Modular Linear Differential Equations and Delign's Exceptional Series Pubblico

Abstract

In 1988, Mathur, Mukhi, and Sen studied rational conformal field theories in terms of differential equations satisfied by their characters. These differential equations are modular invariant, and the solutions they obtain for order 2 equations have relationships with certain Lie algebras. In fact, the Lie algebras in the Deligne Exceptional series appear, whose study is motivated by uniformities which appear in their representation theory. This thesis studies the Deligne Exceptional Series from these two perspectives, and gives a sequence of finite groups which has analogies with the Deligne series.

Table of Contents

Contents

1 Introduction 1

2 Lie groups/Lie algebras 5

2.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Classication of Simple Complex Lie Algebras . . . . . . . . . . . . . . . . . 13

3 Deligne Exceptional Series 19

3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Deligne's observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Rational Conformal Field Theory 22

4.1 Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 Modular Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . 25

4.3 Order 2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 The Finite Group Series 34

5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6 Conclusion 40

List of Figures

 

1 The Classication of simple complex Lie Algebras . . . . . . . . . . . . . . . 18

About this Master's Thesis

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School	Laney Graduate School
Department	Mathematics
Degree	M.S.
Submission	Master's Thesis
Language	English
Research Field	Mathematics
Parola chiave	Lie groups, Physics, Representation Theory
Committee Chair / Thesis Advisor	Duncan, John, Emory University
Committee Members	Prior, Richard, Emory University Ono, Ken, Emory University

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