Conformal Field Theory II (MAT 1739, Spring 2008)
Objectives
This is an introduction to conformal field theory (CFT)
in two dimensions.
This subject is important in many areas in
theoretical physics and mathematical physics.
In string theory,
CFT is the starting point
for perturbative formulation and therefore is an absolute minimum.
CFT also plays some important roles in some areas in
condensed matter physics and quantum computer research.
In mathematical physics,
CFT is a background for many of recent development, such as
loop groups, geometric representation theory,
mirror symmetry, geometric Langlands duality, etc.
Part-I
Hours (Rooms): Tuesdays 5:00- (BA6183) and
Thursdays 5:30- (BA3000)
Special Announcement:
Homework 5 is posted. This is going to be the final homework.
Instructor:
Kentaro Hori
McLennan Laboratory 1113
(416)978-4784, hori _at_ physics.utoronto.ca
Course Plan:
Introduction to QFT
Free Field Theories in Two Dimensions
2d Ising Model
Ward Identity for Infinite Conformal Symmetry
Wess-Zumino-Witten Models and Current Algebras
Textbook:
No text book. But references will be indicated
at appropriate time.
The reference for the main part (Ward identity
for conformal symmetry) is the classical paper:
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov,
Infinite Conformal Symmetry In Two-Dimensional
Quantum Field Theory
Nucl. Phys. B241 (1984)
333.
References:
Grading Scheme:
Homeworks
Homeworks and Exercises:
hw1 (posted Sep. 25):
some correlation functions and some operator identities in 0d theory.
The final homework for Part-I
hw2 (posted Jan. 10, no deadline):
Kramers-Wannier duality, an RG transform.
hw3 (posted Jan. 31, deadline February 5):
GSO-projected Majorana fermion.
hw4 (posted Mar. 4, deadline March 11):
Comparison of some characters.
hw5 (posted Apr. 1, deadline Apr. 8):
Computation of a four-point amplitude.
The precise
problem set is written in page 9-10 of the
note.
Solution is in page 9-12 of the Apr. 1 note.
Handouts:
Progress:
Sep. 18:
Ising model. Partition function, magnetization, correlation functions.
Solution in 0-d, 1-d, 2-d (2-d solution is just a quote).
Correlation length, phase transitions, and critical exponents.
Sep. 20:
Mean field theory, and Landau's theory of phase transitions.
Effective potential. The idea of renormalization group.
Sep. 25:
0-d QFT. Gaussian integrals, Wick contraction rule, normal ordering,
operator product identity;
Fermionic path integrals, calculus of anticommuting variables,
Sep. 27:
Fermionic integrals continued: Wick contraction, determinant and Pfaffian.
1-d QFT (i.e. Quamtum Mechanics).
Action and equation of motion.
Path integrals. Defing measure. Wick rotation.
Oct. 2:
Partition function on a circle, Z(beta). Example: Harmonic oscillator,
zeta function regularization. Example: "Fermionic oscillator".
From path integral to operator formalism. time ordered product.
The operator H.
Oct. 5:
Symmetry and Ward identity.
Change of integration variables.
Symmetry in classical mechanics. Noether's theorem.
Ward identity in quantum mechanics.
The operator H and energy.
Oct. 9:
Derivation of canonical (anti)commutation relation.
Example: Harmonic oscillator and "fermionic oscillator".
Z(beta)=Tr(exp( - beta H)).
Sigma model on a circle. quantization by both operator and path integral.
Oct. 11:
2-d QFT. Action and Wick rotation. 2d surfaces and classification.
Symmetries and Ward identities: Noether current and conservation law.
Ward identities and contour integrals.
Oct. 16:
Target space momentum p, Energy-momentum tensor, and its trace.
Hamiltonian H and (worldsheet)
momentum P.
Quantization of massless scalar: mode expansion, canonical quantization.
the eigenvalues of p,H,P. The sum of zero point energies.
Oct. 18:
Sigma model with circle target and T-duality:
quantized momenum and winding sector.
Topological charge and current.
Canonical quantization.
T-duality: momentum vs winding (Noether vs topological),
EOM vs identity.
Analogy: electric-magnetic duality in 3+1 dimensions.
Digression: Open string and D-branes.
Oct. 23:
Partition function of massless scalar and
circle sigma models. Computation in operator formalism.
2d-torus and modular invariance.
Correlation functions: path-integral on infinite
cylinder vs <0|T(O(1)O(2))|0>.
Computation in operator formalism.
Oct. 25:
Computation of the X two point function in path integral.
(the note includes the detail not shown in the class.)
Other correlation functions including derivatives.
Normal ordered product in operator formalism.
Oct. 30:
The operators exp(ikX) and correlation functions.
Normal ordering (operator). Point split regularization and
multiplicative renormalization (path-integral).
The vortex operators exp(ik'X'). Mixed wave-vortex correlators.
Nov. 1:
Dirac fermion. Action, equations of motion -- right and left movers.
Symmetries and currents.
Canonical quantization
on a circle with anti-periodic boundary condition (NS-NS sector).
Nov. 6:
Dirac fermion continued:
Dirac's sea and hole/particle interpretation.
Quantization on a circle with twisted boundary conditions.
Spectral flow.
Nov. 8:
Dirac fermion continued:
Partition functions. Twists and coupling to gauge fields.
[Digression: Chiral fermion and anomaly.]
Modular transformation property.
