ar X iv : h ep - t h / 97 08 02 9 v 1 6 A ug 1 99 7 F and M Theories as Gauge Theories of Area Preserving Algebra

F theory and M theory are formulated as gauge theories of area preserving diffeomorphism algebra. Our M theory is shown to be 1-brane formulation rather than 0-brane formulation of M theory of Banks, Fischler, Shenker and Susskind and the F theory is shown to be 1-brane formulation rather than -1-brane formulation of type IIB matrix theory of Ishibashi, Kawai, Kitazawa and Tsuchiya. Area preserving diffeomorphism algebra Our starting point is the following 2 + 1 dimensional energy-momentum algebra of Dirac, Schwinger and DeWitt. [T00(σ), T00(σ ′)] = −i (Ti0(σ) + Ti0(σ)) ∂iδ(σ − σ′) (1) [Ti0(σ), T00(σ ′)] = −iT00(σ)∂iδ(σ − σ′), (2) [Ti0(σ), Tj0(σ ′)] = −i(Ti0(σ)∂jδ(σ − σ′) + Tj0(σ)∂iδ(σ − σ′)), (3) where σ = (σ, σ), (i, j = 1, 2) and ∂iδ(σ − σ′) = ∂ ∂σi [δ(σ1 − σ′ 1)δ(σ2 − σ′ 2)]. These commutation relations are valid in an arbitrary Riemannian space where the indices 1 and 2 describe a time-like plane and the index 0 corresponds to the normal direction to the plane. The area preserving diffeomorphism is defined as T (σ) = 1 √ g (∂1T20(σ) − ∂2T10(σ)), (4) where g = detgij(σ)