The Dirac Spectrum on Manifolds with Gradient Conformal Vector Fields Andrei Moroianu, Sergiu Moroianu
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Arxiv:1908.09677V4 [Math.AG] 13 Jul 2021 6 Hr Ro Fterm12.1 Theorem of Proof Third a 16
AN ANALYTIC VERSION OF THE LANGLANDS CORRESPONDENCE FOR COMPLEX CURVES PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN In memory of Boris Dubrovin Abstract. The Langlands correspondence for complex curves is traditionally for- mulated in terms of sheaves rather than functions. Recently, Langlands asked whether it is possible to construct a function-theoretic version. In this paper we use the algebra of commuting global differential operators (quantum Hitchin Hamilto- nians and their complex conjugates) on the moduli space of G-bundles of a complex algebraic curve to formulate a function-theoretic correspondence. We conjecture the existence of a canonical self-adjoint extension of the symmetric part of this al- gebra acting on an appropriate Hilbert space and link its spectrum with the set of opers for the Langlands dual group of G satisfying a certain reality condition, as predicted earlier by Teschner. We prove this conjecture for G = GL1 and in the simplest non-abelian case. Contents 1. Introduction 2 Part I 8 2. Differential operators on line bundles 8 3. Differential operators on BunG 11 4. The spectrum and opers 15 5. The abelian case 19 6. Bundles with parabolic structures 26 1 7. The case of SL2 and P with marked points 28 arXiv:1908.09677v4 [math.AG] 13 Jul 2021 8. Proofs of two results 30 Part II 33 9. Darboux operators 34 10. EigenfunctionsandmonodromyforDarbouxoperators 38 11. Essentially self-adjoint algebras of unbounded operators 43 12. The main theorem 49 13. Generalized Sobolev and Schwartz spaces attached to the operator L. 49 14. Proof of Theorem 12.1 60 15. -
Arxiv:2107.01242V1 [Math.OA] 2 Jul 2021 Euneo Ievle ( Eigenvalues of Sequence a Eetdacrigt T Agbac Utpiiy Y[ by Multiplicity
CONNES’ INTEGRATION AND WEYL’S LAWS RAPHAEL¨ PONGE Abstract. This paper deal with some questions regarding the notion of integral in the frame- work of Connes’s noncommutative geometry. First, we present a purely spectral theoretic con- struction of Connes’ integral. This answers a question of Alain Connes. We also deal with the compatibility of Dixmier traces with Lebesgue’s integral. This answers another question of Alain Connes. We further clarify the relationship of Connes’ integration with Weyl’s laws for compact operators and Birman-Solomyak’s perturbation theory. We also give a ”soft proof” of Birman-Solomyak’s Weyl’s law for negative order pseudodifferential operators on closed mani- fold. This Weyl’s law yields a stronger form of Connes’ trace theorem. Finally, we explain the relationship between Connes’ integral and semiclassical Weyl’s law for Schr¨odinger operators. This is an easy consequence of the Birman-Schwinger principle. We thus get a neat link between noncommutative geometry and semiclassical analysis. 1. Introduction The quantized calculus of Connes [18] aims at translating the main tools of the classical infin- itesimal calculus into the operator theoretic language of quantum mechanics. As an Ansatz the integral in this setup should be a positive trace on the weak trace class L1,∞ (see Section 2). Natural choices are given by the traces Trω of Dixmier [23] (see also [18, 34] and Section 2). These traces are associated with extended limits. Following Connes [18] we say that an operator A L is measurable when the value of Tr (A) is independent of the extended limit. -
WEYL's LAW on RIEMANNIAN MANIFOLDS Contents 1. the Role of Weyl's Law in the Ultraviolet Catastrophe 1 2. Riemannian Manifol
