# 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.*

▛ **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 onJul 22, 2024

References