Physical Degrees of Freedom in 2-D String Field Theories

States in the absolute (semi-relative) cohomology but not in the relative cohomology are examined through the component decomposition of the string field theory action for the 2-D string. It is found that they are auxiliary fields without kinetic terms, but are important for instance in the master equation for the Ward-Takahashi identities. The ghost structure is analyzed in the Siegel gauge, but it is noted that the absolute (semi-relative) cohomology states are lost. Recent studies on matrix models and continuum Liouville approach showed rich structures in the two-dimensional quantum gravity coupled to conformal matter. Especially interesting case is the two-dimensional gravity coupled to the c = 1 conformal matter, which can be regarded as a string theory in two-dimensional target space and is often called the 2-D string. The 2-D string theory possesses infinitely many discrete states as remnants of stringy states, and a large symmetry corresponding to the area-preserving diffeomorphisms which is formulated as an algebraic structure called the ground ring [1], [2], [3]. It seems that the complete informations on the 2-D string theory may be obtained from such an algebraic approach. In particular, it has been shown that the entire content of the WardTakahashi identities from the ground ring can be summarized as a “master equation” of the Batalin-Vilkovisky type [4]. It has been shown that nontrivial cohomology elements appear for more than one ghost numbers in the two-dimensional gravity coupled to conformal matter [5]. They seem to be closely related to the rich structure such as discrete states and the ground ring. We need to distinguish two types of cohomologies: relative and absolute. Absolute cohomology is defined by the states annihilated by the BRST charge QB but not expressed as the BRST charge acting on other states. Relative cohomology is defined by restricting the BRST cohomology to the subspace satisfying b0 |Φ〉 = 0. (1) In the case of closed string, it has been stressed that another cohomology called semi-relative cohomology is more appropriate instead of the absolute cohomology [6]. It has recenrly been found that the states in the semi-relative cohomology played a key role in formulating the master equation for the Ward-Takahashi identities [4]. More recently, the full account of the nonpolynomial formulation of the string field theory has appeared [7]. The master equation was also formulated in this string field theory regarding the elements of semi-relative cohomology as physical. In spite of these important role played by the states in the absolute cohomology or the semirelative cohomology, only the states in the relative cohomology are usually regarded as physical. The purpose of our note is to examine the role played by the nontrivial elements of the absolute cohomology in the case of open string and the semi-relative cohomology in the case of the closed string theory. By writing down component decomposition of the D=2 string field theories, we find that the states in the absolute cohomology 2 (semi-relative cohomology) but not in the relative cohomology have no kinetic terms and hence are auxiliary fields. We also discuss the ghost number structure of the gauge fixed action in the Siegel gauge, and point out that the nontrivial elements of the absolute (semi-relative) cohomology are lost in this particular gauge. Let us first consider the open string theory. The BRST charge is given by