String Theory I Introduction

These pages develop the material of Professor Igor R. Klebanov’s one-semester course, “Introduction to String Theory,” taught 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.

String theory connects a two-dimensional quantum field theory on a worldsheet to particles and interactions in spacetime. This course develops that connection from the classical string actions through quantization, conformal field theory, scattering amplitudes and superstrings. It ends with T-duality and the dynamics and charges of D-branes. String Theory II continues toward string dualities, black branes and holography.

Perturbative strings and worldsheet physics

The recurring question is how a worldsheet calculation acquires a spacetime meaning. Oscillator constraints determine particle polarizations; conformal invariance constrains backgrounds; vertex correlators become scattering amplitudes; and boundary conditions describe D-branes. The pages develop these connections through derivations, checks of limiting cases and normalizations, explicit statements of scope, and solved exercises.

The bosonic string is the first working example. Its tachyon and critical dimension are physical features of that model, not conclusions that can be transferred unchanged to the superstring. The later RNS construction adds worldsheet fermions, projects the spectrum, and leads to Type I and Type II theories. Low-energy actions are identified with their approximation regimes. Holography is developed in the second course after these foundations.

Prerequisites

You should be comfortable with special relativity, Lagrangian and Hamiltonian mechanics, elementary quantum field theory, path integrals, complex analysis and basic Lie algebra representations. Two-dimensional conformal field theory is developed within the course. The relativistic particle introduces reparametrization constraints in a simpler setting, while the free-boson CFT begins the operator methods used later.

How to read the course

The numbered pages form one main sequence. The following stages describe how its questions develop.

Pages 01–09: classical strings and quantization. Derive the actions, constraints and mode expansions, then compare covariant and light-cone descriptions of the spectrum. State counting leads to the Hagedorn scale.
Pages 10–19: conformal fields and physical states. Learn OPEs, conformal representations, ghosts and moduli. Anomaly cancellation and BRST cohomology explain which gauge-fixed states are physical.
Pages 20–24: amplitudes and spacetime fields. Compute tree amplitudes and factorization, identify the massless fields, and derive their leading effective equations from scattering and worldsheet renormalization.
Pages 25–33: the RNS superstring. Construct NS and Ramond states, derive the GSO projection and Type II spectra, and use spin fields and picture changing to compute open-string couplings. Orientation projection and Chan–Paton factors lead to Type I theory.
Pages 34–36: duality and D-brane dynamics. Exchange momentum and winding, turn Wilson lines into brane positions, and connect stretched strings to gauge fields, the DBI action and RR charge.

On a first pass, follow the physical question and reconstruct the central derivation before opening an exercise solution. On a return pass, use the exercises to test a changed assumption or a new example. The global conventions below are the reference for signs and normalizations throughout the course. Polchinski’s two volumes and Tong’s lectures provide complementary treatments; complete references appear at the end of this page.

Five consecutive stages connect string quantization, worldsheet CFT, amplitudes, superstrings and D-brane dynamics — The main reading sequence and the question addressed at each stage. Page ranges are consecutive; the arrows indicate progression through the course, not a claim that one result alone proves the next stage. This is a schematic guide with no physical scale.

