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

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10pick branchγ = 1 √12( √ 25 − d − √ 1 − d) = Q 2 − 1 2√Q2 − 8so that in d → −∞ limit, γ → 0, so thatQ = 2/γ − γ → 2/γ, giving rise to “classical” Liouvilletheory∫S = d 2 ξ √ ĝ(ĝ ab ∂ a ϕ∂ b ϕ + Q ˆRϕ + µ )γ 2eγϕwhich is Weyl invariant under ĝ → e 2ρ ĝ, ϕ → ϕ − 2 γ ρ
11These can be coupled to d free bosons (or CFT withcentral charge c) so thatc matter + c liouv − 26 = 0c ≤ 1 and c ≥ 25 picked out because γ is real. At c = 25the signature of ϕ coordinate flips. We will focus onc ≤ 1 for which ϕ coordinate is Eucledean.• c < 1: 1 linear dilaton dimension• c = 1: 1 linear dilaton dimension, one flat direction
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: 9T (z)T (w) ∼ 1 c+ ..., c = 1 + 3
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 and 26: 25Solution:ρ(m) = 1 π√2N − m2
Page 27 and 28: 27Such generalization is interestin
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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Introduction to Matrix Models and Non-Critical String Theory Akikazu ...

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