【摘要】：本文是一篇介紹數(shù)學物理中鏡對稱的綜述類文章.鏡對稱最初于1990年左右從弦論中提出,之后便引起了物理學家和數(shù)學家的關(guān)注,除了物理學家對其的應(yīng)用外,數(shù)學家對猜想本身的具體刻畫與證明產(chǎn)生了濃厚興趣,并發(fā)展了代數(shù)和幾何兩大類研究方法,取得了不少進展.如今,鏡對稱已成為聯(lián)系理論物理,辛幾何,代數(shù)幾何的最主要的交叉領(lǐng)域,也是數(shù)學物理中最富生機的領(lǐng)域之一.本文從多個方面對鏡對稱進行介紹,以給讀者對其初步的了解:第一章,介紹鏡對稱的概況和發(fā)展歷史,和相關(guān)學習資料.第二章,簡短介紹物理中sigma模型和其中一個鏡對稱的例子.第三章,介紹1994年由Maxim Kontsevich建立使用代數(shù)方法的的同調(diào)鏡對稱[K],它對鏡對稱數(shù)學方面的早期研究有很大的影響.第四章,簡短介紹1996年的由Strominger-Yau-Zaslow建立的用使用幾何方法的SYZ鏡對稱[SYZ],并給出T-對偶的一個例子.第五章,介紹同調(diào)鏡對稱中Fakaya范疇所屬的A∞-范疇的基本概念.第六章,介紹根據(jù)同調(diào)鏡對稱發(fā)展出的帶角辛流形的基本概念.
[Abstract]:This paper is an overview of mirror symmetry in mathematical physics. Mirror symmetry was first proposed from string theory in about 1990, and then aroused the attention of physicists and mathematicians. In addition to physicists' application to it, mathematicians were interested in the concrete characterization and proof of conjecture itself. Two kinds of research methods, algebra and geometry, have been developed, and a lot of progress has been made. Nowadays mirror symmetry has become one of the most dynamic fields in mathematical physics as well as the main cross-fields of theoretical physics symplectic geometry and algebraic geometry. This paper introduces mirror symmetry from many aspects in order to give readers a preliminary understanding of it. Chapter one introduces the general situation and development history of mirror symmetry and related learning materials. In the second chapter, we briefly introduce the sigma model in physics and an example of mirror symmetry. In the third chapter, we introduce the homotopy symmetry [K] which was established by Maxim Kontsevich in 1994 using algebraic method, which has a great influence on the early study of mirror symmetry mathematics. In chapter 4, we briefly introduce the SYZ mirror symmetry with geometric method [SYZ], which was established by Strominger-Yau-Zaslow in 1996, and give an example of T-duality. In chapter 5, we introduce the basic concept of A 鈭，

本文編號：2447520