Conformal Field Theory (w/ string theory and criticality) Oct 26, 2009 @ MIT
CFT s application Points of view from RG and QFT in d-dimensions in 2-dimensions N point func in d-dim OPE, stress tensor and central charge
CFT s application Points of view from RG and QFT Application of CFT 1. critical phenomena and statistical mechanics 2. math interests, connecting diverse branches 3. ground states of string theory
CFT s application Points of view from RG and QFT 1. In RG, R b describes evolution operation on the Hamiltonian of the coupling parameter space. A fixed point S =R b S. Its correlation lengths ξ = 0 or Scale Invariance ξ = describes a critical point(t = T c ). ξ = effectively implies lattice constant a 0, easily impose Translational Invariance and Rotational Invariance to the lattice system, which now enjoys Conformal Invariance 2. In QFT, consider massive QFT lattice model mass scale m 1/ξ, for ξ m 0 massless QFT
CFT s application Points of view from RG and QFT Scale Covariant φ 1 (br 1 )φ 2 (br 2 )... φ n(br n) bd = b P j x j φ 1 (r 1 )φ 2 (r 2 )... φ n(r n) D. (1) with a suggestive form φ j (br) = b x j φ j (r), (2) Conformal Covariant (more tempting) φ 1 (r 1 )φ 2(r 2 )... φn(r n) D = ny b(r j ) x j φ 1 (r 1 )φ 2 (r 2 )... φ n(r n) D, (3) j=1 where b(r) = r / r is the local jacobian of the transformation r r For what transformations do conformal covariant form hold?
in d-dimensions in 2-dimensions In d-dims Conformal covariant holds if the transformation locally looks like scale transformation. i.e. possibly dilation(scaling), rotation, translation(all locally preserve angle), but no shear. Generally, metric satisfies g µν(r )=Ω(r)g µν (r) set of conformal transformations forms the conformal group. 1. translation r r = r + a 2. rotation r r = Λr 3. dilation r r = br 4. special conformal transformation r r = r 2 r + a 2
in d-dimensions in 2-dimensions In d-dims Alternatively, consider r r = r + α(r) anti-symmetric part ν α µ µ α ν rotation diagonal part λ α λ g µν dilation traceless symmetric part ν α µ + µ α ν (2/d) λ α λ g µν = 0 shear = 0 d > 2, this equation strongly restricts on the standard 4 transformations exhaust all possibilities. d = 1, the equation is trivial. Only translation and dilation. d = 2?
Example in 2-dim Outline in d-dimensions in 2-dimensions
In 2-dims Outline in d-dimensions in 2-dimensions Easier in complex coordinates z r 1 + ir 2, z r 1 ir 2 hence ds 2 = dzd z α z / z = α z / z = 0 so α z is holomorphic, and α z is antiholomorphic. loop algebra L n = z (n+1) z, L n = z (n+1) z [L m, L n ] = (m n)l m+n [ L m, L n ] = (m n) L m+n [L m, L n ] = 0 Indeed, L 1, L 0, L 1 are sufficient to generates the earlier 4 transformation. Möbius transformations or Projective conformal transformations. ( z z = ) f (z) by f (z) = (az + b)/(cz + d) with ad bc = 1 a b c d forms SL(2, C)/Z 2 (later)in general, consider TT term, gives a central extension, 1 12 cn(n2 1)δ n, m for [L m, L n ] and [ L m, L n ]. This leads to Virasoro algebra.
N point func in d-dim OPE, stress tensor and central charge N point correlation func in d-dim 2-point func φ i (r 1 )φ j (r 2 ) = δ ij r 1 r 2 2x j 3-point func φ 1 (r 1 )φ 2 (r 2 )φ 3 (r 3 ) = c 123 r 1 r 2 x 1 +x 2 x 3 r 2 r 3 x 2 +x 3 x 1 r 3 r 1 x 3 +x 1 x 2 4-point func there is 1 arbitrary function undetermined degree of freedom(dof). N-point func there is N-3 arbitrary function undetermined DOF. (think of fixing 3 DOF by projective transformation mapping 3 points to 0, 1, )
N point func in d-dim OPE, stress tensor and central charge Operator Product Expansion φ i (r i )φ j (r j )... = k C ijk(r i r j ) φ k ((r i + r j )/2)... for massless QFT C ijk (r j r k ) = c ijk r i r j x i +x j x k useful for computing correlation func with r i r j, can extract info of stress tensor(energy momentum tensor), such as Conformal Wald Identity T (z)φ 1 (z 1, z 1 )φ 2 (z 2, z 2 )... = X j T (z) T (z 1 ) = j (z z j ) 2 + 1 «zj φ 1 (z 1, z 1 )φ 2 (z 2, z 2 ).... z z j c/2 (z z 1 ) 4 + 2 (z z 1 ) 2 T (z 1) + 1 z z 1 z1 T (z 1 ). (5) c is known as central charge, conformal anomaly number. (4)
N point func in d-dim OPE, stress tensor and central charge coformal modular invariant system with c=1 classification of 2-dim critical phenomena
1. Know the aspects of CFT from RG(critical fixed pt ξ = ), from QFT(m = 0), and its application on stat mech and criticality. 2. Conformal transformation of metric, covariant of correlators and conformal invariance of Hamiltonian. 3. How CFT restricts the form of N-pt correlators. 4. Learn how 2-dim CFT is special and useful for studying 2-dim criticality.. 5. OPE, stress tensor. conformal Wald identity and central charge. THANK YOU FOR YOUR ATTENTION!!
Ref: 1. M. Henkel, Conformal invariance and critical phenomena 2. Polchinski Chap 2, Chap 15. 3. E. Brezin and J. Zinn-Justin, Fields, strings and critical phenomena course 1 - P. Ginsparg, Applied conformal field theory arxiv:hep-th/9108028 course 2 - J. Cardy, Conformal invariance and statistical mechanics arxiv:0807.3472 4. J. Cardy, Scaling and renormalization in statistical physics