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

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86Stoke’s phenomenonAi(˜x) = 12π∫d˜s e −i (˜s 3 3 +˜s˜x )solution of ( ) ∂2∂z − z f(z) = 02Two solutions: Ai(z) and Bi(z). Different s contour:different linear combination of homogeneous solution.Pick the solution which gives rise to Ai(z) (decay forpositive real z)
87Airy integral has three saddle pointsSteepest descent contour hits only one of the saddles0pi/12 7pi/12 9pi/12 11pi/12 12pi/12231402314021 1100223 443304But as z is taken off axis, different saddle points appearand disappear (along the steepest descent contour)The locus on parameter space where contributing saddlesre-arrange themselves is called “stoke’s line”
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1Introduction to Matrix Modelsand N
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3The subject is reviewed extensivel
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5Non-critical string theoryBosonic
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7S L =∫dξ 2√ g( 12 gab ∂ a
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9T (z)T (w) ∼ 1 c+ ..., c = 1 + 3
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11These can be coupled to d free bo
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13Other interesting example: (p, q)
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15Gravitational dressing[e αϕ O]
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17Under this transformation∫Qd 2
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19Physical observables are correlat
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21ParametrizingM = U † ΛU,dM = d
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23Simplest Matrix Model: Gaussian e
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25Solution:ρ(m) = 1 π√2N − m2
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27Such generalization is interestin
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29So we arrive at an identity( ) 1
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31• Constant c fixed by requireme
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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: 85• Airy function: Non-perturbati
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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