Very brief introduction to Conformal Field Theory

1. Very brief introduction to Conformal Field Theory Germán Sierra Instituto de Física Teórica CSIC-UAM, Madrid Talk at the 4Th GIQ Mini-workshop February 2011

2. CFT and all that • String theory • Critical phenomena in 2D Statistical Mechanics • Low D-strongly correlated systems in Condensed Matter • Fractional quantum Hall effect • Quantum information and entanglement

3. Feymann diagrams s-channel t-channel u-channel Mandelstam variables Scattering amplitude

4. Birth of String Theory- Veneziano 1968 Regge trayectory s-t duality

5. String action where D= space-time dimension

6. is a 1+1 field that satisfies the equations of motion Open Closed

7. Quantization String=zero modes (x,p)+infinite number of harmonic oscillators Vertex operators: insertions of particles on the world-sheet (Fubini and Veneziano 1970)

8. The energy-momentum tensor Generator of motions on the string world-sheet T is a symmetric, conserved and traceless tensor For closed string T splits into left and right components In light cone variables

9. Virasoro operators Make the Wick rotation Fourier expansion of the energy momentum tensor Where are called the Virasoro operators

10. Virasoro algebra The Virasoro operators satisfy the algebra where c = central charge of the Virasoro algebra Classical version of the Virasoro algebra This contains the conformal transformations of the plane: translations dilatations special conformal

11. In 2D the conformal group is infinite dimensional !! Classical generators of conformal transformations Quantum generators of conformal transformations “c” represents an anomaly of conformal transformations Physical meaning of “c” Bosonic string: X-fields + Faddev-Popov ghost c = D - 26 Superstring: X-fields + fermionic fields + Faddev Popov ghost c = D + D/2 - 26 + 11 = 3D/2 -15 String theory does not have a conformal anomaly!! c = 0 -> D = 26 (bosonic string) and 10 (superstring)

12. c gives a measure of the total degrees of freedom in CFT c= 1 (boson) c= 1/2 (Majorana fermion/Ising model) c= 1 (Dirac fermion/1D fermion) c= 3/2 (boson+Majorana or 3 Majoranas) c=…. Fractional values of c reflect highly non perturbative effects

13. The Belavin-Polyakov-Zamolodchikov (1984) Infinite conformal symmetry in two-dimensional quantum field theory

14. Conformal transformations Covariant tensors are characterized by two numbers Conformal weights

15. Dilation

16. General framework of CFT • T is a symmetric, conserved and traceless tensor • with central charges (no need of an action) • There is a vacuum state |0> which satisfies -There is an infinite number of conformal fields in one-to-one correspondence with the states -There are special fields (and states) called primary satisfying

18. -The remaing fields form towers obtained from the primary fields acting with the Virasoro operators (they are called descendants) Verma module: -The primary fields form a close operator product expansion algebra For chiral (holomorphic fields) OPE constants

20. - Fusion rules (generalized Clebsch-Gordan decomposition) - Rational Conformal Field Theories (RCFT): finite nº primary fields - Minimal models A well known case is the Ising model c=1/2 (m=3)

21. - Conformal invariance determines uniquely the 2 and 3-point correlators normalization - Higher order chiral correlators: their number given by the fusion rules

22. Conformal blocks for the Ising model Fusion rules There are four conformal blocks: The non-chiral correlators (the ones in Stat Mech) Must be invariant under Braiding of coordinates

23. Conformal blocks give a representation of the Braid group Yang-Baxter equation Related to polynomials for knots and links, Chern-Simon theory, Anyons, Topological Quantum Computation, etc

24. Characters and modular invariance Conformal tower of a primary field : number of states at level n=0,1,2,… Upper half of the complex plane Moduli parameter of the torus states propagation

25. Modular group Fundamental region Generators Characters transforms under modular transformations as Partition function of CFT must be modular invariant

26. Verlinde formula (1988) Fusion matrices and S-matrix and related!! Example: Ising model Check

27. Axiomatic of CFT Moore and Seiberg (1988-89) • Algebra: Chiral antichiral Virasoro left right ( c ) + others • Representation: primary fields • Fusion rules: • B and F matrices : BBB =BBB (Yang-Baxter) FF = FFF (pentagonal) • Modular matrices T and S Sort of generalization of group theory-> Quantum Groups

28. Wess-Zumino-Witten model (1971-1984) CFT with “colour” Field is an element of a Group manifold Conformal invariance-> Currents

29. OPE of currents Kac-Moody algebra (1967) k= level (entero) Sugawara construction (1967) g: dual Coxeter number of G

30. Primary fields and fusion rules (Gepner-Witten 1986) G=SU(2) Knizhnik- Zamolodchikov equations (1984)

31. Applications of CFT

32. Spin chains Heisenberg-Bethe spin 1/2 chain Low energy physics is described by the WZW SU(2)@k=1 But the spin 1 chain is not a CFT (Haldane 1983) -> Haldane phase and gap

33. Finite size formulas

34. FQHE/CFT correspondence Laughlin wave function quasihole -> Basis for Topological Quantum Computation (braids -> gates) electron =

35. The entanglement entropy in a bipartition A U B scales as (1D area law) In a critical system described by a CFT (periodic BCs) hence one needs very large matrices to describe critical systems Another alternative is to choose infinite dimensional matrices:

36. MPS state iMPS state physical degrees auxiliary space (string like)

37. Example 5: level k=2, spins =1/2 and 1, D=2 SU(2)@2 = Boson + Ising c=3/2 = 1 + 1/2 spin j=1 field spin j=1/2 field The chiral correlators can be obtained from those of the Ising model (general formula Ardonne-Sierra 2010) N spins 1 The Pfaffian comes from the correlator of Majorana fields

38. Similar chiral correlators have been considered in the Fractional Quantum Hall effect at filling fraction 5/2. This is the so called Pfaffian state due to Moore and Read. FQHE/CFT correspondence electron = quasihole -> Quasiholes are non abelian anyons because their wave functions (chiral correlators) mix under braiding of their positions. Basis for Topological Quantum Computation (braids -> gates)

39. An analogy via CFT FQHE CFT Spin Models Electron Majorana spin 1 Quasihole field spin 1/2 Braid of Monodromy Adiabatic quasiholes of correlators change of H Then if one could get Topological Quantum Computation in the FQHE and the Spin Models. Holonomy = Monodromy

41. Bibliography Applied Conformal Field Theory Paul Ginsparg, arXiv:hep-th/9108028 Non-Abelian Anyons and Topological Quantum Computation C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. Das Sarma, arXiv:0707.1889

42. Thank You