Colored five-vertex models and Demazure atoms

Type A Demazure atoms are pieces of Schur functions, or sets of tableaux whose weights sum to such functions. Inspired by colored vertex models of Borodin and Wheeler, we will construct solvable lattice models whose partition functions are Demazure atoms; the proof of this makes use of a Yang-Baxter equation for a colored five-vertex model. As a biproduct, we construct Demazure atoms on Kashiwara's $\mathcal{B}_\infty$ crystal and give new algorithms for computing Lascoux-Sch\"utzenberger keys

Euclidean quantum field formulation of p-adic open string amplitudes

We study in a rigorous mathematical way p-adic quantum field theories whose N-point amplitudes are the expectation of products of vertex operators. We show that this type of amplitudes admit a series expansion where each term is an Igusa\u27s local zeta function. The lowest term in this series is a regularized version of the p-adic open Koba-Nielsen string amplitude

Multi-Black-Hole Configurations as Models for Inhomogeneous Cosmologies

On the largest scales, the Universe is homogeneous and isotropic, whereas on smaller scales, various structures immediately begin to emerge. The transition from an inhomogeneous spacetime to the homogeneous and isotropic Friedmann universe is not sufficiently understood yet. Modern cosmology rests on the hypothesis that the LambdaCDM-model applies and, indeed, this model is very successful. On the other hand, as the precision of observations steadily increases, it is more than likely that inhomogeneities will no longer be negligible in the future. For this reason, the study of inhomogeneous cosmological models is reasonable. In this thesis, we consider the question which Friedmann universe is the best fit to a particular given inhomogeneous spacetime, which is known as the fitting problem. We consider models in which matter is replaced by a discrete configuration of black holes, that is, we concentrate on vacuum solutions to Einstein's equations. Since the full system of the field equations is too complicated to find an exact time-dependent solution for the whole spacetime, we restrict ourselves to approximative models as well as solutions to the initial value problem. In the former case, we reconsider Swiss-cheese and Lindquist-Wheeler models. In both models, the spacetime around a mass is described by the Schwarzschild metric. In the latter case, we determine the spatial metric of a space-like hypersurface. We limit our attention to time-symmetric initial data characterised by the vanishing of the extrinsic curvature. In this case, we are able to find a solution for an arbitrary number of black holes using the conformal method. Clearly, it is not reasonable to assume that every configuration of black holes leads to a spacetime which may be approximated well by a Friedmann solution. Such an approximation should be possible if the masses are distributed somehow uniformly. The aim of this thesis is to clarify this statement and to provide criteria which allow quantitative statements about the degree of uniformity. We determine the parameters of the fitted dust universe, in particular the scale factor. Our considerations are supported by several example configurations. In particular, we provide a new method based on Lie sphere geometry to construct various configurations with a high degree of uniformity in a surprisingly simple fashion. Moreover, we provide a generalisation to an approximative inhomogeneous model given by Lindquist and Wheeler. In this case, it is possible to determine the parameters of the fitted Friedmann universe even if we do not know the exact solution. Under certain conditions, this model becomes similar to a Swiss-cheese model, allowing us to formulate first expectations on the time evolution, which is otherwise mostly disregarded within the framework of this thesis

Supersymmetric Scattering Amplitudes and Algebraic Aspects of Holography from the Projective Line

