CaltechTHESIS advisor: Monograph
https://feeds.library.caltech.edu/people/Schwarz-J-H/combined_advisor.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 12 Nov 2024 13:23:45 -0800Extended Supergravity with a Gauged Central Charge
https://resolver.caltech.edu/CaltechETD:etd-12022004-160245
Year: 1979
DOI: 10.7907/X6KP-V369
<p>We construct a Lagrangian for the massive scalar multiplet, locally invariant under two types of spinorial transformations (N=2 supersymmetry). Our theory is based on the coupling of the global supermultiplet to N=2 supergravity and corrections generated iteratively in powers of Newton's constant. Consistency of the theory requires the vector field of supergravity to gauge the central charge represented in the massive sector of the multiplet. The same vector may alternatively gauge the internal 0(2) symmetry of the two supersymmetry generators. Furthermore, it may even gauge a linear combination of the generators of these two groups; we indicate the grounds for this compatibility.</p>
<p>We discuss the hierarchy of internal symmetries characterizing each sector of the theory, ranging from U(1)xSU(2)xSU(2) down to 0(2)x0(2). This internal symmetry imposes tight constraints on the system. For instance, the nonpolynomial structure of the spinless fields at hand is considerably more restricted than that present in the general simple supersymmetric (N-1) theory. Furthermore, the vector field is forced to couple to the matter fields with gravitational strength, to the effect that the resulting Coulomb potential exactly cancels against the Newtonian potential of gravity, in the static limit.</p>
<p>Our theory may be also viewed as a truncation of N=8 supergravity theory, compatibly with the SO (8) breakup scheme into SU(3)xU(1)xU(1). The potential of the spinless fields present has a local minimum at the origin, but further off it is not even bounded from below. However, we point out some indications that the tunneling out of the supersymmetric, metastable vacuum is negligibly small.</p>https://resolver.caltech.edu/CaltechETD:etd-12022004-160245The Ultraviolet Divergences of Einstein Gravity
https://resolver.caltech.edu/CaltechTHESIS:04052019-155701481
Year: 1986
DOI: 10.7907/zntd-4x91
<p>We discuss a two-loop calculation showing that the S matrix of Einstein's theory of gravity contains nonrenormalizable ultraviolet divergences in four dimension. We discuss the calculation in both background field and normal field theory. We describe a new method for dealing with ghost fields in gauge theories by combining them with suitable extensions of the gauge fields in higher dimensions. We show how using subtracted integrals in the calculation of higher loop graphs simplifies the calculation in the background field method by eliminating the need for "mixed" counterterms. Finally, we make some remarks about the implications of our result for supergravity theories.</p>https://resolver.caltech.edu/CaltechTHESIS:04052019-155701481G/H Conformal Field Theory
https://resolver.caltech.edu/CaltechETD:etd-08202008-083708
Year: 1988
DOI: 10.7907/YNEZ-8X17
<p>We show that for every affine Lie algebra G and subalgebra H there exists an exactly solvable two-dimensional conformal field theory, and give a procedure for explicitly determining its correlation functions and partition function given those for the Wess-Zumino-Witten models with symmetry algebras G and H.</p>https://resolver.caltech.edu/CaltechETD:etd-08202008-083708The Singular Mechanics of Particles and Strings
https://resolver.caltech.edu/CaltechTHESIS:01172013-111439406
Year: 1988
DOI: 10.7907/hvjy-be63
<p>The quantum mechanics of singular systems is a topic of considerable importance for all the theories of elementary particle physics in which gauge invariance is a universal attribute. This is especially true for string theories which are gauge theories <i>par excellence</i>.</p>
<p>This thesis begins with a brief exposition of singular Hamiltonian mechanics. This tool is applied principally to manifestly supersymmetric particle and string theories. The Dirac particle and the bosonic particle and string are briefly examined. In particular, a method is shown for quantizing the point superparticle in four and ten dimensions. The two actions proposed for describing the manifestly supersymmetric string are shown to be essentially equivalent. The problems of their quantization are briefly discussed.</p>https://resolver.caltech.edu/CaltechTHESIS:01172013-111439406Applications of current algebra in conformal field theory
https://resolver.caltech.edu/CaltechTHESIS:04112011-105355285
Year: 1991
DOI: 10.7907/1tw7-4v34
In this work, two topics concerning the interplay between current algebra and conformal symmetry in two dimensions are discussed. The construction of a conformal algebra from a current algebra, the Virasoro Master Equation, is presented with analytic and perturbative solutions. Second, N = 2 superconformal models based on supersymmetric current algebras with c > 3 are coupled to two dimensional topological gravity.
