A modern treatment of this subject can be found in Segreev's book on the Kahler geometry of loop spaces also available online. This line of research started with the seminal work of Bowick and Rajeev: The holomorphic geometry of the closed bosonic string theory and $Diff S^1/S^1$ (Spires) (and independently Kirillov and Yuriev (Please see the reference in Sergeev's book)).

The basic idea is that loop spaces are Kahler manifolds carrying a $Diff S^1$ invariant closed two form (Kahler form). ( These manifolds are quantizeable by selection of a complex structure compatible with the Kahler form and performing Kahler (holomorphic) quantization.)

In the present problem, one seeks to define a $Diff S^1$ equivariant quantizations on them, i.e., to find a representation of $Diff S^1$ invariant functions on these loop spaces as $Diff S^1$ invariant subspace of vectors on a Hilbert space.
The invariant vectors are obtained as null vectors of the quantized algebra.

Now, as many other cases in quantum mechanics, one cannot find a faithful representation at the quantum level but only a projective one.
This means that the $Diff S^1$ invariant subspace can only be trivial.

This is the main reason that the original solution was found by imposition of half the constraints which is reminiscent of the Kahler quantization itself where one selects holomorphic sections.

The modern solution lies in the fact that a given complex structure on the loop space is not invariant under $Diff S^1$. Thus also a given Kahler quantization is not invariant as well.

The vacuum of the Kahler quantization Hilbert space corresponding to a given complex structure is not invariant under $Diff S^1$. The transformed vacuum is the vacuum corresponding to the Kahler quantization associated with the transformed complex structure.

This leads to the notion of the "Vacuum bundle" which is a line bundle over $Diff S^1/S^1$ comprising all the individual vacua. This bundle has a connection associated with the projective representation of $Diff S^1$, which becomes flat when one takes into account all the fields of the theory (in the case of the bosonic string at $d=26$).

The main result is that the flatness of the connection is equivalent to the unitary equivalence of the quantization Hilbert spaces under the action of $Diff S^1$, which is the required result expressing the equivariance of the quantization.

This treatment has many connections to many contemporary research areas. for example the bundle of complex structures over the loop space is a twistor space as was noticed by Bowick and Rajeev in their original work.

Update:

Sorry for never mentioning it in the original answer (It appears in the references), the loop space under consideration is $\Omega \mathbb{R}^{d-1,1}$ (based loops), (becoming $\Omega \mathbb{R}^{d-1,1}//diff S1$ after the Hamiltonian reduction).

The (unitary) equivalence of the three mentioned quantization methods Goddard Goldstone Rebbi Thorn (GGRT- half of the constraints method), BRST and the Bowick-Rajeev (twistor) can be explained as follows:

GGRT-BRST: The BRST invariance condition of the physical states is equivalent by explicit construction to the GGRT half number of constraints, please see Green Schwarz Witten equation: 3.2.31

BRST - Bowick-Rajeev: The inclusion of the ghost sector of the string is equivalent to tensoring with the anticanonical bundle to the Fock bundle of the bosonic string.
This can be understood from the fact that the Ricci tensor of $Diff S1/S1$ is exactly equal to the contribution of the ghost sector to the vacuum bundle curvature (for example by a direct calculation). Thus the half form quantization is equivalent to the introduction of ghosts. This can be further seen in the fact that the physical states have a ghost number of $-\frac{1}{2}$. This is also consistent with the path integral picture, where one must divide by the redundant gauge orbit measure.

In addition to the fact that in this method, one is not required to introduce the ghost fields explicitly, the flatness of the connection establishes the equivalence between the quantizations with respect to different Kahler polarizations.

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