WEYL'S LAW ON RIEMANNIAN MANIFOLDS SETH MUSSER Abstract. We motivate Weyl's law by demonstrating the relevance of the distribution of eigenvalues of the Laplacian to the ultraviolet catastrophe of physics. We then introduce Riemannian geometry, define the Laplacian on a general Riemannian manifold, and find geometric analogs of various concepts in Rn, along the way using S2 as a clarifying example. Finally, we use analogy with Rn and the results we have built up to prove Weyl's law, making sure at each step to demonstrate the physical significance of any major ideas. Contents 1. The Role of Weyl's Law in the Ultraviolet Catastrophe 1 2. Riemannian Manifolds 3 3. Weyl's Law 11 4. Acknowledgements 20 References 20 1. The Role of Weyl's Law in the Ultraviolet Catastrophe In the year 1900 Lord Rayleigh used the equipartition theorem of thermodynamics to de- duce the famous Rayleigh-Jeans law of radiation. Rayleigh began with an idealized physical concept of the blackbody; a body which is a perfect absorber of electromagnetic radiation and which radiates all the energy it absorbs independent of spatial direction. We sketch his proof to find the amount of radiation energy emitted by the blackbody at a given frequency. Take a blackbody in the shape of a cube for definiteness, and assume it is made of con- ducting material and is at a temperature T . We will denote the cube by D = [0;L]3 ⊆ R3. Inside this cube we have electromagnetic radiation that obeys Maxwell's equations bouncing around. The equipartition theorem of thermodynamics says that every wave which has con- stant spatial structure with oscillating amplitude, i.e. -
Geometry of Noncommutative Three-Dimensional Flat Manifolds
Philosophiae Doctor Dissertation GEOMETRY OF NONCOMMUTATIVE THREE-DIMENSIONAL FLAT MANIFOLDS Piotr Olczykowski Krakow´ 2012 Dla moich Rodzic´ow Contents Introduction ix 1 Preliminaries 1 1.1 C∗−algebras . .1 1.2 Gelfand-Naimark-Seagal Theorem . .3 1.2.1 Commutative Case . .4 1.2.2 Noncommutative Case . .4 1.3 C∗-dynamical systems . .7 1.3.1 Fixed Point Algebras of C(M)..............8 1.4 K−theory in a Nutshell . .8 1.5 Fredholm Modules in a Nutshell . 10 1.5.1 Pairing between K−theory and K−homology . 12 1.5.2 Unbounded Fredholm Modules . 12 2 Spectral Triples 15 2.1 Spin Structures . 15 2.1.1 Clifford Algebras . 15 2.1.2 SO(n) and Spin(n) Groups . 16 2.1.3 Representation of the Clifford Algebra . 17 2.1.4 Spin Structures and Bundles . 18 2.2 Classical Dirac Operator . 20 2.3 Real Spectral Triple { Definition . 23 2.3.1 Axioms . 24 2.3.2 Commutative Real Spectral Triples . 27 3 Noncommutative Spin Structures 29 3.1 Noncommutative Spin Structure . 29 3.2 Equivariant Spectral Triples - Definition . 31 3.3 Noncommutati Tori . 32 3.3.1 Algebra . 33 3.3.2 Representation . 34 v vi CONTENTS T3 3.3.3 Equivariant real spectral triples over A( Θ)...... 35 3.4 Quotient Spaces . 36 3.4.1 Reducible spectral triples . 36 Z 3.4.2 Spectral Triples over A(T1) N .............. 38 3.4.3 Summary . 40 4 Noncommutative Bieberbach Manifolds 41 4.1 Classical Bieberbach Manifolds . 42 4.2 Three-dimensional Bieberbach Manifolds . 44 4.2.1 Spin structures over Bieberbach manifolds . -
Dirac Spectra, Summation Formulae, and the Spectral Action
Dirac Spectra, Summation Formulae, and the Spectral Action Thesis by Kevin Teh In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2013 (Submitted May 2013) i Acknowledgements I wish to thank my parents, who have given me their unwavering support for longer than I can remember. I would also like to thank my advisor, Matilde Marcolli, for her encourage- ment and many helpful suggestions. ii Abstract Noncommutative geometry is a source of particle physics models with matter Lagrangians coupled to gravity. One may associate to any noncommutative space (A; H; D) its spectral action, which is defined in terms of the Dirac spectrum of its Dirac operator D. When viewing a spin manifold as a noncommutative space, D is the usual Dirac operator. In this paper, we give nonperturbative computations of the spectral action for quotients of SU(2), Bieberbach manifolds, and SU(3) equipped with a variety of geometries. Along the way we will compute several Dirac spectra and refer to applications of this computation. iii Contents Acknowledgements i Abstract ii 1 Introduction 1 2 Quaternionic Space, Poincar´eHomology Sphere, and Flat Tori 5 2.1 Introduction . 5 2.2 The quaternionic cosmology and the spectral action . 6 2.2.1 The Dirac spectra for SU(2)=Q8.................... 6 2.2.2 Trivial spin structure: nonperturbative spectral action . 7 2.2.3 Nontrivial spin structures: nonperturbative spectral action . 9 2.3 Poincar´ehomology sphere . 10 2.3.1 Generating functions for spectral multiplicities . 10 2.3.2 The Dirac spectrum of the Poincar´esphere . -
