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

1Introduction to Matrix Modelsand Non-Critical String TheoryAkikazu HashimotoTaipei, July 2005

2These lectures are meant to provide pedagogicalintroduction to continuum and matrix model formulationof non-critical string theoryThey typically describe strings in 1+0 or 1+1 dimensionswith a linear dilaton. They are important because:• Exactly Solvable• Context to explore non-perturbative issues(open/closed duality)• Appears to describe a sector of critical string theory• Have many connections to topics in mathematics

3The subject is reviewed extensively in the literature• Brezin, Wadia (Collection of Reprints)• Di Francesco, Ginsparg, Zinn-Justin• Dijkgraaf• Ginsparg and Moore• Klebanov• Morozov• Seiberg, ShihWarning: “These theories are exactly solvable. Thatdoes not necessarily mean we understand them.”

4Plan1) Review of Liouville Theory and Matrix Model2) Methods of Orthogonal Polynomials and KdVheirarchy3) Branes and open/closed string duality

5Non-critical string theoryBosonic string in d dimensions∫ Dg DXZ =1 ∫Vol(diff) e− 4πα ′ d 2 ξ √ gg µν ∂ µ X i ∂ ν X i , i = 0...d − 1Measure:∫∫∫Dge −1 2 |δg2| = 1, |δg 2 | = d 2 ξ √ g(g ac g bd − 2g ab g cd )δg ab δg cd∫DXe −|δX2| = 1, |δX 2 | = d 2 ξ √ gδX i δX iAction and measure are diffeomorphism invariant

6Gauge fix by chosing conformal gaugeg = e ϕ ĝ(τ)by introducing ghosts∫DbD¯bDcD¯c e −S ghS gh =∫d 2 ξ √ gb zz ∇¯z c z −b¯z¯z ∇ z c¯zUnder g → e σ g, the action is invariant, but the measuretransformsD g XD g (gh) → D e σ gXD e σ g(gh) = e d−2624πα ′S L(σ,g) D g XD g (gh)

7S L =∫dξ 2√ g( 12 gab ∂ a σ∂ b σ + Rσ + µe σ )This is the conformal anomaly which is cancelled bychosing d = 26Alternatively, make ϕ dynamical∫Dϕe − 1 ∫4πα ′ d 2 ξ √ ĝ( 25−d12 ĝab ∂ a ϕ∂ b ϕ+ 25−d6Then δĝ = ɛ(ξ)ĝ, δϕ = −ɛ(ξ) is a symmetryˆRϕ)

Rescale√25−d12 ϕ → ϕ∫Dϕe − 14πα ′ ∫d 2 ξ √ ĝ(ĝ ab ∂ a ϕ∂ b ϕ+Q ˆRϕ)8whereThe stress tensorQ =√25 − d3T = − 1 2 ∂ϕ∂ϕ + Q 2 ∂2 ϕ

9T (z)T (w) ∼ 1 c+ ..., c = 1 + 3Q22(z − w)4Finally, the µe γϕ term adjusted so that the operator hasdimension[e γϕ ] = − 1 γ(γ − Q) = 12So the bottom line∫Dϕe − 1 ∫4πα ′ d 2 ξ √ ĝ(ĝ ab ∂ a ϕ∂ b ϕ+Q ˆRϕ+µe γϕ )γ = 1 √12( √ 25 − d ∓ √ 1 − d) = Q 2 ± 1 2√Q2 − 8

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

¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡12linear dilaton: c.f. NS5-braneX¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡φµe γϕ marginal tachyon condensateno X for c < 1, c = 1 is a barrier.

13Other interesting example: (p, q) Minimal CFT⎡⎤O 11 · · · O 1(p−1)O 21 O 2(p−1)6(p − q)2⎢⎥ c = 1 −⎣ . . ⎦pqO (q−1)1 · · · O (q−1)(p−1)• (p, q) = (2, 3): c = 0 pure gravity• (p, q) = (3, 4): c = 1/2 Ising Model• (p, q) = (2, 5): c = −22/5 Yang-Lee• Other (p, q): critical limits of various statistical models

14[O r,s ] = (pr − qs)2 − (p − q) 24pq• (3,2): ( 0 0 )• (4,3):( 1 12 1600 1 116 2)• (5,2): ( 0 − 1 5− 1 50 )

15Gravitational dressing[e αϕ O] = 1[O] = ∆ 0 , [e αϕ ] = − 1 √25 − d2 α(α − Q), Q = 3soα = √ 1 [ √25 √ ]− d − 1 − d + 24∆012α < Q 2 ,Seiberg Bound

16Some simple observables: String Susceptibility ΓLetZ(A) =∫DϕDX e −S δwhich for large A scales as(∫Z(A) ∼ A (Γ−2)χ/2−1String suceptibility is this scaling exponent.d 2 ξ √ )ĝe γϕ − ASimple scaling argument: Dϕ invariant underϕ → ϕ + ρ/γ

