QFT III

Overview

This course is about the places where quantum field theory stops looking like a formalism for elementary particles and starts looking like a language for phases, defects, geometry, and universality. It begins with the two-dimensional Ising model, where the high-temperature expansion turns the partition function into a sum over closed loops and the low-temperature expansion turns it into a sum over domain walls. Those elementary graphical expansions already contain many of the themes of the course: duality, nonlocal variables, continuum limits, and the emergence of fields from sums over geometrical objects.

From there the course moves through the continuum field theory of critical phenomena, renormalization-group fixed points, scaling dimensions, conformal invariance, the operator product expansion, two-dimensional conformal field theory, Virasoro symmetry, null states, and minimal models. The Ising model returns not as a lattice system but as a conformal field theory with spin, disorder, energy, and fermion operators fitting into a rigid algebraic structure.

The later parts of the course widen the lens. Gauge fields, Wilson loops, worldline and worldsheet path integrals, conformal anomalies, random lattices, random surfaces, superfluidity, compact phases, vortices, monopoles, solitons, and confinement appear as different ways in which quantum fields encode extended objects. The final pages point toward strings, branes, sigma models, and geometric beta functions. The goal is not to make all of these subjects look identical; it is to make their shared mechanisms visible.

The organizing principle is: a quantum field theory is understood by its observables, its defects, and its scaling limits. Local Lagrangians are important, but they are not the whole story. Dual variables, disorder operators, Wilson loops, monopole gases, random surfaces, and worldsheet degrees of freedom often reveal the physics more directly than the original microscopic variables.

Acknowledgement

These webpages are based on handwritten notes taken by Jie Ren from Alexander M. Polyakov’s one-semester course Selected Topics in High-Energy Physics. The notes have been edited, expanded, typeset, supplemented with derivations and figures, and adapted for QFT.org.

Any errors in transcription, interpretation, exposition, convention choices, or notation are the responsibility of the editor of these webpages.

Scope of the reconstruction

The handwritten source is compact: it records formulas, diagrams, changes of subject, and short physical prompts rather than a self-contained transcript. The forty numbered lessons on this site are therefore editorial teaching units, not a claim that the source records forty class meetings. Their order preserves the notebook’s main progression—from lattice expansions and critical fields through conformal methods, extended observables, random surfaces, compact defects, and string geometry—but some recurring topics are gathered so that prerequisites appear before they are used.

Material added for exposition includes intermediate algebra, convention checks, figures, exercises with solutions, and links between ideas that are only juxtaposed in the source. Such additions are intended to explain the course’s reasoning, not to attribute new wording or claims to Polyakov. In particular, the string, brane, and AdS material near the end should be read at the level stated on the relevant pages: some results are derived, while others are explicitly structural hints.

Editorial principles

The course is designed as a serious graduate-level bridge between statistical field theory, conformal field theory, gauge theory, and string-theoretic ideas. The guiding principles are:

• Lattice and continuum descriptions are kept in contact. The Ising model, compact phases, random lattices, and Wilson loops are not just motivational examples; they are laboratories in which continuum QFT ideas become concrete.
• Dual variables are treated as observables, not tricks. Disorder fields, vortex variables, Wilson loops, monopoles, and worldsheet sums are used to reorganize the same physics in a better coordinate system.
• Critical data are separated from microscopic details. Scaling dimensions, OPE coefficients, central charges, beta functions, and anomaly coefficients are emphasized because they survive coarse graining.
• Algebra and geometry are developed together. Virasoro symmetry, Ward identities, conformal maps, random surfaces, and sigma-model beta functions are presented as mutually reinforcing viewpoints.
• Speculative hints are labeled as hints. The final pages gesture toward strings, branes, and AdS-like geometry. They are written as structural intuition unless a precise derivation is given.

Reliability guide for readers

These notes deliberately move between exact lattice identities, continuum scaling arguments, conformal algebra, semiclassical defect sums, and string-theoretic intuition. Those modes of reasoning should not be read with the same level of literalness. The following dictionary is used throughout the course.

