UC Berkeley UC Berkeley Electronic Theses and Dissertations Title MONOIDAL STRUCTURE IN MIRROR SYMMETRY AND NONCOMMUTATIVE GEOMETRY Permalink https://escholarship.org/uc/item/39b741zt Author Do, Hanh Duc Publication Date 2013 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California MONOIDAL STRUCTURE IN MIRROR SYMMETRY AND NONCOMMUTATIVE GEOMETRY by Hanh Duc Do A dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley. Committee in charge: Professor Alan Weinstein, Chair Professor Marc Rieffel Professor Ori Ganor Fall 2013 MONOIDAL STRUCTURE IN MIRROR SYMMETRY AND NONCOMMUTATIVE GEOMETRY Copyright 2013 by Hanh Duc Do Abstract Monoidal Structure in Mirror Symmetry and Noncommutative Geometry by Hanh Duc Do Doctor of Philosophy in Mathematics at the University of California, Berkeley Professor Alan Weinstein, Chair In the thesis, we initial first steps in understanding Quantum mirror symmetry and noncommutative compactification of moduli spaces of tori. To obtain a global invariant of noncommutative torus bundle, we study the monodromy of Gauss- Manin connection on periodic cyclic homology groups of Heisenberg group. Then the global monodromy map is developed, and provides a criterion to detect when a noncommutative torus bundle is dequantizable. A process to construct the de- quantizing Poisson manifold is given, when the dequantization criterion is satisfied. Naively, it seems that the Morita theory for noncommutative torus bundles can be developed naturally as in Rieffel theory for rotation algebras. However, this assumption turns out to be wrong; the Morita class of a non-dequantizable non- commutative torus bundle is not a classical object in the category of C∗−algebras, and we call them C∗−stacks. Even with the extended notion of C∗−stacks, the Morita theory is still incompatible with noncommutative torus bundles with the infinite Poisson limit. There is no rotation algebra that can be used to compact- ify the moduli space of rotation algebras, even though we know that the “infinite Poisson rotation algebra” is strongly Morita equivalent to the classical torus. This subtlety is completely solved by a new mathematical structure hidden behind the quantization of constant Dirac structures on the tori. We develop a new theory of quantum spaces called spatial structure to give a better understanding of quantum spaces and torus fibrations. We clarify some examples of spatial algebras and construct a monoidal structure from a given spatial structure. Using Hilsum-Skandalis maps between groupoids, we find that a groupoid presentation of a C∗- algebra implies a monoidal structure on the category of representations. We decompose the spatial product of the cyclic modules over the rotation algebras as an example, and propose a conjecture that a quantum mirror symmetry lies behind the spatial structure and the Hopfish structure in the sense of Tang and Weinstein. 1 to Ha, Anan and my parents i Contents Contents ii List of Figuresv List of Tables vi 1 Introduction1 2 Fields of noncommutative two-tori and Dequantization8 2.1 Noncommutative two-tori....................9 2.1.1 Definitions of Noncommutative Two Tori........9 2.1.2 Automorphisms of Noncommutative two-torus..... 12 2.2 Morita Equivalence of Rings and Algebras........... 15 2.3 Continuous Fields of C∗−Algebras............... 19 2.3.1 Definitions of Continuous Fields............ 19 2.3.2 Strict Deformation Quantization and Continuous Fields 23 2.4 Moduli Stack........................... 25 2.4.1 A review of moduli stacks................ 26 2.4.2 Homotopically Trivial Family of Noncommutative two- tori............................. 30 2.4.3 Relationship with Echterhoff approach......... 43 2.4.4 Statement of the monodromy theorem......... 47 2.5 DG Algebras........................... 49 2.6 Periodic Cyclic Homology.................... 54 2.6.1 Concrete Approach.................... 54 2.6.2 Abstract approach.................... 56 2.6.3 Periodic cyclic homology................. 59 2.7 Kontsevich formulation and formality.............. 61 ii CONTENTS 2.7.1 Semi-classical side.................... 61 2.7.2 Algebraic side....................... 62 2.7.3 Formality for the Hochschild cochains.......... 63 2.7.4 Twisting Procedure and tangent cohomology...... 64 2.8 Character Map and Monodromy Theorem........... 67 2.8.1 Gauss-Manin connection................. 67 2.8.2 Main proof for the Formal Case............. 71 2.8.3 Proof for the smooth case:................ 75 2.9 A global invariant......................... 76 2.9.1 Monodromy map..................... 76 2.9.2 Dequantization...................... 80 2.9.3 Morita equivalence.................... 