In this thesis, we consider two topics in string theory and quantum field theory which are related by the common appearance of one-dimensional projective geometry. In the first half of the thesis, we study six-dimensional (6D) supersymmetric quantum field theories and supergravity at the leading (tree) approximation and compute the complete S-matrix for these theories as world-sheet integrals over the punctured Riemann sphere. This exploits the analytic structure of tree amplitudes which are rational and holomorphic in the kinematics and naturally related to the geometry of points on the complex projective line. The 6D n-particle S-matrix makes many symmetries and hidden properties manifest and generalizes the well-studied formulas for four-dimensional amplitudes in the form of twistor string theory and the rational curves formalism. While the systems we study are all field theories, they are in essence low-energy effective field theory limits of string theory and M-theory backgrounds. This includes theories such as those with 6D (2,0) supersymmetry which contain U(1) self-dual tensor fields which are difficult to treat from a Lagrangian point of view. Our formulas circumvent this difficulty and allow a generalization and unification of a large class of 6D scattering amplitudes which permit a sensible classical limit, including the abelian world-volume of the M-theory Five-brane. Dimensional reduction to four dimensions is also possible, leading to new formulas for 4D physics from 6D. In the second half of the thesis, we discuss the projective algebraic and geometric structure of the AdS3/CFT2 correspondence. In the usual statement of this correspondence, two-dimensional conformal field theory (CFT) on the Riemann sphere or a higher-genus surface is holographically dual to features of topological gravity in three dimensions with negative curvature. Since every compact Riemann surface is a projective algebraic curve, many constructions of interest in physics (which a priori depend on the analytic structure of the spacetime) can be formulated in purely algebraic language. We generalize the AdS (anti-de Sitter space)/CFT correspondence according to this principle using projective geometry over the p-adic numbers, Qp. The result is a formulation of holography in which the bulk geometry is discrete---the Bruhat--Tits tree for PGL(2,Qp)---but the group of bulk isometries nonetheless agrees with that of boundary conformal transformations and is not broken by discretization. Parallel to the usual holographic correspondence, semi-classical dynamics of fields in the bulk compute the correlation functions of local operators on the boundary. Beyond correlators on the p-adic line, we propose a tensor network model in which the patterns of entanglement on the boundary are computed by discrete geometries in the bulk. We suggest that this forms the natural geometric setting for tensor networks that have been proposed as models of bulk reconstruction via quantum error correcting codes. The model is built from tensors based on projective geometry over finite fields, Fp, and correctly computes the Ryu-Takayanagi formula, holographic entanglement and black hole entropy, and multiple interval entanglement inequalities. In Chapter 2, we present tree-level n-particle on-shell scattering amplitudes of various brane theories with 16 conserved supercharges which are generalizations of Dirac--Born--Infeld theory. These include the world-volume theory of a probe D3-brane or D5-brane in 10D Minkowski spacetime as well as a probe M5-brane in 11D Minkowski spacetime, which describes self interactions of an abelian tensor supermultiplet with 6D (2,0) supersymmetry. We propose twistor-string-like formulas for tree-level scattering amplitudes of all multiplicities for each of these theories, and the amplitudes are written as integrals over the moduli space of certain rational maps localized on the (n-3)! solutions of the scattering equations. The R symmetry of the D3-brane theory is shown to be SU(4) x U(1), and the U(1) factor implies that its amplitudes are helicity conserving. Each of 6D theories (D5-brane and M5-brane) reduces to the D3-brane theory by dimensional reduction. As special cases of the general M5-brane amplitudes, we present compact formulas for examples involving only the self-dual B field with n=4,6,8. In Chapter 3, we extend this formalism to n-particle tree-level scattering amplitudes of six-dimensional N=(1,1) super Yang--Mills (SYM) and N=(2,2) supergravity (SUGRA). The SYM theory arises on the world volume of coincident D5-branes, and the supergravity is the result of toroidal compactification of string theory. These theories have non-abelian interactions which allow for both even and odd-point amplitudes, unlike the branes of Chapter 2. Due to the properties of spinor-helicity variables in six dimensions, the even-n and odd-n formulas are quite different and have to be treated separately. We first propose a manifestly supersymmetric expression for the even-n amplitudes of N=(1,1) SYM theory and perform various consistency checks. By considering soft-gluon limits of the even-n amplitudes, we deduce the form of the rational maps and the integrand for n odd. The odd-n formulas obtained in this way have a new redundancy that is intertwined with the usual SL(2,C) invariance on the Riemann sphere. We also propose an alternative form of the formulas, analogous to the Witten--RSV (Roiban, Spradlin, and Volovich) formulation, and explore its relationship with the symplectic (or Lagrangian) Grassmannian. Since the amplitudes are formulated in a way that manifests double-copy properties, formulas for the six-dimensional N=(2,2) SUGRA amplitudes follow. These six-dimensional results allow us to deduce new formulas for five-dimensional SYM and SUGRA amplitudes, as well as massive amplitudes of four-dimensional N=4 SYM on the Coulomb branch. In Chapter 4, we consider half-maximal supergravity and present a twistor-like formula for the complete tree-level S matrix of chiral 6D (2,0) supergravity coupled to 21 abelian tensor multiplets. This is the low-energy effective theory that corresponds to Type IIB superstring theory compactified on a K3 surface. As in previous chapters, the formula is expressed as an integral over the moduli space of certain rational maps of the punctured Riemann sphere; the new ingredient is an integrand which successfully incorporates both gravitons and multiple flavors of tensors. By studying soft limits of the formula, we are able to explore the local moduli space of this theory, SO(5,21)/(SO(5) x SO(21)). Finally, by dimensional reduction, we also obtain a new formula for the tree-level S-matrix of 4D N=4 Einstein--Maxwell theory. In Chapter 5, we introduce p-adic AdS/CFT and discuss several physical and mathematical features of the holographic correspondence between conformal field theories on P1(Qp) and lattice models on the Bruhat--Tits tree of PGL(2,Qp), an infinite tree of p+1 valence which has the p-adic projective line as its boundary. We review the p-adic numbers, the Bruhat--Tits tree, and some of their applications to physics including p-adic CFT. A key feature of these constructions is the discrete and hierarchical nature of the tree and the corresponding field theories, which serve as a toy model of holography in which there are no gravitons and no conformal descendants. Standard holographic results for massive free scalar fields in a fixed background carry over to the tree; semi-classical dynamics in the bulk compute correlation functions in the dual field theory and we obtain a precise relationship between the bulk mass and the scaling dimensions of local operators. It is also possible to interpret the vertical direction in the tree a renormalization-group scale for modes in the boundary CFT. Higher-genus bulk geometries (the BTZ black hole and its generalizations) can be understood straightforwardly in our setting and their construction parallels the story in AdS_3 topological gravity. In Chapter 6, we consider a class of holographic quantum error-correcting codes, built from perfect tensors in network configurations dual to Bruhat--Tits trees and their quotients by Schottky groups corresponding to BTZ black holes. The resulting holographic states can be constructed in the limit of infinite network size. We obtain a p-adic version of entropy which obeys a Ryu--Takayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genus-zero and genus-one p-adic backgrounds, along with a Bekenstein--Hawking-type formula for black hole entropy. We prove entropy inequalities obeyed by such tensor networks, such as subadditivity, strong subadditivity, and monogamy of mutual information (which is always saturated). In addition, we construct infinite classes of perfect tensors directly from semi-classical states in phase spaces over finite fields, generalizing the CRSS algorithm. These codes and the resulting networks provide a natural bulk geometric interpretation of non-Archimedean notions of entanglement in holographic boundary states.</p

