On the ghost sector of OSFT

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# On the ghost sector of OSFT - PowerPoint PPT Presentation

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)

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### 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?

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

• 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

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

The oscillators are accordingly redefined

• On the vacua we have
• 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,

At gh=0 we have

• This very small simplification gives to squeezed states in the kernel of K1 the commuting properties that one would naively expect

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))

...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
• 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

• 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
• Once zero modes are (mysteriously) reconstructed, we can use the properties of the midpoint basis to get (and analytically compute)
• Zero modes
• Only for N=2 this coincides with the discrete spectrum of G (that’s the reason of the violation of commutativity)
The norm of wedge states
• 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.

Infinitely many rank 1 orthogonal projectors (RSZ, BMS) can be shown to have UNIT norm, see Ellwood talk, CP- factors (CM)

n=3,m=30

n=1,m=1

n=3, m=3

Sliver

n=1, m=7