Open–closed strings: Two-dimensional extended TQFTs and Frobenius algebras
Open–closed strings: Two-dimensional extended TQFTs and Frobenius algebras
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开闭弦：二维扩展 TQFT 和 Frobenius 代数
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
H. Pfeiffer
中科院分区：
文献类型：
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作者：
Aaron D. Lauda;H. Pfeiffer
本文研究一类特殊的二维扩展拓扑量子场论。这些定义在开闭协边上，我们指的是光滑紧致定向的2-流形，其角具有特定的全局结构，以模拟开弦和闭弦世界单的光滑拓扑。我们证明了开闭TQFT范畴等价于知识丰富的Frobenius代数范畴。一个有知识的Frobenius代数（A，C，，）由一个对称Frobenius代数A，一个交换Frobenius代数C，和一个代数同态：C → A与对偶：A → C在一定条件下构成。这一结果是通过提供一个描述的开闭配边范畴的发电机和著名的摩尔-西格尔关系。为了证明我们的关系的充分性，我们提供了一个标准形式的共边，其特征在于拓扑不变量。从一个任意的这样的配边，我们构造一个序列的移动（广义手柄滑动和手柄取消），将给定的配边转换成正常的形式。利用开闭配边范畴的生成元和关系描述，证明了开闭配边范畴等价于由一个有知识的Frobenius代数自由生成的对称monoidal范畴.我们的形式主义，然后推广到开闭协边与标记的自由边界组件的上下文中，即开闭弦世界表与D-膜标签在其自由边界。© 2007爱思唯尔有限公司版权所有。MSC：57R56; 57M99; 81T40; 58E05; 19D23; 18D35
We study a special sort of 2-dimensional extended Topological Quantum Field Theories (TQFTs). These are defined on open– closed cobordisms by which we mean smooth compact oriented 2-manifolds with corners that have a particular global structure in order to model the smooth topology of open and closed string worldsheets. We show that the category of open–closed TQFTs is equivalent to the category of knowledgeable Frobenius algebras. A knowledgeable Frobenius algebra (A,C, ı, ı∗) consists of a symmetric Frobenius algebra A, a commutative Frobenius algebra C, and an algebra homomorphism ı :C → A with dual ı∗ :A → C, subject to some conditions. This result is achieved by providing a description of the category of open–closed cobordisms in terms of generators and the well-known Moore–Segal relations. In order to prove the sufficiency of our relations, we provide a normal form for such cobordisms which is characterized by topological invariants. Starting from an arbitrary such cobordism, we construct a sequence of moves (generalized handle slides and handle cancellations) which transforms the given cobordism into the normal form. Using the generators and relations description of the category of open–closed cobordisms, we show that it is equivalent to the symmetric monoidal category freely generated by a knowledgeable Frobenius algebra. Our formalism is then generalized to the context of open–closed cobordisms with labeled free boundary components, i.e. to open–closed string worldsheets with D-brane labels at their free boundaries. © 2007 Elsevier B.V. All rights reserved. MSC: 57R56; 57M99; 81T40; 58E05; 19D23; 18D35