Label or phrase	How to read it	Typical examples
Exact identity	An equality inside a stated model and convention.	Kramers–Wannier duality, finite-lattice disorder-line deformations, Ward identities, contour-mode commutators.
Controlled approximation	A result with a small parameter or expansion scheme.	One-loop RG near $4-\epsilon$, large-$N$ genus counting, sigma-model beta functions at small $\alpha'$.
Semiclassical mechanism	A saddle-point or dilute-gas explanation whose regime must be checked.	Kinks, vortices, monopole plasmas, dual-photon mass generation, area-law estimates.
Structural hint	A conceptual bridge rather than a derivation from earlier pages.	Gauge/string analogies, random-surface intuition, brane language, AdS-like geometry.

Local convention notes matter. In particular, factors of $2\pi$, signs in Euclidean continuation, normalizations of $T(z)$, and definitions of compact gauge fields vary across the literature. When a formula is convention-sensitive, the page states the convention before using it.

Course roadmap

A schematic path through QFT III. The course begins with statistical systems and critical phenomena, develops conformal data and two-dimensional CFT, and then turns to gauge-invariant observables, random surfaces, defects, confinement, and string geometry.

Prerequisites

Required background. QFT I supplies the needed command of path integrals, Green functions, perturbation theory, conserved currents, gauge fields, Ward identities, and one-loop renormalization. Readers should also be comfortable with dimensional analysis, Wick rotation, and spontaneous symmetry breaking. The opening lessons develop the Ising expansions themselves, so prior mastery of the Ising model is not required.

Helpful background. Selected parts of QFT II provide a useful first encounter with Wilsonian RG, operator products, two-dimensional field theory, sigma models, instantons, monopoles, and confinement. QFT II and QFT III deliberately overlap, so it is reasonable to consult those topics in parallel rather than treating every QFT II lesson as a strict prerequisite. Statistical mechanics—especially partition functions, correlation length, critical exponents, and transfer-matrix reasoning—is also helpful. Readers unsure of their preparation can use the site’s readiness guide before beginning.

The mathematical tools used most often are complex analysis, distributions, Gaussian integration, saddle-point methods, representation theory of Lie algebras, contour integrals, homotopy intuition, and elementary differential geometry. The notes review the relevant pieces when they become central, but readers will get more out of the course if they are comfortable moving between algebraic, analytic, and geometric arguments.

Convention lock

These notes inherit the site-wide natural units $c=\hbar=1$ and mostly-minus Lorentzian metric,

$$\eta_{\mu\nu}=\operatorname{diag}(+,-,-,\ldots,-).$$

Euclidean calculations use a positive metric and the weight $e^{-S_E}$; pages that cross between signatures state the continuation locally. Statistical-mechanics systems are usually written with Boltzmann weights

$$Z=\sum_{\text{states}} e^{-\beta H}$$

or, after passing to a Euclidean field description,

$$Z=\int \mathcal D\phi\, e^{-S_E[\phi]}.$$

For the ferromagnetic nearest-neighbor Ising model, the default convention is

$$H=-J\sum_{\langle ij\rangle}\sigma_i\sigma_j, \qquad \sigma_i=\pm 1, \qquad K=\beta J>0,$$

so

$$Z(K)=\sum_{\{\sigma\}}\exp\left(K\sum_{\langle ij\rangle}\sigma_i\sigma_j\right).$$

The square-lattice Kramers–Wannier dual coupling is written as

$$e^{-2K^*}=\tanh K, \qquad \text{equivalently}\qquad \sinh(2K)\sinh(2K^*)=1.$$

The continuum scalar theory near an Ising-type critical point is normalized as

$$S_E=\int d^d x\, \left[ {1\over2}(\partial\phi)^2+{1\over2}r\phi^2+{u\over4!}\phi^4+\cdots \right],$$

where $r$ is the temperature-like relevant coupling. The dots denote all operators compatible with the symmetries; most are irrelevant near the Wilson–Fisher fixed point.

