II SUMMER SCHOOL IN MODERN MATHEMATICAL PHYSICS September 1-12, 2002 Kopaonik, (SERBIA) YUGOSLAVIA

1. I.Ya. Aref'eva Steklov Mathematical Institute Super String Field Theory and Vacuum Super String Field Theory II SUMMER SCHOOL INMODERN MATHEMATICAL PHYSICSSeptember 1-12, 2002Kopaonik, (SERBIA) YUGOSLAVIA

2. Why String Field Theory? main motivations: gauge invariance principle behind string interactions non-perturbative phenomena • What we can calculate in String Field Theory? (analog of Higgs and Goldstone Phenomena) Condensation of Tachyon Brane tensions Rolling Tachyon • How we can do calculations in SFT? String Field Theory and Noncommutative Geometry String Field Theory and CFT

3. Local space-time fields String Field Theory Second Quantized String Theory • Infinite number of local fields • String FieldA[X(σ)] - functional, or state inFockspace • Ghosts A[X(σ), c(σ), b (σ)] • Example:

4. Pointwise multiplication of functions Multiplication of matrices Moyal product INTERACTION ? Associative Product of String Fields Examples of associative multiplications:

5. Witten’s String Product Associative Product of String Fields -- Coordinate representation

7. A - string field - BRST charge, derivation of the star algebra - inner product - associative non-commutative product - open string coupling constant SFT Action E.Witten (1986) Gaugetransformations:

8. SFT Action is given

9. Tachyon Condensation in SFT • Bosonic String - Tachyon Kostelecky,Samuel (1989)

10. SFT Sen’s conjecture (1999) Vacuum Energy = Brane Tension Strings Branes

11. = NO OPEN STRING EXCITATIONS CLOSED STRING EXCITATIONS Sen’s conjectures (1999)

12. RollingTachyon Motivations:cosmology,... • Anharmonic oscillator • alpha’ corrections • p-adic strings I.Volovich(1987); P.Frampton, Nishino(1988); Brekke,Freund,Witten(1988), I.A,B.Dragovic,I.Volovich(1988) Moeller, Zwiebach, hep-th/0207107 p-adic and NC space-time I.A.,I.Volovich, 1990 • SFT Sen, Strings2002 I.A, A.Giryavets, A.Koshelev

13. RollingTachyon Spatially homogeneous field configurations: E.O.M. where

14. RollingTachyon Anharmonic oscillator approximation E.O.M.

15. If resonance i.e. Anharmonic oscillator

16. Two regimes: Rolling Tachyon Initial condition near the top Initial condition near the bottom

17. Rolling Tachyon (bosonic case) Initial condition near the top Initial condition near the bottom

18. Alpha ‘ corrections (boson case) • First order Solutions

19. RollingTachyon E.O.M.

20. RollingTachyon E.O.M. X-dependence

21. RollingTachyon E.O.M. Two analogues: i) p-adic strings ii) non-commutative solitons in strong coupling regime Gopakumar, Minwalla,Strominger

22. RollingTachyon Time evolution in the «sliver-tachyon» and p-adic strings p=2,3,5,.. (*) There is no solution such that with BUT this contradiction does not take place for (*) decreasing monotonically in time

23. -- + + … Solutions to SFT E.O.M. Analog of Yang-Feldman eq. Resonance = Sen Problems!!!

24. OUTLOOK String Field Theory L.Bonora Noncommutative Field Theories and (Super) String Field Theories,hep-th/0111208, I.A., D. Belov, A.Giryavets, A.Koshelev,P.Medvedev SuperString Field Theory • Cubic SSFT action 2-nd • Tachyon Condensation in SSFT • RollingTachyon 3-d • Vacuum SuperString Field Theory i)New BRST charge ii) Special solutions - sliver, lump, etc.: algebraic; surface states; Moyal representation