We construct the ghost number 9 three strings vertex for OSFT in the natural normal ordering. We find two versions, one with a ghost insertion at z = i and a twistconjugate one with insertion at z = −i. For this reason we call them midpoint vertices. We show that the relevant Neumann matrices commute among themselves and with the matrix G representing the operator K1. We analyze the spectrum of the latter and find that beside a continuous spectrum there is a (so far ignored) discrete one. We are able to write spectral formulas for all the Neumann matrices involved and clarify the important role of the integration contour over the continuous spectrum. We then pass to examine the (ghost) wedge states. We compute the discrete and continuous eigenvalues of the corresponding Neumann matrices and show that they satisfy the appropriate recursion relations. Using these results we show that the formulas for our vertices correctly define the star product in that, starting from the data of two ghost number 0 wedge states, they allow us to reconstruct a ghost number 3 state which is the expected wedge state with the insertion at the midpoint, according to the star recursion relation.

Ghost story. II. The midpoint ghost vertex / Bonora, L.; Maccaferri, C.; Scherer Santos, R. J.; Tolla, D. D.. - In: JOURNAL OF HIGH ENERGY PHYSICS. - ISSN 1029-8479. - 2009:11(2009), pp. 1-61. [10.1088/1126-6708/2009/11/075]

### Ghost story. II. The midpoint ghost vertex

#### Abstract

We construct the ghost number 9 three strings vertex for OSFT in the natural normal ordering. We find two versions, one with a ghost insertion at z = i and a twistconjugate one with insertion at z = −i. For this reason we call them midpoint vertices. We show that the relevant Neumann matrices commute among themselves and with the matrix G representing the operator K1. We analyze the spectrum of the latter and find that beside a continuous spectrum there is a (so far ignored) discrete one. We are able to write spectral formulas for all the Neumann matrices involved and clarify the important role of the integration contour over the continuous spectrum. We then pass to examine the (ghost) wedge states. We compute the discrete and continuous eigenvalues of the corresponding Neumann matrices and show that they satisfy the appropriate recursion relations. Using these results we show that the formulas for our vertices correctly define the star product in that, starting from the data of two ghost number 0 wedge states, they allow us to reconstruct a ghost number 3 state which is the expected wedge state with the insertion at the midpoint, according to the star recursion relation.
##### Scheda breve Scheda completa Scheda completa (DC)
2009
2009
11
1
61
075
https://arxiv.org/abs/0908.0055
Bonora, L.; Maccaferri, C.; Scherer Santos, R. J.; Tolla, D. D.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/20.500.11767/30383`
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