Irina Arefeva Steklov Mathematical Institute, RAS

String Field Theory: from High Energy Physics

to Cosmology

Quantization

ClassicalMechanics

Classical Field

QuantumMechanics

Second quantizationInfinite # of the same particles

n classicalFields

Question: what happens with infinite # of types of classical fields?

….. How to arrange all these fields ?

Looks like а rope, or a string

Quantum Field

3 bodies problem, L.Faddeev

if s<2

New problems:We have but do not have H, wave operators, but we have renormalized S-matrix

Quantization

Experiments gave the Regge asymptotic behavior of the scattering amplitude of hadrons (1957)

Veneziano amplitude

Regge poles

It was proposed by Numbu, Gato, Nielsen, Susskind (1970) to use string to describe infinite number of Regge states

0

1( , ) 2 ' 2 ' cos( ) in

nn

X x p i n en

• First quantized approach, Polyakov's approach• SFT

• M. Kaku and K. Kikkawa, Phys. Rev. D 10 (1974) 1110,1823;• E. Cremmer and J.L. Gervais, Nucl. Phys. B 76 (1974)209;B 90 (1974)

410.• Action=functional of such that it reproduces the Veniziano amplitude• Specific feature -- the string interaction by gluing strings by ends• Green-Schwarz Light-Cone SSFT cancelation of anomalies for SO(32)

• heterotic string

Light-Cone SFT

Wanted(1985): Gauge Invariant Principle behind String Field Interaction

Many attempts:

Various gauge covariant free string actions were given in:

E. Witten, (1985), J.-L. Gervais (1985) L. Baulieu and S. Ouvry, Phys. Lett. B 171 (1986) 57. M. Kaku, Phys. Lett. B 162 (1985) 97; D. Friedan, Chicago University preprint EF 185-27 (1985); T. Banks and M.E. Peskin, SLAC preprint 3740 (1985); A. Neveu, H. Nicolai and P.C. West, Phys. Lett. B 167 (1986) 307; W. Siegel and B. Zwiebach, Berkeley preprint UCBPTH- 85130 (1985);

A.Neveu and P.West have given the gauge-invariant interaction of a bosonic open string at the lowest levels, Phys. Lett. B 168 (1986) 192.

Recent developments • began with Siegel’s formulation of a covariantly gauge-fixed bosonic string

based on Kato and Ogawa’s BRST formalism of a first quantized string W. Siegel, Phys. Lett. B 151 (1985) 391,396 Kato and K. Ogawa, Nucl. Phys. B 212 (1983) 443• Hata, Itoh, Kugo, Kunitomo and Ogawa have constructed the gauge-fixed BRST

invariant field theory for the interacting bosonic string based on the string-ends interaction. Problem with an extra parameter (string length),

Phys. Lett. B 172 (1986) 186 • The gauge-invariant formulation of the classical interacting bosonic open string

related with HIKKO BRST invariant SFT was given in I.A., Volovich,TMF 67(1986) 320,460 = Phys.Lett.182 (1986) 159

Witten’s SFT, 1986

• No extra-parameters, the gluing is different, now this is called Witten's gluing

• Gauge invariant

• Reproduces the Veneziano amplitudes

SuperSFT, 1990• Picture

a=0 AMZ, 1990I.A., Medvedev, Zubarev

PTY and AMZ, 1990,Preitschopf, Thorn, Yost

Non-polynomial action, Berkovits (1995)

SFT E.O.M

• This equation has the form similar to the Chern-Simons

Chern-Simons

• M is 3-manifold, principle G-bundle over M and A is one -form with values in G,

• action

E.O.M.

Th. Flat connections of principal G-bundles over M entirely determined by holonomies around noncontractive cycles on the base M or

Flat connections of principal G-bundles over M are in one to one correspondence with equivalence classes of homomorphism from the fundamental group of M to G up to conjugation, i.e.

Abelian case, QA=0, or dA=0

• Poincare Lemma. On the contractive manifold, all closed forms are exact, i.e.

Closed forms represent cohomology classes(de Rham cohomology).

All solutions of dA=0 are given by the de Rham theory.

Tachyonic (Higgs-Like) Solutions to SFT

• This activity was initiated by the Sen conjectures

• Bosonic string has tachyon.• There is tachyon in the Higgs model as well. In the Higgs model we start from finding the non-zero

expectation value for the Higgs field

Sen's 1-st conjecture:

Numerical Solutions to SFT(2000-2003)

• Bosonic case. Record calculations by Moeller, Rastelli, Zwiebach• Fermionic case,• B(+,-) theory; Berkovits,Sen, Zwiebach (2000) ABKM = IA, Belov,Koshelev, Medvedev (2001)

2005-now M.Schnabl and the following

Schnabl; Okawa; Erler; I.A., Gorbachev, Medvedev, Fuchs,Kroyter; Takahashi

Kishimoto and T. Takahashi; Erler;EllwoodSchnabl; Kiermaier, Zwiebach,…..

Hashimoto, Itzhaki; Gaiotto, Rastelli, Sen, Zwiebach; Kawano, Kishimoto,Takahashi

Main intrigue

T.Erler; R.Gorbachev; D.Grigoriev, P.Khromov;Ellwood

String in non-trivial background

• We do not have SFT in an arbitrary non-trivial background

• we do have in pp-waves, due to R.Metsaev study of the string spectrum in

pp-waves• we try to guess how can look level-truncated

SFT in non-trivial background. (minimal interaction)

Rolling tachyon solutions in level truncated theories

• Rolling solutions in DBI theories,....• Rolling solutions in p-adic theories• p=2 no-go theorems by Moeller and Zwiebach• p=3 existence theorems by Vladimirov,Ya.Volovich

Motivations

I.A.,Joukovskaya, Koshelev;Ya.Volovich2 2 32

( 1) e

Rolling tachyon solutions in level truncated theories in FRW metric

• Sen in 2003 proposed to use the bosonic string tachyon to

describe pressure less Dark Matter.

• Tachyon related with nonBPS brane has been considers also as a model of the dynamical Dark Energy (IA,2005)

Talks by L.Joukovskaya, A.Koshelev, S.Vernov

No Conclusion!

M Ф