String Theory I Introduction

These pages are based on handwritten notes I took during Professor Igor R. Klebanov’s one-semester course, “Introduction to String Theory,” in Spring 2010. The initial drafts were prepared with the assistance of AI and have since been reviewed and edited by me. The site is still under construction; although I have tried to ensure accuracy, some errors may remain. If you notice any significant mistakes, I would greatly appreciate your feedback.

Welcome to String Theory I. The aim of this section is to give a coherent graduate-level path through perturbative string theory: why one-dimensional extended objects are natural, how their worldsheet dynamics is quantized, why two-dimensional conformal field theory becomes the central language, how ghosts and BRST quantization impose gauge invariance, how string amplitudes and spacetime effective actions emerge, and how the NSR superstring leads to Type I/II theories, T-duality, and D-branes. The guiding principle is that every important formula should carry its convention, every conceptual jump should be made explicit, and every calculation should be placed in the larger architecture of the subject.

The course has three layers. First come classical and quantum strings. Then worldsheet CFT, ghosts, BRST, amplitudes, and effective actions provide the perturbative framework. Finally the NSR formalism, Type I/II theories, T-duality, and D-branes give the modern superstring picture.

Scope

This section stops at D-branes, the DBI action, and R—R charges. AdS/CFT is not included here; it belongs naturally in a later group of notes after D-branes and near-horizon limits have been developed.

The notes keep $\alpha'$ explicit whenever possible. Some formulas become shorter in special conventions such as $\alpha'=2$, but such choices are not made globally.

Prerequisites

A reader should be comfortable with special relativity, Lagrangian and Hamiltonian mechanics, basic quantum field theory, path integrals, and elementary Lie algebra representation theory. Prior exposure to two-dimensional conformal field theory is helpful but not required; the necessary CFT tools are developed in the middle part of the course.

Global conventions

Unless stated otherwise, these notes use units

$$\hbar=c=1.$$

The spacetime metric is mostly plus:

$$\eta_{\mu\nu}=\mathrm{diag}(-,+,\ldots,+), \qquad p^2=\eta_{\mu\nu}p^\mu p^\nu.$$

For a massive particle,

$$p^2=-m^2.$$

The string tension is

$$T=\frac{1}{2\pi\alpha'},$$

and the string length is

$$\ell_s=\sqrt{\alpha'}.$$

Thus $\alpha'$ has dimension of length squared, while the characteristic string mass scale is

$$M_s\sim \frac{1}{\sqrt{\alpha'}}.$$

The parameter $\alpha'$ controls the string length, string tension, and tower of massive oscillator states. Keeping it explicit makes comparison with different textbooks safer.

Worldsheet coordinates are usually

$$\sigma^\alpha=(\tau,\sigma), \qquad \alpha=0,1.$$

In conformal gauge we often use light-cone coordinates

$$\sigma^\pm=\tau\pm \sigma, \qquad \partial_\pm=\frac{1}{2}\left(\partial_\tau\pm \partial_\sigma\right),$$

or Euclidean complex coordinates

$$z=\sigma^1+i\sigma^2, \qquad \bar z=\sigma^1-i\sigma^2, \qquad \partial=\partial_z, \qquad \bar\partial=\partial_{\bar z}.$$

For the free bosons $X^\mu$ in flat target space, the Euclidean worldsheet action is normalized as

$$S_X=\frac{1}{4\pi\alpha'}\int d^2 z\, \partial X^\mu \bar\partial X_\mu.$$

The basic OPE is

$$X^\mu(z,\bar z)X^\nu(w,\bar w) \sim -\frac{\alpha'}{2}\eta^{\mu\nu}\ln |z-w|^2.$$

The holomorphic stress tensor of the free bosons is

$$T_X(z)=-\frac{1}{\alpha'}:\partial X^\mu\partial X_\mu:(z),$$

and the closed-string bulk exponential operator

$$:e^{i k\cdot X(z,\bar z)}:$$

has weights

$$h=\bar h=\frac{\alpha' k^2}{4}.$$

Boundary operators for open strings have a different normalization, derived when needed.

How to read the course

There are three natural reading paths.

First, pages 01—10 give the basic bosonic-string construction: actions, symmetries, constraints, spectra, light-cone quantization, Hagedorn growth, and the first worldsheet-CFT ideas.

Second, pages 11—24 develop the worldsheet machinery: OPEs, Virasoro symmetry, radial quantization, null states, ghosts, BRST quantization, tree amplitudes, massless vertices, effective actions, and sigma-model beta functions.

Third, pages 25—36 develop the superstring and D-brane part of the course: NSR worldsheet supersymmetry, GSO projection, Type I/II theories, spin fields, picture changing, T-duality, DBI theory, and R—R charges.

