# 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 :)

**Progress**

… 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.

**Book Recommendations:**

▛ 1997 - Conformal Field Theory - Francesco, Mathieu, Sénéchal $\mid$ so-called the “Yellow Book (YB)”

▟ 2008 - A mathematical introduction to conformal field theory - Schottenloher

▛ 2009 - Introduction to Conformal Field Theory - Blumenhagen, Plauschinn

▛ 2013 - Conformal Invariance and Critical Phenomena - Henkel

**People**

Boris Feigin (M), Dmitry Fuchs (M), Israel Gelfand (M), Edward Frenkel (MP), Edward Witten (MP)$^\dagger$, Miguel Ángel Virasoro (P), David Ridout (MP), Justine Fasquel (M), Zachary Fehily (M), Christopher Raymond (MP), Leszek Hadasz (P/MP), Paulina Suchanek (P/MP), … yet to be updated.

**Note:** P = Physicist, M = Mathematician, MP = Mathematical Physicist

## Formalism and Reviews

- 1998 - Applied Conformal Field Theory - Ginsparg $\mid$ Lecture Note $\mid$ Good introduction to the subject $\mid$ Paul Ginsparg developed the arXiv.org e-print archive.
- 1995 - Conformal Field Theory - Schellekens
- 2009 - Introducing Conformal Field Theory - Tong $\mid$ Part of “Lectures on String Theory”

## 2D CFT

- 1992 - Meromorphic $c=24$ Conformal Field Theories - Schellekens
- 1986 - Operator content of two-dimensional conformally invariant theories - Cardy $\mid$ First pointed out the mathematical implications of modular invariance for CFTs $\mid$ Modular invariance of the partition function (character of reps.) poses constraints on the operator content. These constraints can be useful for the classification of CFTs.
- 1984 - Infinite conformal symmetry in two-dimensional quantum field theory - Belavin, Polyakov, Zamolodchikov $\mid$ CFT became popular in physics with their seminal paper $\mid$ Laid the mathematical foundations of axiomatic CFT $\mid$ Showed the physical significance in statistical physics.
- 1970 - Subsidiary Conditions and Ghosts in Dual-Resonance Models - Virasoro

## Connection to Maths

- 1992 - Monstrous moonshine and monstrous Lie superalgebras - Borcherds $\mid$ Proved the moonshine conjecture $\mid$ He was awarded the 1998 Fields medal for this work.
- 1986 - Vertex algebras, Kac-Moody algebras, and the Monster - Borcherds $\mid$ Introduced vertex operator algebras $\mid$ Motivated by the construction of an infinite-dimensional Lie algebra due to Igor Frenkel $\mid$ This algebraic structure that plays an important role in 2D CFT and string theory. $\mid$ Wiki
- 1979 - Monstrous Moonshine - Conway, Norton $\mid$ Introduced Monstrous Moonshine (finite) group $\mid$ Conjectured the bridge between finite groups and modular forms;
*the (monstrous) moonshine conjecture*$\mid$ Wiki $\mid$ Videos from 3Blue1Brown , Numberphile

## Ideas from Standard QFT

- 1969 - Non-Lagrangian Models of Current Algebra - Wilson $\mid$ Introduced operator product expansion (OPE) $\mid$ Wiki

## 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

Permalink athttps://www.physicslog.com/physics-notes/cft-papers

Published onMay 22, 2023

Last revised onMay 27, 2024

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