Some exact results on the matter star-product in the half-string formalism

TL;DR

This paper analytically investigates the matter star-product in string field theory's half-string formalism, demonstrating factorization properties, matrix representations, and confirming conjectures about projectors related to the sliver state.

Contribution

It provides exact analytical results on the matter star-product, including the factorization of the D25 sliver wavefunction and the validation of projector properties of certain matrices.

Findings

Factorization of the D25 sliver wavefunction in half-string coordinates

Analytical calculation of the star-product as matrix multiplication

Confirmation that specific matrices are projectors on the sliver state

Abstract

We show that the D25 sliver wavefunction, just as the D-instanton sliver, factorizes when expressed in terms of half-string coordinates. We also calculate analytically the star-product of two zero-momentum eigenstates of $\hat{x}$ using the vertex in the oscillator basis, thereby showing that the star-product in the matter sector can indeed be seen as multiplication of matrices acting on the space of functionals of half strings. We then use the above results to establish that the matrices $\rho_{1,2}$, conjectured by Rastelli, Sen and Zwiebach to be left and right projectors on the sliver, are indeed so.