Outline and References for
Physics 230B - Quantum Field Theory II,
with Emphasis on Dualities
Week 1: Introduction.
Review of relations between quantum field theory, classical and quantum
statistical mechanics; QFT at finite temperature. We followed mostly
[Zee, Ch. V.2], in order to establish contact with the previous semester,
of 230A. Additional ideas about quantum critical phenomena were also
presented, a good reference is
[S. Sachdev, Quantum Phase Transitions (Cambridge U.P., 1999)]
Discussion: The exact solution of the 1d Ising model was presented,
to illustrate the concepts of universality and scaling. The presentation
followed [Sachdev, see Ref. above].
Week 2: Dualities on the lattice.
Topics: Kramers-Wannier duality of the Ising model in 2 dimensions;
duality between the Ising model and the Z(2) gauge theory on the
lattice in 3 dimensions.
Some rather old but very good references for dualities of lattice models
[R. Savit, Duality in Field Theory and Statistical Systems,
Rev. Mod. Phys. 52 (1980) 453] (this is the reference that we
primarily followed in the lectures).
[J.B. Kogut, An Introduction to Lattice Gauge Theory and Spin Systems,
Rev. Mod. Phys. 51 (1979) 659] (another nice review, stressing
the Hamiltonian framework and RG).
[J.B. Kogut, A Review of the Lattice Gauge Theory Approach to Quantum
Rev. Mod. Phys. 55 (1983) 775] (this review goes beyond the scope
of what was discussed in lectures, but it represents a natural, readable
continuation of the material covered in the previous two reviews listed
Additional wealth of information on various exact solutions of lattice
models can be found in the classic text
[R.J. Baxter, Exactly Solved Models in Statistical Mechanics
(Academic Press, 1989)]
Week 3: More dualities on the lattice.
continuing with dualities on the lattice, following primarily Savit's review.
Prof. Ori Ganor as a guest lecturer. Topics: Duality of lattice models with
Z(N) symmetry, and with U(1) symmetry.
Week 4: Dualities in 2+1 and 1+1 dimensions; bosonization
Duality between U(1) gauge theory and the XY model in 2+1 dimensions,
in the continuum limit. Role of instantons; "proof" of confinement from
Good references are:
[X.-G. Wen, Quantum Field Theory of Many-Body Systems (Oxford U.P.,
2004), Ch. 6.3.]
[A.M. Polyakov, Gauge Fields and Strings (Harwood, 1987).]
Bosonization in two dimensions: This is a subject with huge amounts of
literature. A very attractively written review is in
[S. Coleman, Aspects of Symmetry, Selected Erice lectures (Cambridge
U.P., 1985)], in particular Section 5.1 of Lecture 6, Classical lumps and
their quantum descendants. The focus is on sine-Gordon/Thirring model
Another good source on sine-Gordon/Thirring duality is
[R. Rajaraman, Solitons and Instantons (North Holland, 1987)].
Bosonization from the condensed-matter perspective is nicely reviewed in
[E. Fradkin, Field Theories of Condensed Matter Systems
(Addison-Wesley, 1991)], in particular, Ch.4.3.
A good extensive review from the "high-energy theory" perspective is Ch.32
[J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (4th
edition: Oxford U.P., 2002)]
Bosonization also plays a very important role on the string worldsheet,
see for example Volume 1 of [Green, Schwarz and Witten].
Discussion: Our Friday discussion sessions resume this week, the paper
to be discussed is
[D. Horn, M. Karliner and S. Yankielowicz, Self-Dual Renormalization
Group Analysis of the Potts Models, Nucl. Phys. B170[FS1] (1980)
467.] (The article is available online via ScienceDirect, you can find the link at Spires, search for: "find a Yankielowicz and t Potts".)
Week 5: More on bosonization.
Continued the discussion of bosonization, the primary reference being Ch. 32
of Zinn-Justin's book mentioned above. Equivalence of massless free theories,
correlation functions, anomalous dimensions, anomalous commutators; turning
on masses for fermions; applications to chiral anomaly in background gauge
Discussion: Was actually postponed to Thursday of Week 6. Witten's
E. Witten, Gauge Theories, Vertex Models, and Quantum Groups Nucl.
Phys. B330 (1990) 285
was discussed. This paper can be viewed as Part III of a sequence, with
Part II being
E. Witten, Gauge Theories and Integrable Lattice Models Nucl.
Phys. B322 (1989) 629
and Part I the extremely influential
E. Witten, Quantum Field Theory and the Jones Polynomial, Comm. Math.
Phys. 121 (1989) 351.
Part I introduces non-Abelian Chern-Simons gauge theory and discusses its
intimate relation to invariants of (knots in) 3-manifolds.
(This entire topic is
closely connected to concepts of 2d CFT; a particularly readable approach
to the quantization of Chern-Simons by conventional canonical methods, and
its relation to 2d CFT, can be found in
S. Elitzur, G. Moore, A. Schwimmer and N. Seiberg, Remarks on the Canonical
Quantization of the Chern-Simons-Witten Theory, Nucl. Phys. B326
Parts II and III go on to present a fascinating web of relations between
integrable lattice models (of the spin, vertex-model, and IRF kinds), 2d
CFT and its dualities, quantum groups, nicely tying things up via the
three-dimensional perspective of Chern-Simons gauge theory.
Week 6: Applications of bosonization; non-Abelian bosonization,
Understanding the Schwinger model by bosonization, deriving confinement,
anomalies, spontaneous chiral symmetry breaking. Finally, establishing
the full Thirring/sine-Gordon duality. The main reference continues to be
Zinn-Justin's book, with an honorable mention to S. Coleman's lectures.
