Lectures on Conformal Field Theory (Krzysztof Gawedzki)
TENTATIVE OUTLINE
1. Simple functional integrals for free theory in geometric background.
2. Circle and toroidal compactifications, also with SUSY.
3. Main properties of CFT, operator product expansion of energy-momentum and
primary fields.
4. The Hilbert space approach and axiomatics a la Segal.
5. Geometric sigma models, the beta function calculation, the SUSY version.
6. WZW via functional integral, current algebra OPE's, the relation to
moduli space of bundles, the free field representations,
7. Coset models, also in the SUSY case (N=2 superconformal cosets).
REFERENCES
A set of Gaussian integration formulae may be found
e.g. in "Quantum Field Theory and Critical Phenomena"
by J. Zinn-Justin (Sects. 1.1 & 1.2). See also Sects. 2.0-2.2
& 2.5 for the discussion of of the path integral and
the Feynman-Kac formula. Of course, for the latter topic,
"Quantum Mechanics and Path Integral" by Feynman-Hibbs is
the physics classic. Rigorous theory of infinite-dimensional
Gaussian integrals may be found e.g. in the 4th volume of
Gelfand-Vilenkin. See also Simon's "The Euclidean $P(\phi)_2$
Quantum Field Theory".
The free field with values in $S^1$ are discussed
briefly e.g. in the Ginspargs contributions to Les Houches 1988 School
(Session XLIV "Fields, Strings and Critical Phenomena", eds.
Brezin-Zinn-Justin) or in "Th\'{e}orie Statistique des Champs"
de Drouffe-Itzykson (Sects. 3.2 & 3.6). For the case of fields
with values in a torus see the paper by Narain-Sarmadi-Wittem
in Nucl. Phys. B 279 (1987) p. 369 and for the case of complex torus
read Vafa's contribution to "Essays on Mirror Symmetry", ed. S.-T. Yau,
International Press, Hong Kong 1992.
The operator product expansion in the 2-dimensional conformal field
theory is a starting point of the famous Belavin-Polyakov-Zamolodchikov
paper in Nucl. Phys. B 241 (1984), p. 333. My discussion will be somewhat
in the spirit of the article of Eguchi-Ooguri in Nucl. Phys. B 282 (1987),
p. 308 and should make the BPZ formalism transparent.
For the Segal's axiomatics of CFT see:
G. Segal, "Two dimensional conformal field tgheory and modular functors"
in IXth International Congress in Math. Phys., Simon, Truman, Davies (eds),
Adam Hilger, Bristol 1989 and G. Segal, "The definition of conformal field
theory," preprint of variable size. A general discussion of CFT
designed for the mathematical audience may be found in: K. Gawedzki,
"Conformal field theory" in Asterisque 177/178 (1989), p.95.
For the rudiments of the perturbative approach to functional
integrals, Feynman graphs etc. see again the book by Zinn-Justin,
Sects. 5.1-5.3. The original reference to the renormalization of
geometric sigma models is Fiedan's thesis published with few
years delay in Ann. Phys. 163 (1985), p. 318. The SUSY case
is discussed in Alvarez-Gaume-Freedman-Mukhi, Ann. Phys. 134 (1981)
p. 85.
The basic papers on the Wess-Zumino-Witten model are:
Witten: "Non-abelian bosonization in 2 dimensions",
CMP 92 (1984), p. 455, Polyakov-Wiegmann: "Goldstone fields in two
dimensions with multivalued actions", Phys. Lett. B 141, (1984), p. 223,
Knizhnik,Zamolodchikov: "Current algebra and Wess-Zumino model in two
dimensions", Nucl. Phys. B 247 (1984), p. 83, Gepner, Witten:
"String theory on group manifolds", Nucl. Phys. B 278 (1986), p. 493,
Verlinde: "Fusion rules and modular transformations in 2d CFT", Nucl.
Phys. B 300 (1988), p. 360, Tsuchiya, Kanie: "Vertex operators
in the conformal field theory on P1 ...", Adv. Sud. Pure Math. 16 (1988),
p. 297, Tsuchiya, Ueno, Yamada: Conformal field theory on universal
family of stable curves ...", Adv. Stud. Pure Math. 19 (1989), p. 459.
I will follow a geometric approach sketched in my Karpacz lectures:
"Constructive conformal field theory" in Functional Integration,
Geometry and Strings, Haba, Sobczyk (eds), Birkhaeuser, Basel 1989, p. 277.
The original papers on the coset models are:
Goddard, Kent Olive in Phys. Lett. B 152 (1985), p. 88 (short version) and
CMP 103 (1896), p. 105. The N=2 SUSY case was discussed in: Kazama-Suzuki,
Y. Kazama, H. Suzuki, Phys. Lett. B 216 (1989), p. 112
and Nucl. Phys. B 321 (1989), p. 232. I will stick to the functional
integral approach of Gawedzki-Kupiainen: Nucl. Phys. B 320 (1989), p. 625.