Physical Properties of W Gravities and W Strings

Abstract

We investigate some basic physical properties of W gravities and W strings, using a free field realization. We argue that the configuration space of W gravities have global characteristics in addition to the Euler characteristic. We identify one such global quantity to be a "monopole" charge and show how this charge appears in the exponents. The free energy would then involve a "θ" parameter. Using a BRST procedure we find all the physical states of W3 and W4 gravities, and show that physical operators are nonsingular composites of the screening charge operators. (The latter are not physical operators for N≥3.) For W strings we show how the W constraints lead to the emergence of a single (and not many) extra dimension coming from the W-gravity sector. By analyzing the resulting dispersion relations we find that both the lower and upper critical dimensions are lowered compared to ordinary two-dimensional gravity. The pure W gravity spectrum reveals an intriguing "numerological" connection with unitary minimal models coupled to ordinary gravity.

W3 string scattering

“…A W 3 string was constructed [6,7] by utilizing the Miura transformation and obeying the consistency conditions arising from the BRST charge [8]. This can be extended, by generalizing the Miura transformation, to have a W 3 string for any number of scalar fields [7].…”

“…It has been noticed as a phenomenological observation that the central charges arising in W N strings are related to those of the minimal models [6,7]. It has been found [4] by explicitly solving the physical state conditions at low levels, that the W 3 string theory has one of the oscillators suppressed and that physical, positive definite norm, states are contained in a subspace of the original oscillator space that obeys L n |ψ = 0, n ≥ 1 and in the a ef f = 1 sector is L −1 |ψ , where L n |ψ = 0 for n ≥ 0, and so scattering in this sector is given by i dz i V N f .…”

The higher spin rectangle

“…This can be achieved as follows [42][43][44]. Let us write down the Miura construction for the W N algebra:…”

Canonical BRST quantisation of worldsheet gravities

“…In the case of the non-critical bosonic string, the worldsheet anomaly gives rise to a propagating Liouville mode whose presence renders the worldsheet "gravity" non-trivial. Analogous anomalies in the W 3 string give rise to a worldsheet W 3 gravity described by an A 2 Toda theory [3].…”