The Fock bundle. (English) Zbl 0721.46040
Analysis and partial differential equations, Coll. Pap. dedic. Mischa Cotlar, Lect. Notes Pure Appl. Math. 122, 301-326 (1990).
[For the entire collection see Zbl 0684.00014.]
The author begins by recalling the classical invariance properties of the Jacobi theta functions and of the (closely related) solution to the heat equation. Although the invariance properties can be verified by direct computation, the author’s aim here is to “understand the invariance... from a higher point of view”.
Roughly, he constructs a Hermitian vector bundle whose fibers are Fock spaces (infinite dimensional Hilbert spaces of entire functions) and whose base is the Siegel generalized upper half plane. The invariance properties for the (many variable) theta function are then seen as the condition for parallelism of the natural connection on this bundle. On the way to this conclusion the author presents background material on symplectic and Hermitian geometry. He also shows how this viewpoint leads to the Bergmann-Siegel representation of the Heisenberg group and the Weil-Shale reprsentation of the metaplectic group.
It will be interesting to see what else flows from this novel and sophisticated point of view.
The author begins by recalling the classical invariance properties of the Jacobi theta functions and of the (closely related) solution to the heat equation. Although the invariance properties can be verified by direct computation, the author’s aim here is to “understand the invariance... from a higher point of view”.
Roughly, he constructs a Hermitian vector bundle whose fibers are Fock spaces (infinite dimensional Hilbert spaces of entire functions) and whose base is the Siegel generalized upper half plane. The invariance properties for the (many variable) theta function are then seen as the condition for parallelism of the natural connection on this bundle. On the way to this conclusion the author presents background material on symplectic and Hermitian geometry. He also shows how this viewpoint leads to the Bergmann-Siegel representation of the Heisenberg group and the Weil-Shale reprsentation of the metaplectic group.
It will be interesting to see what else flows from this novel and sophisticated point of view.
Reviewer: R.Rochberg (St.Louis)
MSC:
| 46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |
| 46E20 | Hilbert spaces of continuous, differentiable or analytic functions |
| 30C40 | Kernel functions in one complex variable and applications |
| 47B38 | Linear operators on function spaces (general) |
| 47F05 | General theory of partial differential operators |
