Field Theory and Topology

Winter and Spring, 2000

- Course Description
- Class Meetings
- Instructors
- Course Requirements
- Prerequisites
- References
- Course Summary
__Lecture Notes__- Homework

**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 supersymmetry, supergravity, 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).

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

email: preskill@theory.caltech.edu

Teaching assistant:

Costin Popescu

404 Downs

Telephone: 626-395-2632

email: popescu@theory.caltech.edu

office hours: Thursdays, 2-4 pm

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

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.

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

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

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

Lecture 11 (2/16). Quantum theory of

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

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

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

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.

**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

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

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.

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

(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