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研究生:李育浚
研究生(外文):Li, Yu-Jun
論文名稱:Construction of Some Topological Quantum Field Theories
論文名稱(外文):一些拓樸場論之構造
指導教授:鄭志豪鄭志豪引用關係
學位類別:碩士
校院名稱:國立清華大學
系所名稱:數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2012
畢業學年度:100
論文頁數:28
中文關鍵詞:拓樸量子場論拓樸場論
外文關鍵詞:Topological Quantum Field Theory
相關次數:
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Topological Quantum Field Theory(TQFT) is an axiomatic theory of quantum field theory.
In this thesis, we start from studying the classical cobordism theory of manifolds, which serves
as the basic model for defining general TQFT, and then construct Euler TQFT and finite total
homotopy TQFT. Furthermore, we show that there is an one-to-one correspondence between
2D-TQFT and Frobenius algebras.
Topological Quantum Field Theory(TQFT) is an axiomatic theory of quantum field theory.
In this thesis, we start from studying the classical cobordism theory of manifolds, which serves
as the basic model for defining general TQFT, and then construct Euler TQFT and finite total
homotopy TQFT. Furthermore, we show that there is an one-to-one correspondence between
2D-TQFT and Frobenius algebras.
Abstract 1
1 Introduction 3
2 Axioms of TQFT 3
2.1 Unoriented and Oriented Cobordism 3
2.2 Cobordism Category of n-Manifolds 5
2.3 Topological Quantum Field Theory 6
3 Euler TQFT 8
4 Finite Total Homotopy TQFT 12
5 2D-Topological Quantum Field Theories and Frobenius Algebras 18
A Algebra 22
A.1 Algebra and Coalgebra 22
A.2 Frobenius Algebra 23
A.3 Monoidal Category 25
References 27

[1] Lowell Abrams, Two-Dimensional Topological Quantum Field Theories and Frobenius
Algebras, J. Knot Theory Ramications 5 (1996), no. 5, 569{587. MR MR1414088(97j:81292)
[2] M.F. Atiyah, Topological quantum eld theories, Publ. Math. de l'IHES, 68 (1989), p.175-186.
[3] S. Axler, F.W. Gehring and K.A. Ribet, Algebraic Topology from a Homotopical Viewpoint,
Universitext, Springer-Verlag, 2002.
[4] John C. Baez and James Dolan, Higher-Dimensional Algebra and Topological Quantum Field
Theory, 1995. Available at arXiv:q-alg/9503002v2.
[5] S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton University Press,
1952.
[6] Daniel S. Freed, Karen K. Uklenbeck, Editors, Geometry and Quantum Field Theory,
IAS/PARK city mathematics series vol. 1, Amer. Math. Soc., 1995.
[7] Allen Hatcher, Algebraic topology, Cambridge University Press, 2002.
[8] Morris W. Hirsch, Dierential Topology, Graduate Texts in Mathematics 33, Springer-Verlag,
1976.
[9] Joachim Kock, Frobenius Algebra and 2D Topological Quantum Field Theories, London Math.
Soc. Stu. Texts 59, Cambridge University Press, 2003.
27
[10] Jacob Lurie, On the Classication of Topological Field Theories, 2009. Available at
arXiv:0905.0465v1.
[11] James R. Munkres, Elementary Dierential Topology, Annals of Mathematics Studies 54,
Princeton University Press, 1961.
[12] S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5,
Springer-Verlag, 1971.
[13] R. Thom, Quelques proprietes globales des varietes dierentiables, Comment. Math. Helv. 28,
p.17-86, 1954.
[14] Vladimir G. Turaev, Quantum invariants of knots and 3-manifolds, de Gruyter Stu. in
Math., de Gruyter, 2010.
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