Aspects of Scattering Amplitudes and Moduli Space Localization

1. Front Matter

   Pages i-xvii

   PDF

2. Introduction

   • Sebastian Mizera

   Pages 1-9

3. Intersection Numbers of Twisted Differential Forms

   • Sebastian Mizera

   Pages 11-47

4. Recursion Relations from Braid Matrices

   • Sebastian Mizera

   Pages 49-76

5. Further Examples of Intersection Numbers

   • Sebastian Mizera

   Pages 77-90

6. Conclusion

   • Sebastian Mizera

   Pages 91-92

7. Back Matter

   Pages 93-134

   PDF

This thesis proposes a new perspective on scattering amplitudes in quantum field theories. Their standard formulation in terms of sums over Feynman diagrams is replaced by a computation of geometric invariants, called intersection numbers, on moduli spaces of Riemann surfaces. It therefore gives a physical interpretation of intersection numbers, which have been extensively studied in the mathematics literature in the context of generalized hypergeometric functions. This book explores physical consequences of this formulation, such as recursion relations, connections to geometry and string theory, as well as a phenomenon called moduli space localization.

 After reviewing necessary mathematical background, including topology of moduli spaces of Riemann spheres with punctures and its fundamental group, the definition and properties of intersection numbers are presented. A comprehensive list of applications and relations to other objects is given, including those toscattering amplitudes in open- and closed-string theories. The highlights of the thesis are the results regarding localization properties of intersection numbers in two opposite limits: in the low- and the high-energy expansion.

 In order to facilitate efficient computations of intersection numbers the author introduces recursion relations that exploit fibration properties of the moduli space. These are formulated in terms of so-called braid matrices that encode the information of how points braid around each other on the corresponding Riemann surface. Numerous application of this approach are presented for computation of scattering amplitudes in various gauge and gravity theories. This book comes with an extensive appendix that gives a pedagogical introduction to the topic of homologies with coefficients in a local system.

• scattering theory
• scattering amplitudes in quantum field theories
• scattering amplitudes in string theory
• moduli spaces of punctured spheres
• twisted forms
• intersection number
• fibred moduli space

• Institute for Advanced Study, Princeton, USA

  Sebastian Mizera

Dr. Sebastian Mizera is a member at the Institute for Advanced Study in Princeton, NJ. He graduated from the University of Cambridge with a bachelor's degree in natural sciences and a master's in mathematics. He obtained a PhD in theoretical physics from the Perimeter Institute and the University of Waterloo in Canada. Dr. Mizera's work focuses on the interconnections between scattering amplitudes in quantum field theories and the mathematics of algebraic geometry and topology.

Policies and ethics