https://resolver.caltech.edu/CaltechTHESIS:04112011-105355285Strings, two-dimensional gravity, and matrix models
https://resolver.caltech.edu/CaltechTHESIS:04112011-114355027
Year: 1991
DOI: 10.7907/e8fb-f051
Two-dimensional models of quantum gravity have been solved using matrix model techniques. Furthermore, these solutions have turned out to be encoded in integrable nonlinear PDEs belonging to the KdV hierarchy. This thesis presents a new KdV recursion relation, distinct from one found previously by Dijkgraaf and Witten, for a certain class of theories known as the two-matrix models. The two recursion relations together are used to relate arbitrary correlation functions containing a puncture operator P (at any genus) to the three basic correlators <PP>, <PQ>, and <QQ> by unique algebraic expressions. (Q is the dilaton operator.) The derivation requires assuming a certain scaling law, whose justification is discussed.
Other KdV recursion relations, given by Virasoro or W-algebra constraints, are possible for multi-matrix models when an infinite number of couplings are added. These constraints have been presented for A_n-type models by Fukuma et al. and Dijkgraaf et ai. We derive analogous Virasoro constraints for the multi-matrix models associated with the other simply-laced Lie algebras D_(2n+1), E_6, E_7, and E_8. As a check, it is verified that the proposed constraints imply operator scaling dimensions identical to those found by Kostov. It is then demonstrated that these Virasoro constraints (or, more generally, W -algebra constraints) can be used to derive expressions for correlation functions containing a non-primary operator in terms of correlation functions that only contain primary operators.
The second subject of this thesis concerns the underlying symmetries of string theory as probed by fixed-angle scattering at very high energy. The asymptotic behavior depends sensitively on the choice of the string vacuum. Therefore, we examine the effect of modifying the vacuum on the behavior of high-energy scattering amplitudes. In particular, high-energy fixed-angle elastic scattering of open-string tachyons is studied explicitly. Tadpole corrections to the tree-level formulas are included. The main conclusion of the analysis is that symmetry relations among amplitudes at high energy seem to be unaffected by modifications of the vacuum, even though the amplitudes themselves do change.
https://resolver.caltech.edu/CaltechTHESIS:04112011-114355027Two topics in 2D quantum field theory
https://resolver.caltech.edu/CaltechTHESIS:04112011-105901546
Year: 1991
DOI: 10.7907/w269-sg43
Two topics in two-dimensional quantum field theory are presented. The first
is a classification of 2- and 3-field rational conformal field theories. Using the fact
that the fusion algebra of a RCFT is specified in terms of integers that are related
to modular transformation properties, we classify 2- and 3-field chiral RCFT's. We
show that the only possibilities for the non-trivial fusion rule in the 2-field case are
φ x φ = 1 or φ x φ = 1 + φ. Similar results are obtained for the 3-field case. A partial
classification of possible conformal dimensions and central charges for these theories
is also obtained. The second topic is in two-dimensional quantum gravity. Explicit
computation of the non-perturbative correlation functions of the (1, q) models of
KdV-gravity is presented. This computation includes contributions from high genus
as well as correlation functions of descendant fields. A ghost number conservation
law for these models is derived from purely algebraic considerations. A hint of further
selection rules is found.
https://resolver.caltech.edu/CaltechTHESIS:04112011-105901546Kac-Moody algebras and string theory
https://resolver.caltech.edu/CaltechETD:etd-03062009-160921
Year: 1993
DOI: 10.7907/FXSK-JK33
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
The focus of this thesis is on (1) the role of Kac-Moody algebras in string theory and the development of techniques for systematically building string theory models based on higher level K ≥ 2) KM algebras and (2) fractional superstrings, a new class of solutions based on SU(2)[subscript K]/U(1) conformal field theories. The content of this thesis is as follows.
