Formal Groups, Witt Vectors and Free Probability
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Problems About Formal Groups and Cohomology Theories Arizona Winter School 2019
PROBLEMS ABOUT FORMAL GROUPS AND COHOMOLOGY THEORIES ARIZONA WINTER SCHOOL 2019 VESNA STOJANOSKA 1. Formal Group Laws In this document, a formal group law over a commutative ring R is always com- mutative and one-dimensional. Specifically, it is a power series F (x; y) 2 R[[x; y]] satisfying the following axioms: • F (x; 0) = x = F (0; x), • F (x; y) = F (y; x), and • F (x; F (y; z)) = F (F (x; y); z). Sometimes we denote F (x; y) by x +F y. For a non-negative integer n, the n-series of F is defined inductively by [0]F x = 0, and [n + 1]F x = x +F [n]F x. (1) Show that for any formal group law F (x; y) 2 R[[x; y]], x has a formal inverse. In other words, there is a power series i(x) 2 R[[x]] such that F (x; i(x)) = 0. As a consequence, we can define [−n]F x = i([n]F x). (2) Check that the following define formal group laws: (a) F (x; y) = x + y + uxy, where u is any unit in R. Also show that this is isomorphic to the multiplicative formal group law over R. x+y (b) F (x; y) = 1+xy . Hint: Note that if x = tanh(u) and y = tanh(v), then F (x; y) = tanh(x + y). (3) Suppose F; H are formal group laws over a complete local ring R, and we're given a morphism f : F ! H, i.e. f(x) 2 R[[x]] such that f(F (x; y)) = H(f(x); f(y)): Now let C be the category of complete local R-algebras and continuous maps between them; for T 2 C, denote by Spf(T ): C!Set the functor S 7! HomC(T;S). -
Generalized Rudin–Shapiro Sequences
ACTA ARITHMETICA LX.1 (1991) Generalized Rudin–Shapiro sequences by Jean-Paul Allouche (Talence) and Pierre Liardet* (Marseille) 1. Introduction 1.1. The Rudin–Shapiro sequence was introduced independently by these two authors ([21] and [24]) and can be defined by u(n) εn = (−1) , where u(n) counts the number of 11’s in the binary expansion of the integer n (see [5]). This sequence has the following property: X 2iπnθ 1/2 (1) ∀ N ≥ 0, sup εne ≤ CN , θ∈R n<N where one can take C = 2 + 21/2 (see [22] for improvements of this value). The order of magnitude of the left hand term in (1), as N goes to infin- 1/2 ity, is exactly N ; indeed, for each sequence (an) with values ±1 one has 1/2 X 2iπn(·) X 2iπn(·) N = ane ≤ ane , 2 ∞ n<N n<N where k k2 denotes the quadratic norm and k k∞ the supremum norm. Note that for almost every√ sequence (an) of ±1’s the supremum norm of the above sum is bounded by N Log N (see [23]). The inequality (1) has been generalized in [2] (see also [3]): X 2iπxu(n) 0 α(x) (2) sup f(n)e ≤ C N , f∈M2 n<N where M2 is the set of 2-multiplicative sequences with modulus 1. The * Research partially supported by the D.R.E.T. under contract 901636/A000/DRET/ DS/SR 2 J.-P. Allouche and P. Liardet exponent α(x) is explicitly given and satisfies ∀ x 1/2 ≤ α(x) ≤ 1 , ∀ x 6∈ Z α(x) < 1 , ∀ x ∈ Z + 1/2 α(x) = 1/2 . -
Fibonacci, Kronecker and Hilbert NKS 2007
Fibonacci, Kronecker and Hilbert NKS 2007 Klaus Sutner Carnegie Mellon University www.cs.cmu.edu/∼sutner NKS’07 1 Overview • Fibonacci, Kronecker and Hilbert ??? • Logic and Decidability • Additive Cellular Automata • A Knuth Question • Some Questions NKS’07 2 Hilbert NKS’07 3 Entscheidungsproblem The Entscheidungsproblem is solved when one knows a procedure by which one can decide in a finite number of operations whether a given logical expression is generally valid or is satisfiable. The solution of the Entscheidungsproblem is of fundamental importance for the theory of all fields, the theorems of which are at all capable of logical development from finitely many axioms. D. Hilbert, W. Ackermann Grundzuge¨ der theoretischen Logik, 1928 NKS’07 4 Model Checking The Entscheidungsproblem for the 21. Century. Shift to computer science, even commercial applications. Fix some suitable