83 2.9.4 Bundle of C∗−Stacks and compactification of the mod- uli spaces......................... 87 3 Spatial Structures 94 3.1 Noncommutative Algebra Motivation.............. 94 3.2 Background on C∗− functor................... 98 3.2.1 C∗−correspondence.................... 98 3.2.2 Groupoid Correspondence................ 98 3.2.3 C∗−Functor........................ 100 3.3 Spatial Structure......................... 102 3.3.1 Definition and examples................. 102 3.3.2 Spatial structure from groupoids............ 104 3.3.3 Spatial structure and Morita equivalence........ 107 3.3.4 Monoidal category.................... 109 3.3.5 Crossed Product of Spatial Algebras.......... 113 3.4 Monoidal Structure For Rotation Algebras........... 116 3.4.1 Case q1 =6 0 and q2 =6 0:................. 118 3.4.2 Only one of q1 or q2 = 0:................ 120 3.4.3 Case q1 = q2 = 0:.................... 120 3.5 Spatial structures and continuous fields............. 123 A Appendix 127 A.1 Groupoids and Stacks....................... 127 A.2 KK−category........................... 128 A.3 Torus bundle........................... 131 iii CONTENTS Bibliography 133 Index 142 iv List of Figures 2.4.1 The Rieffel Projection...................... 36 2.4.2 Moduli space of zigzags...................... 41 2.4.3 Constant noncommutative bundle............... 42 2.4.4 Zigzag of the discrete Heisenberg group............. 42 2.4.5 The zigzag of roof functions................... 43 2.4.6 Trivial monodromy noncommutative two-tori bundles..... 46 2.6.1 The product of Hochschild cochains............... 57 2.9.1 Decomposition of monodromy.................. 77 2.9.2 Normalized monodromy..................... 78 2.9.3 S and T .............................. 80 2.9.4 Cut and paste........................... 80 2.9.5 Compactification of the moduli space of noncommutative two- tori................................. 89 2.9.6 Moduli space of quantization of the Dirac structures...... 92 3.3.1 Spatial product.......................... 113 3.4.1 Spatial monoidal product..................... 123 v List of Tables 1.0.1 Triangle of areas.........................3 2.0.1 Main approach..........................9 2.6.1 Noncommutative Geometry Correspondence.......... 61 3.1.1 Stack to Category Correspondence............... 95 3.1.2 Monoidal structures and Quantization............. 97 vi List of Symbols Tθ Noncommutative torus Φλ,µ Action of classical torus τ Canonical invariant trace on noncommutative tori Γ(E) space of sections of the vector bundle E ∞ Tθ Smooth noncommutative torus h., .iA Rieffel inner product ZM(A) center of multiplier algebra of A {., .} Poisson bracket (A, G, φ) C∗-dynamical system A o G Full crossed product A or G Reduced crossed product P rim(A) Primitive spectrum ∗~ Star product MC(L) Goldman-Millson groupoid HH•(A) Hochschild cohomology of algebra A HH•(A) Hochschild homology of algebra A α CCp,q,n Coefficent of Cyclic module vii LIST OF TABLES AoG IndA ∆ Induced spatial algebra α Tp,q Cyclic module Ai Ai (Ai, ∆ , ) Spatial algebra (E, ρ, F ) KK-cocycle KK•(A, B)) KK-group HP• periodic cyclic homology PC•(A) periodic cyclic complex viii Acknowledgments I am heartily thankful to my supervisor, Alan Weinstein, whose encour- agement, guidance and support from the initial to the final level enabled me to develop an understanding of the subject. His knowledge, mathematical intuition and especially his continuous encouragement were always beside me, even when I felt hopeless in research. I would like to thank Professor Marc Rieffel for teaching me C∗−algebra and giving me lots of helpful discussions to enlighten me to the subject of Noncommutative Geometry, which becomes a very essential part in the thesis. I also want to thank Nathan, Aaron, Benoit, Santiago and Sobhan... my beloved colleagues. Their points of views in many discussions also help me a lot in the thesis. I also thank Jaeyoung, Shenghao, Matt and Kevin for very helpful mid- night seminar on algebraic geometry and algebraic stacks, the areas although seeming unrelated, but lying behind everything here. Jaeyoung, my beloved roommate is a very wonderful guy, walking home together with me every midnight. An Huang is a wonderful friend who told me everything about QFTs. There are a lot of people helping me with polishing the thesis, and I thank them a lot, especially Nathan, Sobhan, Trinh, Lan, Shillin, Jenny, Kevin, Hung, Santiago, Benoit... I also have to thank Izzy Tran, Giang Nguyen and Hung Tran for helping me with paperwork. Without their helps, the thesis could not be submitted. Thank my wife for constant support, and giving me Anan. Finally, thanks to Ying-Yang theory for being awesome and being my great