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

2. Why What How 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. String Field Theory  Second Quantized String Theory
Infinite number of local fields
String Field A[X(σ)] - functional, or state in Fock space
Local space-time fields
Ghosts A[X(σ), c(σ), b (σ)]
Example:

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

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

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

8. SFT Action is given

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

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

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

12. Rolling Tachyon 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/
p-adic and NC space-time
I.A.,I.Volovich, 1990
SFT
Sen, Strings2002
I.A, A.Giryavets, A.Koshelev

13. Rolling Tachyon Spatially homogeneous field configurations: E.O.M.
where

14. Rolling Tachyon
Anharmonic oscillator approximation
E.O.M.

15. Anharmonic oscillatorIf
resonance
i.e.

16. Rolling Tachyon Two regimes: 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. Rolling Tachyon
E.O.M.

20. Rolling Tachyon
E.O.M.
X-dependence

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

22. Rolling Tachyon
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. Sen Problems!!! Analog of Yang-Feldman eq. --+
+ …
Resonance =
Sen
Problems!!!

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