logic L and collection of structures A. Find efficient algorithms to determine A |= ϕ for any structure A ∈ A and sentence ϕ in L. Variants: fix ϕ, fix A. NKS’07 5 CA as Structures Discrete dynamical systems, minimalist description: Aρ = hC, i where C ⊆ ΣZ is the space of configurations of the system and is the “next configuration” relation induced by the local map ρ. Use standard first order logic (either relational or functional) to describe properties of the system. NKS’07 6 Some Formulae ∀ x ∃ y (y x) ∀ x, y, z (x z ∧ y z ⇒ x = y) ∀ x ∃ y, z (y x ∧ z x ∧ ∀ u (u x ⇒ u = y ∨ u = z)) There is no computability requirement for configurations, in x y both x and y may be complicated. -
Harmonic and Complex Analysis and Its Applications
Trends in Mathematics Alexander Vasil’ev Editor Harmonic and Complex Analysis and its Applications Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference. Proposals for volumes can be submitted using the Online Book Project Submission Form at our website www.birkhauser-science.com. Material submitted for publication must be screened and prepared as follows: All contributions should undergo a reviewing process similar to that carried out by journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any significantly new results, should be rejected. High quality survey papers, however, are welcome. We expect the organizers to deliver manuscripts in a form that is essentially ready for direct reproduction. Any version of TEX is acceptable, but the entire collection of files must be in one particular dialect of TEX and unified according to simple instructions available from Birkhäuser. Furthermore, in order to guarantee the timely appearance of the proceedings it is essential that the final version of the entire material be submitted no later than one year after the conference. For further volumes: http://www.springer.com/series/4961 Harmonic and Complex Analysis and its Applications Alexander Vasil’ev Editor Editor Alexander Vasil’ev Department of Mathematics University of Bergen Bergen Norway ISBN 978-3-319-01805-8 ISBN 978-3-319-01806-5 (eBook) DOI 10.1007/978-3-319-01806-5 Springer Cham Heidelberg New York Dordrecht London Mathematics Subject Classification (2010): 13P15, 17B68, 17B80, 30C35, 30E05, 31A05, 31B05, 42C40, 46E15, 70H06, 76D27, 81R10 c Springer International Publishing Switzerland 2014 This work is subject to copyright. -
Formal Group Rings of Toric Varieties 3
FORMAL GROUP RINGS OF TORIC VARIETIES WANSHUN WONG Abstract. In this paper we use formal group rings to construct an alge- braic model of the T -equivariant oriented cohomology of smooth toric vari- eties. Then we compare our algebraic model with known results of equivariant cohomology of toric varieties to justify our construction. Finally we construct the algebraic counterpart of the pull-back and push-forward homomorphisms of blow-ups. 1. Introduction Let h be an algebraic oriented cohomology theory in the sense of Levine-Morel [14], where examples include the Chow group of algebraic cycles modulo rational equivalence, algebraic K-theory, connective K-theory, elliptic cohomology, and a universal such theory called algebraic cobordism. It is known that to any h one can associate a one-dimensional commutative formal group law F over the coefficient ring R = h(pt), given by c1(L1 ⊗ L2)= F (c1(L1),c1(L2)) for any line bundles L1, L2 on a smooth variety X, where c1 is the first Chern class. Let T be a split algebraic torus which acts on a smooth variety X. An object of interest is the T -equivariant cohomology ring hT (X) of X, and we would like to build an algebraic model for it. More precisely, instead of using geometric methods to compute hT (X), we want to use algebraic methods to construct another ring A (our algebraic model), such that A should be easy to compute and it gives infor- mation about hT (X). Of course the best case is that A and hT (X) are isomorphic. -
