Non-conformal geometry on noncommutative two tori Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Chao Xu, B.A. Graduate Program in Department of Mathematics The Ohio State University 2019 Dissertation Committee: Henri Moscovici, Advisor Ovidiu Costin Michael Davis David Penneys c Copyright by Chao Xu 2019 Abstract On the spectral triple of a noncommutative manifold (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−tD2 . 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 Aθ is the study of quantized Gaussian curvature under noncommutative conformal change of \metric" by a positive operator-valued Weyl factor k = eh; 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 k1; k2. The first part of this paper, inspired by work by L.Dabrowski and S.Andrzej, contains ex- tension of the rearrangement lemma that was systematized by M.Lesch, to non-conformal P 2 2 operators, by which we mean the elliptic operators with principal symbol j kj ξj with distinct k1; :::; km. By adapting the technique used by Y.Liu, we interpret the result of rearrangement as generalized hyper-geometric functions on Grassmannians, generalizing the conformal ii results of Y.Liu , namely when k1 = k2. Second part of this paper consists of calculation of scalar curvature density associated m to a non-conformal Laplacian operator ∆k1;:::;km on a m-torus AΘ . Third part is calculation of index density of a non-conformal Dirac operator Dk1;k2;E(g,θ) on the Heisenberg module E(g; θ). 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 End o (T (A )) of the cotangent bundle together with a Aθ θ 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 gener- alization, and propositions that will be needed in the statement of the rearrangement lemma. iii Acknowledgments I would like to show my most sincere gratitude to my advisor Henri Moscovici for his generosity to share his insightful ideas with me and his great support to my academic career through out these years. I also thank my colleague and good friend Yang Liu for guiding me in pseudo-differential calculus, and the other meaningful discussion we had during his stay in OSU, and during several other noncommutative geometry conferences. It is his method of handling the \rearrangement integral" differently that enable me to extend rearrangement lemma to this non-conformal case. I would like to thank David Penneys for the noncommutative geometry seminars, for the lessons on Von Neumann algebra and subfactor theory. Also I thank Tao Yang, Joseph Migler for helpful discussion about various topic on noncommutative geometry. In addition, my academic life would not have been possible without all the uncondi- tional love and support from my wife, my parents, my grandparents. iv Vita 2012 . B.A. Zhejiang University 2013-present . Graduate Teaching Associate, The Ohio State University. Fields of Study Major Field: Department of Mathematics v Table of Contents Page Abstract . ii Acknowledgments . iv Vita............................................v 1. Introduction . .1 1.1 What are the objects in noncommutative geometry . .1 1.2 Noncommutative conformal geometry on two tori . .4 1.3 Metric change beyond conformal . .5 1.4 Noncommutative scalar curvature, heat expansion, Zeta functions . .8 1.5 Heat kernel, symbolic calculus and rearrangement lemma . 11 1.5.1 Resolvent expansion . 12 1.5.2 Trace formula . 12 2. Notation and preliminaries . 14 2.1 noncommutative tori Aθ ........................... 14 2 2.1.1 Heisenberg modules E(g; θ) over a two-torus Tθ .......... 15 2.2 Multiindex . 18 2.3 Functional calculus . 18 2.3.1 A⊗n+1 as an alternating multiplier functional on A⊗n ....... 20 2.3.2 Fubini condition, substitution lemma . 23 2.4 Gaussian average . 25 2.5 Partitions . 26 2.6 Hypergeometric integral and hypergeometric functions from rearrangement 27 2.6.1 Hypergeometric integral . 28 2.6.2 Hypergeometric functions on Grassmannian . 28 vi 3. Main technical lemma . 31 3.1 Rearrangement lemma . 33 3.2 Proof of rearrangement lemma . 36 4. Main result . 42 4.1 Application on untwisted module over m-torus . 42 4.1.1 Pseudodifferential calculus on torus . 