1. p-adic Strings: Thermal Duality & the Cosmological Constant Tirthabir Biswas Loyola University, New Orleans PRL 104, 021601 (2010) JHEP 1010:048, (2010) PRD 82:085028, (2010) with J. Kapusta and J.A. R. Cembranos.

2. The N-point tree amplitudes of the open string can be generated from a non-local Lagrangian of a single scalar field. Volovich, Brekke, Freund, Olson, Witten, Frampton, Okada, late 80’s open string coupling prime number string tension p-adic Action

3. Generalizations  Nonlocal Infinite derivative Actions  p-adic theories  Strings on Random lattice (Douglas&Shenker,Gross&Migdal, 1990) Regge trajectories (TB, Siegel, Grisaru 2004)  String Field Theories  Related cousins appear in Noncommutative Field theory  Theory of unparticles

4. What can we gain?  Insights into string theory  Hagedorn physics  Brane Physics  Applications to Cosmology  Novel kinetic energy dominated non-slow-roll inflationary mechanisms (TB, Barnaby, Cline, 2006), large nongaussianities (Barnaby,Cline, 2007)  Dark Energy (Arefeva et.al.)  Thermal Cosmology in the Early Universe  Applications to Particle Physics [Moffat et.al.]

5. Interesting Properties (?)  Usually higher derivative theories are plagued by ghosts: But padic type theories have no extra states!  Initial value problem may be well defined:  Studied by Russian mathematicians, 2 degrees of freedom for every pole, rigorously established for free theories. [Barnaby, Kamran]  Diffusion equation formulation in one higher dimension suggest finite number of IC’s. [Calcagni, Nardelli...]  No perturbative states, free theory is trivial, quantum contributions only arise when interactions are present.  Field Equations can be recast as integral equations and hence numerical progress/tests can be made.

6. Rescale the fields to put the action in the form with dimensionful coupling constant and non-local propagator Thermal Field Theory

7. p=3 2-loop & Thermal Duality  Compute Feynman diagrams

8.  Partition Function  Analogous to t-duality  This was conjectured using “real string theory” arguments (momentum modes and winding modes)  It was also conjectured that non-perturbative corrections will violate the duality In the p-adic case, this happens at higher loops

10.  Thermodynamics Low Temperature: Behaves as pressureless dust (Deo et. al.,Vafa,Tseythlin, Brandenberger) High Temperature (Atick & Witten): Behaves as stiff fluid

11.  Vacuum Energy  Vacuum energy appears only at 2-loops  It is -ve  Ghost appears due to self-energy  Adding counter-term makes the vacuum energy positive and hierarchically suppressed

12.  Planck Mass & Cosmological Constant hierarchical suppression known dimensionless function of p volume of extra-dimensional compactified space

13. p=7

14. Necklace diagrams and sunset diagrams Low T High T (Atick & Witten) OK Higher Loops

15. Solitons at Finite temperature Classical Equation:

16. even solution p=3

17. odd solution p=3

18. Even solitons are important for high temperatures

19. Future Applications  Thermal fluctuations may dominate the early phase of inflation…any signatures (?)  Thermal solitons suggest existence of branes in extra compact directions  Exponential cut-off acts like a regularization parameter – can it be made physical?  What about bound states?  What about Gravity? (TB,Mazumdar,Siegel’05)

20. In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of absolute value. Wikipedia String Theories over p-adic Fields* Quantum fields valued in the field of complex numbers Space-time coordinates valued in the field of real numbers World-sheet coordinates valued in the field of p-adic numbers *Freund & Olson (1987), Freund & Witten (1987)

21. vacuum energy: low T: high T: no particle degrees of freedom

22. Solitons at Finite Temperature Soliton solutions in Euclidean space at zero temperature were found by Brekke, Freund, Olson & Witten (1988).

23. Key Results There are no particle degrees of freedom so there is no one-loop contribution to the partition function The lowest order contribution arises from interactions A counter-term must be added to avoid the appearance of a ghost in a loop expansion which has the consequence that … The vacuum energy is positive and hierarchically suppressed Perturbation theory breaks down at a temperature of order Soliton solutions exist at all temperatures and become important when