Congruences Between the Coefficients of the Tate Curve Via Formal Groups
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 3, Pages 677{681 S 0002-9939(99)05133-3 Article electronically published on July 6, 1999 CONGRUENCES BETWEEN THE COEFFICIENTS OF THE TATE CURVE VIA FORMAL GROUPS ANTONIOS BROUMAS (Communicated by David E. Rohrlich) 2 3 Abstract. Let Eq : Y + XY = X + h4X + h6 be the Tate curve with canonical differential, ! = dX=(2Y + X). If the characteristic is p>0, then the Hasse invariant, H, of the pair (Eq;!) should equal one. If p>3, then calculation of H leads to a nontrivial separable relation between the coefficients h4 and h6.Ifp=2orp= 3, Thakur related h4 and h6 via elementary methods and an identity of Ramanujan. Here, we treat uniformly all characteristics via explicit calculation of the formal group law of Eq. Our analysis was motivated by the study of the invariant A which is an infinite Witt vector generalizing the Hasse invariant. 1. Introduction The Tate curve in characteristic either zero or positive is an elliptic curve with canonical Weierstrass model: 2 3 (1) Eq : Y + XY = X + h4X + h6 5S +7S nk where h = 5S and h = 3 5 for S = : 4 − 3 6 − 12 k 1 qn n 1 X≥ − Note that the denominator appearing in the expression of h6 is simply a nota- tional convenience and in fact Eq is defined over Z[[q]]. Hence, since all coefficients are integral, we can specialize to positive characteristic p for any prime p of our choice and obtain Eq defined over Fp[[q]]. In all positive characteristics the Hasse invariant of the pair (Eq;!)for!the canonical invariant differential, ! = dX=(2Y + X), is equal to 1. -
Lectures on Conformal Field Theory and Kac-Moody Algebras
hep-th/9702194 February 1997 LECTURES ON CONFORMAL FIELD THEORY AND KAC-MOODY ALGEBRAS J¨urgen Fuchs X DESY Notkestraße 85 D – 22603 Hamburg Abstract. This is an introduction to the basic ideas and to a few further selected topics in conformal quantum field theory and in the theory of Kac-Moody algebras. These lectures were held at the Graduate Course on Conformal Field Theory and Integrable Models (Budapest, August 1996). They will appear in a volume of the Springer Lecture Notes in Physics edited by Z. Horvath and L. Palla. —————— X Heisenberg fellow 1 Contents Lecture 1 : Conformal Field Theory 3 1 Conformal Quantum Field Theory 3 2 Observables: The Chiral Symmetry Algebra 4 3 Physical States: Highest Weight Modules 7 4 Sectors: The Spectrum 9 5 Conformal Fields 11 6 The Operator Product Algebra 14 7 Correlation Functions and Chiral Blocks 16 Lecture 2 : Fusion Rules, Duality and Modular Transformations 19 8 Fusion Rules 19 9 Duality 21 10 Counting States: Characters 23 11 Modularity 24 12 Free Bosons 26 13 Simple Currents 28 14 Operator Product Algebra from Fusion Rules 30 Lecture 3 : Kac--Moody Algebras 32 15 Cartan Matrices 32 16 Symmetrizable Kac-Moody Algebras 35 17 Affine Lie Algebras as Centrally Extended Loop Algebras 36 18 The Triangular Decomposition of Affine Lie Algebras 38 19 Representation Theory 39 20 Characters 40 Lecture 4 : WZW Theories and Coset Theories 42 21 WZW Theories 42 22 WZW Primaries 43 23 Modularity, Fusion Rules and WZW Simple Currents 44 24 The Knizhnik-Zamolodchikov Equation 46 25 Coset Conformal Field Theories 47 26 Field Identification 48 27 Fixed Points 49 28 Omissions 51 29 Outlook 52 30 Glossary 53 References 55 2 Lecture 1 : Conformal Field Theory 1 Conformal Quantum Field Theory Over the years, quantum field theory has enjoyed a great number of successes. -
The BMS Algebra and Black Hole Information
