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Books > Science & Mathematics > Physics > Quantum physics (quantum mechanics)
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Aspects of Scattering Amplitudes and Moduli Space Localization (Hardcover, 1st ed. 2020)
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Aspects of Scattering Amplitudes and Moduli Space Localization (Hardcover, 1st ed. 2020)
Series: Springer Theses
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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 to scattering 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.
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