42 P 2 2 4.1.2 Asymmetric Laplacian P = j kj δj ................. 44 4.1.3 Density R2(P )............................ 51 4.1.4 Gauss-Bonnet function relations . 56 4.2 Application on twisted module over 2-torus . 57 4.2.1 Symbol calculus on Heisenberg module . 58 γ 4.2.2 Index density R2 ........................... 61 Appendices 71 2 A. non-conformal metric from conformal change on End(T (Tθ)) . 71 A.1 Conformal twist . 71 A.2 Complex differential forms Ω(p;q)(A).................... 74 A.3 Construction of asymmetric Dirac . 76 A.3.1 motivation from Riemannian geometry . 76 A.3.2 twisted Ω0 ............................... 77 A.3.3 twisted Ω1 ............................... 78 A.3.4 twisted Ω(0;1) and twisted Dolbeault operator . 78 A.4 Twisted Dolbeault operator on E(g; θ)................... 80 B. Gaussian integral . 82 C. Generalized hypergeometric function . 87 Bibliography . 92 vii Chapter 1: Introduction 1.1 What are the objects in noncommutative geometry In noncommutative geometry, cf.[7], one studies geometric spaces (topological, differ- entiable, measurable, etc.) in terms of coordinates, in the form of algebra of operators. For example, by a noncommutative topological space one means a C∗ algebra A. In the commutative case A = C(X) for some topological space X by Gelfand-Naimark Theorem. By a noncommutative measure space one means a Von Neumann algebra B. In the commutative case B = L1(X; µ) for some σ-finite measure space (X; µ). A spectral triple (A; H; D) is, by definition, collection of information containing ∗-algebra A , a Hilbert space H , self-adjoint unbounded operator D 2 L(H) , and represenatation of A ! B(H), such that [D; A] ⊂ B(H). This will be a very important object in noncommutative geometry, because many of the geometric property will be characterized in terms of spectral property of D and [D; A]. 1 The Riemannian metric g on a manifold M, if spinc structure V ! E ! M exist, can be captured by the corresponding spectral triple (C1(M);L2(M; E);D) as distg(p; q) = sup jf(p) − f(q)j; [D; f] = df f;jdf|≤1 In [8, 9], Connes proved that a commutative spectral triple will recover the spinc structure if the following axioms, firstly listed in [10], are satisfied. 1. D has compact resolvent, namely (D−λ)−1 is a compact operator on H and therefore this self-adjoint operator D has eigenvalues λn ! 1; n ! 1. 1 1 m 2. The eigenvalues of D are of growth rate m , namely λn = O(n ) as n ! 1. Here m is the dimension of the underlying manifold. 3. D should be an \order one" operator, in the sense that [[D; a]; JbJ] = 0, for any a; b 2 A, assuming the existence of a C-skewlinear operator J : A!A such that JAJ −1 being an Ao action. 4. δ = [jDj; ·] should defines a derivation on A, and both A and [D; A] are δ-smooth, l namely A; [D; A] ⊂ \lDom(δ ) 1 l 5. The smooth part of H, H = \lDom(D ) should be finitely generated projective Hilbert A-module, with A-valued inner product (; ) , and −m < ξ; η >H= T r!((ξ; η)jDj ) where T r! on the right hand side is the Dixmier trace associated to taking limit along ultra filter ! 2 L1(R)∗, which detect logarithm growth in partial sum: PN 1 λj(A) T r!(A) = lim ! log(N) 2 6. Moreover, there exists an orientation class [c] in Hochschild Homology Hn(A), cf.[7, 8, 9, 10, 38], such that if (A; H;D) is an odd spectral triple, π(c) = 1; if (A; H;D) is an even spectral triple, there should be a grading map γ : H!H, with γ2 = 1; γ∗ = γ; γD = Dγ; π(c) = γ Example 1.1.1. A natural noncommutative example of spectral triple is from a noncom- 2 mutative torus Aθ, see Def. 2.1, together with Dq = δ1 + qδ2; I(q) 6= 0 and H = L (A; τ) being the Hilbert space closure w.r.t. the tracial state τ on A by the GNS construc- n1 n2 n1 n2 tion. Note that δj(U1 U2 ) = 2πinjU1 U2 . It is not hard to verify that (Aθ; H;Dq) is a spectral triple. Example 1.1.2 (even spectral triple of Dolbeault operator on Aθ, cf.[6]). A natural even spectral triple can also be constructed from a noncommutative torus. As the cotangent bundle on a two torus T2 is trivializable, one can identify (0; 1)-forms (0;1) 2 ∼ 1 2 Ω (T ) = C (T ) by dividing by dz = dx − qdy.