A local scaling operator $\mathcal O_i$ has scaling dimension $\Delta_i$ if, at a fixed point,

$$\mathcal O_i(x)\mapsto \lambda^{-\Delta_i}\mathcal O_i(\lambda x).$$

For scalar primaries in a Euclidean conformal field theory,

$$\langle \mathcal O_i(x)\mathcal O_j(0)\rangle ={C_{ij}\over |x|^{\Delta_i+\Delta_j}}$$

when the dimensions match and the operators have compatible quantum numbers. The OPE convention is

$$\mathcal O_i(x)\mathcal O_j(0) \sim \sum_k C_{ij}{}^k\,|x|^{\Delta_k-\Delta_i-\Delta_j}\mathcal O_k(0)+\cdots,$$

with spin-dependent tensor structures suppressed when the context is scalar.

In two Euclidean dimensions we use

$$z=x^1+ix^2, \qquad \bar z=x^1-ix^2, \qquad \partial={1\over2}(\partial_1-i\partial_2), \qquad \bar\partial={1\over2}(\partial_1+i\partial_2).$$

The holomorphic stress tensor is expanded as

$$T(z)=\sum_{n\in\mathbb Z} L_n z^{-n-2},$$

and the Virasoro algebra is

$$[L_m,L_n]=(m-n)L_{m+n}+{c\over12}m(m^2-1)\delta_{m+n,0}.$$

For a primary field $\Phi(w,\bar w)$ of weights $(h,\bar h)$,

$$T(z)\Phi(w,\bar w) \sim {h\,\Phi(w,\bar w)\over (z-w)^2} +{\partial_w\Phi(w,\bar w)\over z-w} +\cdots.$$

In the charged Abelian sections, the convention is

$$\psi\mapsto e^{iq\alpha}\psi, \qquad A_\mu\mapsto A_\mu+\partial_\mu\alpha, \qquad D_\mu=\partial_\mu-iqA_\mu,$$

with

$$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.$$

With Hermitian generators, Wilson loops are written as

$$W_R(C)={1\over\dim R}\operatorname{tr}_R\,\mathcal P \exp\left(i\oint_C A_\mu^aT_R^a dx^\mu\right),$$

where the connection $A_\mu^aT_R^a$ includes the coupling in this formula. Pages using a canonical gauge field, a dimensionless compact connection, or adjoint-vector notation state the corresponding local normalization before use.

A perimeter law means $\langle W(C)\rangle\sim e^{-\mu\,\operatorname{Perimeter}(C)}$, while an area law means

$$\langle W(C)\rangle\sim e^{-\sigma\,\operatorname{Area}(C)}.$$

The latter is the Wilson-loop diagnostic of confinement in a pure gauge theory.

Compact scalar phases obey

$$\varphi\sim \varphi+2\pi, \qquad \oint d\varphi=2\pi n, \qquad n\in\mathbb Z.$$

For a bosonic worldsheet embedded by $X^\mu(\sigma)$, the Polyakov action convention is

$$S_P={\mathcal T\over2}\int d^2\sigma\,\sqrt h\,h^{ab} \partial_aX^\mu\partial_bX_\mu,$$

where $h_{ab}$ is the worldsheet metric and the string tension is

$$\mathcal T=T_s={1\over2\pi\alpha'}.$$

How to read these notes

The safest route is linear. Lessons 01–08 build the statistical-mechanics core: Ising expansions, Kramers–Wannier duality, continuum limits, disorder operators, and Ising fermions. Lessons 09–16 connect that core to relativistic fields, scaling, the OPE, and finite conformal transformations. Lessons 17–20 are a bridge rather than a single CFT block: they develop real-time correlators, currents, Ward identities, stress tensors, Goldstone reasoning, QED, and the two-dimensional Thirring model. Virasoro theory begins in earnest at lesson 21; lessons 21–29 then build through null states and minimal models to the Ising CFT.