Scalar Curvature for Noncommutative Four-Tori
SCALAR CURVATURE FOR NONCOMMUTATIVE FOUR-TORI FARZAD FATHIZADEH AND MASOUD KHALKHALI Abstract. In this paper we study the curved geometry of noncommutative 4 4-tori Tθ. We use a Weyl conformal factor to perturb the standard volume form and obtain the Laplacian that encodes the local geometric information. We use Connes' pseudodifferential calculus to explicitly compute the terms in the small time heat kernel expansion of the perturbed Laplacian which corre- 4 spond to the volume and scalar curvature of Tθ. We establish the analogue of Weyl's law, define a noncommutative residue, prove the analogue of Connes' trace theorem, and find explicit formulas for the local functions that describe 4 the scalar curvature of Tθ. We also study the analogue of the Einstein-Hilbert action for these spaces and show that metrics with constant scalar curvature are critical for this action. Contents 1. Introduction 1 2. Noncommutative Tori 3 2.1. Noncommutative real tori. 3 2.2. Noncommutative complex tori. 4 3. Laplacian and its Heat Kernel 5 4 3.1. Perturbed Laplacian on Tθ. 5 4 3.2. Connes' pseudodifferential calculus for Tθ. 7 3.3. Small time asymptotic expansion for Trace(e−t4' ). 8 4. Weyl's Law and Connes' Trace Theorem 10 4.1. Asymptotic distribution of the eigenvalues of 4'. 10 4 4.2. A noncommutative residue for Tθ. 12 4 4.3. Connes' trace theorem for Tθ. 15 5. Scalar Curvature and Einstein-Hilbert Action 16 5.1. Scalar curvature for 4. 17 arXiv:1301.6135v1 [math.QA] 25 Jan 2013 Tθ 4 5.2. -
Spectra of Orbifolds with Cyclic Fundamental Groups
SPECTRA OF ORBIFOLDS WITH CYCLIC FUNDAMENTAL GROUPS EMILIO A. LAURET Abstract. We give a simple geometric characterization of isospectral orbifolds covered by spheres, complex projective spaces and the quaternion projective line having cyclic fundamen- tal group. The differential operators considered are Laplace-Beltrami operators twisted by characters of the corresponding fundamental group. To prove the characterization, we first give an explicit description of their spectra by using generating functions. We also include many isospectral examples. 1. Introduction Let M be a connected compact Riemannian manifold, and let Γ1 and Γ2 be cyclic subgroups of Iso(M). In this paper we will consider the question: Under what conditions the (good) orbifolds Γ1\M and Γ2\M are isospectral? Here, isospectral means that the Laplace-Beltrami operators on Γ1\M and on Γ2\M have the same spectrum. Such operators are given by the Laplace-Beltrami operator on M acting on Γj-invariant smooth functions on M. Clearly, the condition on Γ1 and Γ2 of being conjugate in Iso(M) is sufficient since in this case Γ1\M and Γ2\M are isometric. However, it is well known that this condition is not necessary due to examples of non-isometric isospectral lens spaces constructed by Ikeda [Ik80a]. Lens spaces are spherical space forms with cyclic fundamental groups, that is, they are the only compact manifolds with constant sectional curvature and cyclic fundamental group. In the recent paper [LMR15], R. Miatello, J.P. Rossetti and the author show an isospectral characterization among lens spaces in terms of isospectrality of their associated congruence lattices with respect to the one-norm k·k1. -
Location and Weyl Formula for the Eigenvalues of Some Non Self-Adjoint Operators
