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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. 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

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  1. On the ghost sector of OSFT Carlo Maccaferri SFT09, Moscow Collaborators: Loriano Bonora, Driba Tolla

  2. 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?

  3. Surface States as squeezed states • Given a map

  4. This is a good representation because • The squeezed form exactly captures all the n-point functions

  5. 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

  6. Invariance On the gh=0 vacuum we have On the gh=3 vacuum K1 invariance does not mean commuting nemann coefficients

  7. 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

  8. 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

  9. 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

  10. The oscillators are accordingly redefined • On the vacua we have • Still we have • And the fundamental

  11. K1 in the midpoint basis • Remember that K1 has the following form • The midpoint basis just kills the spurious 3’s,

  12. 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

  13. 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

  14. 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

  15. 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)...

  16. Then we decompose • Insertion functions • Again, LT shows a diverging eigenvalue in the U’s

  17. As for reduced gh=3 wedges we observe • And therefore define • Which very easily generalizes to N strings (3 N)

  18. Properties • Twist/bpz covariance • K1 invariance • Non linear identities (of Gross/Jevicki type) thanks to the “chiral” insertion

  19. 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

  20. 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

  21. 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

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

  23. Normalizations • Completeness relation

  24. 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)

  25. 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

  26. Gh=3 • Remembering the neumann function for Continuous spectrum Reduced gh=3 wedges Needed for BRST invariance

  27. 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)

  28. 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

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