Course Information for Physics 230
Field Theory and Topology

John Preskill
Winter and Spring, 2000



Course Description

Ph 230abc. Elementary Particle Theory.
9 units (3-0-6); first, second, third terms.
Prerequisite: Ph 205abc or equivalent.

Advanced methods in quantum field theory.

First term:  Introduction to supersymmetry, including the minimal supersymmetric extension of the standard model, supersymmetric  grand unified theories, extended supersymmetrysupergravity, and supersymmetric theories in higher dimensions.

Second and third terms: Nonperturbative phenomena in nonabelian gauge field theories, including quark confinement, chiral symmetry breaking, anomalies, instantons, the 1/N expansion, lattice gauge theories, and topological solitons.

Instructors: Schwarz (first term), Preskill (second and third terms).

Class Meetings

Wednesdays 4:30 to 6:00 and Fridays 4:00-5:30 in 469 Lauritsen, second and third terms.


John Preskill
459 Lauritsen Laboratory
Telephone: 626-395-6691

Teaching assistant:
Costin Popescu
404 Downs
Telephone: 626-395-2632
office hours: Thursdays, 2-4 pm

Course Requirements

There will be regularly assigned problem sets. The grading is pass/fail.


An introductory course in quantum field theory, covering quantization of gauge theories, path integrals, and Feynman diagram perturbation theory.


The recommended textbook is Quantum Theory of Fields II: Modern Applications, by Steven Weinberg. However, we will not be following it closely.

Some references on topological defects:

S. Coleman, Aspects of Symmetry (1985). Especially Chapter 6: Classical lumps and their quantum descendants, and Chapter 7: The uses of instantons.

F. Wilczek, Fractional Statistics and Anyon Superconductivity (1990). Lectures and reprints covering anyons and Chern-Simons theory.

J. Preskill, "Chromatic aberrations: Yang and Mills meet Aharonov and Bohm (1993)." Physics colloquium about the non-abelian Aharonov-Bohm effect.

M. Bucher, H.-K. Lo, and J. Preskill, "Topological approach to Alice electrodynamics (1991)," On transfer of electric and magnetic Cheshire charge.

J. Preskill, "Vortices and Monopoles," in Architecture of Fundamental Interactions at Short Distances (1987), P. Ramond and R. Stora, editors.

J. Preskill, "Magnetic Monopoles," Ann. Rev. Nucl. Part. Sci. 34, 461 (1984).

S. Coleman, "The magnetic monopole 50 years later" in The Unity of the Fundamental Interactions (1983), A. Zichichi, editor.

M. Atiyah and N. Hitchin, The Geometry and Dynamics of Magnetic Monopoles (1988). On the moduli space of multimonopoles.

Course Summary

Part I:  Vortices and Anyons

Lecture 1 (1/5).  The formulation of quantum chromodynamics. The analogy between Riemannian geometry and Yang-Mills theory. The interpretation of the Yang-Mills potential as a connection that determines parallel transport of color, and of the Yang-Mills field strength as the curvature that characterizes the path dependence of parallel transport.  The Aharonov-Bohm effect. The concept of Cheshire charge -- color electric charge with no localized source that can be carried by a closed loop of color magnetic flux.

Lecture 2 (1/6).  The history of the gauge principle and of the Aharonov-Bohm effect. The static potential between colored objects.  The abelian Higgs model and its topological conservation law. Sectors classified by winding number and magnetic flux.

No class 1/12 and 1/14 due to String Theory at the Millennium Conference.

Lecture 3 (1/19).  Energetics and structure of the vortices of the abelian Higgs model. The Bogomol'nyi bound. Spontaneous breaking of SO(3) and vortices classified by Z_2.

Lecture 4 (1/21).  Z_N vortices in the Higgs phase of an SU(N) theory. The general classification of topologically stable vortices. Local discrete symmetry and superselection sectors. Fractional angular momentum of flux-charge composites.

Lecture 5 (1/26).  Fractional statistics. Violation of P and T. The spin-statistics connection for systems with antiparticles. The braid group and its one-dimensional unitary representations. Composites of anyons. Anyons in abelian Chern-Simons theory.