Foundations of General Relativity

This book, dedicated to Roger Penrose, is a second, mathematically oriented course in general relativity. It contains extensive references and occasional excursions in the history and philosophy of gravity, including a relatively lengthy historical introduction. The book is intended for all students of general relativity of any age and orientation who have a background including at least first courses in special and general relativity, differential geometry, and topology. The material is developed in such a way that through the last two chapters the reader may acquire a taste of the modern mathematical study of black holes initiated by Penrose, Hawking, and others, as further influenced by the initial-value or PDE approach to general relativity. Successful readers might be able to begin reading research papers on black holes, especially in mathematical physics and in the philosophy of physics. The chapters are: Historical introduction, General differential geometry, Metric differential geometry, Curvature, Geodesics and causal structure, The singularity theorems of Hawking and Penrose, The Einstein equations, The 3+1 split of space-time, Black holes I: Exact solutions, and Black holes II: General theory. These are followed by two appendices containing background on Lie groups, Lie algebras, & constant curvature, and on Formal PDE theory

Foundations of General Relativity

This book, dedicated to Roger Penrose, is a second, mathematically oriented course in general relativity. It contains extensive references and occasional excursions in the history and philosophy of gravity, including a relatively lengthy historical introduction. The book is intended for all students of general relativity of any age and orientation who have a background including at least first courses in special and general relativity, differential geometry, and topology. The material is developed in such a way that through the last two chapters the reader may acquire a taste of the modern mathematical study of black holes initiated by Penrose, Hawking, and others, as further influenced by the initial-value or PDE approach to general relativity. Successful readers might be able to begin reading research papers on black holes, especially in mathematical physics and in the philosophy of physics. The chapters are: Historical introduction, General differential geometry, Metric differential geometry, Curvature, Geodesics and causal structure, The singularity theorems of Hawking and Penrose, The Einstein equations, The 3+1 split of space-time, Black holes I: Exact solutions, and Black holes II: General theory. These are followed by two appendices containing background on Lie groups, Lie algebras, & constant curvature, and on Formal PDE theory