The Zero Curvature Formulation of TB, sTB Hierarchy and Topological Algebras

A particular dispersive generalization of long water wave equation in 1 + 1 dimensions, which is important in the study of matrix models without scaling limit, known as two–Boson (TB) equation, as well as the associated hierarchy has been derived from the zero curvature condition on the gauge group SL(2, R) ⊗ U(1). The supersymmetric extension of the two–Boson (sTB) hierarchy has similarly been derived from the zero curvature condition associated with the gauge supergroup OSp(2|2). Topological algebras arise naturally as the second Hamiltonian structure of these classical integrable systems, indicating a close relationship of these models with 2d topological field theories. ∗E-mail address: [email protected] Integrable models [1–4] in 1 + 1 dimensions play very interesting and sometimes mysterious roles in the study of conformal field theories [5], matrix model formulation of string theories [6], 2d topological field theories [7,8] and the intersection theory on the moduli space of Riemann surfaces [9,10]. These models contain very rich mathematical structures and deserve to be studied on their own right. Recently, a particular dispersive generalization of long water wave equation in narrow channel, known as the TB equation [11] has attracted some attention due to its relation with the matrix models without the continuum limit [12]. This relation has been interpreted as an indication of the topological nature of the latter models. The TB–hierarchy as well as their close relatives have been studied quite extensively in the literature [11,13,14]. In particular, TB–hierarchy was constructed in ref.[11] in a non–standard Lax operator approach and has been shown to possess a tri–Hamiltonian structure. The supersymmetric extension of TB–hierarchy or sTB–hierarchy and many of their interesting properties have also been studied recently [15,16] in the non–standard superLax operator approach. Integrable models, on the otherhand, are also known [17] to be obtained from the group theoretical point of view where the dynamical equations are obtained from the zero curvature condition associated with some symmetry group. While some properties of an integrable model can be understood in the Lax operator approach, the bi–Hamiltonian structure, integrabilty property and the geometry of the phase–space become more transparent in the zero curvature formulation [18]. Moreover, ordinary 2d gravity theories and their underlying current algebra structure in the light cone gauge can be better understood by connecting them to integrable models in the zero curvature approach [19,20]. Thus, these two approaches play complementary roles with each other. In this paper we study both TB and sTB–hierarchy in the zero curvature approach. We derive TB–hierarchy from the zero curvature condition associated with the gauge group SL(2, R) ⊗ U(1) and similarly sTB–hierarchy is obtained from the gauge supergroup OSp(2|2). We point out that both these integrable systems possess two bi–Hamiltonian structures. The second Hamiltonian structure of the TB–hierarchy is shown to be U(1) extended Virasoro algebra with zero central charge and that of sTB–hierarchy is the twisted N = 2 superconformal algebra. This observation, therefore, indicates that these integrable models are closely related to 2d topological field theories. The dispersive long water wave equation (TB equation) with which we will be con-