Table of contents

Page	Title	Main topics
01	Why Strings? Regge Behavior and Dual Resonance Models	Motivation, open and closed strings, Regge trajectories, compact dimensions, Veneziano duality.
02	The Relativistic Point Particle	Reparametrization invariance, einbein action, mass-shell constraint, proper-time propagator.
03	The Nambu—Goto and Polyakov Actions	Induced metric, string tension, static-gauge expansion, Lüscher term, Polyakov action.
04	Worldsheet Symmetries, Boundary Conditions, and Conformal Gauge	Diffeomorphisms, Weyl symmetry, conformal gauge, open-string boundary conditions, D-branes.
05	Classical Rotating Strings and the Regge Slope	Spinning open string, Poincare currents, conserved charges, classical Regge slope.
06	Mode Expansions and Canonical Quantization	Closed/open string modes, oscillator algebra, Virasoro generators, cylinder-plane map.
07	Covariant Virasoro Constraints and Bosonic Spectra	Normal ordering, intercepts, open/closed spectra, level matching, negative-norm states.
08	Light-Cone Quantization and the Critical Dimension	Light-cone gauge, transverse oscillators, Lorentz algebra, $D=26$.
09	Hagedorn Growth and String Thermodynamics	Oscillator degeneracies, eta-function asymptotics, Hagedorn temperature.
10	String Interactions and the Free Boson CFT	Worldsheet interactions, radial ordering, Green functions, first OPEs.
11	Stress Tensor, Primaries, and Conformal Transformations	Holomorphic stress tensor, primary fields, Ward identities, global conformal maps.
12	Vertex Operators, OPEs, and the Virasoro Algebra	Exponential vertices, OPEs, central charge, Virasoro modes and algebra.
13	Radial Quantization and the State—Operator Correspondence	Cylinder-plane map, Schwarzian shift, highest-weight states, oscillator dictionary.
14	Highest-Weight Modules, Null States, and the Kac Determinant	Verma modules, Gram matrices, null states, Kac determinant.
15	Minimal Models, Ward Identities, and Correlators	Unitary minimal models, BPZ equations, global constraints, cross-ratios, conformal blocks.
16	Bosonic Physical States and the No-Ghost Structure	Old covariant physical states, null states, gauge redundancy, no-ghost theorem.
17	The $bc$ Ghost System and Ghost Zero Modes	Ghost weights, central charge, ghost number, ghost vacua, sphere zero modes.
18	The Polyakov Path Integral and Moduli	Gauge fixing, conformal Killing vectors, moduli, Beltrami differentials, $b$-ghost insertions.
19	Weyl Anomaly, Liouville Theory, and BRST Quantization	Critical dimension, Liouville mode, BRST transformations, BRST cohomology, vertex descent.
20	Tree-Level String Amplitudes	Sphere amplitudes, vertex insertions, Möbius fixing, Koba—Nielsen factors.
21	The Virasoro—Shapiro Amplitude and Factorization	Four-tachyon amplitude, gamma functions, pole towers, OPE factorization, Regge behavior.
22	Massless Closed-String Vertices and Gauge Invariance	Graviton, $B$-field, dilaton, transversality, worldsheet total derivatives.
23	The Low-Energy Effective Action	String-frame action, dilaton genus weight, Einstein frame, low-energy expansion.
24	Sigma-Model Beta Functions and the Linear Dilaton	Background fields, RG flow, spacetime equations, linear-dilaton CFT.
25	Worldsheet Supersymmetry and the NSR Action	Worldsheet fermions, NSR matter action, superconformal gauge, stress tensor, supercurrent.
26	Superconformal Algebra and NS/R Sectors	Super-Virasoro algebra, spin structures, NS/R modes, normal-ordering constants.
27	The Open Superstring Spectrum	NS/R Hilbert spaces, physical constraints, NS vector, Ramond ground state.
28	Ten-Dimensional Spinors and Ramond Ground States	Clifford algebra, spinor Fock construction, Majorana-Weyl spinors, $SO(8)$ little group.
29	The GSO Projection and Open-String Supersymmetry	Tachyon removal, Ramond chirality, vector multiplet, Bose/Fermi matching.
30	Closed Superstring Sectors and Type II Theories	NS-NS, NS-R, R-NS, R-R sectors, Type IIA/IIB spectra, R—R forms.
31	Bosonization, Spin Fields, and Superghosts	Fermion bosonization, spin fields, branch cuts, $\beta\gamma$ superghosts, picture number.
32	NSR Vertex Operators and Picture Changing	NS/R vertices, picture-changing operator, sphere and disk picture bookkeeping.
33	Type I Strings, Orientifolds, and Chan—Paton Factors	Type 0, worldsheet parity, Type I projection, Chan—Paton labels, $SO/Sp$ groups.
34	Closed-String T-Duality and Enhanced Symmetry	Momentum and winding, radius inversion, self-dual radius, enhanced current algebra.
35	Open-String T-Duality and D-Branes	Neumann/Dirichlet exchange, Wilson lines, brane positions, stretched strings, gauge enhancement.
36	DBI Action, Tachyon Condensation, and R—R Charges	DBI action, Wess—Zumino couplings, tachyon condensation, Type II brane charges.

Suggested references

The main references for comparison are Green—Schwarz—Witten, Polchinski, Polyakov, Becker—Becker—Schwarz, and Zwiebach. These notes often keep the derivations more explicit than standard textbooks, but the textbooks remain essential for additional examples, exercises, and historical context.