A Note on the Zakharov-shabat Topological Model

In this note I discuss some features of the topological theory obtained from the ZakharovShabat (or general sl(2,C)) hierarchy, and comment on some possible physical and/or mathematical interpretations of it. † Research supported by an INFN fellowship. In some recent papers (see for example refs. [1,2,3,4]) it has been shown that the double scaling limit of hermitian and anti-hermitian matrices in the oneand two-arc sector, leads to two dimensional quantum gravity models based on the following sl(2,C) hierarchies: Korteweg-de Vries (KdV), modified KdV (mKdV), non-linear Schrödinger (NLS) and Zakharov-Shabat (ZS). Of particular interest are the two dimensional quantum gravity theories constructed from the ZS hierarchy. Indeed, the ZS hierarchy contains both the KdV and mKdV hierarchies as reductions, and its first (odd) critical point gives rise to a new “topological theory” [3], different from the one obtained from the KdV hierarchy [5,6,7]. (Notice that only the even critical points of the ZS hierarchy transform, under reduction, into those of the KdV hierarchy, whereas the odd critical points don’t.) The purpose of this note is to discuss some features of the topological theory, which will be called “ZS topological model”, corresponding to the first critical point of the ZS hierarchy [3], and to make some comments on its possible physical and/or mathematical interpretations. The Zakharov-Shabat hierarchy is simply defined by introducing two real, independent functions, ψ and ψ, of the parameters (t−1, x, t1, . . .) 1 and by defining the flows as ∂ψ ∂tk = 1 2 (Fk+1 −Gk+1) ∂ψ ∂tk = 1 2 (Fk+1 +Gk+1) (k ≥ −1) (1) where the polynomials Fk and Gk are given by Fk+1 = G ′ k + (ψ − ψ)Hk , Gk+1 = F ′ k + (ψ + ψ)Hk (2) H ′ k = ψ(Gk − Fk)− ψ(Gk + Fk) (3) with F0 = ψ − ψ , G0 = ψ + ψ , H0 = 0 . (4) The connexion with the anti-hermitian 1-matrix models is given by the following formulæ: 〈PP 〉 = −ψψ , 〈OiPP 〉 = ∂ ∂ti 〈PP 〉 , 〈OiP 〉 = 1 2 Hi+1 (5) 1 I have adopted the conventions of ref. [4]. 1 〈PP 〉 = ∂ x log Z = −F , t0 = x. The string equations are ∞ ∑ k=0 (k + 1)tkFk = 0 ∞ ∑ k=0 (k + 1)tkGk = 0 . (6) The string equations and the hierarchy equations can be written as Virasoro constraints acting on the partition function of the matrix model (or on the tau-function of the ZS hierarchy since Z = τ). These are the Virasoro constraints of an untwisted scalar field Ln(α)Z = 0 , n ≥ −1 (7) where α is an a-priori arbitrary integration constant (see refs. [3,4] for a discussion of the Virasoro constraints in the ZS hierarchy). The topological point of the ZS hierarchy is given by t1 = β tn = 0 (n 6= 1) (8) where β is a number to be determined. For convenience, it is better to shift t1 by −β in such a way that the topological point is given by tn = 0 for any n. I will also introduce the following notation for the correlation functions: 〈 On1 . . .Onp 〉 t will denote a correlation function computed not at the ZS topological point (which means that it is a function of the t’s), 〈 On1 . . .Onp 〉 will, instead, denote a correlation function computed at the ZS topological point (i.e. at tn = 0). The Virasoro constraints, eq. (7), can then be written in the following way