Lattice Study of the Simplified Model of M - Theory

Lattice discretization of the supersymmetric Yang-Mills quantum mechanics is reviewed and results of the Monte Carlo simulations of the simplified model are presented. The D=4, N=2, quenched system, studied at finite temperature, reveals existence of the two distinct regions which may correspond to a black hole and the elementary 0-branes phases of the M-theory conjectured in the literature. New results for higher gauge groups N < 9 lead to the similar picture, however the nature of the transition between the two phases and its precise scaling with N is yet unresolved. 1 Matrix quantum mechanics and its lattice formulation Recent developments in nonperturbative string theories have lead to an exciting hypothesis of the existence of a single theory (M-theory) which encompasses five known superstring theories and eleven dimensional supergravity in a unified scheme (for a recent review see e.g. 1). Even though its details are not known, the M-theory has a tantalizing potential to unify all interactions and particles. In particular it may offer a topological explanation of such fundamental features as three families and fractional charges. It may lead to a standard model gauge group with N =1 supersymmetry. It provides understanding of the Bekenstein-Hawking entropy puzzle and much more. Moreover, Banks, Fishler, Susskind and Shenker 2 suggested that the spectrum of M-theory is equivalent to that of a supersymmetric Yang-Mills quantum mechanics (SYMQM) which results from the dimensional reduction of the 10 dimensional supersymmetric Yang-Mills theory. This allows to use a host of nonperturbative methods to quantitatively study both systems. Accordingly we have constructed the Wilson discretization of the above quantum mechanics and proposed to investigate it with the standard lattice techniques 3. The ultimate goal is to study D=10 SYMQM for the large size of the SU(N) matrices. However, even for this relatively simple, one dimensional quantum mechanical system this is a formidable task, the main difficulty being the complex fermionic determinant (and pfaffian) in D=10, and time