Lecture 6 (1/28).  The link invariant from abelian Chern-Simons theory. Self-linking and regularization. The quantum Hall effect. Incompressibility of a filled Landau level, and the integer quantum Hall effect. The need for an incompressible collective state at fractional filling to explain the fractional effect. Phenomenological C-S theory of the FQHE states. Cancellation of the applied magnetic flux by the statistical flux in the Chern-Simons theory, and vortices as quasiparticles with fractional charge and statistics.

Lecture 7 (2/2).  Topological degeneracy for anyonic systems on Riemann surfaces. Canonical quantization of gauge theories in the A_0=0 gauge. The Gauss law constraint and the gauge-invariant physical subspace. Little gauge transformations, global gauge transformations, charge quantization, and the charge superselection rule. Canonical quantization of (pure) Chern-Simons theory on the torus. Commutators of Wilson lines. Large gauge transformations on the torus,  the anomaly in their algebra, and quantization of mass.

Lecture 8 (2/4).  The relation between the large gauge transformations on the torus and quantum tunneling by vortices. The connection between topological degeneracy and spontaneous breaking of global symmetry. Lifting of the degeneracy by finite size effects. "Theta vacua" and the theta-dependence of the vacuum energy. The topological conservation law of the nonlinear sigma model in two-spatial dimensions. The topological charge density and the Bogomol'nyi bound. Explicit construction of "skyrmion" solutions. Scale invariance.

Lecture 9 (2/9).  The conserved topological current of the nonlinear sigma model, its interpretation as the dual of the pullback of the volume form on the two-sphere, and the associated gauge potential. Turning skyrmions into anyons with a Chern-Simons term or Hopf term. Linking number interpretation of the Hopf invariant, and the connection with the spin and statistics of skyrmions. The theta term for the (3+1)-dimensional abelian Higgs model, and its implications for processes involving linking and unlinking of strings.

Lecture 10 (2/11). The Alice vortex associated with SO(3)/O(2) = RP^2. The global unrealizability of  SO(2) gauge transformations on the vortex background. The singular gauge and classical solutions with "Cheshire charge." Transfer of charge from point particles to pairs of vortices (or loops of string). Gauge-invariant description of charge transfer.

Lecture 11 (2/16). Quantum theory of Cheshire charge. The flux and charge of a string (or vortex pair) as complementary observables. Vortices associated with unbroken discrete nonabelian gauge symmetry. Measuring a flux via Aharonov-Bohm scattering. The braid operator acting on a pair of vortices. Nonabelian statistics of indistinguishable vortices.

Lecture 12 (2/18). Holonomy interactions of string loops. Entanglement of colliding noncommuting strings. Charge-flux composites: The charge of the flux is an irreducible representation of the normalizer of the flux. A spin-statistics connection for nonabelian vortices. Discrete Cheshire charge carried by vortex pairs.

Part II:  Monoples, Dyons, and Instantons

Lecture 13 (2/23).
Magnetic monopoles and the Dirac quantization condition. Magnetic charge as an element of the first homotopy group of the (unbroken) gauge group. Monopoles with Z_N charge.

Lecture 14 (2/25). Magnetic charge and the global structure of the gauge group. Dirac quantization condition in the standard model. Monopoles as solitons: the 't Hooft-Polyakov model. Winding number and magnetic charge of a hedgehog. The size and mass of the monopole core.

Lecture 15 (3/1). Bogomol'nyi bound on the masses of monopoles and dyons. BPS multimonopoles. Monopole scattering at low velocity in the moduli space approximation.

Lecture 16 (3/3). Winding number and magnetic charge: the general case. The exact homotopy sequence. Examples: SU(3) --> [SU(2) X U(1)]/Z_2 and SU(5) -- > [SU(3) X SU(2) X U(1)]/Z_6. Monopoles in grand unified theories.

Lecture 17 (3/8). Monopoles in the Einstein-Maxwell theory: Reissner-Nordstrom black holes. Magnetically charged black holes in Yang-Mills theory. Can magnetically charged black holes be pair produced? Topological solitons with Z_2 and Z_N magnetic charge. Monopoles in the SO(10) model. Monopoles and Alice strings.