The BMS algebra and black hole information Han van der Ven1 supervised by Stefan Vandoren2, Johan van de Leur3 July 12, 2016 1Graduate student, Utrecht University, [email protected] 2Institute for Theoretical Physics and Spinoza Institute, Utrecht University, [email protected] 3Mathematical Institure, Utrecht University, [email protected] Abstract As a means of working towards solving the information paradox, we discuss conformal diagrams of evaporating black holes. We generalize the Schwarzschild solution to the asymptotically flat Bondi-metric and give a derivation of its asymptotic symmetry algebra, the BMS algebra. We discuss the interpretation as a charge algebra of zero-energy currents. Finally, we establish the centrally extended BMS-algebra as the semi-simple product of the Virasoro algebra acting on a representation. Contents 1 Introduction 4 1.1 Introduction . .4 1.1.1 Black hole information . .4 1.1.2 Lumpy black holes . .4 1.1.3 Asymptotic Symmetries . .5 2 Black holes and evaporation 7 2.1 Black hole evaporation . .7 2.1.1 Evaporation . .7 2.2 Conformal diagrams . .8 2.2.1 Time direction . 11 3 Introduction to asymptotic symmetry 12 3.1 Asymptotic Symmetries of the plane . 12 3.2 The BMS group . 14 3.2.1 BMS in three dimensions . 14 3.2.2 The BMS algebra in three dimensions . 15 3.2.3 BMS in higher dimensions . 17 3.3 Spherical Metrics and Conformal Killing vectors . 17 3.3.1 Riemann sphere . 17 3.3.2 Conformal transformations . 18 3.4 Local vs Global transformations . 19 3.4.1 Global transformations: Lorentz group . -
FORMAL SCHEMES and FORMAL GROUPS Contents 1. Introduction 2
FORMAL SCHEMES AND FORMAL GROUPS NEIL P. STRICKLAND Contents 1. Introduction 2 1.1. Notation and conventions 3 1.2. Even periodic ring spectra 3 2. Schemes 3 2.1. Points and sections 6 2.2. Colimits of schemes 8 2.3. Subschemes 9 2.4. Zariski spectra and geometric points 11 2.5. Nilpotents, idempotents and connectivity 12 2.6. Sheaves, modules and vector bundles 13 2.7. Faithful flatness and descent 16 2.8. Schemes of maps 22 2.9. Gradings 24 3. Non-affine schemes 25 4. Formal schemes 28 4.1. (Co)limits of formal schemes 29 4.2. Solid formal schemes 31 4.3. Formal schemes over a given base 33 4.4. Formal subschemes 35 4.5. Idempotents and formal schemes 38 4.6. Sheaves over formal schemes 39 4.7. Formal faithful flatness 40 4.8. Coalgebraic formal schemes 42 4.9. More mapping schemes 46 5. Formal curves 49 5.1. Divisors on formal curves 49 5.2. Weierstrass preparation 53 5.3. Formal differentials 56 5.4. Residues 57 6. Formal groups 59 6.1. Group objects in general categories 59 6.2. Free formal groups 63 6.3. Schemes of homomorphisms 65 6.4. Cartier duality 66 6.5. Torsors 67 7. Ordinary formal groups 69 Date: November 17, 2000. 1 2 NEIL P. STRICKLAND 7.1. Heights 70 7.2. Logarithms 72 7.3. Divisors 72 8. Formal schemes in algebraic topology 73 8.1. Even periodic ring spectra 73 8.2. Schemes associated to spaces 74 8.3. -
Finite Dimensional Grading of the Virasoro Algebra 1
FINITE DIMENSIONAL GRADING OF THE VIRASORO ALGEBRA RUBEN´ A. HIDALGO1), IRINA MARKINA2), AND ALEXANDER VASIL’EV2) Abstract. The Virasoro algebra is a central extension of the Witt algebra, the complexified Lie algebra of the sense preserving diffeomorphism group of the circle Diff S1. It appears in Quantum Field Theories as an infinite dimen- sional algebra generated by the coefficients of the Laurent expansion of the analytic component of the momentum-energy tensor, Virasoro generators. The background for the construction of the theory of unitary representations of Diff S1 is found in the study of Kirillov’s manifold Diff S1/S1. It possesses a natural K¨ahlerian embedding into the universal Teichm¨uller space with the projection into the moduli space realized as the infinite dimensional body of the coefficients of univalent quasiconformally extendable functions. The dif- ferential of this embedding leads to an analytic representation of the Virasoro algebra based on Kirillov’s operators. In this paper we overview several inter- esting connections between the Virasoro algebra, Teichm¨uller theory, L¨owner representation of univalent functions, and propose a finite dimensional grad- ing of the Virasoro algebra such that the grades form a hierarchy of finite dimensional algebras which, in their turn, are the first integrals of Liouville partially integrable systems for coefficients of univalent functions. 1. Introduction The Virasoro-Bott group vir appears in physics literature as the space of reparametrization of a closed string. It may be represented as the central ex- tension of the infinite dimensional Lie-Fr´echet group of sense preserving diffeo- morphisms of the unit circle. -
