University of Michigan
Quantization of symplectic cobordisms.
Degree: PhD, Pure Sciences, 1999, University of Michigan
In this work we construct a unitary operator acting between Spin c
quantizations of compact integral symplectic manifolds which are symplectically cobordant. The construction is based on a recent proof of the cobordism invariance of the index of Dirac operators, due to L. Nicolaescu. We prove that, away from a small set on the boundary manifolds, this quantized cobordism operator is an Hermite-Fourier integral operator in the sense of L. Boutet de Monvel and V. Guillemin. The proof consists of two parts. In the first we show that the Spin c
Szego&huml; projector, II, which defines the quantization, is itself an Hermite-Fourier integral operator just as in the well studied case of Kahler quantization. In the second part we explicitly construct the fundamental solution, K, of a boundary value problem for a transversally elliptic Dirac operator on a principal circle bundle over the cobordism. This construction is carried out within the category of Hermite-Fourier distributions and relies on their symbolic properties. For example, the transport equations are evolutions equations on <blkbd>R2n,</blkbd> in contrast with the case of Lagrangian distributions where they are ordinary differential equations. The microlocal structure of the quantized cobordism operator is the same as that of the composition P&j0;g&j0;K, where gamma is the boundary restriction operator. From the composition theorems for Hermite-Fourier integral operators it follows that P&j0;g&j0;K is an Hermite-FIO. In the last part of this work we present examples where our construction applies.
Advisors/Committee Members: Uribe, Alejandro (advisor).
Subjects/Keywords: Cobordisms; Dirac Operators; Quantization; Symplectic Manifolds
to Zotero / EndNote / Reference
APA (6th Edition):
Korpas, L. (1999). Quantization of symplectic cobordisms. (Doctoral Dissertation). University of Michigan. Retrieved from http://hdl.handle.net/2027.42/131922
Chicago Manual of Style (16th Edition):
Korpas, Levente. “Quantization of symplectic cobordisms.” 1999. Doctoral Dissertation, University of Michigan. Accessed January 23, 2021.
MLA Handbook (7th Edition):
Korpas, Levente. “Quantization of symplectic cobordisms.” 1999. Web. 23 Jan 2021.
Korpas L. Quantization of symplectic cobordisms. [Internet] [Doctoral dissertation]. University of Michigan; 1999. [cited 2021 Jan 23].
Available from: http://hdl.handle.net/2027.42/131922.
Council of Science Editors:
Korpas L. Quantization of symplectic cobordisms. [Doctoral Dissertation]. University of Michigan; 1999. Available from: http://hdl.handle.net/2027.42/131922