On the ghost sector of OSFT

Carlo MaccaferriSFT09, 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

InvarianceOn 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• “Adapting” a trick by Okuyama (see also Gross-Erler) we can define a convenient gh=3 vacuum

• We need to redefine the oscillators on the new gh=0/gh=3 doublet by means of the unitary operator

• Reality

Potentially dangerousSame as in gh=1/gh=2

• 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• 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

• 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• 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=3n=3,m=30

n=1, m=7n=1,m=1

Sliver

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