This book studies using string-net models to accomplish a direct, purely two-dimensional, approach to correlators of two-dimensional rational conformal field theories. The authors obtain concise geometric expressions for the objects describing bulk and boundary fields in terms of idempotents in the cylinder category of the underlying modular fusion category, comprising more general classes of fields than is standard in the literature. Combining these idempotents with Frobenius graphs on the world sheet yields string nets that form a consistent system of correlators, i.e. a system of invariants under appropriate mapping class groups that are compatible with factorization. The authors extract operator products of field objects from specific correlators; the resulting operator products are natural algebraic expressions that make sense beyond semisimplicity. They also derive an Eckmann-Hilton relation internal to a braided category, thereby demonstrating the utility of string nets for understanding algebra in braided tensor categories. Finally, they introduce the notion of a universal correlator. This systematizes the treatment of situations in which different world sheets have the same correlator and allows for the definition of a more comprehensive mapping class group.
Author(s): Jürgen Fuchs, Christoph Schweigert, Yang Yang
Series: SpringerBriefs in Mathematical Physics, 45
Publisher: Springer
Year: 2023
Language: English
Pages: 128
City: Cham
Preface
References
Contents
1 Introduction
1.1 A New Construction of RCFT Correlators
1.2 Advantages of the String-Net Construction
1.3 Organization of the Book
References
2 Correlators in Rational Conformal Field Theory
2.1 Open-Closed Modular Functor
2.2 World Sheets
2.3 Field Contents
2.3.1 Fields in the Bulk
2.3.2 Fields on Physical Boundaries
2.4 Correlators as Vectors in Spaces of Conformal Blocks
References
3 Correlators from String Nets
3.1 String Nets
3.2 Idempotent Completion
3.3 The String-Net Modular Functor
3.4 String Nets from World Sheets
3.5 String-Net Correlators
References
4 Correlators of Particular Interest
4.1 Vertical Operator Product
4.2 Horizontal Operator Products
4.3 Bulk Algebras
4.4 Torus Partition Function
4.5 Boundary Operator Product
4.6 Bulk-Boundary Operator Product
References
5 Internal Eckmann–Hilton Relation
5.1 An Internalized Eckmann–Hilton Argument
5.2 Three Braided Colored Operads
5.3 Proof of the Relation
References
6 Outlook: Universal Correlators
6.1 String Nets Colored by a Pivotal Bicategory
6.2 Universal Correlators from String Nets Colored by mathcalFr(mathcalC)
References
Appendix
A.1 Spherical Fusion Categories
A.2 Frobenius Algebras
A.3 Modules and Bimodules
A.4 Module Categories
A.5 Drinfeld Center
A.6 The Central Monad and Comonad
A.7 Coends for Generating Subcategories
A.8 Graphical Calculus for Spherical Fusion Categories
A.9 Three Useful Lemmas
Appendix References