Lessons 30–35 pivot from local operators to extended objects: worldlines, worldsheets, conformal gauge, anomalies, gauge-invariant observables, and random surfaces. Lessons 36–39 develop superfluid phases, vortices, compact gauge fields, solitons, monopole plasmas, and confinement. Lesson 40 is a synthesis: it connects those mechanisms to strings, branes, sigma-model beta functions, and explicitly labeled AdS hints rather than beginning a separate complete string-theory course.

A first reading should follow the lead, subject-specific sections, and stated checks on each page; use the local prerequisite notes when a lesson points backward. A second reading should include the exercises and the derivations of Ward identities, OPE constraints, duality relations, and area-law estimates. Many formulas become more transparent after drawing the object they count: loops, contours, worldsheets, vortices, or Wilson surfaces.

Reading routes

The linear route is best for a first pass, but the course also supports targeted reading.

Goal	Suggested route	What to watch
Learn critical phenomena and RG	01–06, then 12–15	Separate exact lattice rewritings, continuum matching, perturbative RG, and fixed-point kinematics.
Learn the Ising-to-CFT story	01–16, then 21–29	Track how $\sigma$, $\mu$, $\varepsilon$, and $\psi$ change meaning from lattice observables to scaling fields.
Understand conformal machinery	12–16, then 21–29	Separate global conformal symmetry, Virasoro symmetry, and null-vector constraints.
Follow gauge-invariant observables	11, 19–20, 30, 34, 38–40	Bare fields are often not physical observables; Wilson loops, currents, asymptotic states, and defects carry the invariant content.
Study defects and confinement	07–08, then 36–39	Compare Ising disorder lines, vortices in compact phases, and monopoles in compact gauge theory; the dimensions and defects are different.
Reach the string/surface viewpoint	30–35, then 39–40	Distinguish a worldline sum, a worldsheet path integral, a random triangulation, and an effective flux-tube string.

Lesson list

Part I — Ising model, duality, and continuum limits

1. Ising Model and Graphical Expansions develops the high-temperature loop expansion, the low-temperature domain-wall expansion, and the entropy–energy competition behind phase transitions.
2. Kramers–Wannier Duality and Mean-Field Theory derives the dual coupling relation, identifies the self-dual point, and compares exact duality logic with mean-field intuition.
3. Hubbard–Stratonovich Transformation and the Continuum Field turns spin interactions into an auxiliary field and explains how the continuum $\phi^4$ description emerges.
4. Critical Propagators and the Upper Critical Dimension studies lattice kernels, correlation length, loop divergences, and the special role of four dimensions.
5. Epsilon Expansion and One-Loop Scaling introduces $4-\epsilon$ dimensions, beta functions, and the Wilson–Fisher fixed point.
6. Fixed Points, Tricriticality, and Order–Disorder Variables organizes relevant perturbations, tricritical behavior, and the first appearance of order and disorder variables.
7. Disorder Lines, Branch Cuts, and Defect Operators explains disorder-line insertions as branch cuts and interprets them as defect operators.
8. Order–Disorder Duality and Ising Fermions shows how order–disorder composites lead naturally to fermionic variables in the Ising continuum limit.

Part II — Relativistic fields, scaling, and conformal symmetry

9. Relativistic Fields, Spinors, and the Ising Continuum Limit connects free scalar, vector, and spinor fields to the continuum Ising description.
10. Lattice Dirac Equations and Euclidean Spinors develops lattice Green functions, Euclidean spinor conventions, and the continuum Dirac equation.
11. Z₂ Gauge Systems, Wilson Loops, and Free Correlators introduces lattice gauge variables, Wilson loops, and basic free-field correlators.
12. Scaling Dimensions, Correlators, and Critical Exponents derives scaling forms for correlation functions and relates anomalous dimensions to critical behavior.
13. Callan–Symanzik Scaling and Emergent Conformal Symmetry explains RG equations for Green functions and the route from scale invariance to conformal invariance.
14. Conformal Transformations in d Dimensions derives conformal Killing equations, inversions, and the covariance of correlators in general dimension.
15. OPE, Associativity, and Conformal Correlators introduces the operator product expansion, conformal three-point data, cross ratios, and associativity constraints.
16. Finite Conformal Maps, Gauge Symmetry, and Free Fields studies finite conformal maps, primary transformations, gauge analogies, and free-field mode setups.