In chapter two we review KM algebras and their role in string theory. In the next chapter, we present two results concerning the construction of modular invariant partition functions for conformal field theories built by tensoring together other conformal field theories. This is based upon our research in ref. [2]. First we show how the possible modular invariants for the tensor product theory are constrained if the allowed modular invariants of the individual conformal field theory factors have been classified. We illustrate the use of these constraints for theories of the type [...], finding all consistent theories for K[subscript A] and K[subscript B] odd. Second we show how known diagonal modular invariants can be used to construct inherently asymmetric invariants where the holomorphic and anti-holomorphic theories do not share the same chiral algebra. Explicit examples are given.
Next, in chapter four we investigate some issues relating to recently proposed fractional superstring theories with D[subscript critical] < 10. Using the factorization approach of Gepner and Qiu, we systematically rederive the partition functions of the K = 4, 8, and 16 theories and examine their spacetime supersymmetry. Generalized GSO projection operators for the K = 4 model are found. Uniqueness of the twist field, [...] as source of spacetime fermions, is demonstrated. Our research was originally presented in refs. [3, 4]
https://resolver.caltech.edu/CaltechETD:etd-03062009-160921Extending the theory of random surfaces
https://resolver.caltech.edu/CaltechTHESIS:01082013-113837650
Year: 1993
DOI: 10.7907/1hwe-nn67
The theory of embedded random surfaces, equivalent to two-dimensional quantum
gravity coupled to matter, is reviewed, further developed and partly generalized
to four dimensions. It is shown that the action of the Liouville field theory that describes
random surfaces contains terms that have not been noticed previously. These
terms are used to explain the phase diagram of the Sine-Gordon model coupled to
gravity, in agreement with recent results from lattice computations. It is also demonstrated
how the methods of two- dimensional quantum gravity can be applied to
four-dimensional Euclidean gravity in the limit of infinite Weyl coupling. Critical
exponents are predicted and an analog of the "c = 1 barrier" of two-dimensional
gravity is derived.
https://resolver.caltech.edu/CaltechTHESIS:01082013-113837650Global Analogue of the Aharonov-Bohm Effect
https://resolver.caltech.edu/CaltechTHESIS:12282012-124421666
Year: 1993
DOI: 10.7907/crgk-1x57
This thesis deals with a global analogue of the Aharonov-Bohm effect previously
pointed out by other authors. The effect was not well understood because the pure
Aharonov-Bohm cross section was thought to be merely an approximate low energy
limit. This thesis provides a detailed analysis and reveals that in the particular model
considered, there is an exact Aharonov-Bohm cross section over the energy range that
a mass splitting occurs. At energies slightly above the mass splitting, the effect has
completely disappeared and there is effectively no scattering at large distances. This
is a curious observation as it was previously thought that a global theory would not
act exactly like a local one over an extended range of energies. It begs the heretical
speculation that experimentally observed forces modelled with Lagrangians possessing
local symmetries may have an underlying global theory.https://resolver.caltech.edu/CaltechTHESIS:12282012-124421666The dimension of spacetime
https://resolver.caltech.edu/CaltechETD:etd-03062009-162225
Year: 1993
DOI: 10.7907/5q06-5105
The implications of string theory for the dimension of spacetime are investigated by two methods. First, a new potential class of string theories is studied, which have critical dimensions 3, 4 and 6. In particular, the partition functions of these theories are derived and interpreted using generalized GSO projection. The possible uniqueness of field assignments, as well as the bosonization of the K = 4 model are also addressed. Second, using recent ideas in string cosmology, a new model is proposed to explain why three spatial dimensions grew large. Unlike the original work of Brandenberger and Vafa, this paradigm uses the theory of random walks. A simple computer model is developed to test the implications of this new approach. It is found that a four-dimensional spacetime can be explained by the proper choice of initial conditions.
