Bose-Fermi equivalence and interacting string field theory

Abstract

The bosonic representation of the half string ghost in the full string basis is examined in full. The proof that the comma 3-vertex (matter and ghost) in the bosonic representation satisfy the Ward-like identities is established thus completing the proof of the Bose Fermi equivalence in the half string theory.

Ward-like Operator in the comma theory

“…They also proved BRST -invariance and showed that BRST -invariance implies both K-invariance and the W L identities. In [6], the K-invariance of the comma theory was established and the BRST -invariance was briefly discussed, but the WL identities were not addressed. Before discussing the WL identities and BRST -invariance in the comma theory one needs, first, to construct the WL operator in the comma theory.…”

“…1 in [3]), it is not obvious that all ingredients of the comma theory employing the comma bosonic fields ϕ L (σ) and ϕ R (σ) are equivalent to those constructed using the original comma fermi fields c L (σ), c R (σ), b L (σ) and b R (σ) appearing in [5]. It has been claimed in [6], that the ghost bosonic vertices in the comma operator formulation obey the same overlap equations as the fermionic vertices and are identical. However a rigorous proof was not presented in [5].…”

“…In 2001, the work of references [8,9,10,11] generated much interest in the comma formulation of Witten's theory of interacting open bosonic strings. Subsequently, interest in the comma theory faded a way and many of the issues in the comma approach raised in [6,7] remain unsolved. In [14], Ward-like (WL) identities were established and used as part of the proof of the equivalence between the bosonic and fermionic ghost realizations of the gauge invariance in Witten's theory.…”

“…In [14], Ward-like (WL) identities were established and used as part of the proof of the equivalence between the bosonic and fermionic ghost realizations of the gauge invariance in Witten's theory. Although K invariance was established in [6], the WL identities were not discussed in the case of the comma theory. Just as in the case of the full string, the proof of equivalence in the comma theory requires that similar WL identities hold in the comma theory.…”

“…One of the reasons that this problem received little attention has to do with the difficulty of constructing the WL operator in the comma approach as we shall see shortly. Once we construct the WL operator, we can complete the proof of equivalence of the comma three-point vertex (coordinate and ghost), and establish the BRST invariance of the full three-point vertex in the bosonized ghost [15], a problem that was briefly discussed in [6].…”