Lectures on Conformal Field Theory Arxiv:1511.04074V2 [Hep-Th] 19
Prepared for submission to JHEP Lectures on Conformal Field Theory Joshua D. Quallsa aDepartment of Physics, National Taiwan University, Taipei, Taiwan E-mail: [email protected] Abstract: These lectures notes are based on courses given at National Taiwan University, National Chiao-Tung University, and National Tsing Hua University in the spring term of 2015. Although the course was offered primarily for graduate students, these lecture notes have been prepared for a more general audience. They are intended as an introduction to conformal field theories in various dimensions working toward current research topics in conformal field theory. We assume the reader to be familiar with quantum field theory. Familiarity with string theory is not a prerequisite for this lectures, although it can only help. These notes include over 80 homework problems and over 45 longer exercises for students. arXiv:1511.04074v2 [hep-th] 19 May 2016 Contents 1 Lecture 1: Introduction and Motivation2 1.1 Introduction and outline2 1.2 Conformal invariance: What?5 1.3 Examples of classical conformal invariance7 1.4 Conformal invariance: Why?8 1.4.1 CFTs in critical phenomena8 1.4.2 Renormalization group 12 1.5 A preview for future courses 16 1.6 Conformal quantum mechanics 17 2 Lecture 2: CFT in d ≥ 3 22 2.1 Conformal transformations for d ≥ 3 22 2.2 Infinitesimal conformal transformations for d ≥ 3 24 2.3 Special conformal transformations and conformal algebra 26 2.4 Conformal group 28 2.5 Representations of the conformal group 29 2.6 Constraints of Conformal -
The Hirzebruch Genera of Complete Intersections 3
THE HIRZEBRUCH GENERA OF COMPLETE INTERSECTIONS JIANBO WANG, ZHIWANG YU, YUYU WANG Abstract. Following Brooks’s calculation of the Aˆ-genus of complete intersections, a new and more computable formula about the Aˆ-genus and α-invariant will be described as polynomials of multi-degree and dimension. We also give an iterated formula of Aˆ-genus and the necessary and sufficient conditions for the vanishing of Aˆ-genus of complex even dimensional spin complete intersections. Finally, we obtain a general formula about the Hirzebruch genus of complete intersections, and calculate some classical Hirzebruch genera as examples. 1. Introduction A genus of a multiplicative sequence is a ring homomorphism, from the ring of smooth compact manifolds (up to suitable cobordism) to another ring. A multiplicative se- quence is completely determined by its characteristic power series Q(x). Moreover, ev- ery power series Q(x) determines a multiplicative sequence. Given a power series Q(x), there is an associated Hirzebruch genus, which is also denoted as a ϕQ-genus. For more details, please see Section 2. The focus of this paper is mainly on the description of the Hirzebruch genus of complete intersections with multi-degree and dimension. n+r A complex n-dimensional complete intersection Xn(d1,...,dr) CP is a smooth complex n-dimensional manifold given by a transversal intersection⊂ of r nonsin- gular hypersurfaces in complex projective space. The unordered r-tuple d := (d1,...,dr) is called the multi-degree, which denotes the degrees of the r nonsingular hypersur- faces, and the product d := d1d2 dr is called the total degree.