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“

ZiF, Bielefeld University, Bielefeld, Germany April 15–19, 2013

Outlook• Holography for p-adic effective action SFT(String Field Theory) Inspired Action

• SFT Inspired Action for Physical Problems

• Cosmology

• Quark Gluon Plasma in Heavy Ion Collisions

• SFT/p-adic Duality

String theory(ies) is UV complete theory

No UV divergences

We ( )

33

4

SFT action for tachyons (boson string)

Kostelecky, Samuel, 1988ABGKM, review arXiv:0111208, 2001

Cosmological Models with SFT Inspired Actions

String inspired

4( )4

V

P-adics non-local models

V.S.Vladimirov

Homogeneous case

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 D76 (2007) 105007; JHEP (2009 )  G. Calcagni, M.Montobbio,G.Nardelli, 0705.3043; 0705.3043; 0712.22370712.2237;; 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, arXiv: 1103.0273, works in progress

I.A., L.Joukovskaya, S.Vernov, JHEP 0707 (2007) 087…….

Holography and duality.

Holography:

D=4

D=5D=5 Susskind, 1994

D=3 ‘t Hooft, 1993

Duality: D=4 Gauge theoryN=4 YM with SU(Nc)

String theoryon AdS5xS5

Gravityin AdS5

Maldacena 1997:

D=3

Simplest Holographic Models. Plane version of GKPW duality

2 21[( ) ( ) ]

2 d

z i

R M

S dzdx

Gubser,Klebanov,Polyakov; Witten, 1998

Proof

IA, I.Volovich, 1998

Holographic Models with SFT Inspired Actions

Simplest Holographic Models

21[( ) ], 0

2 d

z

R M

S dzdx A A

0 0 0 0( ) ( , ) ( )2 2d dM M

c cS dx A dxdy x G x y y

GKPW Holographic Model

1

2 21

1[( ) ( ) ],

2d

z xdAdS

dzdxS

z

4 2 40 0( ln )

d

ren

M

S dx a p p b p

Nonlocal Holographic Model

1

21

1[( ) ], 0

2d

zdAdS

dzdxS A A

z

/2/2 0( , ) ( ) ( ) ,ddz x c z A K z A

2 20 0( ln )

d

ren

M

S dx a A A b A

d=4

p-adic as Holography for SFT inspired Model

2 20 0( ln )

d

ren

M

S dx aA A bA SFT inspired action

P-adic action

Theories with Infinite # of derivatives

Conclusion

• P-adic effective models in AdS produce the SFT-inspired models on the holograpic boundary of AdS

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, arXiv: 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

( )t

The heat equation of the half-line

( ) ( )t

( , )D t

Solutions to Linearized Nonlocal Equation on the Half-line

• Dirichlet's daemon without source

0

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