https://resolver.caltech.edu/CaltechETD:etd-03062009-162225Three-Dimensional Gauge Theories and Gravitational Instantons from String Theory
https://resolver.caltech.edu/CaltechTHESIS:08072017-161302326
Year: 1998
DOI: 10.7907/1c59-jj22
Various realizations of gauge theories in string theory allow an identification of their
spaces of vacua with gravitational instantons. Also, they provide a correspondence of
vacua of gauge theories with nonabelian monopole configurations and solutions of a
system of integrable equations called Nahm equations. These identifications make it
possible to apply powerful techniques of differential and algebraic geometry to solve
the gauge theories in question. In other words, it becomes possible to find the exact
metrics on their moduli spaces of vacua with all quantum corrections included. As
another outcome we obtain for the first time the description of a series of all D<sub>k</sub>-type
gravitational instantons.https://resolver.caltech.edu/CaltechTHESIS:08072017-161302326String Theory on Calabi-Yau Manifolds: Topics in Geometry and Physics
https://resolver.caltech.edu/CaltechTHESIS:08072017-110249276
Year: 1999
DOI: 10.7907/jtpn-rn07
We study aspects of the geometry and physics of type II string theory compactification
on Calabi-Yau manifolds. The emphasis is on non-perturbative phenomena which arise when the compactification manifold develops singularities, and the implications
on quantum geometry of the the Calabi-Yau spaces. We use both the methods of low energy
supergravity and the complementary approach via D brane probes. Applications to the study of four-dimensional N = 1 and N = 2 supersymmetric gauge theories are considered as well.https://resolver.caltech.edu/CaltechTHESIS:08072017-110249276Branes, Brane Actions and Applications to Field Theory
https://resolver.caltech.edu/CaltechETD:etd-12082006-104418
Year: 2001
DOI: 10.7907/qr6g-tp18
This thesis describes the construction of supersymmetric world-volume actions for various kinds of extended objects that appear in string theory, the so-called p-branes, D-branes and M-branes. We also present an application of branes to computing the spectrum of a conformal field theory in the context of the AdS-CFT correspondence.
https://resolver.caltech.edu/CaltechETD:etd-12082006-104418Testing Gauge/Gravity Duality: The Eleven-Dimensional PP-Wave
https://resolver.caltech.edu/CaltechETD:etd-05252004-230238
Year: 2004
DOI: 10.7907/0JV2-RT78
The gauge/gravity duality in the interaction between M theory objects has taught us a lot about quantum gravity. The eleven-dimensional PP-wave background provides a new arena for exploring this duality beyond flat and almost flat, i.e., weakly curved, backgrounds. In this thesis we discuss the gauge theories that describe the dynamics of interacting M theory objects, the supergravity calculations that capture these dynamics, the comparison of the two sides, and various objects (such as gravitons and membranes) in the eleven-dimensional PP-wave background. We only consider the one-loop gauge theory and linearized supergravity approximations.
https://resolver.caltech.edu/CaltechETD:etd-05252004-230238D-Brane Actions and N=2 Supergravity Solutions
https://resolver.caltech.edu/CaltechETD:etd-06022004-125935
Year: 2004
DOI: 10.7907/WSR1-2S04
<p>Among the most remarkable recent developments in string theory are the AdS/CFT duality, as proposed by Maldacena, and the emergence of noncommutative geometry. It has been known for some time that for a system of almost coincident D-branes the transverse displacements that represent the collective coordinates of the system become matrix-valued transforming in the adjoint representation of U(N). From a geometrical point of view this is rather surprising but, as we will see in Chapter 2, it is closely related to the noncommutative descriptions of D-branes.</p>
<p>A consequence of the collective coordinates becoming matrix-valued is the appearance of a dielectric effect in which D-branes can become polarized into higher-dimensional fuzzy D-branes. This last aspect has inspired Polchinski and Strassler to find a nonsingular string dual of a confining four-dimensional gauge theory. The nonsingular geometry is sourced by an extended brane arising from Myers' dielectric effect. Following the spirit of the Polchinski-Strassler paper, we find N = 2 supergravity solutions with polarized branes and a field-theory dual. In our case we are able to present exact supergravity solutions by using M-theory reductions to type IIA supergravity.</p>https://resolver.caltech.edu/CaltechETD:etd-06022004-125935Gauge Theory and Supergravity Duality in the PP-Wave Background
