II SUMMER SCHOOL INII SUMMER SCHOOL INMODERN MATHEMATICAL PHYSICSMODERN MATHEMATICAL PHYSICS

September 1-12, 2002Kopaonik, (SERBIA) YUGOSLAVIA

and Vacuum Super String Field Theory

Super String Field Theory

I.Ya. Aref'evaSteklov Mathematical Institute

• Why

main motivations: gauge invariance principle behind string interactions non-perturbative phenomena

String Field Theory?

• What we can calculate in String Field Theory?

• How we can do calculations in SFT?

String Field Theory and Noncommutative Geometry

Condensation of Tachyon (analog of Higgs and Goldstone Phenomena)

Rolling Tachyon

Brane tensions

String Field Theory and CFT

String Field Theory Second Quantized String Theory

• Infinite number of local fields

• String Field A[X(σ)] - functional, or state in Fock space

• Example:

• Ghosts A[X(σ), c(σ), b (σ)]

n

ppdpA ,)( nn

Local space-time fields

INTERACTION ?

Associative Product of String Fields

?3)(),(),( bcxA

xxxgxf

xxixgf

)()(exp))((

)()( xgxf

jlijj

KM

Examples of associative multiplications:

Pointwise multiplication of functions

Multiplication of matrices

Moyal product

Witten’s String Product

Coordinate representation AxxA )()(

20

22

2121

)()()]()([

)]([)]([)]()[(

dydxyx

yAxAzAA

2)(

20)(

)(fory

forxz

Associative Product of String Fields --

SFT Action E.Witten (1986)

Gauge transformations:

A - string field

BQ - BRST charge, derivation of the star algebra

- inner product

- open string coupling constant0g- associative non-commutative product

SFT Action is given

Tachyon Condensation in SFT

• Bosonic String - Tachyon

Kostelecky,Samuel (1989)

int2

20

)1)(()(211 Spptptdp

gS

332

0int

~311 tdx

gS

textt log~

334

Sen’s conjecture (1999)

0gSFT

braneT

E

E

Branes

Strings

Brane Tension=Vacuum Energy

Sen’s conjectures (1999)

E braneT=

NO OPEN STRING EXCITATIONS

CLOSED STRING EXCITATIONS

• Anharmonic oscillator• alpha’ corrections• p-adic strings

Rolling Tachyon

p-adic and NC space-time I.A.,I.Volovich, 1990

Motivations:cosmology,...

Moeller, Zwiebach, hep-th/0207107

I.Volovich(1987); P.Frampton, Nishino(1988);Brekke,Freund,Witten(1988),I.A,B.Dragovic,I.Volovich(1988)

Sen, Strings2002

I.A, A.Giryavets, A.Koshelev

• SFT

Rolling Tachyon

)~31

21

2(1 3

32

1320

ttttdxg

S

tt logexp~

334

Spatially homogeneous field configurations: )()( 0xxt tx 0

E.O.M.0)(1 2

32

2

tt DDdtd

where

2

2

lnexpdtdDt

Rolling Tachyon

Anharmonic oscillator approximation

E.O.M.01 2

32

2

dtd

1lnexp 2

2

dtdDt

Anharmonic oscillator

tdtjttGtxtx

tjtxm

ret

dtd

)(),()()(

),()()(

0

202

2

)(3 txj ...cos 03

433

0 tmaxIf resonance

tmax 00 cos

0m0

2

83

0 mam tmax m

a )cos(0

2

83

00i.e.

)()()( 3202

2 txtxmdtd tmax m

a )cosh(0

2

83

00

Rolling Tachyon

Initial condition near the top Initial condition near the bottom

tmax ma )cosh(0

2

83

00tmax m

a )(2cos0

2

83

00

Two regimes:

Rolling Tachyon (bosonic case)

Initial condition near the top Initial condition near the bottom

Alpha ‘ corrections (boson case)

• First order

0)()()()()1( 2log221log4332

2

3 dtdx

dtd txtxtxx

Solutions33

4

Rolling Tachyon

E.O.M.0)(1 2

32

2

tt DDdtd

)(lnexp)( 2

2

tdtdtDt

tdtea

t att

a

)(

21)( 2

)( 2

23

2 1)1( 2

2 dt

d

0ln2 a

Rolling Tachyon

E.O.M.

X-dependence0)(1 2

32

2

xx DDdxd

)()(lnexp)( 2

2

xxdxdxD xax

tdtea

t att

a

)(

21)( 2

)( 2

0)(1 223

2

2

dtd

2)(

0)(123

2/1 a ln2 a

Rolling Tachyon

E.O.M.0)(1

232/1 a

Two analogues:

i) p-adic strings

ii) non-commutative solitons in strong coupling regime

0)(ln pp

0 0)(1 223

a

Gopakumar, Minwalla,Strominger

Rolling Tachyon

0)(2/1 0)( p

Time evolution in the «sliver-tachyon» and p-adic strings

p=2,3,5,..

decreasing monotonically in time

,1)( There is no solution such that

0)( )(twith BUT this contradictiondoes not take place for (*)

(*)

Solutions to SFT E.O.M.

0 AAQA AAAA Lb 0

00

...))(())(()( 000000000 0

0

0

0

0

0

0

0 AAAAAAAAAA Lb

Lb

Lb

Lb

= -- + + …Resonance

tinew aeA 0 02)4(22 am

Problems!!!

Sen

Analog of Yang-Feldman eq.

0A

000 AL

OUTLOOK

i) New BRST chargeii) Special solutions - sliver, lump, etc.: algebraic; surface states; Moyal representation

• Cubic SSFT action

• Vacuum SuperString Field Theory

• Tachyon Condensation in SSFT

• Rolling Tachyon

String Field Theory L.Bonora

SuperString Field Theory

2-nd

3-d

Noncommutative Field Theories and (Super) String Field Theories, hep-th/0111208,I.A., D. Belov, A.Giryavets, A.Koshelev,P.Medvedev