Page	Title	Main topics
01	Why Strings? Regge Behavior and Dual Resonance Models	Regge trajectories, string scales, compact dimensions and Veneziano duality.
02	The Relativistic Point Particle	Reparametrization, the einbein, the mass-shell constraint and the proper-time propagator.
03	The Nambu–Goto and Polyakov Actions	Worldsheet area, transverse fluctuations, Casimir energies and the auxiliary metric.
04	Worldsheet Symmetries, Boundary Conditions, and Conformal Gauge	Diffeomorphisms, Weyl symmetry, residual conformal maps and endpoint conditions.
05	Classical Rotating Strings and the Regge Slope	Energy and angular momentum, the classical Regge bound and folded strings.
06	Mode Expansions and Canonical Quantization	Open and closed modes, oscillator commutators and the cylinder-to-plane map.
07	Covariant Virasoro Constraints and Bosonic Spectra	Virasoro constraints, null-state quotients, level matching and low-lying spectra.
08	Light-Cone Quantization and the Critical Dimension	Transverse oscillators, the intercept and the Lorentz-anomaly calculation.
09	Hagedorn Growth and String Thermodynamics	Physical-state degeneracies, eta-function asymptotics and thermal limits.
10	String Interactions and the Free Boson CFT	Worldsheet topology, Gaussian correlators, zero modes and normal ordering.
11	Stress Tensor, Primaries, and Conformal Transformations	Stress-tensor contours, primaries, descendants and Ward identities.
12	Vertex Operators, OPEs, and the Virasoro Algebra	Exponential and tensor vertices, conformal families and the central extension.
13	Radial Quantization and State–Operator Correspondence	States from local insertions, cylinder evolution and the Schwarzian term.
14	Highest-Weight Modules, Null States, and the Kac Determinant	Verma modules, Gram forms, singular vectors and null-state equations.
15	Minimal Models, Ward Identities, and Correlators	Minimal models, Ising conformal blocks, monodromy, crossing and fusion.
16	Bosonic Physical States and the No-Ghost Structure	Physical matter vertices, gauge descendants and the no-ghost theorem.
17	The bc Ghost System and Ghost Zero Modes	Graded contractions, central charge, ghost number and sphere zero modes.
18	The Polyakov Path Integral and Moduli	Faddeev–Popov zero modes, Beltrami pairings, punctures and the modular quotient.
19	Weyl Anomaly, Liouville Theory, and BRST Quantization	Weyl response, the Liouville mode, BRST cohomology and vertex descent.
20	Tree-Level String Amplitudes from Worldsheet Correlators	Sphere correlators, Möbius fixing, Koba–Nielsen factors and the moduli integral.
21	The Virasoro–Shapiro Amplitude and Factorization	The complex beta integral, closed-string poles, residues and the Regge limit.
22	Massless Closed-String Vertices and Gauge Invariance	Tensor polarizations, graviton, two-form and dilaton states, and gauge equivalence.
23	The Low-Energy Effective Action	Cubic amplitudes, string-frame interactions, genus weights and Einstein frame.
24	Sigma-Model Beta Functions and the Linear Dilaton	The Ricci beta function, Weyl invariance and the exact linear-dilaton CFT.
25	Worldsheet Supersymmetry and the NSR Action	Worldsheet fermions, local supersymmetry, superspace and the supercurrent.
26	Superconformal Algebra and NS/R Sectors	Super-Virasoro modes, spin structures, Clifford zero modes and intercepts.
27	The Open Superstring Spectrum	Endpoint doubling, NS and Ramond constraints, and unprojected physical states.
28	Ten-Dimensional Spinors and Ramond Ground States	Clifford modules, Majorana–Weyl reality and massless Spin(8) polarizations.
29	The GSO Projection and Open-String Supersymmetry	GSO projectors, massive multiplets and all-level Bose–Fermi matching.
30	Closed Superstring Sectors and Type II Theories	Level matching, mixed-sector spinors, RR form polarizations and Type IIA/IIB.
31	Bosonization, Spin Fields, and Superghosts	Spin fields, cocycles, superghost dressing, picture number and its cohomology scope.
32	NSR Vertex Operators and Picture Changing	BRST-complete vertices, picture changing, descent and fermion/vector disk couplings.
33	Type I Strings, Orientifolds, and Chan–Paton Factors	Worldsheet parity, endpoint projections, color traces and the Type I tadpole condition.
34	Closed-String T-Duality and Enhanced Symmetry	Canonical radius duality, the self-dual spectrum, current algebra and radius deformations.
35	Open-String T-Duality and D-Branes	Wilson-line holonomy, brane positions, stretched masses and worldvolume fields.
36	D-brane dynamics, tachyon condensation, and RR charges	DBI dynamics, electric flux, tachyon energy cancellation and electric/magnetic RR charge.

