Loading...
Deformation Quantization over a Z-graded base
Altinay-Ozaslan, Elif
Altinay-Ozaslan, Elif
Citations
Altmetric:
Genre
Thesis/Dissertation
Date
2017
Advisor
Committee member
Group
Department
Mathematics
Subject
Permanent link to this record
Collections
Files
Research Projects
Organizational Units
Journal Issue
DOI
http://dx.doi.org/10.34944/dspace/661
Abstract
We investigate the problem how to describe the equivalence classes of formal deformations of a symplectic manifold $M$ in the case when we have several deformation parameters $\ve_1, \ve_2, \dots, \ve_g$ of non-positive degrees. We define formal deformations of $M$ over the base ring $\bbC[[\ve, \ve_1, \dots, \ve_g]]$ as Maurer-Cartan elements of the differential graded Lie algebra $(\ve, \ve_1, \dots, \ve_g) \sPD^\bullet(M)[[\ve, \ve_1, \dots, \ve_g]]$ where $\sPD^\bullet(M)$ denotes the algebra of polydifferential operators on $M$. The interesting feature of such deformations is that, if at least one formal parameter carries a non-zero degree, then the resulting Maurer-Cartan element corresponds to a $\bbC[[\ve, \ve_1, \dots, \ve_g]]$-multilinear $A_\infty$-structure on the graded vector space $\cO(M)[[\ve, \ve_1, \dots, \ve_g]]$ with the zero differential, where $\cO(M)$ is the algebra of smooth complex-valued functions $M$. This dissertation focuses on formal deformations of $\cO(M)$ with the base ring $\bbC[[\ve, \ve_1, \dots, \ve_g]]$ such that corresponding MC elements $\mu$ satisfy these two conditions: The Kodaira-Spencer class of $\mu$ is $\ve \al$ and $\mu$ satisfies the equation $\mu \rvert_{\ve=0} =0$. The main result of this study gives us a bijection between the set of isomorphism classes of such deformations and the set of all degree 2 vectors of the graded vector space \, $\bigoplus_{q \geq 0} \, (\ve, \ve_1, \dots, \ve_g) \, H^q(M, \bbC)[[\ve, \ve_1, \dots, \ve_g]]$ where $H^\bullet(M, \bbC)$ is the singular cohomology of $M$ with coefficients in $\bbC$ and every vector of $H^q(M, \bbC)$ carries degree $q$.
Description
Citation
Citation to related work
Has part
ADA compliance
For Americans with Disabilities Act (ADA) accommodation, including help with reading this content, please contact scholarshare@temple.edu
Embedded videos
License
IN COPYRIGHT- This Rights Statement can be used for an Item that is in copyright. Using this statement implies that the organization making this Item available has determined that the Item is in copyright and either is the rights-holder, has obtained permission from the rights-holder(s) to make their Work(s) available, or makes the Item available under an exception or limitation to copyright (including Fair Use) that entitles it to make the Item available.