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Eisenstein Series and Automorphic Representations: With Applications in String Theory [Kietas viršelis]

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(Stanford University, California), (Max-Planck-Institut für Gravitationsphysik, Germany), , (Chalmers University of Technology, Gothenberg)
  • Formatas: Hardback, 584 pages, aukštis x plotis x storis: 235x157x38 mm, weight: 920 g, Worked examples or Exercises; 3 Halftones, black and white; 17 Line drawings, black and white
  • Serija: Cambridge Studies in Advanced Mathematics
  • Išleidimo metai: 05-Jul-2018
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1107189926
  • ISBN-13: 9781107189928
Kitos knygos pagal šią temą:
Eisenstein Series and Automorphic Representations: With Applications in String TheoryDidesnis paveikslėlis
  • Formatas: Hardback, 584 pages, aukštis x plotis x storis: 235x157x38 mm, weight: 920 g, Worked examples or Exercises; 3 Halftones, black and white; 17 Line drawings, black and white
  • Serija: Cambridge Studies in Advanced Mathematics
  • Išleidimo metai: 05-Jul-2018
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1107189926
  • ISBN-13: 9781107189928
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781107189928.html
Raktažodžiai:
This introduction to automorphic forms on adelic groups G(A) emphasises the role of representation theory. The exposition is driven by examples, and collects and extends many results scattered throughout the literature, in particular the Langlands constant term formula for Eisenstein series on G(A) as well as the Casselman–Shalika formula for the p-adic spherical Whittaker function. This book also covers more advanced topics such as spherical Hecke algebras and automorphic L-functions. Many of these mathematical results have natural interpretations in string theory, and so some basic concepts of string theory are introduced with an emphasis on connections with automorphic forms. Throughout the book special attention is paid to small automorphic representations, which are of particular importance in string theory but are also of independent mathematical interest. Numerous open questions and conjectures, partially motivated by physics, are included to prompt the reader's own research.

Aimed at advanced students and active researchers in mathematics or theoretical physics, this book provides a detailed exposition of automorphic forms and representations, from the basics up to cutting-edge research topics at the interface between number theory and string theory.

Recenzijos

'This book provides a bridge between two very active and important parts of mathematics and physics, namely the theory of automorphic forms on reductive groups and string theory. The authors have masterfully presented both aspects and their connections, and have provided examples and details at all levels to make the book available to a large readership, including non-experts in both fields. This is a valuable contribution and a welcoming text for graduate students as well.' Freydoon Shahidi, Purdue University, Indiana 'The book is a valuable addition to the literature, and it may inspire more exchange between mathematics and physics at an advanced level.' Anton Deitmar, MathSciNet 'The prerequisites for a protable reading this book are enormous. Readers without a solid background in algebraic and analytic number theory, classeld theory, modular forms and representation theory will only be able to read a couple of sections. Researchers in these elds will be grateful to the authors and the publisher for providing access to some rather advanced mathematics. The material is presented in a very clear and lucid way; there is an extensive index and a list of references containing 634 items.' Franz Lemmermeyer, zbMATH