Global conventions

These conventions apply throughout String Theory I unless an explicit conversion is being made.

Units, spacetime metric, and string scale

We use $\hbar=c=1$ and a mostly-plus spacetime metric:

\eta_{\mu\nu}=\mathrm{diag}(-,+,\ldots,+), \qquad p^2=\eta_{\mu\nu}p^\mu p^\nu=-m^2 \quad\text{on shell}.

We keep $\alpha'$ explicit. It has dimension of length squared and determines the tension, string length, and characteristic mass scale:

T=\frac{1}{2\pi\alpha'}, \qquad \ell_s=\sqrt{\alpha'}, \qquad M_s\sim\frac{1}{\sqrt{\alpha'}}.

Special choices such as $\alpha'=2$ are used only when explicitly identified.

Spacetime curvature, dilaton, and gravitational normalization

Our curvature convention is

\begin{aligned} R^\rho{}_{\sigma\mu\nu} &=\partial_\mu\Gamma^\rho_{\nu\sigma} -\partial_\nu\Gamma^\rho_{\mu\sigma} +\Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} -\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma},\\ R_{\sigma\nu}&=R^\rho{}_{\sigma\rho\nu}. \end{aligned}

In particular, a round sphere has positive scalar curvature. For expansions about a constant asymptotic dilaton, distinguish the total field, its dimensionless fluctuation, and the string coupling:

\Phi=\Phi_\infty+\phi, \qquad g_s=e^{\Phi_\infty}.

In $D>2$ dimensions, the Einstein metric is normalized to agree with the string metric at infinity. The gravitational couplings and the canonical scalar are

\begin{aligned} g_{\mu\nu}&=e^{-4\phi/(D-2)}G^{\rm S}_{\mu\nu}, &\kappa_D&=g_s\kappa_{D,0},\\ \varphi&=\frac{2\phi}{\kappa_D\sqrt{D-2}}, &S_\varphi&=-\frac12\int d^Dx\sqrt{-g}\,(\partial\varphi)^2. \end{aligned}

The unshifted string-frame prefactor is $e^{-2\Phi}/(2\kappa_{D,0}^2)$ ; equivalently it is $e^{-2\phi}/(2\kappa_D^2)$ . Thus $[\kappa_D]=\mathrm{length}^{(D-2)/2}$ , whereas $\Phi$ and $\phi$ are dimensionless. In a constant background, $\Phi_0$ also denotes the background value. These are spacetime fields: the worldsheet Weyl parameter $\varphi$ and the bosonized superghost $\phi$ are separate objects. These definitions do not assign a numerical value to the bosonic gravitational coupling or determine its value after compactification. For $D=10$ , they agree with the String II normalization. The effective-action page derives the frame transformation and scalar kinetic term.

Matrix gauge fields

Chan–Paton generators $T^a$ and the matrix gauge potential $A_\mu=A_\mu^aT^a$ are Hermitian, with $[T^a,T^b]=if^{ab}{}_cT^c$ . With the gauge coupling in front of the action, use

\begin{aligned} D_\mu&=\partial_\mu-iA_\mu, &[D_\mu,D_\nu]&=-iF_{\mu\nu},\\ F_{\mu\nu}&=\partial_\mu A_\nu-\partial_\nu A_\mu-i[A_\mu,A_\nu]. \end{aligned}

For an adjoint field $Y$ , $D_\mu Y=\partial_\mu Y-i[A_\mu,Y]$ . The representation trace and its normalization must be specified with the action; the D3-brane matrix normalization is used in the holographic dictionary.

