| Summary: | <p>This thesis presents an overview of recent developments in the applications of twistor theory to the study of the gravitational S-matrix in flat as well as curved space-times. We begin by introducing the novel geometric tool of twistor sigma models. These are two-dimensional chiral sigma models governing holomorphic maps from the Riemann sphere into twistor spaces of self-dual vacuum space-times. What follows is a concise list of their main highlights, in the order that the reader will encounter them:</p>
<p>i) Solutions to the equations of motion of our sigma models provide the incidence relations of the twistor correspondence.</p>
<p>ii) On-shell actions of our models compute Kahler potentials for hyperkahler metrics on the associated space-times.</p>
<p>iii) Connected, tree-level correlators of our models give rise to tree-level, maximally helicity violating (MHV) graviton amplitudes of general relativity in flat space.</p>
<p>iv) The chiral algebra of operators in our models enjoys an action of the loop algebra of the wedge subalgebra of w(1+infinity). This is associated with the soft sector of celestial holography.</p>
<p>v) By coupling them to background twistor spaces of self-dual, vacuum space-times, our models can be used to derive tree-level MHV graviton amplitudes in self-dual, radiative space-times.</p>
<p>The self-dual space-times mentioned in the last point provide an ideal laboratory for studying amplitudes in curved backgrounds. The corresponding formulae for MHV graviton amplitudes are built out of Hodges' determinants familiar from flat space, but also exhibit exciting new structures like gravitational wave tails.</p>
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