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Arithmetic and Geometry Around Quantication (Progress In Progress in Mathematics Volume 279 Series Editors Hyman Bass Joseph Oesterlé Alan Weinstein F or other titles in this series, go to http://www.springer.com/series/4848 Arithmetic and Geometry Around Q uantization Özgür Ceyhan Yuri I. Manin Matilde Marcolli Editors Birkhäuser Boston • Basel • Berlin Editors Ö zgü r Ceyhan Yuri I. Manin MPI fü r Mathematik MPI fü r Mathematik V ivatsgasse 7 V ivatsgasse 7 53111 Bonn 53111 Bonn Germany Germany [email protected] [email protected] Matilde Marcolli California Institute of Technology Department of Mathematics 1200 E . California Blvd. Pasadena, CA 91125 USA [email protected] ISBN 978-0-8176-4830-5 e-ISBN 978-0-8176-4831-2 DOI 10.1007/978-0-8176-4831-2 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009942422 Mathematics Subject Classification (2000): 53C38, 17B67 , 11F 27, 18D05 ,18E 30, 14N35 © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhäuser Boston, c/o Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Birkhäuser Boston is part of Springer Science+Business Media (www.birkhauser.com) Preface Quantization has been a potent source of interesting ideas and problems in various branches of mathematics. A European Mathematical Society activity, the Arithmetic and Geometry around Quantization (AGAQ) conference has been organized in order to present hot topics in and around quantization to younger mathematicians, and to highlight possible new research directions. This volume comprises lecture notes, and survey and research articles orig- inating from AGAQ. A wide range of topics related to quantization is covered, thus aiming to give a glimpse of a broad subject in very different perspectives • Symplectic and algebraic geometry, in particular, mirror symmetry and related topics by S. Akbulut, G. Ben Simon, O.¨ Ceyhan, K. Fukaya, S. Salur. • Representation theory, in particular quantum groups, the geometric Langlands program and related topics by S. Arkhipov, D. Gaitsgory, E. Frenkel, K. Kremnizer. • Quantum ergodicity and related topics by S. Gurevich, R. Hadani. • Non-commutative geometry and related topics by S. Mahanta, W. van Suijlekom. In their chapter, Akbulut and Salur introduce a new construction of certain ‘mirror dual’ Calabi–Yau submanifolds inside of a G2 manifold. The question of constructing central extensions of (2-)groups using (2-)group actions on cat- egories is addressed by Arkhipov and Kremnizer. In his chapter, Ceyhan intro- duces quantum cohomology and mirror symmetry to real algebraic geometry. Frenkel and Gaitsgory discuss the representation theory of affine Kac–Moody algebras at the critical level and the local geometric Langlands program of the authors. Motivated by constructions of Lagrangian Floer theory, Fukaya explains the fundamental structures such as A∞-structure and operad struc- tures in Lagrangian Floer theory using an abstract and unifying framework. Ben Simon’s chapter provides the first results about the universal covering vi Preface of the group of quantomorphisms of the prequantization space. In their two chapters, Hadani and Gurevich give a detailed account of self-reducibility of the Weil representation and quantization of symplectic vector spaces over fi- nite fields. These two subjects are the main ingredients of their proof of the Kurlberg–Rudnick Rate Conjecture. The expository chapter of Mahanta is a survey on the construction of motivic rings associated to the category of differential graded categories. Finally, van Suijlekom introduces and discusses Connes–Kreimer type Hopf algebras of Feynman graphs that are relevant for quantum field theories with gauge symmetries. Yuri Tschinkel Yuri Zarhin Contents Preface ........................................................ v Mirror Duality via G2 and Spin(7) Manifolds Selman Akbulut and Sema Salur ................................... 1 2-Gerbes and 2-Tate Spaces Sergey Arkhipov and Kobi Kremnizer ............................... 23 The Geometry of Partial Order on Contact Transformations of Pre-Quantization Manifolds Gabi Ben Simon ................................................. 37 Towards Quantum Cohomology of Real Varieties Ozg¨¨ ur Ceyhan ................................................... 