Part III — Free fields, currents, and Ward identities

17. Free Fields, Wightman Functions, and the iε Prescription derives Wightman functions, Feynman functions, analyticity, and the $i\epsilon$ prescription.
18. Commutators, Currents, and Ward Identities relates causal commutators, current conservation, and Ward identities.
19. Stress Tensors, Goldstone Modes, and QED Ward Identities develops stress tensors, spacetime symmetries, Goldstone logic, and transversality in QED.
20. Two-Dimensional QED, the Thirring Model, and Currents studies two-dimensional gauge dynamics, current–current interactions, and chiral-current structures.

Part IV — Virasoro symmetry, null states, and minimal models

21. Stress-Tensor OPE and Conformal Ward Identities derives the singular terms in stress-tensor OPEs and translates contour integrals into conformal Ward identities.
22. Primary Fields, Cylinder Maps, and Mode Expansions maps the plane to the cylinder and relates primary-field transformations to stress-tensor modes.
23. Virasoro Generators, Descendants, and Central Charge constructs descendant states, the $T T$ OPE, and the central extension of conformal symmetry.
24. Schwarzian Derivative and the Virasoro Algebra explains the anomalous transformation of the stress tensor and derives the Schwarzian term.
25. Global Conformal Generators and Descendant States isolates the global conformal subgroup and studies descendant towers under $L_{-1},L_0,L_1$.
26. Null States and BPZ Differential Equations derives null-vector decoupling and the resulting differential equations for correlation functions.
27. Degenerate Fields, Kac Labels, and Constraints introduces degenerate representations, Kac labels, and restrictions on dimensions and central charge.
28. Minimal Models, Fusion, and the Ising Operator Algebra builds the minimal-model spectrum, fusion rules, and the Ising operator algebra.
29. Ising CFT, Majorana Fermions, and Tricritical Extensions connects the Ising CFT to Majorana fermions and previews tricritical and superconformal extensions.

Part V — Worldlines, worldsheets, anomalies, and random surfaces

30. Superconformal Currents, Maxwell Lines, and Worldline Actions transitions from conformal currents to Maxwell line intuition and relativistic worldline actions.
31. Worldlines, Worldsheets, and Reparametrization Gauge compares point-particle and string actions, gauge fixing, induced metrics, and stress-tensor constraints.
32. Conformal Gauge, Two-Dimensional Gravity, and Vacuum Polarization develops Weyl symmetry, conformal gauge, minimal area, and polarization effects.
33. Conformal Anomalies, Liouville Action, and Nonlocal Effective Actions derives anomaly logic and explains how local and nonlocal effective actions encode stress-tensor physics.
34. Gauge-Invariant Observables and Asymptotic Amplitudes distinguishes gauge-dependent fields from physical observables and motivates asymptotic amplitudes and random-lattice ideas.
35. Random Lattices, Matrix Models, and Random Surfaces introduces random triangulations, ribbon graphs, planar diagrams, resolvents, and area-counting criticality.

Part VI — Defects, confinement, strings, and geometry

36. Superfluidity, Landau Criterion, and Accelerated Frames studies phonons, broken Galilean symmetry, the Landau criterion, and accelerated-frame intuition.
37. Compact Phases, Vortices, and Duality derives winding sectors, vortex Coulomb gases, Debye screening, dual gauge fields, and elasticity analogies.
38. Compact Gauge Fields, Higgsing, Solitons, and Topological Defects studies compact QED, charge quantization, Higgs phases, kinks, zero modes, vortices, and skyrmion-type defects.
39. Wilson Loops, Worldlines, Monopole Plasma, and Confinement derives area-law intuition from monopole instanton gases, worldline sums, and compact gauge dynamics.
40. Strings, Branes, Sigma Models, and AdS Hints synthesizes strings, branes, nonlinear sigma-model beta functions, vertex operators, open/closed strings, warped geometries, and AdS-like hints.

