Transcript from High Energy Physics to Cosmology

String Field Theory:
from High Energy Physics
to Cosmology
Irina Arefeva
Steklov Mathematical Institute, RAS
…..
Question: what happens with infinite # of
types of classical fields?
n classical
Fields
Classical
Field
if s<2
How to arrange all these fields ?
Looks like а rope, or a string
New problems:
We have
but do not have H,
wave operators,
but we have renormalized S-matrix
Quantization
3 bodies problem, L.Faddeev
Quantum Field
Quantization
Classical
Mechanics
Second quantization
Quantum
Mechanics
Infinite # of the same
particles
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
1 
 in
X ( , )  x  2 ' p   i 2 '   n cos(n )e
n 0 n



• First quantized approach,
Polyakov's approach
• SFT
Light-Cone 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
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, 1990
I.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
Hashimoto, Itzhaki; Gaiotto, Rastelli, Sen,
Zwiebach; Kawano, Kishimoto,Takahashi
Kishimoto and T. Takahashi; Erler;Ellwood
Schnabl; Kiermaier, Zwiebach,…..
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
2

3
(    1) e   
2 2
I.A.,Joukovskaya,
Koshelev;Ya.Volovich
Motivations
• 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
Rolling tachyon solutions in level
truncated theories in FRW metric
Talks by L.Joukovskaya, A.Koshelev, S.Vernov
• 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)
No Conclusion!
M
Ф