Berezin-Toeplitz quantization of vector bundles over Kahler manifolds

概要

One of the approaches of the quantum theory of gravity or the theory of everything is string
theory. String theory is a theory in which the fundamental constituents of the universe are onedimensional strings rather than zero-dimensional point particles. This simple modification makes
a huge difference from theories of fundamental point particles. The spectrum of a single firstquantized string contains various different types of states with particular masses and spins. More
surprisingly, there is a particular state corresponding to the graviton, which is a quantum of
gravitational interaction, and the string theory is naturally incorporate the quantum theory of
gravity.
Various studies of superstring theory and M-theory [1–3] suggests that the noncommutative
geometry may play an essential role in the description of spacetime in Planck scale. In the Einstein’s
classical theory of gravity, we assume that the spacetime is a smooth manifold, which means that
the spacetime coordinates are a set of real numbers. On the contrary, in the noncommutative
geometry, we assume that the spacetime coordinates are noncommutative operators on a suitable
Hilbert space. When the Hilbert space is finite-dimensional, the spacetime coordinates are finite
dimensional square matrices. We call this kind of noncommutative geometry as fuzzy geometry or
matrix geometry and it is deeply related to matrix models of superstring theory and M-theory.
In the study of the fuzzy geometry, a theory called the matrix regularization [4] plays an essential role to uncover the relationship between the commutative geometry and the fuzzy geometry.
The matrix regularization is a map from functions on a manifold to corresponding matrices on a
fuzzy geometry. Using this map, one can construct a matrix model of superstring theory (or Mtheory) from a world-sheet theory of a single string (or world-volume theory of a single membrane).
Therefore, this theory is important to study the relationship between string (or a membrane) states
to the corresponding matrix states. Let us briefly discuss a mathematical aspect of matrix regularization. Let us consider a symplectic manifold (M, ω), which is an even-dimensional manifold
equipped with a nondegenerate closed two-form ω. Let 2n be the dimension of M . From the
symplectic structure ω, one can naturally define a volume form µω := ω ∧n /n! and the Poisson
bracket {f, g} = Xf g. Here, f, g are smooth functions on M and Xf is the Hamiltonian vector
field. In this sense, a symplectic manifold is a mathematical generalization of the phase space.
Now, let us also assume M is closed, i.e. M is a compact manifold without boundary. ...