Rolling Tachyon and Vacuum SuperString Field Theory

I.Ya. Aref'evaSteklov Mathematical Institute

Based on : I.A., D. Belov, A.Giryavets, A.Koshelev, hep-th/0112214, hep-th/0201197, hep-th/0203227, hep-th/0204239

and

OUTLOOK

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

• Cubic SSFT action

• Vacuum SuperString Field Theory

• Conclusion

• Tachyon Condensation in SSFT

i) Field theory (anharmonic oscillator)

ii) correctionsiii) p-adic stringsiv) SFT

• Rolling Tachyon

Tachyon Condensation in SFT

• Bosonic String - Tachyon

• Tachyon Condensation in SFT

• Tachyon in GSO ( - ) sector of NS string

• Level truncation

V

it

Kostelecky,Samuel (1989)

String Field Theory on a non-BPS brane

I.A.,Belov,Koshelev,Medvedev(2001)

3ˆ BQQ

]|ˆ3

2ˆ|[4

1222

0

AAAYAQAYTrg

S

23 iAAA

Parity GSO odd +

even -A

A

E.Witten (1986)

I.A., Medvedev, Zubarev (1990)Preitschopf, Thorn, Yost (1990)

2Y

Vertex operators in pictures –1 and 0

N.B.,Sen,Zwiebach (2000)Berkovits (1995))(z

I.A., Belov, Koshelev, P.M. (2001)6,...1,,, ivtu i

+2

s-3/2

r+1

t-1/2

-u+0

Picture 0Picture -1NameGSOLevel10 L F)1(

iv

ce

2 ecc

ec

22

,,

cec

ceTF

c

e cc,

eebcecTF ,),(,

eTccT

cTcTc

F

B

,,

,,,2

2

Tachyon Condensation in SSFT

242/12

0

)1(

4

1

1024

811tt

gV p

242/12

0

)4(

4

1

69120

50531tt

gV

p

257.1)1( ct

308.1)4( ct

FAQ: cubic unbounded 24

A.: Auxiliary fields

23

222 )( ag )(21 222 ag

u, t fields

]3

1

4

1[

1 22

222/12

0

)1( uttug

Lp

33

4

242/12

0

)1(

4

1

1024

811tt

gV p

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

Our calculations:

97.5%

105.8%

Rolling Tachyon

• Anharmonic oscillator• Alpha ‘ corrections• p-adic strings• SFT (for bosonic string Sen,hep-th/0203211)• SSFT for non-BPS branes

Anharmonic oscillator

tdtjttGtxtx

tjtxm

ret

dtd

)(),()()(

),()()(

0

202

2

)(3 txj ...cos 0433

0 tmaxIf resonance

tmax 00 cos

0m0

2

83

0 mam tmax m

a )cos(0

2

83

00i.e.

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( 2log231log4332

2

3 dtdx

dtd txtxtxx

Solutions

Alpha ‘ corrections (non-BPS case)

• First order

)1(

)(2

9

log20

1

)()(

2

9

log203

922

9

log20

249

log204249

log20

349

2

2

2

442

2

4

)()(

)()()()()()1(

x

tx

dtdx

x

txtx

dtd

dtdx

dtd

tx

txtxtxtxx

Solutions

Solutions to SFT E.O.M.

0 AAQA00 Ab 0)(00 AAbALSiegel gauge

Usual pert.theory

AAAA Lb 0

0

0

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

0

0

0

0

0

0

0 AAAAAAAAAA Lb

Lb

Lb

Lb

= -- + + …Resonance

Sen,hepth/020715

)(0

0

0 AAAA Lbnew newnew AcAAAA 000 ),( 0),( 0 newAA

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

0

0

0

0

0

0

0 newnewLbnewnewnewnew

Lb

Lbnewnew

Lbnew AAAAAAAAAA

tinew aeA 0 02)4(22 am Problems!!!

Solutions to SSFT E.O.M.

0 AAQA No picture changing operatorNS sector

)(0

0

0 AAAA Lbnew newnew AcAAAA 000 ),( 0),( 0 newAA

tinew aeA 0 02)4(22 am Problems!!!

02)4(2 anmdefines the fold AMZ,1990

NO OPEN STRING EXCITATIONS

VSFT

Vacuum String Field Theory on a non-BPS brane

I.A., Belov, Giryavets (2002)

23 iAAA

oddeven

evenodd

QQ

QQQ

B

BB Q

QQ

0

0ˆ

AAAYA |3

22 AYTr

gS |[

4

122

0

Q

)(

)()( 221

iQ

dzzbicQ

even

iodd

Structure of new Q

)()(

)()(

)()()(

1

2211

iiQ

iiQ

dzzbicicQ

even

ieven

iiodd

BRST

BRST

Q

QQ

0

0

,...}{ 0AQBRST Q

solution to E.O.M0A

SFT in the background field

Q redef.field-Q

A

A

000 , AAA

AAA 0

Ohmori

Tests

Solution to VSFT E.O.M

E.O.M.

Analog of Noncommutative Soliton in Strong Coupling Limit

Gopakumar, Minwalla,Strominger

0 AAAQ 0 mmm AAA

Methods of solving

• Algebraic method

AAA

• Surface states method

• Moyal representation

• Half-strings

I.Bars,M.Douglas,G.Moore

Algebraic Method

I.A., Giryavets, Medvedev;Marino, Schiappa

Identities for squeezed states

Bosonic sliver

Rastelli, Sen, Zwiebach; Kostelecky, Potting...

)1('))()(('' )()( ii eiceic

Twisted SuperSliver

• Superghost twisted sliver

00'0'0' )0(1

ecUUbcfbcf

• Superghost twisted sliver equation

~))()((

~~ii

0,'000)0(0)(~

))(( )0(1

ecUiYUiY

bcfbcf

• Sliver with insertion

• Picture changing

0)0(0

~ bcfU

)1()()( iYiY

Sliver in the Moyal representation

Identity

Sliver

Conclusion

• What we know

• What we get

• Open problems

What we have got in cubic SSFT

Tachyon condensation

Rolling tachyon near the top

Vacuum SSFT and some solutions

What we know

Two sets of basis:

SFT proposes a hard, but a surmountable way to get answers concerning non-perturbative phenomena

i) related with spectrum of free string

ii) related with "strong coupling “ regime (may be suitable for study VSFT)

Open Problems

More tests for checking validity of VSSFT

Other solutions (lump, kink solutions); especially with time dependence

Classification of projectors in open string field algebra and its physical meaning

Use the Moyal basis to construct the tachyon condensate and other solutions

Closed string excitations in VSSFT