Conformal Field Theory

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Autumn '08

10/3 - Conformal symmetry and correlation functions
  • Jon Walsh
  • Abstract - I'll be talking about conformal symmetry and the consequences on correlation functions.
  • Refs - A good reference is the Conformal Field Theory book by Di Francesco, Mathieu, and Senechal, Chapter 4 and section 5.1. Also, Ginsparg, hep-th/9108028, up to section 2.2 gives a good overview.
10/10 - Virasoro algebra and representations
  • Chris Vermilion
  • Abstract - A lot of formalism leading up to the Virasoro algebra: Radial quantization, Operator Product Expansions, etc. OPE's for more than two operators were NOT covered.
  • Refs - The talk is pulled mostly from Ginsparg (supra). Other good references include Di Francesco et al. and Polchinski's String Theory, v.1. OPE's are also covered in Peskin and Schroeder, Ch. 18.
10/17 - Minimal models c < 1
  • Kristan Jensen
  • Abstract - We will develop the machinery to study minimal models, that is a class of solvable 2d conformal field theories. we will begin by classifying the unitary representations of the virasoro algebra and then develop a set of self-consistent conditions for the correlation functions of primary fields. these conditions manifest themselves as the 'bootstrap equations,' which are completely solvable in the case of a minimal model.
  • Refs -
    • Polchinski, chapters 2 & 15
    • Belavin, Polyakov, and Zamolodchikov, Nucl. Phys. B241 (333-380)
    • Difrancesco, et al., CFT
10/24 - Minimal Models II
  • Andrew Lytle
  • Abstract - I will finish up our investigation of 2d CFTs by discussing minimal models, a special class of theories composed of a finite number of primaries. I will show how this property comes about by considering reducible representations of the Virasoro algebra, and identify the simplest minimal model with the critical Ising model.
  • Refs -
    • Belavin, Polyakov, and Zamolodchikov, Nucl. Phys. B241 (333-380)
    • Difrancesco, et al., CFT
10/31 - Wilson-Fisher fixed point and epsilon expansion
  • Steve Paik
  • Abstract - Conformal field theories arise at the fixed points of the renormalization group equations. We show how the renormalization group constrains the form of beta functions for the parameters of a theory near a given fixed point. If this fixed point is infrared unstable along a certain direction and flows to an infrared stable fixed point, then we may use the scaling dimensions and OPE coefficients at the first fixed point to predict the scaling dimensions at the second fixed point. This requires that the two fixed points be close together. We illustrate these concepts in the standard example of the Ising model near 4 dimensions.
  • Refs -
    • Callan, Coleman, Jackiw, A New Improved Energy-Momentum Tensor, Ann. Phys. 59, 42 (1970). Towards the end of the paper you can find their discussion of conformal symmetry arising from scale and Poincare symmetry.
    • Wilson, Fisher, Critical Exponents in 3.99 Dimensions, PRL 28, 240 (1972). In this letter they introduce the epsilon expansion and present their results. It's quite dense.
    • Cardy, Scaling and Renormalization in Statistical Physics, Cambridge Univ. Press (1996). A very good book which quickly gets to the renormalization group derivation of critical exponents. He uses lots of heuristic and physical arguments. Can be challenging to fully digest. Read multiple times.
11/7 - Banks-Zaks fixed point
  • Jong-Wan Lee
  • Abstract - In this talk I will discuss Banks-Zaks fixed point and the simple phase diagrams of the vector-like SU(N) gauge theories with massless fermions in N_f - g plane.
  • Refs -
    • T.Banks, A, Zaks, "on the phase structure of vector like gauge theories with massless fermions", Nucl.Phys.B 196 (1982) 189 (Link is to an 8 MB pdf; don't be alarmed if it's slow.)
    • T.Appelquist et al, "Zero temperature chiral phase transition", PRL 77, 1214 (1996)
    • V.A.Miransky, Koichi Kamawaki, "Conformal phase transition, beta function, and Infrared dynamics in QCD ", hep-ph/0003137 (2000)
11/14 - Unitary constraints from superconformal symmetry
  • Alan Jamison
  • Abstract - Conformal symmetry places strong constraints on the form of correlation functions. It also constrains the possible values of operator dimensions, depending upon how they transform under Lorentz transformations. We'll derive these constraints, making use of a cute trick from intro quantum mechanics. Then, we'll add in supersymmetry and see how the bounds on operator dimensions tighten for fields transforming in different representations of the R symmetry group.
  • Refs -
    • Minwalla, Shiraz "Restrictions Imposed by Superconformal Invariance on Quantum Field Theories", hep-th/9712074 (1997)
11/21 - Nonrelativistic CFT
  • Ethan Thompson
  • Abstract - We will investigate the extension of the Galilean algebra by scale and conformal generators and investigate how much of the machinery we have developed for the relativistic case can be applied to the non-relativistic case. We will define primary operators and find the constraints put on their two-point functions by the algebra. We will develop a mapping between the primary operators and states in a harmonic potential. And we will look to what extent scale invariance implies conformal invariance. Time permitting, we will look at example systems.
  • Refs -
    • C.R. Hagen, "Scale and Conformal Transformations in Galilean-Covariant Field Theory", PRD 5 No. 2, 1972
    • Mehen, Stewart and Wise, "Conformal invariance for non-relativistic field theory", Phys. Lett. B 474, 145 (2000)
    • Nishida and Son, "Non-relativistic conformal field theories", PRD 76, 086004, 2007
11/28 - Thanksgiving
12/5 - Unparticles
  • Chris Spitzer
  • Abstract - An unintroduction to unparticles.
  • Refs - Two unpapers by Georgi:
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