2018/02/01 by Hirohiko Shimada
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Amplitude #Black Holes and Theoretical Physics #Conformal field theory #Conformal map #Congruence subgroup #Farey sequence #Geodesic #Logarithm #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Waves and Solitons #Physics #Piecewise #Pure mathematics #Quantum mechanics #cond-mat.stat-mech #hep-th #math-ph #math.MP
paper · pdf · doi:10.1088/1742-6596/965/1/012036
published as J. Phys.: Conf. Ser. 965 (2018) 012036 · 13 pages, 2 figures. see this version; corrections made; references added
openalex publication_date 2018/02/01 · arxiv created 2018/03/16 · arxiv updated 2018/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The amplitude for the singlet channels in the 4-point function of the fundamental field in the conformal field theory of the 2d O ( n ) model is studied as a function of n . For a generic value of n , the 4-point function has infinitely many amplitudes, whose landscape can be very spiky as the higher amplitude changes its sign many times at the simple poles, which generalize the unique pole of the energy operator amplitude at n = 0. In the stadard parameterization of n by angle in unit of π , we find that the zeros and poles happen at the rational angles, forming a hierarchical tree structure inherent in the Poincaré disk. Some relation between the amplitude and the Farey path, a piecewise geodesic that visits these zeros and poles, is suggested. In this hierarchy, the symmetry of the congruence subgroup Γ(2) of SL (2, ) naturally arises from the two clearly distinct even/odd classes of the rational angles, in which one respectively gets the truncated operator algebras and the logarithmic 4-point functions.