Aspects of Scattering Amplitudes and Moduli Space Localization

A new perspective on scattering amplitudes in quantum field theories is proposed, which replaces the standard formulation in terms of sums over Feynman diagrams with a computation of geometric invariants called intersection numbers on moduli spaces of Riemann surfaces. This physical interpretation of intersection numbers gives rise to recursion relations, connections to geometry and string theory, and a phenomenon called moduli space localization. The thesis introduces recursion relations to facilitate efficient computations of intersection numbers and applies them to compute scattering amplitudes in various gauge and gravity theories. An extensive appendix provides a pedagogical introduction to homologies with coefficients in a local system.

Format: Paperback / softback
Length: 134 pages
Publication date: 25 September 2021
Publisher: Springer Nature Switzerland AG

This thesis presents a novel perspective on scattering amplitudes in quantum field theories, revolutionizing their traditional formulation in terms of sums over Feynman diagrams. Instead, it proposes a computation of geometric invariants, known as intersection numbers, on moduli spaces of Riemann surfaces. This groundbreaking approach provides a physical interpretation of intersection numbers, which have been extensively studied in the mathematics literature within the context of generalized hypergeometric functions.

In this comprehensive book, we delve into the physical consequences of this formulation, exploring topics such as recursion relations, connections to geometry and string theory, and the phenomenon of moduli space localization. After reviewing the necessary mathematical background, including topology of moduli spaces of Riemann spheres with punctures and its fundamental group, we introduce the definition and properties of intersection numbers. We provide a comprehensive list of applications and relations to other objects, including scattering amplitudes in open- and closed-string theories.

The thesis's highlights revolve around the results regarding localization properties of intersection numbers in two opposite limits: the low- and the high-energy expansion. To facilitate efficient computations of intersection numbers, the author introduces recursion relations that leverage the fibration properties of the moduli space. These recursion relations are formulated in terms of braid matrices, which encode the information of how points braid around each other on the corresponding Riemann surface. Numerous applications of this approach are presented for computing scattering amplitudes in various gauge and gravity theories.

Furthermore, this book includes an extensive appendix that serves as a pedagogical introduction to the topic of homologies with coefficients. This appendix provides a clear and concise explanation of the necessary concepts, making it accessible to readers with a limited background in mathematics.

In summary, this thesis offers a transformative perspective on scattering amplitudes, providing a physical interpretation of intersection numbers and exploring their profound physical consequences. Through the application of recursion relations and braid matrices, it enables efficient computations of intersection numbers in various gauge and gravity theories. This book is a valuable resource for researchers and students interested in quantum field theories, topology, and mathematical physics.

Weight: 244g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030530129
Edition number: 1st ed. 2020