65 Weyl Modules and Opers without Monodromy Edward Frenkel and Dennis Gaitsgory ..............................101 Differentiable Operads, the Kuranishi Correspondence, and Foundations of Topological Field Theories Based on Pseudo-Holomorphic Curves Kenji Fukaya ....................................................123 Notes on the Self-Reducibility of the Weil Representation and Higher-Dimensional Quantum Chaos Shamgar Gurevich and Ronny Hadani ..............................201 Notes on Canonical Quantization of Symplectic Vector Spaces over Finite Fields Shamgar Gurevich and Ronny Hadani ..............................233 viii Contents Noncommutative Geometry in the Framework of Differential Graded Categories Snigdhayan Mahanta .............................................253 Multiplicative Renormalization and Hopf Algebras Walter D. van Suijlekom ..........................................277 Mirror Duality via G2 and Spin(7) Manifolds Selman Akbulut∗ and Sema Salur Dept. of Mathematics, Michigan State University, East Lansing MI, 48824 [email protected] Dept. of Mathematics, University of Rochester, Rochester NY, 14627 [email protected] Summary. The main purpose of this chapter is to give a construction of certain “mirror dual” Calabi–Yau submanifolds inside of a G2 manifold. More specifically, we explain how to assign to a G2 manifold (M,ϕ,Λ), with the calibration 3-form ϕ andanoriented2-planefieldΛ, a pair of parametrized tangent bundle valued 2- and 3-forms of M. These forms can then be used to define different complex and symplectic structures on certain 6-dimensional subbundles of T (M). When these bundles are integrated they give mirror CY manifolds. In a similar way, one can 8 8 define mirror dual G2 manifolds inside of a Spin(7) manifold (N ,Ψ). In case N admits an oriented 3-plane field, by iterating this process we obtain Calabi–Yau submanifold pairs in N whose complex and symplectic structures determine each other via the calibration form of the ambient G2 (or Spin(7)) manifold. AMS Classification Codes (2000). 53C38, 53C29 1 Introduction 7 Let (M ,ϕ)beaG2 manifold with a calibration 3-form ϕ.Ifϕ restricts to be the volume form of an oriented 3-dimensional submanifold Y 3,thenY is called an associative submanifold of M. Associative submanifolds are very interesting objects as they behave very similarly to holomorphic curves of Calabi–Yau manifolds. In [AS], we studied the deformations of associative submanifolds of (M,ϕ) in order to construct Gromov-Witten-like invariants. One of our main obser- vations was that oriented 2-plane fields on M always exist by a theorem of Thomas [T], and by using them one can split the tangent bundle T (M)= E ⊕ V as an orthogonal direct sum of an associative 3-plane bundle E and a complex 4-plane bundle V. This allows us to define “complex associative submanifolds” of M, whose deformation equations may be reduced to the ∗ partially supported by NSF grant 0805858 DMS 0505638 O.¨ Ceyhan et al. (eds.), Arithmetic and Geometry Around Quantization, Progress in Mathematics 279, DOI: 10.1007/978-0-8176-4831-2 1, c Springer Science + Business Media LLC 2010 2 S. Akbulut and S. Salur Seiberg–Witten equations, and hence we can assign local invariants to them, and assign various invariants to (M,ϕ,Λ), where Λ is an oriented 2-plane field on M. It turns out that these Seiberg–Witten equations on the submanifolds are restrictions of global equations on M. In this chapter, we explain how the geometric structures on G2 mani- folds with oriented 2-plane fields (M,ϕ,Λ) provide complex and symplectic structures to certain 6-dimensional subbundles of T (M). When these bun- dles are integrated we obtain a pair of Calabi–Yau manifolds whose complex and symplectic structures are remarkably related into each other. We also study examples of Calabi–Yau manifolds which fit nicely in our mirror set- up. Later, we do similar constructions for Spin(7) manifolds with oriented 3-plane fields. We then explain how these structures lead to the definition of “dual G2 manifolds” in a Spin(7) manifold, with their own dual Calabi–Yau submanifolds. In the main part of this chapter we give a geometric approach to the “mirror duality” problem. One aspect of this problem is about finding a pair of Calabi–Yau 3-folds whose complex and symplectic structures are related to each other, more specifically their Hodge numbers are related by hp,q ↔ h3−p,q. We propose a method of finding such pairs in 7-dimensional 6 3 3 2 4 G2-manifolds. The basic examples are the six-tori T = T × T ↔ T × T 6 1 in the G2 manifold T × S . Also, by this method we pair the
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