https://resolver.caltech.edu/CaltechETD:etd-05262005-151056
Year: 2005
DOI: 10.7907/8QS9-Q326
We test the matrix theory conjecture in the pp-wave by studying two-body interactions between gravitons and membranes. We compute the one-loop effective potential of matrix theory and compare it to the light cone Lagrangian of linearized supergravity. We have exact agreement in the absence of M-momentum transfer. We also find the effective potential that smoothly interpolates between the spherical membrane result and the graviton result. We also collect here partial results from our investigation of interactions with M-momentum transfer.https://resolver.caltech.edu/CaltechETD:etd-05262005-151056Superstring Holography and Integrability in AdS₅ x S⁵
https://resolver.caltech.edu/CaltechETD:etd-05042005-144547
Year: 2005
DOI: 10.7907/QD85-9603
The AdS/CFT correspondence provides a rich testing ground for many important topics in theoretical physics. The earliest and most striking example of the correspondence is the conjectured duality between the energy spectrum of type IIB superstring theory on AdS₅ x S⁵ and the operator anomalous dimensions of N=4 supersymmetric Yang-Mills theory in four dimensions. While there is a substantial amount of evidence in support of this conjecture, direct tests have been elusive. The difficulty of quantizing superstring theory in a curved Ramond-Ramond background is compounded by the problem of computing anomalous dimensions for non-BPS operators in the strongly coupled regime of the gauge theory. The former problem can be circumvented to some extent by taking a Penrose limit of AdS₅ x S⁵, reducing the background to that of a pp-wave (where the string theory is soluble). A corresponding limit of the gauge theory was discovered by Berenstein, Maldacena and Nastase, who obtained successful agreement between a class of operator dimensions in this limit and corresponding string energies in the Penrose limit. In this dissertation we present a body of work based largely on the introduction of worldsheet interaction corrections to the free pp-wave string theory by lifting the Penrose limit of AdS₅ x S⁵. This provides a new class of rigorous tests of AdS/CFT that probe a truly quantum realm of the string theory. By studying the correspondence in greater detail, we stand to learn not only about how the duality is realized on a more microscopic level, but how Yang-Mills theories behave at strong coupling. The methods presented here will hopefully contribute to the realization of these important goals.https://resolver.caltech.edu/CaltechETD:etd-05042005-144547The Near-Penrose Limit of AdS/CFT
https://resolver.caltech.edu/CaltechETD:etd-05252005-175001
Year: 2005
DOI: 10.7907/EHS6-QF38
The conjectured duality between type IIB string theory on AdS5 x S5 and N = 4 SU(Nc) super Yang-Mills theory in four dimensions simplifies in the Penrose limit or, in other words, when the string's angular momentum is large. As the string action in this limit is solvable, it is possible to go beyond the supergravity approximation and compare exact string energies with the anomalous dimensions of a sector of large R-charge operators. This equivalence should of course extend to the full AdS5 x S5 space and to operators of finite R-charge. We take some modest steps in this direction by expanding the full string action in inverse powers of the angular momentum and finding the first order perturbative corrections to the energy spectrum. These corrections reproduce the gauge theory anomalous dimensions for a range of different operators to two-loops in the 't Hooft parameter but disagree at three-loops. Furthermore, these near-plane wave results are useful in studying the recently discovered integrability in this AdS/CFT system and can be used to motivate the form of quantum string scattering matrices.https://resolver.caltech.edu/CaltechETD:etd-05252005-175001On Quantum Interacting Embedded Geometrical Objects of Various Dimensions
https://resolver.caltech.edu/CaltechETD:etd-06072006-174745
Year: 2006
DOI: 10.7907/1VN2-VZ71