Thematic map

Question	Where it appears	Core idea
How does a lattice spin model become a field theory?	01–05	Graphical expansions, auxiliary fields, and critical propagators reveal the continuum degrees of freedom.
What does duality do to observables?	02, 06–08, 37–39	Duality exchanges local and nonlocal descriptions: spins become disorder fields, phases become vortices, and compact gauge fields become defect gases.
Why are fixed points so powerful?	04–15	At a fixed point, microscopic details collapse into scaling dimensions, OPE coefficients, and symmetry constraints.
Why is two-dimensional CFT special?	21–29	Local conformal transformations become infinite-dimensional, producing Virasoro symmetry, null-state equations, and minimal models.
How do gauge theories force us to rethink observables?	11, 19–20, 30, 34, 38–40	Gauge-invariant information lives in Wilson loops, currents, asymptotic amplitudes, and topological sectors rather than bare gauge potentials.
Why do random surfaces appear in QFT?	30–35, 40	Worldline sums, worldsheet actions, ribbon graphs, and random triangulations turn field-theory expansions into geometry.
How do defects change phases?	07–08, 36–39	Disorder lines, vortices, monopoles, kinks, and instantons reorganize long-distance physics and can drive screening, Higgsing, or confinement.
Where do strings enter?	30–35, 39–40	Area laws, random surfaces, and sigma models suggest string-like descriptions of gauge dynamics and extended objects.

Formula and convention quick index

This table is meant for readers returning to the course as a reference. It points to the first page where each convention or formula is developed with context.

Topic	First detailed appearance
High-temperature closed-loop expansion	01
Kramers–Wannier duality relation	02
Hubbard–Stratonovich effective action	03
Ornstein–Zernike critical propagator	04
Wilson–Fisher beta function and $\epsilon$ expansion	05
Disorder-line endpoint and branch-cut convention	07
Ising Majorana continuum dictionary	08, 09
Scaling dimensions and thermodynamic exponents	12
Conformal Killing equation and finite conformal maps	14, 16
OPE and crossing/associativity	15
Wightman, Feynman, commutator, and retarded functions	17, 18
Stress-tensor OPE and Virasoro algebra	21, 23, 24
BPZ null-state equation and Kac labels	26, 27
Ising and tricritical Ising operator algebras	28, 29
Worldline reparametrization and proper time	30
Nambu–Goto and Polyakov worldsheet actions	31, 40
Anomaly-induced $R\Box^{-1}R$ action and Liouville mode	32, 33
Matrix-model genus counting and random surfaces	35
BKT vortices and compact-phase duality	37
Compact gauge fields, monopoles, and Wilson-loop area laws	38, 39
Sigma-model beta functions and string geometry	40

Diagnostic warm-ups

These short problems test the background assumptions for the course. They are not barriers to entry; they simply flag mechanisms that will return often.

Warm-up 1: closed loops in the high-temperature Ising expansion

Use

$$e^{K\sigma_i\sigma_j}=\cosh K\left(1+\sigma_i\sigma_j\tanh K\right)$$

for each nearest-neighbor bond of the Ising model. Show that after summing over all spins, only subgraphs in which every vertex has even degree contribute.

Solution

Expanding the product over bonds chooses some set $\Gamma$ of occupied bonds. The contribution of $\Gamma$ contains

$$\prod_{\langle ij\rangle\in\Gamma}\sigma_i\sigma_j.$$

At a site $i$, the spin $\sigma_i$ appears once for every occupied bond ending at $i$. If that number is $d_i(\Gamma)$, then the spin sum contains

$$\sum_{\sigma_i=\pm1}\sigma_i^{d_i(\Gamma)}.$$

This equals $2$ when $d_i(\Gamma)$ is even and $0$ when $d_i(\Gamma)$ is odd. Therefore a graph contributes only if all vertices have even degree. On the square lattice this means the occupied bonds form closed loops, possibly with intersections.

