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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)

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on the ghost sector of osft

On the ghost sector of OSFT

Carlo Maccaferri

SFT09, Moscow

Collaborators: Loriano Bonora, Driba Tolla

  • 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 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

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


  • 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
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
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

The oscillators are accordingly redefined

  • On the vacua we have
  • Still we have
  • And the fundamental
k1 in the midpoint basis
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
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
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
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)
  • Twist/bpz covariance
  • K1 invariance
  • Non linear identities (of Gross/Jevicki type) thanks to the “chiral” insertion
the vertex in the midpoint basis
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 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



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 discrete spectrum of G
  • The zero mode matrix has eigenvalues
  • Important to observe that


  • Completeness relation
spectroscopy in the midpoint basis
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
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
  • Remembering the neumann function for

Continuous spectrum

Reduced gh=3 wedges

Needed for BRST invariance

  • 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
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=3


n=1, m=7