String Theory and Its Applications

The following sections are included:

• Preface

• Contents

These lectures are an introduction to gauge/gravity duality. The first three sections present the basics, focusing on AdS₅×S⁵. The last section surveys a variety of ways to generate duals of reduced symmetry.

I argue that the conventional field theoretic notion of vacuum state is not valid in quantum gravity. The arguments use gravitational effective field theory, as well as results from string theory, particularly the AdS/CFT correspondence. Different solutions of the same low energy gravitational field equations correspond to different quantum systems, rather than different states in the same system. I then introduce holographic space-time a quasi-local quantum mechanical construction based on the holographic principle. I argue that models of quantum gravity in asymptotically flat space-time will be exactly super-Poincare invariant, because the natural variables of holographic space-time for such a system, are the degrees of freedom of massless superparticles. The formalism leads to a non-singular quantum Big Bang cosmology, in which the asymptotic future is required to be a de Sitter space, with cosmological constant (c.c.) determined by cosmological initial conditions. It is also approximately SUSic in the future, with the gravitino mass KΛ^¼.

Experiments on the Large Hadron Collider at CERN represent our furthest excursion yet along the energy frontier of particle physics. The goal of probing physical processes at the TeV energy scale puts strict requirements on the performance of accelerator and experiment, dictating the awe-inspiring dimensions of both. These notes, based on a set of five lectures given at the 2010 Theoretical Advanced Studies Institute in Boulder, Colorado, not only review the physics considered as part of the accelerator and experiment design, but also introduce algorithms and tools used to interpret experimental results in terms of theoretical models. The search for new physics beyond the Standard Model presents many new challenges, a few of which are addressed in specific examples.

These lectures give a brief overview of the topics of which every student of modern hadron collider physics must be aware, and which are not easily found organized elsewhere. They begin with a basic review of the Higgs mechanism and discussion of the associated unitarity and hierarchy problems. Then the subject turns to the basics of a proton-proton collider: the rates for various standard model and new-physics processes at the LHC, the nature of triggers and reconstruction, and the kinematical variables used at hadron colliders. Two discussions — one preliminary, one advanced — of how QCD impacts hadron colliders is given, touching on confinement and the running coupling, parton distribution functions and luminosities, hadrons, and jets thereof. The role of and limitations of theory in calculations of backgrounds is covered, as is the experimental role of the third generation of fermions. Finally, the lectures conclude with an extremely and overly brief (and, I hope, interestingly idiosyncratic) review of new physics to look for at the LHC, and the challenges involved.

In these notes we review aspects of semi-realistic particle physics from the point of view of type II orientifold compactifications. We discuss the appearance of gauge theories on spacetime filling D-branes which wrap non-trivial cycles in the Calabi-Yau. Chiral matter can appear at their intersections, with a natural interpretation of family replication given by the topological intersection number. We discuss global consistency, including tadpole cancellation and the generalized Green-Schwarz mechanism, and also the importance of related global U(1) symmetries for superpotential couplings. We review the basics of D-instantons, which can generate superpotential corrections to charged matter couplings forbidden by the global U(1) symmetries and may play an important role in moduli stabilization. Finally, for the purpose of studying the landscape, we discuss certain advantages of studying quiver gauge theories which arise from type II orientifold compactifications rather than globally defined models. We utilize the type IIa geometric picture and CFT techniques to illustrate the main physical points, though sometimes we supplement the discussion from the type IIb perspective using complex algebraic geometry.

These lectures describe some classes of known supersymmetric string vacua in various dimensions greater than 4, and how these sets of theories fit into the larger space of possible quantum supergravity theories. Anomalies and other quantum consistency conditions, combined with supersymmetry, place strong limitations on the space of physical theories which can be coupled to gravity in space-times with a large number of dimensions. There is a unique supersymmetric theory of gravity in eleven dimensions, which is realized in the regime of string theory known as “M-theory”. As the dimensionality of space-time decreases, the range of possible supergravity theories expands, as does the set of string vacua. We focus on theories in 10, 8, and 6 dimensions, and develop the basic technology for describing a variety of string vacua, including heterotic, intersecting brane, and F-theory constructions. The resulting picture of the space of supergravity and string vacua provides a number of lessons for understanding the space of theories in four dimensions.