Location and Weyl formula for the eigenvalues of some non self-adjoint operators Vesselin Petkov Abstract We present a survey of some recent results concerning the location and the Weyl formula for the complex eigenvalues of two non self-adjoint operators. We study the eigenvalues of the generator G of the contraction semigroup etG, t ≥ 0, related to the wave equation in an unbounded domain Ω with dissipative boundary conditions on ∂Ω. Also one examines the interior transmission eigenvalues (ITE) in a bounded domain K obtaining a Weyl formula with remainder for the counting function N(r) of complex (ITE). The analysis is based on a semi-classical approach. 1 Introduction ∞ Let P(x,Dx) be a second order differential operator with C (K) real-valued coef- d ficients in a bounded domain K ⊂ R , d ≥ 2, with C∞ boundary ∂K. Consider a boundary problem ( P(x,D )u = f in K, x (1.1) B(x,Dx)u = g on ∂K, where B(x,Dx) is a differential operator with order less or equal to 1 and the princi- 2 pal symbol P(x,ξ) of P(x,Dx) satisfies p(x,ξ) ≥ c0|ξ| , c0 > 0. Assume that there exists 0 < ϕ < π such that the problem ( (P(x,D ) − z)u = f in K, x (1.2) B(x,Dx)u = g on ∂K. is parameter-elliptic for every z ∈ Γψ = {z : argz = ψ}, 0 < |ψ| ≤ ϕ. Then following a classical result of Agranovich-Vishik [1] we can find a closed operator A with Vesselin Petkov Institut de Mathematiques´ de Bordeaux, 351, Cours de la Liberation,´ 33405 Talence, France , e- mail: [email protected] 1 2 Vesselin Petkov domain D(A) ⊂ H2(K) related to the problem (1.1). -
An Introduction to Pseudo-Differential Operators
An introduction to pseudo-differential operators Jean-Marc Bouclet1 Universit´ede Toulouse 3 Institut de Math´ematiquesde Toulouse [email protected] 2 Contents 1 Background on analysis on manifolds 7 2 The Weyl law: statement of the problem 13 3 Pseudodifferential calculus 19 3.1 The Fourier transform . 19 3.2 Definition of pseudo-differential operators . 21 3.3 Symbolic calculus . 24 3.4 Proofs . 27 4 Some tools of spectral theory 41 4.1 Hilbert-Schmidt operators . 41 4.2 Trace class operators . 44 4.3 Functional calculus via the Helffer-Sj¨ostrandformula . 50 5 L2 bounds for pseudo-differential operators 55 5.1 L2 estimates . 55 5.2 Hilbert-Schmidt estimates . 60 5.3 Trace class estimates . 61 6 Elliptic parametrix and applications 65 n 6.1 Parametrix on R ................................ 65 6.2 Localization of the parametrix . 71 7 Proof of the Weyl law 75 7.1 The resolvent of the Laplacian on a compact manifold . 75 7.2 Diagonalization of ∆g .............................. 78 7.3 Proof of the Weyl law . 81 A Proof of the Peetre Theorem 85 3 4 CONTENTS Introduction The spirit of these notes is to use the famous Weyl law (on the asymptotic distribution of eigenvalues of the Laplace operator on a compact manifold) as a case study to introduce and illustrate one of the many applications of the pseudo-differential calculus. The material presented here corresponds to a 24 hours course taught in Toulouse in 2012 and 2013. We introduce all tools required to give a complete proof of the Weyl law, mainly the semiclassical pseudo-differential calculus, and then of course prove it! The price to pay is that we avoid presenting many classical concepts or results which are not necessary for our purpose (such as Borel summations, principal symbols, invariance by diffeomorphism or the G˚ardinginequality). -
What Is Noncommutative Geometry ? How a Geometry Can Be Commutative and Why Mine Is Not
What is Noncommutative Geometry ? How a geometry can be commutative and why mine is not Alessandro Rubin Junior Math Days 2019/20 SISSA - Scuola Internazionale Superiore di Studi Avanzati This means that the algebra C∞(M) contains enough information to codify the whole geometry of the manifold: 1. Vector fields: linear derivations of C∞(M) 2. Differential 1-forms: C∞(M)-linear forms on vector fields 3. ... Question Do we really need a manifold’s points to study it ? Do we really use the commutativity of the algebra C∞(M) to define the aforementioned objects ? Doing Geometry Without a Geometric Space Theorem Two smooth manifolds M, N are diffeomorphic if and only if their algebras of smooth functions C∞(M) and C∞(N) are isomorphic. 