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研究生：	陳琮方 Chen, Tsung-Fang
論文名稱：	規範理論中朗蘭茲對偶的研究 Survey of Langlands Duality in Gauge theory
指導教授：	劉之中 Liu, Chih-Chung
學位類別：	碩士 Master
系所名稱：	理學院 - 數學系應用數學碩博士班 Department of Mathematics
論文出版年：	2025
畢業學年度：	113
語文別：	英文
論文頁數：	45
中文關鍵詞：	朗蘭茲對偶群、磁對偶群、規範理論、希欽模空間、鏡像對稱
外文關鍵詞：	Langlands dual group, magnetic dual group, gauge theory, Hitchin moduli space, mirror symmetry
相關次數：	點閱：41 下載：1
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本論文探討物理中的磁對偶群與數學中的朗蘭茲對偶群之關係。我們首先介紹朗蘭茲對偶群的概念，並透過常見李群的實例加以說明。接著，證明在非阿貝爾規範理論中出現的磁對偶群與朗蘭茲對偶群完全對應。最後，提出本研究的未來發展方向，特別著重於朗蘭茲對偶群在希金方程之模空間與鏡像對稱背景下的出現，揭示此對偶性背後深刻的幾何和物理架構。

In this thesis, we survey the relation between the magnetic dual group in physics and the Langlands dual group in mathematics. We begin by introducing the notion of Langlands dual groups, illustrated through examples from common Lie groups. We then show that the magnetic dual group arising in non-abelian gauge theory corresponds precisely to the Langlands dual group. Finally, we outline future prospects of this work, with particular emphasis on the appearance of the Langlands dual group in the context of Hitchin moduli spaces and mirror symmetry, revealing deep geometric and physical frameworks underlying this duality.

Introduction 1
Introduction to Root Systems and Langlands Duality 2
1 Definition of Root Systems 2
2 Dual Root Systems 6
3 Dynkin Diagrams 7
4 Classification of Semisimple Lie Algebras 9
5 Classification of Compact Lie Groups 10
6 Definition of Langlands Duality 11
7 Root Data of Common Lie Groups 11
7.1 GL(n) 12
7.2 SL(n) v.s. PGL(n) 13
7.3 Sp(2n) v.s. SO(2n + 1) 14
7.4 SO(2n) 15
Summary of Goddard, Nuyts and Olive (1977) 17
1 Mathematical Preliminaries 17
2 Physical Motivation 19
2.1 Gauge Symmetry 20
2.2 U(1) Symmetry of Dirac Magnetic Monopole 20
2.3 Non-Abelian Gauge Symmetry and Magnetic Dual Group 21
2.4 Charges and Representations 21
3 The Construction of the Magnetic Dual Group 22
3.1 Magnetic Dual Group of H˜ 23
3.2 Structure of Z(H˜ ) 23
3.3 The Magnetic Group of H 24
4 Physical Significance of the Magnetic Dual Group 27
Future Prospects 29
1 Higgs Bundle and Hitchin Equation 29
2 Hitchin Fibration and Mirror Symmetry 30
3 Conclusion 31
A Physical Interpretation of Charge Classification 32
A.1 Electric Charges and Hom(H, U(1)) 32
A.2 Quantization Condition and Magnetic Charges 33
A.2.1 Magnetic Charges as Topological Classes 33
A.2.2 The Interpretation of Hom(U(1), T) ∼= π1(T) 33
A.2.3 Quantization Condition 34
B Lattice Theory for Duality 35
B.1 Introduction to Lattice 35
B.2 Lattice Pairings and Weight-Type Extensions 36
B.3 Pairing Criterion via the Exponential Map 37
                                    

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Additional References
[17] E. Witten, Duality, the Langlands Correspondence, and the Influence of Michael Atiyah, Lecture at Isaac Newton Institute, 2021.
[18] S. Makisumi, Structure Theory of Reductive Groups through Examples, 2011. https://makisumi.com/math/old/reductivegroups.pdf
[19] W. Fulton and J. Harris, Representation Theory: A First Course, Springer, New York, 2004.

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