Conformal Field Theory

This webpage is dedicated to curating papers on (the physical & mathematical aspect of) conformal field theory (CFT). For Lie algebra and its representation, visit here. I will only add the papers that are relevant to me.

$\quad$ In general, typical quantum field theories are arduous to analyze unless they are free. One of the prominent examples is quantum chromodynamics (QCD), where this theory becomes asymptotically free, thus allowing us to do some perturbative calculations. It is known as asymptotic freedom, discovered in 1973 by the 2004 Nobel physics laureates: Gross and Wilczek , and independently by Politzer in the same year. It is interesting to look for an exactly solvable quantum field theory (QFT) that would be non-trivial. It would be useful to find a direct physical application. What do we mean by exactly solvable? Here are three natural answers:
$\quad$ 1. What is the (energy) spectrum of the theory? In QCD, this would be the masses of all mesons, baryons, and gluons.
$\quad$ 2. Can I exactly compute the correlation functions?
$\quad$ 3. Can I find an exact scattering amplitude (or the S-Matrix)?

A particular arena of exactly solvable theories appears in two dimensions. Why do two dimensions require special attention? One key point here is the local conformal invariance which means the transformation preserves angles and has no mass scale because of dilation (scale) invariance. For example :
$\quad$ 1. Conformal transformations in dimension $>2$ form a finite-dimensional algebra.
$\quad$ 2. Conformal transformations in dimension $=2$ form an infinite-dimensional algebra, and is known as the Witt algebra. On the quantum level, it’s called the Virasoro algebra.

The QFT which deals with the invariance of such transformations is known as the conformal field theory (CFT). The two-dimensional CFT garnered much attention after the 1984 seminal work of Belavin, Polyakov, and Zamolodchikov (BPZ) . Since CFT is a massless theory, we look forward to answering mainly the first two questions above. Why can some QFTs be solved? Sometimes the theory is too simple. For example, Yang-Mills in two dimensions, and two-dimensional pure quantum gravity due to no local degree of freedom (exception) . Or, it can have an exceptionally large number of symmetries. For example:
$\quad$ 1. Virasoro algebra, due to infinite-dimensional symmetry algebra (synonymously known as CFT)
$\quad$ 2. Higher spin-conserved charges
$\quad$ 3. Supersymmetry.

See the Influence Map based on the 1984 BPZ paper:
Read the most recent papers on this topic in arXiv.

Open Knowledge Map for CFT Research :)


… yet to be updated.

  • 2014: Bosonic ghost system of central charge 2 was addressed by Ridout and Wood
  • 2008: Grumiller and Johansson suggested that the conformal field theories dual to certain topological gravity theories on $AdS_3$ are logarithmic.
  • 1998: Guruswamy and Ludwig realized the $c=-1$ bosonic ghost systems (also known as $\beta\gamma$ systems; a logarithimic CFT) exhibits an $\hat{\mathfrak{sl}}(2)_{-1/2}$ symmetry.
  • 1998: Verlinde showed fusion coefficient is related to modular S-matrices of character which is now known as Verlinde’s formula. It connects the local and global properties of CFT.
  • 1996: Gaberdiel and Kausch applied Nahm algorithm to explicitly construct (chiral) representations upon which the energy operator acts non-diagonalisably
  • 1994: Nahm introduced a method for computing the fusion product of representations
  • 1993: Link between the non-diagonalisability of the energy operator (i.e. Virasoro zero-mode $L_0$) and logarithmic singularities in correlators by Gurarie. He first coined “Logarithmic CFT“
  • 1992: Rozensky and Saleur noted in the study of the $U(1|1)$ Wess-Zumino-Witten model that some correlation functions will possess logarithmic branch-cuts and reducible representation
  • 1990: The importance of topological excitation was shown in 2D quantum gravity (or 2D CFT) by Witten 1988.
  • 1987: Knizhnik noted that the correlation function can have logarithmic singularities.
  • 1986: First introduced the concept of ghost system by Friedan, Martinee, and Shenker
  • 1984: Formally introduced by Belavin, Polyakov and Zamolodchikov
  • 1974: Proposed Conformal bootstrap program by Polyakov and to a large extend realized by Belavin, Polyakov and Zamolodchikovn in 1984.
  • 1970: Polyakov demonstrated that the conformal invariance arises at the critical points
  • 1969: The short-distance expansion of the product of fields known as the Operator product expansion (OPE) was originally proposed in the context of standard quantum field theory (QFT) by Wilson . It’s a useful tool in CFT.

Formalism and Reviews


Connection to Maths

Ideas from Standard QFT

Noting “Yellow Book”

▛ YB Erratas: first printing and second printing

  • Virasoro modes $L_n$ have $L^{\dagger}_n = L_{-n}$ which is true in most CFTs. For eg. the free boson. YB stated on page 202 to be a general result. $\mid$ Note

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Published on May 22, 2023

Last revised on May 27, 2024