Warm-up 2: the upper critical dimension of φ⁴ theory

In $d$ Euclidean dimensions, use

$$S_0={1\over2}\int d^d x\,(\partial\phi)^2$$

to find the engineering dimension of $\phi$. Then find the engineering dimension of $u$ in

$$S_{\mathrm{int}}={u\over4!}\int d^d x\,\phi^4.$$

What dimension makes $u$ marginal by power counting?

Solution

The Euclidean action is dimensionless in natural units. Since $[d^d x]=-d$ and $[\partial]=1$, the kinetic term gives

$$-d+2+2[\phi]=0.$$

Thus

$$[\phi]={d-2\over2}.$$

For the interaction,

$$[u]-d+4[\phi]=0,$$

so

$$[u]=d-4\left({d-2\over2}\right)=4-d.$$

The coupling is marginal at $d=4$. This is why $d=4$ is the upper critical dimension of the Ising universality class described by scalar $\phi^4$ theory.

Warm-up 3: a contour generator in two-dimensional CFT

Let

$$T(z)\Phi(w,\bar w) \sim {h\Phi(w,\bar w)\over(z-w)^2} +{\partial_w\Phi(w,\bar w)\over z-w} +\cdots$$

and define

$$L_n=\oint_{0}{dz\over2\pi i}\,z^{n+1}T(z).$$

Assuming the contour can be moved to encircle $w$, derive the infinitesimal action of $L_n$ on $\Phi$.

Solution

Moving the contour around the insertion gives

$$[L_n,\Phi(w,\bar w)] =\oint_w {dz\over2\pi i}\,z^{n+1}T(z)\Phi(w,\bar w).$$

Using the singular terms in the OPE,

$$[L_n,\Phi(w,\bar w)] =\oint_w {dz\over2\pi i}\,z^{n+1} \left({h\Phi\over(z-w)^2}+{\partial_w\Phi\over z-w}\right).$$

The simple pole gives $w^{n+1}\partial_w\Phi$. The double pole gives the derivative of $z^{n+1}$ at $z=w$:

$$\oint_w {dz\over2\pi i}\,{z^{n+1}\over(z-w)^2} =(n+1)w^n.$$

Therefore

$$[L_n,\Phi(w,\bar w)] =\left(w^{n+1}\partial_w+(n+1)h w^n\right)\Phi(w,\bar w).$$

This is the local form of the conformal transformation generated by $L_n$.

Further reading

The lesson pages cite sources where they are used. The following uncited companions are useful for the course as a whole:

• John Cardy, Scaling and Renormalization in Statistical Physics (Cambridge University Press, 1996).
• Sidney Coleman, Aspects of Symmetry: Selected Erice Lectures (Cambridge University Press, 1985).
• Philippe Di Francesco, Pierre Mathieu, and David Sénéchal, Conformal Field Theory (Springer, 1997).
• Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, 1995).
• Joseph Polchinski, String Theory, Volume 1: An Introduction to the Bosonic String (Cambridge University Press, 1998).
• Alexander M. Polyakov, Gauge Fields and Strings (Harwood Academic Publishers, 1987).
• Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations (Cambridge University Press, 1995) and Volume II: Modern Applications (Cambridge University Press, 1996).
• Jean Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th ed. (Clarendon Press, 2002).

Where to begin

For a first pass, start with Ising Model and Graphical Expansions. The Ising model may look elementary, but it introduces the loops, domain walls, duality, scaling limits, disorder variables, and fermions that organize the rest of the course. If one of the diagnostic warm-ups is unfamiliar, review the linked QFT I or QFT II background before the corresponding lesson rather than postponing the entire course.