VORTEX SOLUTIONS IN THE MODEL OF 3He FILMS AND THE 0(3) o-MODEL

1. Recently there have been experimental studies of thin 3He films one or two atoms in thickness [1]. In ref. [2] a microscopic model o f the 3He film was suggested, based on the 3-dimensional model of 3He developed earlier [3]. As shown in ref. [2] , in two dimensions there may exist several superfluid phases, two of them being energetically preferable and stable under small fluctuations (the so-called a and b phases). The Bose-spectrum of the system in question contains both phonon and nonphonon branches. The order parameter in such a theory is a 2 × 3 matrix. The Bosecondensate cannot exist in two-dimensional systems at f'mite temperatures. Nevertheless we can expect such a system to be a superfluid because the long-range correlations decrease more slowly than exponentially at large distances. In ref. [2] it was demonstrated that if we introduce an analog of polar coordinates for tensor Bose-fields, we can show that the long-range behaviour o f the field correlator is determined by the behaviour o f the correlator of angle variables, as in the two-dimensional superfluidity theory [4]. The number of such angle variables is equal to the number o f phonon modes of the system. The angle variables in the a-phase are the angle ¢ E S 1 and the vector n E S 2 C R 3. The action functional for "slow" fields is