Strings

During the Fall 2008 I took the course on string theory taught by Professors Rastelli, Rocek, Siegel and van Nieuwenhuizen. Here is a rough outline of the topics discussed:

Weeks 01 02 & 03 (lectures by van Nieuwenhuizen) – Bosonic strings

• Actions: Polyakov and Nambu-Goto form
• Worldsheet metric: Virasoro constraint and symmetries
• Symmetries of actions: Einstein, Weyl, Lorentz
• Field equations and solutions: mode expansions, boundary conditions
• Light-cone gauge and conformal gauge
• Energy-momentum tensor of the worldsheet: Virasoro generators, commutation relations and other properties
• Quantization: canonical, light-cone, covariant
• Spectrum of the bosonic string: tachyonic, massless, massive states
• Covariant quantization: gauge-fixing, ghost and antighost; Weyl ghost
• BRST rules for the bosonic string, BRST charge and current, nilpotency and critical dimension, normal-ordering anomaly and intercepts

Weeks 03 04 & 05 (lectures by Rocek) – Spinning strings

• Worldsheet spinors/spacetime vectors
• Action and symmetries: superconformal symmetry, Einstein, Weyl
• Worldsheet objects: metric, vielbein, gravitino as gauge field for local supersymmetry
• Field equations and solutions: mode expansions, boundary conditions, zero-modes
• Clifford vacuum, spectrum of open and closed spinning strings; Neveu-Schwarz and Ramond sectors and their tensor products
• GSO projection operator, physical states; type IIA and IIB strings; d=10 supergravity coincidence
• Covariant quantization of spinning string: ghost and antighost, gauge-fixing, mode expansions
• BRST charge and current
• Critical dimension

Weeks 06 & 07 (lectures by van Nieuwenhuizen) – Conformal field theory

• Conformal transformations of fields
• Complex coordinate basis, complex plane and cylinder
• Normal ordering
• Wick’s theorem and contractions
• Operator product expansions
• Propagators: string coordinates, ghosts, worldsheet spinors
• Energy-momentum tensor as generator of conformal transformations
• Schwartzian derivative
• Conformal normal ordering
• BRST charge and current in the conformal plane

Weeks 07 08 09 & 10 (lectures by Siegel) – Vertex operators and trees

• Two-dimensional quantum field theory
• Bosonization and fermionization
• Vertex operators: integrated and otherwise
• Picture changing
• …

Week 11 (lectures by Rocek) – T-duality

• T-duality for closed strings: winding and momentum states, compact dimensions
• Spectrum
• T-duality for open strings: Wilson lines, string charge
• Mode expansions
• T-duality and spinning strings

Weeks 12 13 14 15 (lectures by Rastelli) – D-branes and boundary state formalism

• Boundary states: bosonic and fermionic
• Duality between open- and closed-string channels
• Born-Infeld term as worldvolume action for branes

The lectures on D-branes followed some notes that can be found online:

1. hep-th/9912161
2. hep-th/0005029

The Spring 2009 view the second part of this course.

Weeks 01 02 03 & 04 (lectures by Siegel) – Loop amplitudes

• Regge theory and duality
• Topology of interactions
• First quantization of particles and scattering amplitudes
• Background field method
• Open and closed string propagators
• Regge behavior of string amplitudes
• Jacobi -functions and their properties
• Partition function for loops
• Cancellation of anomalies and divergences

Weeks 05 & 06 (lectures by van Nieuwenhuizen) – Heterotic strings

• Lagrangian
• Model with 16 scalars
• Model with 32 fermions
• Gauge symmetry of and
• Lattices
• Vertex operator representation of simply laced Lie algebras

Weeks 06 07 08 09 10 (lectures by Rocek and van Nieuwenhuizen) – Supergravity, extended solutions and anomalies

• Four-dimensional simple supergravity: field contents, action and symmetries
• Eleven-dimensional supergravity: field contents, action, supersymmetry transformations, field equations
• Supersymmetry backgrounds and BPS conditions
• Explicit calculation of M2 brane solution
• M5 solution
• Near-horizon limit of solutions; superposition; smearing
• Taub-NUT solution, KK6 solution, KK1 solution
• Ten-dimensional supergravities: type IIB, IIA and I, field content, lagrangians,
• Einstein and string frames
• Ten-dimensional super Yang-Mills
• Gravitational and gauge anomalies; cancellation
• Extended solutions and the relations among them; ten-dimensional reduction
• T-duality and the Buscher rules
• Tension of branes

Weeks 12 13 14 15 (lectures by Rastelli) – Anti-de Sitter/Conformal Field Theory correspondence

• String and field theory sides, symmetries, identification of parameters
• Conformal group, correlation functions, representations
• Geometry of anti-de-Sitter spacetime, Penrose diagrams, coordinate systems
• Masses in gravity theory and conformal dimensions in the field theory side
• Superconformal group and representations
• Long, short and semi-short multiplets
• Supergravity side
• Correlation function

The lectures on adS/CFT followed roughly the first three chapters of MAGOO and also some of this article by Dolan and Osborn.