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

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

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

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

A splicing-dependent transcriptional checkpoint associated with prespliceosome formation

There is good evidence for functional interactions between splicing and transcription in eukaryotes, but how and why these processes are coupled remain unknown. Prp5 protein (Prp5p) is an RNA-stimulated adenosine triphosphatase (ATPase) required for prespliceosome formation in yeast. We demonstrate through in vivo RNA labeling that, in addition to a splicing defect, the prp5-1 mutation causes a defect in the transcription of intron-containing genes. We present chromatin immunoprecipitation evidence for a transcriptional elongation defect in which RNA polymerase that is phosphorylated at Ser5 of the largest subunit’s heptad repeat accumulates over introns and that this defect requires Cus2 protein. A similar accumulation of polymerase was observed when prespliceosome formation was blocked by a mutation in U2 snRNA. These results indicate the existence of a transcriptional elongation checkpoint that is associated with prespliceosome formation during cotranscriptional spliceosome assembly. We propose a role for Cus2p as a potential checkpoint factor in transcription

Characteristic polynomials of random Hermitian matrices and Duistermaat-Heckman localisation on non-compact Kaehler manifolds

We reconsider the problem of calculating a general spectral correlation function containing an arbitrary number of products and ratios of characteristic polynomials for a N x N random matrix taken from the Gaussian Unitary Ensemble (GUE). Deviating from the standard "supersymmetry" approach, we integrate out Grassmann variables at the early stage and circumvent the use of the Hubbard-Stratonovich transformation in the "bosonic" sector. The method, suggested recently by one of us, is shown to be capable of calculation when reinforced with a generalization of the Itzykson-Zuber integral to a non-compact integration manifold. We arrive to such a generalisation by discussing the Duistermaat-Heckman localization principle for integrals over non-compact homogeneous Kaehler manifolds. In the limit of large $N$ the asymptotic expression for the correlation function reproduces the result outlined earlier by Andreev and Simons.Comment: 34 page, no figures. In this version we added a few references and modified the introduction accordingly. We also included a new Appendix on deriving our Itzykson-Zuber type integral following the diffusion equation metho

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

Theta dependence of SU(N) gauge theories in the presence of a topological term

We review results concerning the theta dependence of 4D SU(N) gauge theories and QCD, where theta is the coefficient of the CP-violating topological term in the Lagrangian. In particular, we discuss theta dependence in the large-N limit. Most results have been obtained within the lattice formulation of the theory via numerical simulations, which allow to investigate the theta dependence of the ground-state energy and the spectrum around theta=0 by determining the moments of the topological charge distribution, and their correlations with other observables. We discuss the various methods which have been employed to determine the topological susceptibility, and higher-order terms of the theta expansion. We review results at zero and finite temperature. We show that the results support the scenario obtained by general large-N scaling arguments, and in particular the Witten-Veneziano mechanism to explain the U(1)_A problem. We also compare with results obtained by other approaches, especially in the large-N limit, where the issue has been also addressed using, for example, the AdS/CFT correspondence. We discuss issues related to theta dependence in full QCD: the neutron electric dipole moment, the dependence of the topological susceptibility on the quark masses, the U(1)_A symmetry breaking at finite temperature. We also consider the 2D CP(N) model, which is an interesting theoretical laboratory to study issues related to topology. We review analytical results in the large-N limit, and numerical results within its lattice formulation. Finally, we discuss the main features of the two-point correlation function of the topological charge density.Comment: A typo in Eq. (3.9) has been corrected. An additional subsection (5.2) has been inserted to demonstrate the nonrenormalizability of the relevant theta parameter in the presence of massive fermions, which implies that the continuum (a -> 0) limit must be taken keeping theta fixe