String-Net Construction of RCFT Correlators

1. Front Matter

   Pages i-x

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2. Introduction

   • Jürgen Fuchs, Christoph Schweigert, Yang Yang

   Pages 1-7

3. Correlators in Rational Conformal Field Theory

   • Jürgen Fuchs, Christoph Schweigert, Yang Yang

   Pages 9-34

4. Correlators from String Nets

   • Jürgen Fuchs, Christoph Schweigert, Yang Yang

   Pages 35-59

5. Correlators of Particular Interest

   • Jürgen Fuchs, Christoph Schweigert, Yang Yang

   Pages 61-84

6. Internal Eckmann–Hilton Relation

   • Jürgen Fuchs, Christoph Schweigert, Yang Yang

   Pages 85-93

7. Outlook: Universal Correlators

   • Jürgen Fuchs, Christoph Schweigert, Yang Yang

   Pages 95-100

8. Back Matter

   Pages 101-123

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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.

• Conformal field theory (CFT)
• String net model
• Skein theory
• CFT correlators
• Two-dimensional rational conformal field theories (RCFTs)
• Topological field theories (TFT)

• Theoretical Physics, Karlstad University, Karlstad, Sweden

  Jürgen Fuchs

• Department of Mathematics, University of Hamburg, Hamburg, Germany

  Christoph Schweigert, Yang Yang

Jürgen Fuchs has received his PhD in 1985 at the University of Heidelberg. Since 2000 he is a professor of theoretical physics at Karlstad University, Sweden. He has been a visiting scientist at ETH Zürich, Paris 6 and 7, the Erwin-Schrödinger Institute, the Australian National University and the University of Alberta.

Christoph Schweigert has received his PhD from the University of Amsterdam in 1995. Since 2003 he is a professor of mathematics at Hamburg University. He was an invited speaker at the ICM 2006 and the ECM 2008.

Yang Yang has received his MSc at the University of Hamburg in 2019. He is a PhD student at the University of Hamburg where he expects to receive his PhD in 2022.

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