Lecture 18 (3/29). Strings ending on monopoles. Walls bounded by strings. A classification of strings: Type-U (which can conceivably end on monopoles) vs. Type-S (which can conceivably be the boundary of a wall).

Lecture 19 (3/31). The topological classification of vector bundles. Dirac quantization on Riemann surfaces. Cohomology with integer coefficients. Classifying U(1) bundles on manifold M with H^2(M,Z). First Chern class. Topologically nontrivial flat connections on nonorientable Riemann surfaces.

Lecture 20 (4/5). Classifying G gauge fields on S^k with Pi_{k-1}(G). G gauge fields on manifold M and H^k(M,Pi_{k-1}(G)). Torsion classes and nonintegral classes.

Lecture 21 (4/7). Spinors on manifolds. The second Stiefel-Whitney class. Spin structures on coset manifolds. CP^2 and SU(3)/SO(3) are not spin manifolds. The Berry phase. Level crossings and the monopole number of the Berry connection on a 2-manifold. Quantization of the Hall conductivity and the Dirac quantization condition. Generic q-fold degeneracy, the U(q) Berry connection, and fractional Hall conductance.

Lecture 22 (4/12). Stable homotopies of SU(n). Connection and curvature in form notation. Integral invariants of  mappings from a closed k-manifold to a compact Lie group. Pi_3(G) and the second Chern class. Instantons and the semiclassical approximation. Topological sectors in Euclidean Yang-Mills theory, and a bound on the Euclidean action.

Lecture 23 (4/14). Instantons and the semiclassical analysis of quantum tunneling. The kink as an instanton in 0+1 dimensions. Lifting of classical degeneracy due to tunneling. The dilute instanton gas. Decay of unstable states -- the "bounce" solution. The negative mode of the bounce and analytic continuation via distortion of the integration contour. Decay of unstable strings and vortices. Beads on strings as instantons.

Lecture 24 (4/19). The Yang-Mills field equations. Temporal gauge quantization, the Gauss law constraint, small gauge transformations, and the physical Hilbert space. Global gauge transformations and charge quantization. "Semiclassically accessible" vacuum states: pure gauge where the gauge transformation is the identity at spatial infinity. Distinct classical vacua distinguished by the winding number of the gauge transformation on S^3. Instantons and semiclassical vacuum tunneling.

Lecture 25 (4/21). Theta vacua and the theta superselection rule. The dilute instanton gas approximation and semiclassical evaluation of the theta-dependent vacuum energy density. The arbitrary scale of the instanton, and the breakdown of semiclassical methods for large instantons. The theta parameter as a coupling constant, and CP nonconservation. Theta dependence on a monopole background: semiclassically accessible gauge transformations, and the theta-dependent dyon charge. Dirac quantization for dyons.

Lecture 26 (4/26). Theta-dependent dyon charge from the canonical viewpoint. Theta-dependent charge of a nonabelian monopole. Global realizability of gauge transformations on a monopole background. Cheshire charge carried by monopole pairs.

Part III:  Phases of Gauge Theories

Lecture 27 (4/28). Realizations of global symmetries: local order parameters, the infinite volume limit, superselection rules, and the lower critical dimension. Realizations of gauge symmetries cannot be distinguished with local gauge-noninvariant order parameters. Coulomb, Higgs, and confinement phases.The Wilson loop: a timelike loop as insertion of a classical charged source, and a spacelike loop as creation of an electric flux tube. The Wilson loop as a criterion for confinement: area law versus perimeter law. Magnetic disorder as an explanation for confinement.

Lecture 28 (5/3). Magnetic disorder due to instantons in the (1+1)-dimensional abelian Higgs model. The theta parameter as a background charge. The vortex operator in the (2+1)-dimensional abelian Higgs model. The topological U(1) global symmetry. The photon as a Goldstone boson in the Coulomb phase.

