p-adic models as holographic description of

1. 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

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

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

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

5. Cosmological Models with SFT Inspired Actions String inspired

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

7. 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, PRD76 (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, arXiv: 1103.0273, • works in progress • I.A., L.Joukovskaya, S.Vernov, JHEP 0707 (2007) 087 • …….

8. Holography and duality. Holography: D=5 Susskind, 1994 D=3 ‘t Hooft, 1993 String theory on AdS5xS5 D=5 D=4 D=3 Gravity in AdS5 Maldacena 1997: Duality: D=4 Gauge theory N=4 YM with SU(Nc)

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

10. Holographic Models with SFT Inspired Actions

11. Simplest Holographic Models

12. GKPW Holographic Model

13. Nonlocal Holographic Model d=4

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

15. Theories with Infinite # of derivatives

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

17. BACKUP slides

18. 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.

19. Cosmological DaemonI. A., I.Volovich, arXiv: 1103.0273 • Nonlocalstringfieldtheoryequationswithinfinitenumberofderivatives. • Weusetheheatequationmethodandshowthatonthehalf-lineinadditiontotheusualinitialdataanewarbitraryfunction (externalsource) occursthatiscalledthedaemonfunction. • Thedaemonfunctiongovernstheevolutionoftheuniversesimilartothe Maxwelldemon. • Inthesimplestcasethenonlocalscalarfieldreducestotheusuallocalscalarfieldcoupledwithanexternalsource.

20. The Heat Equation on the Whole Line

21. The heat equation of the half-line

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

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

24. Dynamical system with Higgs potential elliptic cosine function of modulus k