The structure of the ground ring in critical W 3 gravity Chuan -

By explicit calculation, I determine the structure of the ground ring of the critical W3 gravity and show that there is an su(3) invariant quadratic relation among the six basic elements. By using this result, I also construct some discrete physical states of the critical W3 gravity. The study of W gravity is surely an interesting extension of the usual two dimensional gravity coupled with matter. The matter part is the minimal W matter. Particularly interesting is the borderline case where the matter part has an integral central charge which is equal to the rank of the associated Lie algebra. (I will call this case the critical W gravity.) The study of the simplest case [1–6], the D = 2 string theory, has revealed a lot of interesting structures: the existence of a ground ring [3] and the infinite dimensional Lie algebra generated by ghost number 0 currents [3–5]. I will report in this letter some results on the extension of these structures to the critical W3 gravity. Several papers [7–12] have studied the cohomology problem of W gravity. Nevertheless the results are still not complete and explicit. The methods used often turn out to be quite transendential, or by using computer to do most of the calculations, the results are quite messy and are of little use. Especially important is what is the analogous ground ring structure and what is the infinite dimensional Lie algebra in these W gravity theories. Many conjectures exist in the literature and very few proofs and explicit results are offered. Here I will study the structure of the ground ring in critical W3 gravity by explicit calculation. Although most of the calculations are also done by computer [13,14], I have try my best to give the results in their simplest analytic forms. I will show that the ground ring in critical W3 gravity is not a free polynomial ring of 6 elements. This ring is actually a polynomial ring of 6 elements module Preprint submitted to Elsevier Preprint 1 February 2008 one quadratic relation. This quadratic relation is su(3) invariant. By exploiting the property of this ground ring, I also construct some discrete physical states. Partial results have also been obtained for the algebra generated by the discrete states and the generalization to other critical W gravities, but I will not report these results here. As usual I deal with only one chiral sector and restrict my attention to the critical case. I will speak only the prime physical states or the relative physical states in a generalized relative cohomology. For the pure W3 gravity, the complete physical states have been obtained in [7] but there is no interesting symmetry structures, just as in the case with the usual pure gravity (i.e. W2 gravity). For the W3 gravity coupled with (the W3) minimal matter, the study and enumeration of all the physical states is complicated by the problem of decoupling all the null matter states, see [12] and refernces therein. The critical W3 gravity is easier and is also more interesting. This theory has been studied in [8,11]. The basic fields of the critical W3 gravity are the two free matter fields X1(z) and X2(z), two Liouville fields φ1(z) and φ2(z), two pairs of ghost anti-ghost fields (b(z), c(z)) and (β(z), γ(z)) associated with the spin 2 and 3 generators of the W3 algebra. From these fields we can construct the following stressenergy tensors and two spin-3 generators: TX =− 1 2 (∂zX1(z)) 2 −− 2 (∂zX2(z)) 2 , (1) Tφ =− 1 2 (∂zφ1(z)) 2 −− 2 (∂zφ2(z)) 2 + √ 2∂ zφ1(z) + √ 6∂ zφ2(z), (2) Tbc= 2 ∂zc(z) b(z) + c(z) ∂zb(z), (3) Tβγ = 3 ∂zγ(z) β(z) + 2γ(z) ∂zβ(z), (4)