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

Citation

Heydeman, Matthew Thomas Edwin (2019) Supersymmetric Scattering Amplitudes and Algebraic Aspects of Holography from the Projective Line. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/HFPD-JX10. https://resolver.caltech.edu/CaltechTHESIS:06102019-125514401

Abstract

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, Q_p. The result is a formulation of holography in which the bulk geometry is discrete---the Bruhat--Tits tree for PGL(2,Q_p)---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, F_p, 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(Q_p) and lattice models on the Bruhat--Tits tree of PGL(2,Q_p), 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.

Thesis Files