Transcript On the ghost sector of Open String Field Theory

On the ghost sector of OSFT
Carlo Maccaferri
SFT09, Moscow
Collaborators: Loriano Bonora, Driba Tolla
Motivations
• We focus on the oscillator realization of the gh=0 star algebra
• Fermionic ghosts have a “clean” 3 strings vertex at gh=1,2
(Gross, Jevicki)
• We need a formulation on the SL(2,R)-invariant vacuum to
be able to do (for example)
•
is a squeezed state on the gh=0 vacuum,
• How do such squeezed states star-multiply?
• Is it possible to have
in critical dimension?
Surface States as squeezed states
• Given a map
• This is a good representation because
• The squeezed form exactly captures all the n-point functions
Surfaces with insertions as
squeezed states
• Surfaces with k c-insertions are also squeezed states on the gh=k vacuum
• With the neumann function given by
• Again 2n-point functions are given by the determinant of n 2-point
function, so the squeezed state rep is consistent
• To reflect a surface to gh=3 we can use the BRST invariant insertion of
Invariance
On the gh=0 vacuum we have
On the gh=3 vacuum
K1 invariance does not mean commuting nemann coefficients
•
The reason is in the vacuum doublet
But
•
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Is it possible to have K1 invariance at gh=3?
The obvious guess is given by
•
But this is not a squeezed state (but a sum of two)
•
(very different from the gh=1/gh=2 doublet , or to the h=(1,0) bc-system)
•
Our aim is to define gh=3 “mirrors” for all wedge states, which are still squeezed
states with non singular neumann coefficients (bounded eigenvalues) and which
are still annihilated by K1
Reduced gh=3 wedges
•
Consider the Neumann function for the states
•
LT analysis shows diverging eigenvalues, indeed
Real and bounded eigenvalues <1
Rank 1 matrix (1 single diverging eigenvalue)
•
We thus define reduced gh=3 wedges as
•
Still we have
Midpoint Basis
•
“Adapting” a trick by Okuyama (see also Gross-Erler) we can define a convenient
gh=3 vacuum
Same as in gh=1/gh=2
Potentially dangerous
•
We need to redefine the oscillators on the new gh=0/gh=3 doublet by means of
the unitary operator
•
Reality
•
We will see that this structure is also encoded in the eigenbasis of K1
•
On the vacua we have
•
The oscillators are accordingly redefined
•
Still we have
•
And the fundamental
K1 in the midpoint basis
•
Remember that K1 has the following form
•
The midpoint basis just kills the spurious 3’s,
•
This very small simplification gives to squeezed states in the kernel of K1 the
commuting properties that one would naively expect
At gh=0 we have
At gh=3 we have
K1 invariance now implies commuting matrices
Gh=3 in the midpoint basis
•
Going to the midpoint basis is very easy for gh=3 squeezed states
•
The “bulk” part (non-zero modes) is unaffected
•
The zero mode column mixes with the bulk
for reduced gh=3 wedges
•
For reduced states we thus have the non trivial identity
Gh=0 in the midpoint basis
•
Here there are non normal ordered terms in the exponent, non linear relations
•
In LT we also observe
•
The midpoint basis is singular at gh=0, nontheless very useful as an intermediate
step, because it effectively removes the difference between gh=0 and gh=3
The midpoint star product
•
We want to define a vertex which implements
•
For a N—strings vertex we choose the gluing functions (up to SL(2,R))
•
We start with the insertion of
on the interacting worldsheet
...It is a squeezed state but not a “surface” state (the surface would be the sum of
2 complex conjugated squeezed)...
•
Then we decompose
•
Insertion functions
•
Again, LT shows a diverging eigenvalue in the U’s
•
As for reduced gh=3 wedges we observe
•
And therefore define
•
Which very easily generalizes to N strings (3
N)
Properties
•
Twist/bpz covariance
•
K1 invariance
•
Non linear identities (of Gross/Jevicki type) thanks to the “chiral” insertion
The vertex in the midpoint basis
•
As for reduced gh=3 wedges, the vertex does not change in the bulk (non-zero modes)
•
And it looses dependence on the zero modes
•
So, even if zero modes are present at gh=0, they completly decouple in such a kind
of product (isomorphism with the zero momentum matter sector)
In particular, using the midpoint basis, it is trivial to show that
•
K1 spectroscopy
•
•
•
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K1 is well known to have a continuos spectrum, which manifests itself in
continuous eigenvalues and eigenvectors of the matrices G and H
Belov and Lovelace found the “bi-orthogonal” continuous eigenbasis of K1 for the
bc system (our neumann coefficients are maps from the b-space to the c-space
and vic.)
Orthogonality
“Almost” completeness
RELATION
WITH
MIDPOINT
BASIS
•
These are left/right eigenvectors of G
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However that’s not the whole spectrum of G
•
The zero mode block has its own discrete spectrum
The discrete spectrum of G
•
The zero mode matrix
has eigenvalues
•Important to observe that
•
Normalizations
•
Completeness relation
Spectroscopy in the midpoint basis
•
Continuous spectrum with NO zero modes (both h=-1,2 vectors start from n=2)
•
Discrete spectrum with JUST zero modes
•
The midpoint basis confines the zero modes in the discrete spectrum (separate
orthonormality for zero modes and bulk)
Reconstruction of BRST invariant
states from the spectrum
•
It turns out that all the points on the imaginary k axis are needed (not just ±2i)
•
Wedge states eigenvalues have a pole in
•
Given these poles, the wedge mapping functions are obtained from the
genereting function of the continuous spectrum
Gh=3
•
Remembering the neumann function for
Continuous
spectrum
Reduced gh=3
wedges
Needed for BRST
invariance
Gh=0
•
Zero modes
•
Only for N=2 this coincides with the discrete spectrum of G (that’s the reason of
the violation of commutativity)
Once zero modes are (mysteriously) reconstructed, we can use the properties of
the midpoint basis to get (and analytically compute)
•
The norm of wedge states
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As a check for the BRST consistency of our gh=0/gh=3 squeezed states, we
consider the overlap (tensoring with the matter sector, so that c=0)
•
Using Fuchs-Kroyter universal regularization (which is the correct way to do
oscillator level truncation), we see that this is perfectly converging to 1 (for all
wedges, identity and sliver included.
n=3,m=30
n=1,m=1
n=3, n=1,
Sliver
m=3 m=7
Infinitely many rank 1
orthogonal projectors
(RSZ, BMS) can be
shown to have UNIT
norm, see Ellwood talk,
CP- factors (CM)