These lecture notes give an introduction to a number of ideas and methods that have been useful in the study of complex systems ranging from spin glasses to D-branes on Calabi-Yau manifolds. Topics include the replica formalism, Parisi's solution of the Sherrington-Kirkpatrick model, overlap order parameters, supersymmetric quantum mechanics, D-brane landscapes and their black hole duals.

If supersymmetry turns out to be a symmetry of nature at low energies, the first order of business to measure the soft breaking parameters. But one will also want to understand the symmetry, and its breaking, more microscopically. Two aspects of this problem constitute the focus of these lectures. First, what sorts of dynamics might account for supersymmetry breaking, and its manifestation at low energies. Second, how might these features fit into string theory (or whatever might be the underlying theory in the ultraviolet). The last few years have seen a much improved understanding of the first set of questions, and at least a possible pathway to address the second.

I present a pedagogical survey of a variety of quantum phases of the Hubbard model. The honeycomb lattice model has a conformal field theory connecting the semi-metal to the insulator with Néel order. States with fractionalized excitations are linked to the deconfined phases of gauge theories. I also consider the confining phases of such gauge theories, and show how Berry phases of monopoles induce valence bond solid order. The triangular lattice model can display a metal-insulator transition from a Fermi liquid to a deconfined spin liquid, and I describe the theory of this transition. The bilayer triangular lattice is used to illustrate another compressible metallic phase, the ‘fractionalized Fermi liquid’. I make numerous connections of these phases and critical points to the AdS/CFT correspondence. In particular, I argue that two recent holographic constructions connect respectively to the Fermi liquid and fractionalized Fermi liquid phases.

We discuss some recent attempts to apply AdS/CFT correspondence to systems with finite temperature and chemical potential, emphasizing the hydrodynamic aspects. We also discuss the system of nonrelativistic fermions at unitarity, the Schrödinger symmetry and possible directions for constructing a holographic dual of this system. This is the write-up of lectures delivered at the TASI school of 2010.

In four lectures, delivered at the TASI 2010 summer school, I cover selected topics in the application of the gauge-string duality to nuclear and condensed matter physics. On the nuclear side, I focus on multiplicity estimates from trapped surfaces in AdS₅, and on the consequences of conformal symmetry for relativistic hydrodynamics. On the condensed matter side, I explain the fermion response to the zero-temperature limit of p-wave holographic superconductors.

The lecture note consists of four parts. In the first part, we review a 2+1 dimensional lattice model which realizes emergent supersymmetry at a quantum critical point. The second part is devoted to a phenomenon called fractionalization where gauge boson and fractionalized particles emerge as low energy excitations as a result of strong interactions between gauge neutral particles. In the third part, we discuss about stability and low energy effective theory of a critical spin liquid state where stringy excitations emerge in a large N limit. In the last part, we discuss about an attempt to come up with a prescription to derive holographic theory for general quantum field theory.

In these lecture notes we review some recent attempts in searching for non-Fermi liquids and novel quantum phase transitions in holographic systems using gauge/gravity duality. We do this by studying the simplest finite density system arising from the duality, obtained by turning on a nonzero chemical potential for a U(1) global symmetry of a CFT, and described on the gravity side by a charged black hole. We address the following questions of such a finite density system:

(1) Does the system have a Fermi surface? What are the properties of low energy excitations near the Fermi surface?

(2) Does the system have an instability to condensation of scalar operators? What is the critical behavior near the corresponding quantum critical point?

We find interesting parallels with those of high T_c cuprates and heavy electron systems. Playing a crucial role in our discussion is a universal intermediate-energy phase, called a “semi-local quantum liquid,” which underlies the non-Fermi liquid and novel quantum critical behavior of a system. It also provides a novel mechanism for the emergence of lower energy states such as a Fermi liquid or a superconductor.

If supersymmetry turns out to be a symmetry of nature at low energies, the first order of business to measure the soft breaking parameters. But one will also want to understand the symmetry, and its breaking, more microscopically. Two aspects of this problem constitute the focus of these lectures. First, what sorts of dynamics might account for supersymmetry breaking, and its manifestation at low energies. Second, how might these features fit into string theory (or whatever might be the underlying theory in the ultraviolet). The last few years have seen a much improved understanding of the first set of questions, and at least a possible pathway to address the second.

The following sections are included:

• Student Seminars at TASI 2010