Daugiau informacijos

Detailed exposition of automorphic representations and their relation to string theory, for mathematicians and theoretical physicists.
List of Definitions and Theorems
xi
List of Examples
xiv
Preface xvii
1 Motivation and Background
1(16)
1.1 Automorphic Forms and Eisenstein Series
1(4)
1.2 Why Eisenstein Series and Automorphic Forms?
5(1)
1.3 Analysing Automorphic Forms and Adelisation
6(3)
1.4 Reader's Guide and Main Theorems
9(8)
PART ONE AUTOMORPHIC REPRESENTATIONS
17(288)
2 Preliminaries on p-adic and Adelic Technology
19(20)
2.1 p-adic Numbers
19(4)
2.2 p-adic Integration
23(2)
2.3 p-adic Characters and the Fourier Transform
25(5)
2.4 p-adic Gaussian and Bessel Functions
30(2)
2.5 Adeles
32(2)
2.6 Adelisation
34(1)
2.7 Adelic Analysis of the Riemann Zeta Function
35(4)
3 Basic Notions from Lie Algebras and Lie Groups
39(19)
3.1 Real Lie Algebras and Real Lie Groups
39(10)
3.2 p-adic and Adelic Groups
49(9)
4 Automorphic Forms
58(29)
4.1 Preliminaries on SL(2, R)
58(3)
4.2 Classical Modular Forms
61(6)
4.3 From Classical Modular Forms to Automorphic Forms
67(8)
4.4 Adelic Automorphic Forms
75(6)
4.5 Eisenstein Series
81(6)
5 Automorphic Representations and Eisenstein Series
87(36)
5.1 A First Glimpse at Automorphic Representations
87(6)
5.2 Automorphic Representations
93(3)
5.3 Principal Series Representations
96(1)
5.4 Eisenstein Series and Induced Representations
97(1)
5.5 Classifying Automorphic Representations
98(2)
5.6 Embedding of the Discrete Series in the Principal Series
100(6)
5.7 Eisenstein Series for Non-minimal Parabolics
106(9)
5.8 Induced Representations and Spherical Vectors*
115(8)
6 Whit taker Functions and Fourier Coefficients
123(32)
6.1 Preliminary Example: SL(2,R) Whittaker Functions
123(5)
6.2 Fourier Expansions and Unitary Characters
128(8)
6.3 Induced Representations and Whittaker Models
136(5)
6.4 Wavefront Set and Small Representations
141(10)
6.5 Method of Piatetski-Shapiro and Shalika*
151(4)
7 Fourier Coefficients of Eisenstein Series on SL(2, A)
155(15)
7.1 Statement of Theorem
155(3)
7.2 Constant Term
158(7)
7.3 The Non-constant Fourier Coefficients
165(5)
8 Langlands Constant Term Formula
170(16)
8.1 Statement of Theorem
170(1)
8.2 Bruhat Decomposition
171(1)
8.3 Parametrising the Integral
172(1)
8.4 Obtaining the a-dependence of the Integral
173(1)
8.5 Solving the Remaining Integral by Induction
174(1)
8.6 The Gindikin-Karpelevich Formula
175(3)
8.7 Assembling the Constant Term
178(1)
8.8 Functional Relations for Eisenstein Series
179(2)
8.9 Expansion in Maximal Parabolics*
181(5)
9 Whittaker Coefficients of Eisenstein Series
186(33)
9.1 Reduction of the Integral and the Longest Weyl Word
186(3)
9.2 Unramified Local Whittaker Functions
189(2)
9.3 The Casselman-Shalika Formula
191(6)
9.4 Whittaker Functions for Generic Characters ψ
197(2)
9.5 Degenerate Whittaker Coefficients
199(5)
9.6 The Casselman-Shalika Formula and Langlands Duality*
204(3)
9.7 Quantum Whittaker Functions*
207(2)
9.8 Whittaker Coefficients on SL(3, A)*
209(10)
10 Analysing Eisenstein Series and Small Representations
219(36)
10.1 The SL (2, R) Eisenstein Series as a Function of s
220(3)
10.2 Properties of Eisenstein Series
223(9)
10.3 Evaluating Constant Term Formulas
232(10)
10.4 Evaluating Spherical Whittaker Coefficients
242(13)
11 Hecke Theory and Automorphic L-functions
255(35)
11.1 Classical Hecke Operators and the Hecke Ring
255(2)
