The momentum amplituhedron

Diese Dissertation befasst sich mit einigen der jüngeren theoretischen Entwicklungen auf dem Gebiet der Streuamplituden. In den letzten Jahren wurde immer mehr der traditionelle Ansatz der Extraktion von Streuamplituden aus Feynman-Diagrammen zugunsten von Techniken, die als On-Shell-Methoden bekannt sind, aufgegeben. Diese Methoden offenbaren eine interessante Beziehung zwischen Streuamplituden und einer Geometrie, die als positive Grassmannsche Geometrie bekannt ist und zu einer radikalen Neuformulierung von Streuamplituden durch so genannte positiven Geometrien geführt hat. Positive Geometrien sind Geometrien mit Rändern aller Kodimensionen und gewissen zugehörigen \emph{kanonischen Formen}, aus denen Streuamplitude extrahiert werden können. Der zentrale Akteur dieser Dissertation ist das Impulsamplituhedron, welches durch die Positive Geometrie gegeben ist und die on-shell Amplituden auf Baumniveau in der maximal supersymmetrischen Yang-Mills-Theorie kodiert, die im Raum der Spinor-Helizitätsvariablen definiert ist. Die canonical Form das Impulsamplituhedron verfügt über eine besondere Singularitätsstruktur, die die physikalischen Singularitäten der Streuamplituden in allen Helizitätssektoren auf Baumniveau kodiert, aus denen die Streuamplituden extrahiert werden können. Dies ermöglicht es, Streuamplituden in maximal supersymmetrischen Yang-Mills Theorie zu bestimmen ohne Bezug auf Felder, Lagrangedichten, Raumzeit oder Feynman-Diagramme zu nehmen. In neueren Arbeiten über das Impulsamplituhedron konnten wir sehen, das seine kanonische Form mit der kanonischen Form - die mit einer Geometrie assoziiert ist, welche die Streuamplituden für bi-adjungierte Skalare - dem kinematischen Associahedron kodiert, in Verbindung gebracht werden kann. Die Definition des Impusamplituhedron auf dem Raum der Spinor-Helizitäts-Variablen ermöglicht einen direkten Vergleich von Geometrien, mit unterschiedlich Farb-geordneten Streuamplituden im selben Raum verbunden sind. Die wird genutzt, um die Kleiss-Kuijf-Relationen -- eine Reihe von Beziehungen zwischen Streuamplituden verschiedener Farbordnungen, wiederherzustellen, die sich aus der Farbzerlegung von Streuamplituden ergeben. Die Kleiss-Kuijf-Relationen manifestieren sich als orientierte Summen von Impulsamplituhedronen verschiedener Farbordnungen ohne Vertices in ihren Rändern. Wir leiten einen homologischen Algorithmus ab, der auf diesem Prinzip basiert, um Kleiss-Kuijf-Beziehungen für Impulsamplituhedronen zu finden.This dissertation focus on some of the modern theoretical developments in the field of scattering amplitudes. Recent years have seen a departure from the traditional approach of extracting scattering amplitudes in terms of Feynman diagrams in favor of techniques known as on-shell methods. These methods reveal a striking relationship between scattering amplitudes and a geometry known as the positive Grassmannian, leading to a radical reformulation of scattering amplitudes in terms of so-called positive geometries. Positive geometries are geometries with boundaries of all codimensions and have a certain associated canonical form. In some special cases, physical observables can be extracted from the canonical forms of positive geometries. The central player in this dissertation is the \emph{momentum amplituhedron} which is the positive geometry encoding on-shell tree-level amplitudes in maximally supersymmetric Yang-Mills theory defined on the space of spinor helicity variables. The momentum amplituhedron is equipped with a canonical form with a particular singularity structure, encoding the physical singularities of scattering amplitudes in all helicity sectors at tree-level, from which scattering amplitudes can be extracted. This allows us to determine scattering amplitudes in maximally supersymmetric Yang-Mills without reference to fields, Lagrangians, space-time, or Feynman diagrams. We will in this dissertation report on the most recent results for the momentum amplituhedron obtained in collaboration with other authors. In particular, we will see that its canonical form can be related to the canonical form associated with a geometry encoding scattering amplitudes for bi-adjoint scalars -- the kinematic associahedron. Furthermore, since we can define the momentum amplituhedron on the space of spinor helicity variables, it allows for a direct comparison of geometries associated with differently color-ordered scattering amplitudes in the same space. This ability to compare momentum amplituhedra of different color orderings will be employed to rederive the Kleiss-Kuijf relations, a set of relations between scattering amplitudes of different color orderings stemming from the color decomposition of scattering amplitudes. The Kleiss-Kuijf relations will appear as oriented sums of momentum amplituhedra of different color orderings with no vertices in their boundary stratifications. We will use this fact to derive a homological algorithm based on this principle to find Kleiss-Kuijf relations for momentum amplituhedra

