오비폴드 특이점 및 매듭에 관한 정수체의 p진법 제타함수

Abstract

In this paper, we consider p-adic zeta function of totally real number field and relate it to string partition function on irregular Sasaki-Einstein CY orbifold. We show that in moduli space of N = 1 string vacua, irregular Sasaki-Einstein manifold is generalized attractor points(genrealized CM points) determined by supersingular/irregular prime for supersingular reduction. By Stark-Heegner unit, we recover extremal volume of Sasaki-Einstein CY and sign from conductor which is the order of the Seiberg duality. Then we obtain algebraicity of p-adic zeta function of totally real number field and AdS dual 4d SCFT index(Hilbert series) of irregular Sasaki-Einstein CY manifold, with integral Stark-Heegner unit in Hilbert class field. We relate the special value of p-adic zeta function of totally real number field F at negative argument as virtual Euler characteristic of orbifold Hurwitz space. We analysis the extremal metric from the Heun equation(Painleve 6-th), and relate to integrable system for elliptic surface with torsion Mordell Weil group. We recover integrable system from the cluster transformation of Poisson algebra(path algebra of Sasaki-Einstein quiver) with symplectic double. For irregular Sasaki-Einstein CY, we have p-adic Galois representation which is the torsion global Galois representation in supersingular locus over sufficiently ramified field, which determine Haken covering of pro-p covering of Bianchi manifold with first Betti number 0. By Dehn twist with punctured torus, we consider exotic 4 manifold by Lens space surgery, and show that Lens space L(p2,pq-1) double bridge knot as ramification knot for irrational parameter of Painleve 6-th equation. By Hitchin moduli space of rank 2 vector bundle on P^1-{0,1,a,infty) with parabolic structure a in K, we obtain compactification of moduli space of stability condition of mirror Landau Ginzburg model of orbifold Fano base of irregular Sasaki-Einstein manifold by SL(2; F) Bruhat- Tits cocycle. We also consider modular Chern-Simons invariant as Stark-Heegner unit from p-adic regulator of cup product of two Siegel unit by second Milnor homomorphism. We relate it to Arason invariant of quadratic form as non-torsion mod 2 algebraic cycle, by Galois cohomology of totallyreal number field. We extend this to p-adic L function of real quadratic number field F with odd character for irregular N = 1 vacua with non-zero Mordell-Weil rank. We introduce supersingular transition in moduli of N = 1 vacua from Z/3 at arithmetic infinity,and provide complete description of moduli of N = 1 vacua.