1/41 Question Do we really need a manifold’s points to study it ? Do we really use the commutativity of the algebra C∞(M) to define the aforementioned objects ? Doing Geometry Without a Geometric Space Theorem Two smooth manifolds M, N are diffeomorphic if and only if their algebras of smooth functions C∞(M) and C∞(N) are isomorphic. This means that the algebra C∞(M) contains enough information to codify the whole geometry of the manifold: 1. Vector fields: linear derivations of C∞(M) 2. Differential 1-forms: C∞(M)-linear forms on vector fields 3. ... 1/41 Do we really use the commutativity of the algebra C∞(M) to define the aforementioned objects ? Doing Geometry Without a Geometric Space Theorem Two smooth manifolds M, N are diffeomorphic if and only if their algebras of smooth functions C∞(M) and C∞(N) are isomorphic. -
Calculation of Supercritical Dirac Resonances in Heavy-Ion Collisions
CALCULATION OF SUPERCRITICAL DIRAC RESONANCES IN HEAVY-ION COLLISIONS EDWARD ACKAD A DISSERTATION SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY arXiv:0809.4256v2 [physics.atom-ph] 26 Sep 2008 GRADUATE PROGRAM IN DEPARTMENT OF PHYSICS AND ASTRONOMY YORK UNIVERSITY TORONTO, ONTARIO 2021 CALCULATION OF SUPERCRITICAL DIRAC RESONANCES IN HEAVY-ION COLLISIONS by Edward Ackad a dissertation submitted to the Faculty of Graduate Stud- ies of York University in partial fulfilment of the require- ments for the degree of DOCTOR OF PHILOSOPHY c 2021 Permission has been granted to: a) YORK UNIVER- SITY LIBRARIES to lend or sell copies of this disserta- tion in paper, microform or electronic formats, and b) LI- BRARY AND ARCHIVES CANADA to reproduce, lend, distribute, or sell copies of this dissertation anywhere in the world in microform, paper or electronic formats and to authorise or procure the reproduction, loan, distribu- tion or sale of copies of this dissertation anywhere in the world in microform, paper or electronic formats. The author reserves other publication rights, and nei- ther the dissertation nor extensive extracts for it may be printed or otherwise reproduced without the author’s written permission. CALCULATION OF SUPERCRITICAL DIRAC RESONANCES IN HEAVY-ION COLLISIONS by Edward Ackad By virtue of submitting this document electronically, the author certifies that this is a true electronic equivalent of the copy of the dissertation approved by York University for the award of the degree. No alteration of the content has occurred and if there are any minor variations in formatting, they are as a result of the coversion to Adobe Acrobat format (or similar software application). -
Notes on the Quantum Harmonic Oscillator Brief Discussion of Coherent States, Weyl's Law and the Mehler Kernel
Notes on the Quantum Harmonic Oscillator Brief Discussion of Coherent States, Weyl's Law and the Mehler Kernel Brendon Phillips 250817875 April 22, 2015 1 The Harmonic Oscillators Recall that any object oscillating in one dimension about the point 0 (say, a mass-spring system) obeys Hooke's law Fx = −ksx, where x is the compression of the spring, and ks is the spring constant of the system. By Newton's second law of motion, we then have the equation of the Classical Harmonic Oscillator @2x m = −k x (1) @t2 s 1 Now, Fx is conservative, so we can recover the potential energy function as follows : @V 1 −F = =) V (x) = k x2 (2) x @x 2 s The solution to (1) is given by r ! k x(t) = A cos s t + (3) m The frequency of oscillation is then r 1 1 k f = = s (4) T 2π m Let the oscillating object be a particle. Now, classical mechanics only provides a good ap- proximation of the system as long as E >> hf, whereE is the total energy of the particle, f is the classical frequency, and h is the Planck constant. Assume that this is not the case; we now see the system as a Quantum Mechanical Os- cillator. As a consequence of the Heisenberg Uncertainty Principle ∆^x∆^p ≥ } (5) 2 1We assume without loss of generality that the constant of integration is 0. 1 (wherex ^ represents the position operator,p ^ represents the momentum operator, and } is the reduced Planck constant), we lose our grasp of the particle's trajectory, since we now switch our focus from the position of the particle to its wavefunction (x; t).