Nov. 13:
Dirac fermion continued:
Correlation functions. Two point functions and general correlators.
Modular invariant set of twists.
Ramond-Ramond sector. GSO projection.
[Digression: Orbifold]
Nov. 15:
Boson fermion correspondence. Twisted partition functions
and precise map between states. Map between operators.
Nov. 20:
Matching of correlators.
Duality between massive Thirring model and sine-Gordon model.
Conformal Symmetry in various dimensions (translations, rotations, dilatation,
special conformal transformations).
Infinite conformal transformations in two dimensions.
The definition of energy-momentum tensor.
Nov. 22:
Ward identity for diffeomorphisms (correction).
Definition of conformal field theory.
Definition of the central charge.
Calculus on a Riemann surface: From metric to complex structure.
Examples. The Levi-Civita connection.
Variation of the metric: Associated variation of complex structure.
Variational formula of tensors, L-C connection and the curvature.
[Digression: the moduli space and the Teichmuler space of
Riemann surfaces]
A general first order system. [Digression: spinors on Riemann surfaces
as half-forms]
Nov. 27:
E-M tensor for the first order system.
Consequence of E-M conservation in a CFT.
OPE of E-M tensors.
A non-covariant but (anti)holomorphic version. Transformation
using Schwarzian derivative. OPE of E-M tensors in free CFTs.
Nov. 28:
Ward identity for conformal transformations in a CFT.
Definition of L_n's and tilde{L}_n's. The Virasoro algebra.
Primary operators and descendants: identity operator as a primary.
Independence of coordinates. Conformal transformation
of primary operators.
Examples in free field theories.
Nov. 29:
L_n's and tilde{L}_n's as operators acting on states.
The Hermiticity. Commutation relations.
Primary states and their descendants.
Unitarity bound on the central charge and conformal weights.
State-operator correspondence. L_n's on states vs L_n's on operators.
The ground state and the identity operator.
Winter Break
Jan. 8:
2D Ising Model. Two representations of Z using loop configurations.
Kramers-Wannier duality.
Jan. 10:
(2d Ising) Order and Disorder operators. The critical temperature.
Jan. 15:
(2d Ising) The transfer matrix: expression using Pauli's matrices,
Jordan-Wigner transform and a fermionic system,
GSO-like projection.
Jan. 17:
(Digression) From operator to path integral, in Fermionic systems.
Kernel functions and formula for partition function.
Jan. 22:
(2d Ising) Onsager's solution.
Jan. 24:
(2d Ising) Property of the free energy.
The critical theory.
Jan. 29:
(2d Ising) Two point functions of the spin operators and
the disorder operators.
Cut for the fermionic systems.
(Critical 2d Ising)
Massless Majorana fermion; central charge,
identification of the energy operator.
Jan. 31:
(Critical 2d Ising)
Partition function of the GSO projected Majorana fermion.
The spin operator.
Feb. 5:
Renormalization group, CFTs as RG fixed points.
(Digression: Zamolodchikov's C-theorem)
Feb. 7:
RG equations, Critical exponents from CFT data.
Feb. 12:
Review of conformal invariance, etc;
CFT on S^2 and R^2. Ward identities on S^2.
Feb. 14:
Projective symmetry of S^2,
Projective Ward identities and
solutions.
[Digression: the moduli space of Riemann surfaces
with marked points, M_{g,s}]
Feb. 19 and 21: reading week (no classes).
Feb. 26:
coordinate/metric dependence of operators,
global form of projective Ward identity,
operator at infinity
state operator correspondence and unitarity.
OPE of primaries (preview).
Feb. 28:
OPE of primaries, structure constants for descendants in terms of
that of primaries, the matrix M,
4 point functions and conformal blocks,
crossing symmetry/associativity constraints, conformal bootstrap.
Mar. 4:
Representation theory of the Virasoro algebra:
quick review of su(2) (spin j representation, highest weight repr,
Verma module, singular vector, characters),
Verma module for Virasoro, its character,
singular vectors, Feigin-Fuchs theorem,
the hermitian form, null vectors, etc.
Mar. 6:
Representation theory of the Virasoro algebra - Unitarity constraints:
elemantary considerations, Kac's determinant formula,
Friedan-Qiu-Shenker's theorem.
Mar. 11:
Friedan-Qiu-Shenker's theorem --- statement and proof.
Mar. 13:
Inclusion relation of Verma modules. Character formula.
Modular transformation property.
No classes for March 18 and 20.
Mar. 25:
Differential equations of correlation functions from null operators.
Constraints on OPE.
Mar. 27:
Fusion rule. Minimal CFT's. Ising model, Lee-Yang edge,
tricritical Ising model and supersymmetry, 3-state Potts model
(off-diagonal modular invariant).
Apr. 1:
Four-point functions of critical Ising model. Determination
of the OPE structure constant. Determination (and check)
of the conformal blocks.
Apr. 3:
Landau-Ginzburg description of unitary minimal CFTs.
Check with the spectrum of relevant opeartors
and their fusion rules. Implication to RG flows.
Apr. 8:
Zamolodchikov's C-Theorem. RG equation for correlation functions.
The functions F, H, G.
Zamolodchikov metric.
Apr. 10:
Conformal perturbation theory. RG flows in position space and OPE.
(1,3)-perturbation of the unitary minimal CFT.
Plan:
Wess-Zumino-Witten models, affine Lie algebras, Loop groups
and representation theory.