11.2 Hecke Operators for SL(2, R)
257(8)
11.3 The Spherical Hecke Algebra
265(3)
11.4 Hecke Algebras and Automorphic Representations
268(3)
11.5 The Satake Isomorphism
271(2)
11.6 The L-group and Generalisation to GL(n)
273(4)
11.7 The Casselman--Shalika Formula Revisited
277(5)
11.8 Automorphic L-functions
282(3)
11.9 The Langlands--Shahidi Method*
285(5)
12 Theta Correspondences
290(15)
12.1 Classical Theta Series
290(3)
12.2 Representation Theory of Classical Theta Functions
293(3)
12.3 Theta Correspondence
296(1)
12.4 Theta Series and the Weil Representation
297(1)
12.5 The Siegel-Weil Formula
298(2)
12.6 Exceptional Theta Correspondences
300(5)
PART TWO APPLICATIONS IN STRING THEORY
305(124)
13 Elements of String Theory
307(60)
13.1 String Theory Concepts
308(16)
13.2 Four-Graviton Scattering Amplitudes
324(4)
13.3 The Four-Graviton Tree-Level Amplitude*
328(5)
13.4 One-Loop String Amplitudes and Theta Lifts*
333(16)
13.5 D-branes*
349(6)
13.6 Non-perturbative Corrections from Instantons*
355(12)
14 Automorphic Scattering Amplitudes
367(28)
14.1 U-duality Constraints in the a'-expansion
367(3)
14.2 Physical Interpretation of the Fourier Expansion
370(12)
14.3 Automorphic Representations and BPS Orbits
382(6)
14.4 Supersymmetry Constraints*
388(7)
15 Further Occurrences of Automorphic Forms in String Theory
395(34)
15.1 Δ6 R4-amplitudes and Generalised Automorphic Forms
395(3)
15.2 Modular Graph Functions
398(7)
15.3 Automorphic Functions and Lattice Sums
405(3)
15.4 Black Hole Counting and Automorphic Representations
408(8)
15.5 Moonshine
416(13)
PART THREE ADVANCED TOPICS
429(78)
16 Connections to the Langlands Program
431(20)
16.1 The Classical Langlands Program
431(3)
16.2 The Geometric Langlands Program
434(5)
16.3 The Langlands Program and Physics
439(3)
16.4 Modular Forms and Elliptic Curves
442(9)
17 Whittaker Functions, Crystals and Multiple Dirichlet Series
451(9)
17.1 Generalisations of the Weyl Character Formula
451(2)
17.2 Whittaker Functions and Crystals
453(3)
17.3 Weyl Group Multiple Dirichlet Series
456(4)
18 Automorphic Forms on Non-split Real Forms
460(27)
18.1 Eisenstein Series on SU/(2,1)
460(5)
18.2 Constant Term and L-functions
465(3)
18.3 Connection with the Langlands-Shahidi Method
468(5)
18.4 Global Whittaker Coefficients
473(1)
18.5 More General Number Fields
474(1)
18.6 String Theory and Enumerative Geometry
475(3)
18.7 Twistors and Quaternionic Discrete Series
478(9)
19 Extension to Kac--Moody Groups
487(20)
19.1 Motivation
488(2)
19.2 Eisenstein Series on Affine Kac--Moody Groups
490(7)
19.3 Extension to General Kac--Moody Groups
497(10)
Appendices
507(20)
Appendix A SL(2, R) Eisenstein Series and Poisson Resumniation
509(4)
Appendix B Laplace Operators on G / K and Automorphic Forms
513(5)
Appendix C Structure Theory of su (2,1)
518(3)
Appendix D Poincare Series and Kloosterman Sums
521(6)
References 527(32)
Index 559
Philipp Fleig is a Postdoctoral Researcher at the Max-Planck-Institut für Dynamik und Selbstorganisation, Germany. Henrik P. A. Gustafsson is a Postdoctoral Researcher in the Department of Mathematics at Stanford University, California. Axel Kleinschmidt is a Senior Scientist at the Max-Planck-Institut für Gravitationsphysik, Germany (Albert Einstein Institute) and at the International Solvay Institutes, Brussels. Daniel Persson is an Associate Professor in the Department of Mathematical Sciences at Chalmers University of Technology, Gothenburg.