Introduction to Matrix Models and Non-Critical String Theory Akikazu ...

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26One can also consider more general potential (BIPZ)e.g. V (M) = 1 2 M 2 + gM 4so that equation of motion becomes∫12 λ + 2gλ3 = dλ ′ 1)λ − λ ′ρ(λ′solved byρ(λ) = 1 π( ) 12 + 4ga2 + 2gλ2 √4a2N − λN2 , 12ga 4 + a 2 − 1 = 0
27Such generalization is interesting because the matrixaction∫dMe −TrM2 2 +gM3 3!can be represented in Feynman expansion↔ triangulation of 2D surface1/N expansion ↔ genus expansiong → g c ↔ continuum limit
Page 1 and 2: 1Introduction to Matrix Modelsand N
Page 3 and 4: 3The subject is reviewed extensivel
Page 5 and 6: 5Non-critical string theoryBosonic
Page 7 and 8: 7S L =∫dξ 2√ g( 12 gab ∂ a
Page 9 and 10: 9T (z)T (w) ∼ 1 c+ ..., c = 1 + 3
Page 11 and 12: 11These can be coupled to d free bo
Page 13 and 14: 13Other interesting example: (p, q)
Page 15 and 16: 15Gravitational dressing[e αϕ O]
Page 17 and 18: 17Under this transformation∫Qd 2
Page 19 and 20: 19Physical observables are correlat
Page 21 and 22: 21ParametrizingM = U † ΛU,dM = d
Page 23 and 24: 23Simplest Matrix Model: Gaussian e
Page 25: 25Solution:ρ(m) = 1 π√2N − m2
Page 29 and 30: 29So we arrive at an identity( ) 1
Page 31 and 32: 31• Constant c fixed by requireme
Page 33 and 34: 33With this ansatz, one hasR(z) =
Page 35 and 36: 35Now, series∑kΓ−3( gg c) 2k=
Page 37 and 38: 37All this was for pure 2D gravity
Page 39 and 40: 39Double Scaling LimitIf one also c
Page 41 and 42: 41Now, define a set of polynomialsP
Page 43 and 44: 43Need f n for large n.This can be
Page 45 and 46: 45Similarly,∫=∫e −V (P ′ n(
Page 47 and 48: 47P n+1soλ : P n (λ) ↗ ↘r n P
Page 49 and 50: 49For generic W (r),gξ = W (r) = g
Page 51 and 52: 51Now, if one is interested in high
Page 53 and 54: 53Summary of Orthogonal Polynomials
Page 55 and 56: 55Along similar lines, define∂∂
Page 57 and 58: 57(2,3) model:1 = [P, Q] =( 34 u2 +
Page 59 and 60: 59Key identity: Itzykson-Zuber inte
Page 61 and 62: 61P = (Q p/q ) + , [P, Q] = 1define
Page 63 and 64: 63∂∂t nQ = [(Q n/q ) + , Q]α n
Page 65 and 66: 65So we haveDouble Scaled Matrix Mo
Page 67 and 68: 67In double scaling limit1M ΦM = 1
Page 69 and 70: 69What does the resolvent measure?1
Page 71 and 72: 71In this limit, one obtainsT 2 (y)
Page 73 and 74: 73CFT side: one has the Liouville B
Page 75 and 76: 75FZZT expectation value probe targ
Page 77 and 78:
77Non-perturbativelyW (x) = log(Φ
Page 79 and 80:
79〈e W (x) 〉 ==∫∫= 12πg= 1
Page 81 and 82:
81Recursion relationλP n (λ) =
Page 83 and 84:
83Multi-FZZT amplitude generalizes
Page 85 and 86:
85• Airy function: Non-perturbati
Page 87 and 88:
87Airy integral has three saddle po
Page 89 and 90:
89• Perturbatively, many possible
Page 91 and 92:
91Open SFTS =∫Ψ ∗ QΨ + Ψ ∗
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