Worldsheet coordinates and integration measure

Lorentzian worldsheet coordinates are

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

In conformal gauge we use light-cone coordinates

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

On a Euclidean worldsheet, write $x=\sigma^1$ , $y=\sigma^2$ , and

z=x+iy, \qquad \partial=\partial_z=\frac12(\partial_x-i\partial_y), \qquad \bar\partial=\partial_{\bar z}=\frac12(\partial_x+i\partial_y).

Our complex integration measure is twice the ordinary coordinate area:

d^2z=2\,dx\,dy=2r\,dr\,d\theta.

The delta function $\delta_z^{(2)}$ is normalized in this measure, so

\int d^2z\,\delta_z^{(2)}(z-w)f(z)=f(w), \qquad \partial\bar\partial\log|z-w|^2 =2\pi\delta_z^{(2)}(z-w).

In real-coordinate covariant integrals, $d^2\sigma\sqrt h$ denotes the ordinary metric area element; it does not include this extra factor of two.

Free matter fields and ghosts

For free coordinates in flat target space, the Euclidean action and two-point singularity are

\begin{aligned} S_X&=\frac{1}{2\pi\alpha'}\int d^2z\,\partial X^\mu\bar\partial X_\mu\\ &=\frac{1}{4\pi\alpha'}\int dx\,dy\,\partial_aX^\mu\partial_aX_\mu, \end{aligned}

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

Accordingly, the bulk exponential $:e^{ik\cdot X}:$ has weights $h=\bar h=\alpha'k^2/4$ . Boundary operators use the doubling construction, derived where it is needed.

The reparametrization ghosts have holomorphic weights $h_b=2$ and $h_c=-1$ :

S_{bc}=\frac{1}{2\pi}\int d^2z\, \bigl(b\bar\partial c+\widetilde b\partial\widetilde c\bigr), \qquad b(z)c(w)\sim\frac{1}{z-w}.

Tildes denote the antiholomorphic sector. Ghost number is $+1$ for $c$ , $-1$ for $b$ , and $0$ for the identity vacuum $|0\rangle_{gh}$ , with the same assignments in the antiholomorphic sector.

Worldsheet fermions

The RNS fermions have canonical normalization: their target-space index is a vector index, and their worldsheet spinor components are dimensionless when the worldsheet coordinates are dimensionless. In Lorentzian signature we use real Majorana components and

\psi^\mu=\begin{pmatrix}\psi_-^\mu\\\psi_+^\mu\end{pmatrix}, \qquad \bar\psi=\psi^{\mathsf T}\rho^0, \qquad \rho^0=\begin{pmatrix}0&-1\\1&0\end{pmatrix}, \quad \rho^1=\begin{pmatrix}0&1\\1&0\end{pmatrix}.

Thus $\{\rho^a,\rho^b\}=2\eta^{ab}_{\rm ws}$ with $\eta_{\rm ws}=\operatorname{diag}(-1,+1)$ . The flat, gauge-fixed matter action is

S_L=-\frac1{4\pi\alpha'}\int d\tau\,d\sigma\, \left(\partial_aX\cdot\partial^aX+i\alpha'\bar\psi\cdot\rho^a\partial_a\psi\right).

Its chiral fermion term is $i(\psi_+\cdot\partial_-\psi_++\psi_-\cdot\partial_+\psi_-)/(2\pi)$ . In particular, $\psi_+$ depends on $\sigma^+$ on shell. With $\tau=-i\tau_E$ and $w=\tau_E+i\sigma$ , it continues to the holomorphic field; $\psi_-$ continues to the antiholomorphic field. No additional fermion phase is needed with these choices.

On a flat Euclidean worldsheet, the fermion action and OPEs are

S_\psi=\frac1{4\pi}\int d^2z\, \left(\psi\cdot\bar\partial\psi+\widetilde\psi\cdot\partial\widetilde\psi\right), \qquad d^2z=2\,dx\,dy,

\psi^\mu(z)\psi^\nu(w)\sim\frac{\eta^{\mu\nu}}{z-w}, \qquad \widetilde\psi^\mu(\bar z)\widetilde\psi^\nu(\bar w) \sim\frac{\eta^{\mu\nu}}{\bar z-\bar w}.