Modern string theory naturally gives rise to an assortment of dynamical geometrical objects of various dimensions (collectively referred to as "branes") embedded into spacetime. The aim of this thesis is to present a series of results (of varying novelty and rigor) pertinent to dynamics of the low-dimensional geometrical objects of this kind. The processes considered are the D0-brane recoil and annihilation, "local recoil" of D1-branes (which is a peculiar effect manifested by one-dimensional topological defects in response to an impact, and closely related to soliton recoil), and D- and F-string loop mixing. Apart from the practical relevance within the formalism of string theory, such considerations are worthwhile in that the quantum dynamics of the geometrical objects involved is complex enough to be interesting, yet simple enough to be tractable. Furthermore, some of the results derived here within the string theory formalism may give valuable insights into the dynamics of low-dimensional field-theoretical topological defects.https://resolver.caltech.edu/CaltechETD:etd-06072006-174745Integrability of N = 6 Chern-Simons theory
https://resolver.caltech.edu/CaltechTHESIS:05122011-105136289
Year: 2011
DOI: 10.7907/QDSM-8448
In 2008, Aharony, Bergman, Jafferis, and Maldacena (ABJM) discovered a three-dimensional Chern-Simons theory with N = 6 supersymmetry and conjectured that in a certain limit, this theory is dual to type IIA string theory on AdS4xCP3. Since then, a great deal of evidence has been accumulated which suggests that the ABJM theory is integrable in the planar limit. Integrability is a very useful property that allows many physical observables, such as anomalous dimensions and scattering amplitudes, to be computed efficiently. In the first half of this thesis, we will explain how to use integrabilty to compute the anomalous dimensions of long, single-trace operators in the ABJM theory. In particular, we will describe how to compute them at weak coupling using a Bethe Ansatz, and how to compute them at strong coupling using string theory. The latter approach involves using algebraic curve and world-sheet techniques to compute the energies of string states dual to gauge theory operators. In the second half of this thesis, we will discuss integrability from the point of view of on-shell scattering amplitudes in the ABJM theory. In particular, we will describe how to parameterize the amplitudes in terms of supertwistors and how to relate higher-point tree-level amplitudes to lower-point tree-level amplitudes using a recursion relation. We will also explain how this recursion relation can be used to show that all tree-level amplitudes of the ABJM theory are invariant under dual superconformal symmetry. This symmetry is hidden from the point of the action and implies that the theory has Yangian symmetry, which is a key feature of integrability. This thesis is mainly based on the material in [94], [76], and [77].https://resolver.caltech.edu/CaltechTHESIS:05122011-105136289Superconformal Chern-Simons Theories and Their String Theory Duals
https://resolver.caltech.edu/CaltechTHESIS:05242011-235017396
Year: 2011
DOI: 10.7907/2D9D-1876
<p>In this thesis, we consider two aspects of the conjectured gauge theory/string theory correspondence between three-dimensional maximal supersymmetric conformal field theories, which describe the world-volume theory of multiple M2-branes in flat space, and M-theory on AdS<sub>4</sub> x S<sup>7</sup>.</p>
<p>First we study three classes of N = 6,8 superconformal Chern-Simons theories that are related to the gauge theory side of the correspondence: the Bagger-Lambert (BL) theories based on 3-algebras, the Lorentzian signature 3-algebra theories, and the Aharony-Bergman-Jafferis-Maldacena (ABJM) theories. We verify the superconformal symmetry of the BL theory, prove that it is parity conserving and conjecture the (by now proven) uniqueness of its SO(4) realization. We then consider the Lorentzian signature 3-algebra theories and show that although the ghosts can be removed to ensure unitarity by gauging certain global symmetries, the resulting theories spontaneously break the conformal symmetry and reduce to maximally supersymmetric three-dimensional Yang-Mills theories. After this, we recast the ABJM theory in a form for which the SU(4) R-symmetry of the action is manifest; then we use this form to verify in complete detail the OSp(6|4) superconformal symmetry of the theory and to express the scalar potential as a sum of squares.</p>
<p>Next, we study the one-loop correction to the energy of a point-particle and circular string solutions to type IIA string theory on AdS<sub>4</sub> x CP<sup>3</sup>. We compute the spectrum of fluctuations for each of these solutions using two techniques, known as the algebraic curve approach and the world-sheet approach. We propose a new prescription for computing the one-loop corrections that gives well-defined results and agrees with the predictions of the all-loop Bethe ansatz for our point-particle and circular string solutions as well as for previous folded-spinning string solutions.</p>https://resolver.caltech.edu/CaltechTHESIS:05242011-235017396Three-Dimensional Superconformal Field Theory, Chern-Simons Theorv, and Their Correspondence