Non-Abelian bosonization. The best reference still is the original paper
by E. Witten,
E. Witten, Non-Abelian Bosonization in Two Dimensions, Comm. Math.
Phys. 92 (1984) 455.
General structure of CFT in 1+1 dimensions. Our primary reference for this
topic has been
P. Ginsparg, Applied
Conformal Field Theory, hep-th/9108028.
The advantage of this review is that it is relatively compact, uses the
conventional notation without going overboard, and reviews all the crucial
basics. Many other useful references for 2d CFT exist, in particular, many
books have been written. The one that I find particularly nice and useful
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory,
Graduate Texts in Contemporary Physics (Springer, 1996).
If you have a particular fascination with 2d CFT I am willing to recommend
other sources, based on your specific interests.
String theory applications of 2d CFT can be glimpsed in several chapters of
Polchinski's String theory books.
Discussion: We heard how bosonization in 1+1 dimensions can be viewed
algorithmically as a duality, by first introducing a gauge field, constraining
its dynamics to be trivial by introducing a Lagrange multiplier, then
integrating out the fermions and the gauge field; the Lagrange multiplier
becomes the dual, bosonic variable. This was based on
C.P. Burgess and F.
Quevedo, Bosonization as Duality, hep-th/9401105.
There are two very interesting follow-up papers worth looking at,
C.P. Burgess and F.
Quevedo, Nonabelian Bosonization as Duality, hep-th/9403173,
which proves the non-Abelian bosonization to the WZW model along the
same lines, and
C.P. Burgess, C.A. Lutken
and F. Quevedo, Bosonization in Higher Dimensions,
in which the same strategy is attempted in the notoriously difficult problem
of bosonizing fermions in higher spacetime dimensions, with some interesting
Week 7: General structure of 2d CFT; classification of c=1
We continue discussing two-dimensional CFT, following primarily excerpts from
Ginsparg's review. Classification of c =1 CFTs illustrates several
interesting dualities: The famous T-duality, which plays a very important
role in string theory (where, in fact, it led to the discovery of D-branes),
as well as a secret relation between two theories of distinct target-space
Week 8: Structure of N=2 SCFT in d=2 spacetime
dimensions; mirror symmetry.
We developed concepts special to extended supersymmetry: the ground ring,
chiral primaries, spectral flow, etc.
Our main reference for this topic is the review article
String Theory on Calabi-Yau Manifolds, hep-th/9702155 (TASI 1996
a slightly modified version of this article can also be found in 1995 Les
Houches Lectures by B. Greene, entitled Lectures on the Quantum Geometry
of String Theory.
The topic of mirror symmetry viewed from the
point of view of the =2 SCFT on the worldsheet of the closed string
might be in full string theory called "perturbative mirror symmetry."
String nonperturbative effects make the story even more fascinating, but
those effects are beyond the scope of this QFT course.
Week 9: Supersymmetric field theories in d=3 and d=4.
The primary source is a beautiful review article
M. Strassler, An
Unorthodox Introduction to Supersymmetric Gauge Theory,
hep-th/0309149 (TASI 2001 lectures).
The advantage is that this review teaches a lot about the quantum dynamics
of theories with various degrees of supersymmetry and in various dimensions,
without smothering the discussion in many technical details.
Most required technicalities can be found in S. Weinberg's Quantum Theory
of Fields, Volume III.
Another review paper worth reading is
K. Intriligator and
N. Seiberg, Lectures on Supersymmetric Gauge Theories and Electric-Magnetic
This week, we developed basic concepts of a classical moduli space of vacua,
classical RG flows, and the nonrenormalization theorem for the
Week 10: Susy QFTs in d=3 and d=4.
Continuing with Strassler's review, we discussed quantum moduli spaces,
quantum RG flows, U(1) gauge theories, Seiberg dualities, mirror
dualities in d=3 etc.
Week 11: Dynamics of non-Abelian N=2 Susy Yang-Mills in
The Seiberg-Witten solution of pure N=2 susy SU(2) Yang-Mills.
The best source on this topic might still be the original papers by
Seiberg and Witten,
N. Seiberg and
E. Witten, Electric-Magnetic Duality, Monopole Condensation, and
Confinement in N=2 Supersymmetric Yang-Mills Theory,
N. Seiberg and
E. Witten, Monopoles, Duality and Chiral Symmetry Breaking in
N=2 Supersymmetric QCD, hep-th/9408099.
In the remainder of the semester, we shall use the following main
Week 12: Quantum gravity as an effective theory.
In preparation for AdS/CFT correspondence, we must consider quantum gravity,
as a low-energy effective theory. One good review is
C.P. Burgess, Quantum
Gravity in Everyday Life: General Relativity as an Effective Theory,
It is also worthwhile to mention S. Weinberg's
S. Weinberg, Ultraviolet Divergences in Quantum Theories of
Gravitation, in: General Relativity. An Einstein Centenary Survey
(eds: S.W. Hawking and W. Israel, Cambridge, 1979)
S. Weinberg, The
Cosmological Constant Problem Rev. Mod. Phys. 61 (1989) 1.
Week 13: Gravity and supergravity as an effective
We shall continue with the topic from the previous week, also adding some
discussion of (primarily Type IIB) supergravity in ten spacetime dimensions,
and its dualities.
Week 14: AdS/CFT correspondence.
The review article that we will use is
J.M. Maldacena, TASI 2003
Lectures on AdS/CFT, hep-th/0309246.
Week 15: AdS/CFT correspondence.
Continuation from previous week, using the same review article.