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Non-conformal geometry on noncommutative two tori

Xu, Chao
http://orcid.org/0000-0001-7627-1315
http://rave.ohiolink.edu/etdc/view?acc_num=osu1566225527101998

Abstract Details

2019, Doctor of Philosophy, Ohio State University, Mathematics.
On the spectral triple of a noncommutative manifold $(\mathcal{A},H,D)$, despite the absence of underlying space of points, one can still consider its scalar curvature in terms of spectral information of the Dirac operator $D$, for example using short-time asymptotic expansion of the heat kernel $e^{-tD^2}$. In the recent decade, the conformal theory on a noncommutative two tori was firstly started by Connes and Tretkoff(nee Cohen), and later greatly developed by Connes, Moscovici and many others. Noncommutative conformal geometry on a noncommutative torus $\mathcal{A}_\theta$ is the study of quantized Gaussian curvature under noncommutative conformal change of ``metric'' by a positive operator-valued Weyl factor $k=e^h,h^*=h$. In this dissertation, by using Lesch and Moscovici's extension of Connes' pseudo-differential calculus to the Heisenberg modules, we will calculate the scalar curvature of a non-conformal change of metric by means of two commuting positive operator-valued factors $k_1,k_2$.\\ The first part of this paper, inspired by work by L.Dabrowski and S.Andrzej, contains extension of the rearrangement lemma that was systematized by M.Lesch, to non-conformal operators, by which we mean the elliptic operators with principal symbol $\sum_j k_j^2\xi_j^2$ with distinct $k_1,...,k_m$. By adapting the technique used by Y.Liu, we interpret the result of rearrangement as generalized hyper-geometric functions on Grassmannians, generalizing the conformal results of Y.Liu , namely when $k_1=k_2$. Second part of this paper consists of calculation of scalar curvature density associated to a non-conformal Laplacian operator $\Delta_{k_1,...,k_m}$ on a $m$-torus $\mathcal{A}_\Theta^m$. Third part is calculation of index density of a non-conformal Dirac operator $D_{k_1,k_2;\mathcal{E}(g,\theta)}$ on the Heisenberg module $\mathcal{E}(g,\theta)$. In appendix A, we will justify our terminology ``non-conformal''. We show that such a non-conformal Dirac operator on Heisenberg module amounts to a conformal change on the endomorphism algebra $\mathrm{End}_{\mathcal{A}^o_\theta}(\mathcal{T}^*(\mathcal{A}_\theta))$ of the cotangent bundle together with a change of complex structure. In appendix B, we put the elementary but crucial lemma of Gaussian averages, which will be used in the proof of extended rearrangement lemma. In appendix C, we list the definitions of hyper-geometric functions and their generalization, and propositions that will be needed in the statement of the rearrangement lemma.
Henri Moscovici (Advisor)
Ovidiu Costin (Committee Member)
Michael Davis (Committee Member)
David Penneys (Committee Member)
96 p.
Mathematics
noncommutative tori, pseudo-differential calculus, noncommutative scalar curvature, noncommutative scalar curvature density, index density, non-conformal change, rearrangement lemma, hypergeometric functions on Grassmannian

Recommended Citations

Citations

  • Xu, C. (2019). Non-conformal geometry on noncommutative two tori [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1566225527101998

    APA Style (7th edition)

  • Xu, Chao. Non-conformal geometry on noncommutative two tori. 2019. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1566225527101998.

    MLA Style (8th edition)

  • Xu, Chao. "Non-conformal geometry on noncommutative two tori." Doctoral dissertation, Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1566225527101998

    Chicago Manual of Style (17th edition)

osu1566225527101998
186
© 2019, some rights reserved.Creative Commons License Non-conformal geometry on noncommutative two tori by Chao Xu is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. Based on a work at etd.ohiolink.edu.
This open access ETD is published by The Ohio State University and OhioLINK.