Lecture 29 (5/5). Magnetic monopoles as instantons in 3 Euclidean dimensions. The global U(1) vortex as a dual description of an electric charge, and the Wilson loop as an insertion of a loop of global string. Intrinsic breaking of the U(1) symmetry due to monopoles drives confinement: the photon becomes a massive pseudo-Goldston boson, and a global string becomes the boundary of a domain wall  (elecric flux tube world sheet). The global Z_N topological symmetry of Yang-Mills theory with G=SU(N)/Z_N. Spontaneous breakdown of the global Z_N symmetry: stable domain walls and confinement. Charge-N magnetic monopoles as the mechanism for confinement and the mass gap in (2+1)-dimensional Yang-Mills theory.

Lecture 30 (5/10). Confinement in (3+1)-dimensional Yang-Mills theory. The 't Hooft loop operator. Either magnetic or electric confinement must occur if the theory has a mass gap. Twisted gauge fields in a box. Sectors with definite magnetic and electric flux. Electric-magnetic duality relation and its solution.

Lecture 31 (5/12). Deconfinement at finite temperature, and its relation to the realization of a Z_N global symmetry. Dynamical quarks. The free-charge phase and the Aharonov-Bohm order parameter. Absence of a phase boundary that separates the Higgs and confinement phases.

Part IV: Anomalies

Lecture 32 (5/17). Chiral anomalies. Creation of chiral pairs in an electric field in 1+1 dimensions. Creation of chiral pairs in parallel electric and magnetic pairs in 3+1 dimensions. Feynman diagram computation of the anomaly in 1+1 dimensions. Impossibility of a local counterterm that restores both the vector and axial Ward identites. Inference from the anomaly that the spectrum contains massless particles.

            No class 5/19.

Lecture 33 (5/24). Chiral anomalies from the path integral viewpoint. Derivation of Ward identities by changing variables in the path integral. Jacobian of a functional change of variable. Heat kernel expansion. Chiral asymmetry of the Dirac operator and the Atiyah-Singer index theorem.

Lecture 34 (5/26). The U(1) problem of QCD: no parity doubling and no fourth light pseudo-Goldstone boson. Symmetry breaking without a Goldston boson via the chiral selection rule. How the Atiyah-Singer index theorem enforces the chiral selection rule. Rotating the theta parameter by redefining phases of the quark fields. The strong CP problem and the axion. Lack of perturbative or nonperturbative corrections to the anomalous Ward identity for three conserved flavor currents. The 't Hooft anomaly condition. Unbroken chiral symmetry in supersymmetric QCD with number of flavors minus number of colors equals one.

Lecture Notes 

Lectures 1-6, pages 1-53 

Lectures 7-10, pages 53-109 

Lectures 11-13, pages 109-152

Lectures 14-17, pages 153-206 

Lectures 18-20, pages 207-251 

Lectures 21-23, pages 252-298 

Lectures 24-26, pages 299-346 

Lectures 27-29, pages 347-402

Lectures 30-31, pages 403-441 

Lectures 32, pages 1-13


Problem Set 1, due Feb. 4, 2000.
    (1.1) Chromostatics
    (1.2) From second to first order
    (1.3) Gauge-invariant mass

Problem Set 2, due Feb. 16, 2000.
    (2.1) Linking number from Wilson loops
    (2.2) "Anyons" in higher dimensions
    (2.3) Quantization of mass

Problem Set 3, due Feb. 25, 2000.
    (3.1) The skyrmion solution
    (3.2) The Hopf invariant and the linking number
    (3.3) Planar topological degeneracy

Problem Set 4, due Mar. 3, 2000.
    (4.1) The CP^N skyrmion
    (4.2) A "semilocal" vortex

Problem Set 5, due Apr. 28, 2000.
    (5.1) Is a dyon a boson?
    (5.2) Electroweak strings and monopoles
    (5.3) The real Berry phase
    (5.4) A bounce for a hole in a wall

Problem Set 6, due May 5, 2000.
    (6.1) Spin manifolds
    (6.2) Maurer-Cartan integral invariants
    (6.3) Quantization of the Chern-Simons coupling
    (6.4) Abelian Chern-Simons theory revisited
    (6.5) Dirac quantization for branes

Problem Set 7, due May 17, 2000.
    (7.1) Perturbative Wilson loop
    (7.2) Instantons in (1+1)-dimensional Yang-Mills theory
    (7.3) Duality and Higgs exotica