Renormalization group & Conformal field theory

Renormalization group & Conformal field theory (Spring '20)

Textbooks: di Francesco, Mathieu, Senechal, "Conformal field theory";   Strogatz, "Nonlinear dynamics & chaos";   Altland & Simons, "Condensed matter field theory";   Peskin & Schroeder, "Quantum field theory";  

Rough outline of content:

Lattice models of quantum field theory, scalars mainly.

Spin systems and the Ising model; General intuition for phase transitions and critical phenomena; Kadanoff scaling and block spins; 1-dim Ising model and the transfer matrix;

Block spin transformations, explicitly, in 1-dim Ising model and 2-dim Ising model on a triangular lattice and the resulting RG flow equations, fixed points.

Digression on 1st order flow equations and dynamical systems: this module is based on Strogatz's book.
2-d flows: fixed points, stability and classification of linear flows (saddle points, nodes, spirals, and relation to eigenvalues/eigenvectors) [all of ch.5];
Nonlinear flows and phase portraits: simple examples, fixed points and linearization in their neighbourhood, extrapolation to full phase portrait, validity thereof [sec.6.1,6.2,6.3];
Simple biological systems: population growth and the logistic equation [sec.2.3]; simple model for the evolution of an epidemic (ex.3.7.6); toy example of Lotka-Volterra model of competition between two species (rabbits vs sheep) and phase portrait [sec.6.4];
Limit cycles --- ruling out closed orbits and gradient flows, Liapunov functions [sec.7.2];
Discrete/iterative maps, fixed points, stability, cobweb diagrams.
Logistic map [x(n+1)=r.xn.(1-xn)], fixed points, bifurcations, period doubling, chaos [sec.10.1-10.4]. Numerical plots on Mathematica and evolution for various values of r, onset of chaotic behaviour, periodic window within etc.

Back to RG: momentum space renormalization. First a blurb on path integrals and quantum field theory, Euclideanizing and the associated statistical physics model partition function; Generalities on operators, dimensions and scaling properties, relevance/irrelevance etc.
Slow and fast modes, integrating out fast modes; scalar field with quartic interaction; blurb on Feynman diagrams.

Scalar field with quartic interaction, integrating out fast modes, RG flow equations and the Wilson Fisher fixed point (cont'd to 2nd lec).

Beta-functions in QFT; brief overview of negative beta-function in QCD; 2-loop beta-fn in QCD and the Banks-Zaks fixed point; Brief overview of supersymmetric theories and non-renormalization of the superpotential and the holomorphy arguments; 2-dim nonlinear sigma models, strings and geometry -- very brief snapshot of gauged linear sigma models, RG flows and closed string tachyon condensation.

Conformal field theory -- general story for higher dimensions. Restrictions on 2- and 3-pt correlation functions. Focus on 2-dimensions and infinite conformal symmetry.

Continuing with 2-dim CFT (online, 24/3/20!): mainly following Polchinski vol.1, chap.2: focussing on 2-dim (Euclidean) massless scalars in complex coordinates -- path integrals and operator equations, normal ordered operators, the operator product expansion organized in terms of singular pieces (descending) etc...

2d CFT: symmetries and Ward identities, conformal transformations and the stress tensor, opes of primary operators etc.

2d CFT: the contour argument and commutators of operators, the Virasoro algebra, free scalars and mode expansions, radial quantization and vertex operators, unitarity etc.

Assignments: 60%

Endsem exam, 40%.

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