PHY 622 & 623 String Theory I & II

String theory is a vast subject, so it is not possible to cover all areas. We give simple introductions to the main areas; the lectures are self-contained. We do not use phrases like "It can be shown that...", but rather do all derivations and calculations explicitly.

This year we will begin teaching a revised version of the string course. As such, the following is a tentative description, subject to revision. The course will be more self-contained, and thus more introductory.

Student participation

Grading: S/U

Textbook: none

Prerequisites: none

• No general relativity: We will cover the minimal amount of general relativity you will need for this course. This is only a small fraction of what is covered in the course on general relativity; but we will emphasize some items it may miss, including the vielbein & Lorentz connection (necessary for coupling to fermions), and Weyl scale transformations.
• No quantum field theory: The only quantum "field theory" appearing will actually be relativistic quantum mechanics (of strings). Even the Scattering-Matrix theory covered in the 2nd semester will be from the point of view of quantum mechanics of particles/strings in an external potential.
• The only quantum mechanics in the first semester will be essentially that of harmonic oscillators & free point particles.

First semester

• Motivation: We will begin with brief descriptions of what a string is, how it relates to (super)gravity, what problems it's supposed to solve, etc.
• Spectra: A significant portion of this semester concerns the quantum mechanics (not field theory) of free particles and strings. It is first-quantization of the relativistic classical mechanics of spacetime coordinates x, as functions of the worldline coordinate τ (particle) or worldsheet coordinates τ & σ (string). This is the relativistic generalization of the familiar quantum mechanics of a point particle. As per the usual correspondence principle, the wave functions of these particles/strings can be identified with spacetime fields. The coordinates x will describe spacetime in arbitrary dimensions for particles (but we will find next semester strings are restricted to certain numbers of dimensions).
• Worldvolume "gravity": The 1-dimensional worldline and 2-dimensional worldsheet are curved spaces in their own right, and "fields" analogous to gravity can be defined on them. (But these should not be confused with the true gravity fields that are functions of spacetime x.) This worldline/worldsheet gravity does not contribute to the spectrum, but constrains it instead.
• Supersymmetry: We will also cover some supersymmetry, which relates bosons & fermions: On the worldline/worldsheet, it relates the spacetime position x to a fermionic partner ψ representing Dirac γ-matrices. Thus fermions are also introduced into the spacetime spectrum.
• Supergravity: Most of the semester will involve supergravity (supersymmetric gravity) in spacetime, the massless part of the spectrum of (super)strings. It describes the low-energy limit of string theory, where many of its properties can be seen: Some dual descriptions in terms of string theory will be seen in the second semester.

1. Why strings? What are strings? (MR)
2. 1st-quantization & spectrum
   1. Klein-Gordon scalar particle in all dimensions (MR)
      1. worldline gravity
      2. 1st-quantization: Klein-Gordon equation
   2. bosonic string (MR)
      1. worldsheet gravity: Weyl scale
      2. 1st-quantization: spectrum
   3. Dirac spinor particle in all dimensions (PvN)
      1. worldline supersymmetry
      2. worldline supergravity: Noether currents
      3. 1st-quantization: Dirac equation
   4. Ramond-Neveu-Schwarz spinning string (PvN)
      1. worldsheet supersymmetry
      2. worldsheet supergravity
      3. 1st-quantization: spectrum
3. supergravity
   1. gravity in all dimensions, coupling to Dirac spinor (PvN)
      1. vielbein & local Lorentz symmetry
      2. Lorentz connection & covariant derivative
   2. 4D supergravity (PvN)
   3. higher-dimensional supergravity (MR)
      1. 11D supergravity
      2. dimensional reduction
      3. 10D supergravity: Types I, IIA & B
      
   4. some solutions of supergravity (MR)
      1. branes
      2. other compactifications
4. Anti-de Sitter/Conformal Field Theory correspondence -- an introduction (WS)
   1. superconformal groups
      1. conformal group
      2. supergroups & superconformal groups
      3. coordinate cosets
   2. 4D classical conformal field theory
      1. N=4 super Yang-Mills
      2. superconformal coset & field strength
   3. 10D IIB supergravity on AdS₅×S⁵
      1. de Sitter & Anti-de Sitter spaces
      2. superAdS coset & field strength
   4. correspondence
      1. 1/N expansion
      2. definition
      3. boundary limit (group contraction) & lightcone gauge

Second semester

1. more supergravity (MR)
   1. more solutions of supergravity
      1. branes
      2. other compactifications
   2. supergravity dualities
      1. T-duality: Buscher rules
      2. S-duality: weak ↔ strong coupling
2. Becchi-Rouet-Stora-Tyutin charge: covariant quantization (PvN)
   1. constrained systems
   2. strings
3. worldsheet (super)conformal quantum "field theory" (PvN)
   1. worldsheet conformal symmetry
   2. 1 "loop" on worldsheet: D = 26 (bosonic) or 10 (spinning)
   3. bosonization
4. scattering amplitudes (WS)
   1. worldsheet symmetries
   2. Regge theory
5. tree graphs (WS)
   1. particles
   2. bosonic strings
6. Green-Schwarz approach (WS)
   1. triality: Ramond-Neveu-Schwarz ↔ Green-Schwarz
   2. superparticle
   3. superstring
7. Introduction to loops (WS)

On-line lecture notes, etc.

van Nieuwenhuizen

Siegel

• Reading & homework
• Fields v4
• Previous semesters
  • Reading & homework
    • F'18
    • F'17, S'18
    • F'16, S'17
    • F'15, S'16
    • F'14, S'15
  • Extra lecture notes
    • Green-Schwarz superstrings
    • vertex operators
    • superconformal symmetry & AdS/CFT

University-required statements