The two chiralities have no singular mixed OPE in the bulk. Euclidean chiral fields are treated as independent Grassmann fields; the real Lorentzian Majorana condition is not imposed separately on each Euclidean component. The matter currents are

T_m=-\frac1{\alpha'}:\partial X\cdot\partial X: -\frac12:\psi\cdot\partial\psi:, \qquad G=i\sqrt{\frac2{\alpha'}}:\psi\cdot\partial X:.

Their weights are $2$ and $3/2$ , and $D$ free coordinate multiplets have $c_m=3D/2$ in each chiral sector. Some pages use the alternate name $T_F=G$ for the supercurrent. These normalizations keep $\alpha'$ explicit; setting $\alpha'=2$ is an explicitly stated specialization. The corresponding rigid transformations and Lorentzian constraints are summarized in the String II overview and derived in the RNS pages.

Bosonization and superghost pictures

Chiral matter bosons obey $H_I(z)H_J(w)\sim-\delta_{IJ}\log(z-w)$ and have stress tensor $T_H=-\frac12\sum_I:(\partial H_I)^2:$ . A normalized complex fermion is represented by $\psi_I^\pm=\kappa_I^\pm:e^{\pm iH_I}:$ , with cocycles $\kappa_I^\pm$ supplying the statistics between different species. Ramond spin fields belong to twisted charge sectors.

For the commuting superghosts, the ordered OPEs are

\beta(z)\gamma(w)\sim-\frac1{z-w},\qquad \gamma(z)\beta(w)\sim\frac1{z-w}.

Their bosonization uses an anticommuting pair of weights $(1,0)$ and a scalar $\phi_{\rm gh}$ , written $\phi$ when unambiguous:

\begin{gathered} \eta(z)\xi(w)\sim\xi(z)\eta(w)\sim\frac1{z-w},\qquad \phi_{\rm gh}(z)\phi_{\rm gh}(w)\sim-\log(z-w),\\ \beta=:e^{-\phi_{\rm gh}}\partial\xi:, \qquad \gamma=: \eta e^{\phi_{\rm gh}}: ,\\ T_{\beta\gamma}=-\frac12:(\partial\phi_{\rm gh})^2: -\partial^2\phi_{\rm gh}-:\eta\partial\xi:. \end{gathered}

Cocycles make $e^{\pm\phi_{\rm gh}}$ odd, so these ordered products are even. Ghost number and picture are distinct gradings, with currents

j_{\rm gh}=-:bc:-:\xi\eta:, \qquad j_P=-\partial\phi_{\rm gh}+:\xi\eta:.

Thus $e^{q\phi_{\rm gh}}$ has ghost number $0$ and picture $q$ ; $(\eta,\xi)$ have ghost numbers $(1,-1)$ and pictures $(-1,1)$ . Both $\beta$ and $\gamma$ have picture zero. Derivatives preserve both charges. The small Hilbert space is $\ker\eta_0$ , with $\{\eta_0,\xi_0\}=1$ . Derivations appear on bosonization, spin fields, and superghosts and ghosts, superghosts, and BRST.

Worldsheet superspace

Euclidean worldsheet superspace has independent odd coordinates $\theta$ and $\bar\theta$ . Odd derivatives act from the left; odd coordinates, fermion fields, and supersymmetry parameters anticommute. The ordered Berezin measure is

d^2\theta=d\theta\,d\bar\theta, \qquad \int d\theta\,\theta=1, \qquad \int d^2\theta\,\theta\bar\theta=-1.