https://resolver.caltech.edu/CaltechTHESIS:06062014-152022421
Year: 2014
DOI: 10.7907/7CA7-9C79
In this thesis, we discuss 3d-3d correspondence between Chern-Simons theory and three-dimensional N = 2 superconformal field theory. In the 3d-3d correspondence proposed by Dimofte-Gaiotto-Gukov information of abelian flat connection in Chern-Simons theory was not captured. However, considering M-theory configuration giving the 3d-3d correspondence and also other several developments, the abelian flat connection should be taken into account in 3d-3d correspondence. With help of the homological knot invariants, we construct 3d N = 2 theories on knot complement in 3-sphere for several simple knots. Previous theories obtained by Dimofte-Gaiotto-Gukov can be obtained by Higgsing of the full theories. We also discuss the importance of all flat connections in the 3d-3d correspondence by considering boundary conditions in 3d N = 2 theories and 3-manifold.https://resolver.caltech.edu/CaltechTHESIS:06062014-152022421Branes and Supersymmetric Quantum Field Theories
https://resolver.caltech.edu/CaltechTHESIS:01242014-104001899
Year: 2014
DOI: 10.7907/Y1VH-E821
<p>Since the discovery of D-branes as non-perturbative, dynamic objects in string theory, various configurations of branes in type IIA/B string theory and M-theory have been considered to study their low-energy dynamics described by supersymmetric quantum field theories.</p>
<p>One example of such a construction is based on the description of Seiberg-Witten curves of four-dimensional N = 2 supersymmetric gauge theories as branes in type IIA string theory and M-theory. This enables us to study the gauge theories in strongly-coupled regimes. Spectral networks are another tool for utilizing branes to study non-perturbative regimes of two- and four-dimensional supersymmetric theories. Using spectral networks of a Seiberg-Witten theory we can find its BPS spectrum, which is protected from quantum corrections by supersymmetry, and also the BPS spectrum of a related two-dimensional N = (2,2) theory whose (twisted) superpotential is determined by the Seiberg-Witten curve. When we don’t know the perturbative description of such a theory, its spectrum obtained via spectral networks is a useful piece of information. In this thesis we illustrate these ideas with examples of the use of Seiberg-Witten curves and spectral networks to understand various two- and four-dimensional supersymmetric theories.</p>
<p>First, we examine how the geometry of a Seiberg-Witten curve serves as a useful tool for identifying various limits of the parameters of the Seiberg-Witten theory, including Argyres-Seiberg duality and Argyres-Douglas fixed points. Next, we consider the low-energy limit of a two-dimensional N = (2, 2) supersymmetric theory from an M-theory brane configuration whose (twisted) superpotential is determined by the geometry of the branes. We show that, when the two-dimensional theory flows to its infra-red fixed point, particular cases realize Kazama-Suzuki coset models. We also study the BPS spectrum of an Argyres-Douglas type superconformal field theory on the Coulomb branch by using its spectral networks. We provide strong evidence of the equivalence of superconformal field theories from different string-theoretic constructions by comparing their BPS spectra.</p>https://resolver.caltech.edu/CaltechTHESIS:01242014-104001899Supersymmetric Scattering Amplitudes and Algebraic Aspects of Holography from the Projective Line
https://resolver.caltech.edu/CaltechTHESIS:06102019-125514401
Year: 2019
DOI: 10.7907/HFPD-JX10
<p>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. </p>
<p>In the second half of the thesis, we discuss the projective algebraic and geometric structure of the AdS<sub>3</sub>/CFT<sub>2</sub> 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<sub>p</sub>. The result is a formulation of holography in which the bulk geometry is discrete---the Bruhat--Tits tree for PGL(2,Q<sub>p</sub>)---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<sub>p</sub>, and correctly computes the Ryu-Takayanagi formula, holographic entanglement and black hole entropy, and multiple interval entanglement inequalities.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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 P<sup>1</sup>(Q<sub>p</sub>) and lattice models on the Bruhat--Tits tree of PGL(2,Q<sub>p</sub>), 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.</p>
<p>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>https://resolver.caltech.edu/CaltechTHESIS:06102019-125514401