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/HFPDJX10. https://resolver.caltech.edu/CaltechTHESIS:06102019125514401
Abstract
In this thesis, we consider two topics in string theory and quantum field theory which are related by the common appearance of onedimensional projective geometry. In the first half of the thesis, we study sixdimensional (6D) supersymmetric quantum field theories and supergravity at the leading (tree) approximation and compute the complete Smatrix for these theories as worldsheet 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 nparticle Smatrix makes many symmetries and hidden properties manifest and generalizes the wellstudied formulas for fourdimensional 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 lowenergy effective field theory limits of string theory and Mtheory backgrounds. This includes theories such as those with 6D (2,0) supersymmetry which contain U(1) selfdual 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 worldvolume of the Mtheory Fivebrane. 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 AdS_{3}/CFT_{2} correspondence. In the usual statement of this correspondence, twodimensional conformal field theory (CFT) on the Riemann sphere or a highergenus 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 (antide Sitter space)/CFT correspondence according to this principle using projective geometry over the padic numbers, Q_{p}. The result is a formulation of holography in which the bulk geometry is discretethe BruhatTits 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, semiclassical dynamics of fields in the bulk compute the correlation functions of local operators on the boundary. Beyond correlators on the padic 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 RyuTakayanagi formula, holographic entanglement and black hole entropy, and multiple interval entanglement inequalities.
In Chapter 2, we present treelevel nparticle onshell scattering amplitudes of various brane theories with 16 conserved supercharges which are generalizations of DiracBornInfeld theory. These include the worldvolume theory of a probe D3brane or D5brane in 10D Minkowski spacetime as well as a probe M5brane in 11D Minkowski spacetime, which describes self interactions of an abelian tensor supermultiplet with 6D (2,0) supersymmetry. We propose twistorstringlike formulas for treelevel 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 (n3)! solutions of the scattering equations. The R symmetry of the D3brane 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 (D5brane and M5brane) reduces to the D3brane theory by dimensional reduction. As special cases of the general M5brane amplitudes, we present compact formulas for examples involving only the selfdual B field with n=4,6,8.
In Chapter 3, we extend this formalism to nparticle treelevel scattering amplitudes of sixdimensional N=(1,1) super YangMills (SYM) and N=(2,2) supergravity (SUGRA). The SYM theory arises on the world volume of coincident D5branes, and the supergravity is the result of toroidal compactification of string theory. These theories have nonabelian interactions which allow for both even and oddpoint amplitudes, unlike the branes of Chapter 2. Due to the properties of spinorhelicity variables in six dimensions, the evenn and oddn formulas are quite different and have to be treated separately. We first propose a manifestly supersymmetric expression for the evenn amplitudes of N=(1,1) SYM theory and perform various consistency checks. By considering softgluon limits of the evenn amplitudes, we deduce the form of the rational maps and the integrand for n odd. The oddn 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 WittenRSV (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 doublecopy properties, formulas for the sixdimensional N=(2,2) SUGRA amplitudes follow. These sixdimensional results allow us to deduce new formulas for fivedimensional SYM and SUGRA amplitudes, as well as massive amplitudes of fourdimensional N=4 SYM on the Coulomb branch.
In Chapter 4, we consider halfmaximal supergravity and present a twistorlike formula for the complete treelevel S matrix of chiral 6D (2,0) supergravity coupled to 21 abelian tensor multiplets. This is the lowenergy 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 treelevel Smatrix of 4D N=4 EinsteinMaxwell theory.
In Chapter 5, we introduce padic AdS/CFT and discuss several physical and mathematical features of the holographic correspondence between conformal field theories on P^{1}(Q_{p}) and lattice models on the BruhatTits tree of PGL(2,Q_{p}), an infinite tree of p+1 valence which has the padic projective line as its boundary. We review the padic numbers, the BruhatTits tree, and some of their applications to physics including padic 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; semiclassical 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 renormalizationgroup scale for modes in the boundary CFT. Highergenus 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 errorcorrecting codes, built from perfect tensors in network configurations dual to BruhatTits 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 padic version of entropy which obeys a RyuTakayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genuszero and genusone padic backgrounds, along with a BekensteinHawkingtype 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 semiclassical states in phase spaces over finite fields, generalizing the CRSS algorithm. These codes and the resulting networks provide a natural bulk geometric interpretation of nonArchimedean notions of entanglement in holographic boundary states.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  String theory; quantum field theory; supersymmetric scattering amplitudes; AdS/CFT; holography; mathematical physics; supergravity; quantum information  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Physics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Group:  Walter Burke Institute for Theoretical Physics  
Thesis Committee: 
 
Defense Date:  9 May 2019  
Funders: 
 
Record Number:  CaltechTHESIS:06102019125514401  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:06102019125514401  
DOI:  10.7907/HFPDJX10  
Related URLs: 
 
ORCID: 
 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  11735  
Collection:  CaltechTHESIS  
Deposited By:  Matthew Heydeman  
Deposited On:  10 Jun 2019 23:10  
Last Modified:  04 Oct 2019 00:26 
Thesis Files

PDF (Full thesis)
 Final Version
See Usage Policy. 3MB 
Repository Staff Only: item control page