-4470/32/7/010

published as J.Phys.A32:1197-1216,1999 · 28 pages, 3 Postscript figures

arxiv created 1998/03/20 · openalex publication_date 1999/01/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We show that the quantum field theoretical formulation of the -function theory has a geometrical interpretation within the classical transformation theory of conjugate nets. In particular, we prove that (i) the partial charge transformations preserving the neutral sector are Laplace transformations, (ii) the basic vertex operators are Lévy and adjoint Lévy transformations and (iii) the diagonal soliton vertex operators generate fundamental transformations. We also show that the bilinear identity for the multicomponent Kadomtsev-Petviashvili hierarchy becomes, through a generalized Miwa map, a bilinear identity for the multidimensional quadrilateral lattice equations.

Citations

• Multidimensional quadrilateral lattices are integrable
• Monodromy perserving deformation of linear ordinary differential equations with rational coefficients. II

Cited by

• Lagrangian formulation of the Darboux system
• Desargues maps and the Hirota–Miwa equation
• Darboux transformations for linear operators on two-dimensional regular lattices
• The τ-function of the quadrilateral lattice
• On τ-Function of Conjugate Nets