The rightmost integration acts first. The covariant odd derivatives and supertranslation generators are

\begin{aligned} D&=\partial_\theta+\theta\partial, &Q&=\partial_\theta-\theta\partial,\\ \bar D&=\partial_{\bar\theta}+\bar\theta\bar\partial, &\bar Q&=\partial_{\bar\theta}-\bar\theta\bar\partial. \end{aligned}

With $a=\sqrt{\alpha'/2}$ , the scalar matter superfield is

\mathbb X^\mu =X^\mu+ia\theta\psi^\mu+ia\bar\theta\widetilde\psi^\mu +\theta\bar\theta F^\mu.

Here $F^\mu$ is an auxiliary field. With dimensionless worldsheet coordinates, $X^\mu$ , $a$ , and $F^\mu$ have dimensions of length, while the canonical fermions and odd coordinates are dimensionless. Parameters are written on the left in $\delta\mathbb X=\epsilon Q\mathbb X+\bar\epsilon\bar Q\mathbb X$ . These odd coordinates are worldsheet spinor coordinates; they are distinct from the target-space spinors used in the Green–Schwarz formulation. The worldsheet supersymmetry page derives the component action and transformations with these definitions.

Stress tensors and normal ordering

The holomorphic stress tensors and central charges are

\begin{aligned} T_X&=-\frac{1}{\alpha'}:\partial X\cdot\partial X:, &&c_X=D,\\ T_{gh}&=-2:b\partial c:-:(\partial b)c:, &&c_{gh}=-26. \end{aligned}

Here $D$ is the number of free coordinates. More generally, $T_m$ and $c_m$ refer to the complete matter sector, including any internal CFT. In the bosonic string, $T_{\rm tot}=T_m+T_{gh}$ ; the RNS string also includes the superghost stress tensor $T_{\beta\gamma}$ .

For variations of the Euclidean worldsheet metric, we use

\mathcal T_{ab} =-\frac{2}{\sqrt h}\frac{\delta S_E}{\delta h^{ab}}, \qquad T(z)=2\pi\mathcal T_{zz} \quad\text{on the flat plane}.

The partition function has weight $e^{-S_E}$ and the Euclidean effective action is $-\log Z$ . A metric-source tensor defined instead by $T^E_{ab}=-(2/\sqrt h)\,\delta\log Z/\delta h^{ab}$ has the opposite sign: $T^E_{ab}=-\langle\mathcal T_{ab}\rangle$ . The worldsheet convention above fixes the signs in Weyl variations.

For classical Lorentzian calculations, distinguish the Hilbert stress tensor from the constraint tensor with the tension removed:

\mathcal T^L_{ab}=-\frac{2}{\sqrt{-h}}\frac{\delta S_L}{\delta h^{ab}}, \qquad C_{ab}=\frac{\mathcal T^L_{ab}}{T}.

For the bosonic Polyakov action, let $H_{ab}=G_{\mu\nu}\partial_aX^\mu\partial_bX^\nu$ be the induced metric. Then

C_{ab}=H_{ab}-\frac12h_{ab}h^{cd}H_{cd}, \qquad C_{ab}=0 \quad\text{is the metric equation}.

This $C_{ab}$ is not the holomorphic CFT stress tensor $T(z)$ . With the Euclidean definition above, the free theory has $\mathcal T_{ab}=-T C^E_{ab}$ and $T_X(z)=-C^E_{zz}/\alpha'$ , with normal ordering understood. Thus a classical constraint and a canonically normalized stress tensor have the same vanishing condition but different normalization factors.

Ordinary colons in local fields denote plane OPE normal ordering. When oscillator normal ordering uses a different vacuum, it is marked separately; its constant shifts are derived explicitly in the state-space calculation.

References

Joseph Polchinski, String Theory, Vol. 1: An Introduction to the Bosonic String (Cambridge University Press, 1998), doi:10.1017/CBO9780511816079; Vol. 2: Superstring Theory and Beyond (Cambridge University Press, 1998), doi:10.1017/CBO9780511618123.
David Tong, String Theory, University of Cambridge Part III lecture notes (2009; revised 2012), arXiv:0908.0333.