padic models as holographic description of String Field

p-adic models as holographic description of String Field Theory inspired models (SFT/p-adic Strings Duality) Irina Aref'eva Steklov Mathematical Institute, RAN, Moscow International Workshop "p -Adic Methods for Modelling of Complex Systems“ Zi. F, Bielefeld University, Bielefeld, Germany April 15– 19, 2013

Outlook • Holography for p-adic effective action SFT(String Field Theory) Inspired Action • SFT/p-adic Duality • SFT Inspired Action for Physical Problems • Cosmology • Quark Gluon Plasma in Heavy Ion Collisions

We String theory(ies) is UV complete theory No UV divergences

SFT action for tachyons (boson string) Kostelecky, Samuel, 1988 ABGKM, review ar. Xiv: 0111208, 2001

Cosmological Models with SFT Inspired Actions String inspired

P-adics non-local models Homogeneous case V. S. Vladimirov

String Field Inspired Nonlocal Models in Cosmology • • Later cosmology w<-1 Inflation steep potential, non-gaussianity • Bouncing solutions • I. A. , astro-ph/0410443 I. A. , L. Joukovkaya, JHEP, 05109 (2005) 087 I. A. , A. Koshelev, JHEP, 07022 (2007) 041 L. Joukovskaya, PR D 76 (2007) 105007; JHEP (2009 ) G. Calcagni, M. Montobbio, G. Nardelli, 0705. 3043; 0712. 2237; Calcagni, Nardelli, 0904. 424 ……. . • N. Barnaby, T. Biswas, J. M. Cline, hep-th/0612230, J. Lidsey, hep-th/0703007; Nunes, Mulryne, 0810. 5471; I. A. , I. Volovich, ar. Xiv: 1103. 0273, works in progress I. A. , L. Joukovskaya, S. Vernov, JHEP 0707 (2007) 087 …….

Holography and duality. Holography: D=5 D=4 D=5 Susskind, 1994 D=3 ‘t Hooft, 1993 Maldacena 1997: Duality: D=4 Gauge theory N=4 YM with SU(Nc) String theory on Ad. S 5 x. S 5 Gravity in Ad. S 5

Simplest Holographic Models. Plane version of GKPW duality Gubser, Klebanov, Polyakov; Witten, 1998 IA, I. Volovich, 1998 Proof

Holographic Models with SFT Inspired Actions

Simplest Holographic Models

GKPW Holographic Model

Nonlocal Holographic Model d=4

p-adic as Holography for SFT inspired Model P-adic action SFT inspired action

Theories with Infinite # of derivatives

Conclusion • P-adic effective models in Ad. S produce the SFT-inspired models on the holograpic boundary of Ad. S

BACKUP slides

Cosmological Singularity • Classical versions of the Friedmann Big Bang cosmological models of the universe contain a singularity at the start of time. Here we restrict ourself with a simple approach by considering the time variable t running over the half-line with regular boundary conditions at t = 0.

Cosmological Daemon I. A. , I. Volovich, ar. Xiv: 1103. 0273 • Nonlocal string field theory equations with infinite number of derivatives. • We use the heat equation method and show that on the half -line in addition to the usual initial data a new arbitrary function (external source) occurs that is called the daemon function. • The daemon function governs the evolution of the universe similar to the Maxwell demon. • In the simplest case the nonlocal scalar field reduces to the usual local scalar field coupled with an external source.

The Heat Equation on the Whole Line

The heat equation of the half-line

Solutions to Linearized Nonlocal Equation on the Half-line • Dirichlet's daemon without source

Linearized Nonlocal Equation on the Half-line • Dirichlet's daemon – one mode Energy and pressure on the solution

Dynamical system with Higgs potential elliptic cosine function of modulus k