N=4 Supersymmetry on a Space-Time Lattice

Maximally supersymmetric Yang--Mills theory in four dimensions can be formulated on a space-time lattice while exactly preserving a single supersymmetry. Here we explore in detail this lattice theory, paying particular attention to its strongly coupled regime. Targeting a theory with gauge group SU(N), the lattice formulation is naturally described in terms of gauge group U(N). Although the U(1) degrees of freedom decouple in the continuum limit we show that these degrees of freedom lead to unwanted lattice artifacts at strong coupling. We demonstrate that these lattice artifacts can be removed, leaving behind a lattice formulation based on the SU(N) gauge group with the expected apparently conformal behavior at both weak and strong coupling

The D=1 Matrix Model and the Renormalization Group

We compute the critical exponents of $d = 1$ string theory to leading order, using the renormalization group approach recently suggested by Br\'{e}zin and Zinn-Justin.Comment: 8 pages, Latex, CERN-TH-6546/9

Kleiss-Kuijf relations from momentum amplituhedron geometry

44 pages, 19 figuresAbstract: In recent years, it has been understood that color-ordered scattering amplitudes can be encoded as logarithmic differential forms on positive geometries. In particular, amplitudes in maximally supersymmetric Yang-Mills theory in spinor helicity space are governed by the momentum amplituhedron. Due to the group-theoretic structure underlying color decompositions, color-ordered amplitudes enjoy various identities which relate different orderings. In this paper, we show how the Kleiss-Kuijf relations arise from the geometry of the momentum amplituhedron. We also show how similar relations can be realised for the kinematic associahedron, which is the positive geometry of bi-adjoint scalar cubic theory.Peer reviewe

Results from lattice simulations of N=4 supersymmetric Yang--Mills

We report recent results and developments from our ongoing lattice studies of $\mathcal N = 4$ supersymmetric Yang--Mills theory. These include a proof that only a single fine-tuning needs to be performed, so long as the moduli space is not lifted by nonperturbative effects. We extend our investigations of supersymmetry restoration in the continuum limit by initiating Monte Carlo renormalization group studies. We present additional numerical evidence that the lattice theory does not suffer from a sign problem. Finally we study the static potential, which we find to be Coulombic at both weak and strong coupling. We compare the static potential Coulomb coefficients to perturbation theory, including initial results for N=3 colors in addition to N=2.Comment: 19-page combined contribution to the proceedings of Lattice 2014, for talks by SC, JG & DS. v2: Eq. 6.3 corrected, reference adde

Spectrum of the Dirac operator coupled to two-dimensional quantum gravity

We implement fermions on dynamical random triangulation and determine numerically the spectrum of the Dirac-Wilson operator D for the system of Majorana fermions coupled to two-dimensional Euclidean quantum gravity. We study the dependence of the spectrum of the operator (epsilon D) on the hopping parameter. We find that the distributions of the lowest eigenvalues become discrete when the hopping parameter approaches the value 1/sqrt{3}. We show that this phenomenon is related to the behavior of the system in the 'antiferromagnetic' phase of the corresponding Ising model. Using finite size analysis we determine critical exponents controlling the scaling of the lowest eigenvalue of the spectrum including the Hausdorff dimension d_H and the exponent kappa which tells us how fast the pseudo-critical value of the hopping parameter approaches its infinite volume limit.Comment: 26 pages, Latex + 23 eps figs, extended analysis of the spectrum, added figure

The Momentum Amplituhedron

In this paper we define a new object, the momentum amplituhedron, which is the long sought-after positive geometry for tree-level scattering amplitudes in N = 4 super Yang-Mills theory in spinor helicity space. Inspired by the construction of the ordinary amplituhedron, we introduce bosonized spinor helicity variables to represent our external kinematical data, and restrict them to a particular positive region. The momentum amplituhedron M n,k is then the image of the positive Grassmannian via a map determined by such kinematics. The scattering amplitudes are extracted from the canonical form with logarithmic singularities on the boundaries of this geometry.Peer reviewedFinal Published versio

S-duality in lattice super Yang-Mills

We present a progress report on studying S-duality in lattice N = 4 super Yang-Mills. This is being done through a computation of 1/2-BPS states on the Coulomb branch, especially the ’t Hooft–Polyakov monopole and the W boson. Key to these calculations is the use of twisted and C-periodic boundary conditions. In addition we describe a variational method to disentangle operators with definite scaling dimension, particularly the Konishi and supergravity operators