What is Conformal field theory: Definition and 19 Discussions

A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified.
Conformal field theory has important applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points.

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  1. josephsanders

    High Energy Literature for introduction to O(N) vector model

    TL;DR Summary: Looking for literature on O(N) vector model Hello, We have been going over the O(N) vector model in my QFT class but the notes are not very detailed and we are not using a textbook. Does anyone know of a good QFT book which goes over this material? I have a copy of Shrednicki...
  2. snypehype46

    Computing OPEs of linear dilaton CFT

    I'm trying to do the following question from David Tong's problem sheets on string theory: > A theory of a free scalar field has OPE $$\partial X(z)\partial X(w) = \frac{\alpha'}{2}\frac{1}{(z-w)^2}+...$$. Consider the putative candidate for the stress energy tensor $$T(z) = \frac{1}{\alpha '}...
  3. John Greger

    A Solving Problems in "Conformal Field Theories" by Francesco

    I have been following the book called "Conformal Field Theories" by Francesco, also known as "the yellow pages". I do this for fun but, of course, sometimes it gets rather technical. Do there exist solutions to the problems in this book? I haven't found a solutions manual available. Many...
  4. J

    Massless theories can be conformally invariant

    Homework Statement The exercise needs us to first show that ##P^2## (with ##P_\mu=i\partial_\mu##) is not a Casimir invariant of the Conformal group. From this, it wants us to deduce that only massless theories could be conformally invariant. Homework Equations The Attempt at a Solution I...
  5. X

    A Quick Question about Conformal Blocks

    When a conformal block has dimensions and spin that violates its unitary bounds, does that make the block equal to zero? I'm asking because I'm trying to calculate 3D conformal blocks via a recursion relation and get blocks in the relation that violate unitary bounds. Thanks!
  6. C

    Field transformation under Conformal transformation

    I am confused about the field transformation under conformal transformation. Consider the scale transformation of field ##\phi## (not necessarily scalar) In CFT of Francesco et al, formula (2.121), the transformation is $$ \vec{x}\rightarrow \vec{x}'=\lambda x,\,\,\,\phi(\vec{x}) \rightarrow...
  7. diegzumillo

    A Conformal Window: Understanding the Basics

    I'm reading about extensions of standard model and this pops up frequently but it's not very clear. I understand it's a region in parameters space so renormalization group naturally becomes relevant and that's about it for my understanding. I can't connect any of this to the beta function of the...
  8. K

    A Primaries, descendents and transformation properties in CFT

    I want to clarify the relations between a few different sets of operators in a conformal field theory, namely primaries, descendants and operators that transform with an overall Jacobian factor under a conformal transformation. So let us consider the the following four sets of...
  9. Ace10

    Conformal Field Theory: Questions & Answers

    Hi all, my question is rather a simple one and regards conformal transformations. On "Applied CFT" by P.Ginsparg, http://arxiv.org/pdf/hep-th/9108028.pdf , on page 10, gives the transformation rule of a quasi primary field and relates the exponent of 1.12 to the one of 1.10. My first question...
  10. K

    Operator state mapping in Conformal Field Theory

    can anyone suggest any good reading material on operator state mapping in conformal field theory? I know only elementary field theory... So it might be helpful ifsomeone suggest a book where it is done in little detailed way..
  11. P

    Ginsparg Applied Conformal Field Theory

    Ginsparg "Applied Conformal Field Theory" I have some questions concerning the first two chapters of Ginsparg's text on CFT, which can be found here 1. In equation (2.1) primary fields of conformal weight (h,\overline{h}) are introduced as fields that transform the following way...
  12. I

    Conformal Field Theory: Evaluating the Vertex Operator on the Vacuum

    Homework Statement Evaluate \lim_{z \to 0}:e^{ik \cdot X(z)}:|0\rangle where X(z) is a free chiral scalar field in the complex plane. Homework Equations In Conformal Field Theory, the free chiral scalar field in the complex plane is given by: \begin{array}{rcl} X(z) &=& \frac{1}{2}q -...
  13. T

    Conformal field theory: vertex operator

    I am getting apparently conflicting statements about the conformal transformation law of the vertex operator appearing in and 2D QFT (such as in bosonic string theory). For example, according to http://en.wikipedia.org/wiki/Conformal_field_theory" (eqn 64 on page 15), the transformation law is...
  14. T

    Mathematics Ph.D Student trying to learn Conformal Field Theory

    Hi everybody, I'm not entirely sure if this should be posted here or in the Quantum Physics section, if a moderator feels it would be more suitable there, please feel free to move it. As the title indicates, I'm a mathematics Ph.D student (studying Vertex Operator Algebras) and I'm interested...
  15. O

    I need an introductory book on Conformal Field Theory

    Can anybody suggest a good introductory book on CFT. I would like to learn String Theory; and this is my basic motivation to learn CFT...
  16. N

    OPEs in conformal field theory

    When people give the rules for the operator product expansion of fields in CFT, they always give the rule for the OPE of a product of two fields. But let's say that we have three fields. To be specific, consider the OPE of T(z_1) T(z_2) \Phi(z_3) where T is the energy-momentum tensor and \Phi...
  17. W

    Discover the Basics of Conformal Field Theory: An Introduction for Beginners

    Hi there, Can anyone explain to me what Conformal Field Theory really is in brief summary? I do not mind if anyone wants to go into technical details as I have some basic knowledge of quantum field theory already. Thank you
  18. malawi_glenn

    What Are the Best Introductory Resources on Conformal Field Theory?

    Hello Anyone got some introductory notes / books on Conformal Field Theory for someone who know Quantum Field Theory at the level of Peskin's book? cheers
  19. J

    Learning Conformal Field Theory after QM

    I will be starting a Masters degree in physics next year - 30% of the assessment will be a thesis (review, not research). I have selected CFT as a topic and my future advisor has pointed me to Francesco's 'Conformal Field Theory' as the book to use. Francesco's book is very good but I am...