17Under this transformation∫Qd 2 ξ √ ĝ8πˆRϕ → Q ∫8πsoZ[A] ==∫∫DϕDXe −S e Qρχ2γ δd 2 ξ √ ĝ ˆRϕ + Q 8π(∫DϕDXe −S e Qρχ2γ −ρ δργ d2 ξ √ ĝ ˆRd 2 ξ √ )ĝe γϕ+ρ − A(∫d 2 ξ √ ĝe γϕ − e −ρ A= e Qρχ2γ −ρ Z[e −ρ A] = A − 2γ−1Z[1], Qχand soΓ = 2 − Q γ = 1 [(d − 1) − √ ](d − 25)(d − 1)12)

18Along similar lines, one can define scaling dimension ∆ r,s1Z[A]∫DϕDXe −S δ(∫d 2 ξ √ ) ∫ĝe γϕ − Ad 2 ξ √ ĝe α r,sϕ O r,sFor ∆ 0 = (pr−qs)2 −(p−q) 24pqand c = 1 − 6(p−q)2pq,αγ = 1 − ∆ rs =p − q − |pr − qs|2q

19Physical observables are correlation functions of BRSTcohomologyFor L ⊗ M(p, q),H = {O n e α nϕ },O n : matter, ghosts, Liouvilleα nγ = p + q − n , n > 1, ≠ 0 mod q2qBRST cohomology has infinite element even thoughnumber of primary matter fields were finite.

20Matrix ModelTheory of random matrces: (applications in disorderedsystems and quantum chaos)∫−TrV (M)Z = dMeM is hermitan N × N matrix. MeasureDM = ∏ ∏dM ii dM ij dMij∗i i

21ParametrizingM = U † ΛU,dM = dU ∏ idλ i ∆(λ) 2whereVandermonde : ∆(λ) = ∏ i

22One way to see this. Let dU = idT U. Line element:TrdM 2= ∑ idM 2 ii + ∑ i

23Simplest Matrix Model: Gaussian ensumble∫∫∏Z = dMe −TrM 2 = dm i (m i − m j ) 2 e − ∑ i m2 i=∫i

24Large N limit described in terms of the eigenvaluedensity ρ(m).∫∫S = − dm m 2 ρ(m)+ dm dm ′ log(m−m ′ )ρ(m)ρ(m ′ )Equation of motionδS = −m 2 + 2∫dm ′ log(m − m ′ )ρ(m ′ ) = 0and differentiating with respect to m,∫m = dm ′ 1)m − m ′ρ(m′

25Solution:ρ(m) = 1 π√2N − m2Wigner semi-circule distributionρm

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

28Resolvent: useful computational tool∫Z =−NV (M)dMeChange integration variable M → M + 1M−ztransformationδdMNV (M)δe == −Tr( 1)M − z− NV ′ (M)M − zTr( 1)M − zUnder thisdM

29So we arrive at an identity( ) 1〈Tr TrM − z( 1M − z)〉 + 〈 NV ′ (M)M − z 〉 = 0In the large N limit, one can factorize( ) ( )11〈Tr 〉〈Tr 〉 + 〈 NV ′ (M)M − z M − z M − z 〉 = 0Schwinger-Dyson equation. A little rewriting:( ) ( )11〈Tr 〉〈Tr 〉 + NV ′ 1′(z)〈M − z M − zM − z 〉 = −〈NV (M) − NV ′ (z)〉M − z= N 2 f(z), Polynomial order p − 2

30Let’s denoteR(z) = 1 N Tr 1M − zThenR(z) 2 + R(z)V ′ (z) = f(z)For the simple case of V (M) = M 2 /2,R 2 + zR(z) = corR(z) = −z + √ 4c + z 22= −z + √ z 2 − 42

31• Constant c fixed by requirement thatR(z) = 1/z + O(z −2 )• R(z) imaginary for −2 < z < 2.ρ(z) = 1 √4 − z22πρm

32For the interacting cubic theoryone hasV = M 22 + gM 3R(z) = −V ′ (z) + √ V ′ (z) 2 + 4f2f = cz + d, and the descriminant is quartic in z. Arrangef so that descriminiant has two real roots and onedouble root.

33With this ansatz, one hasR(z) = − z + 3gz22+ (1 + 3g(a + b) + 3gz)√ (z − 2a)(z − 2b)2where3g(a−b) 2 +2(a+b)(1+3g(a+b)) = 0, (b−a) 2 (1+6g(a+b)) = 4so thatR(z) = 1 z + O(z−2 )This determines the eigenvalue distribution ρ(z) and itsolves the equation of motion.

34∫ 2b( )1F (0) = dλρ(λ)2a 2 λ2 + gλ 3 − 1 ∫dλdλ ′ ρ(λ)ρ(λ ′ ) ln(λ − λ ′ )2= − 1 σ(3σ 2 + 6σ + 2)3(1 + σ)(1 + 2σ) 2 + 1 ln(1 + 2σ)2whereσ = 3g(a + b)is a solution of18g 2 − σ(1 + σ)(1 + 2σ)with a bit more massagingF (0) = − ∑ k√1 (72g 2 ) k Γ(3k/2) 22(k + 2)! Γ(k/2 + 1) ≈ 3πk 7(108√ 3g 2 ) 2k

35Now, series∑kΓ−3( gg c) 2k= (g c − g) 2−ΓThe area scales likeA = g ∂ ∂g F (0)F (0) ∼ 1g c − gsoZ ∼ A Γ−2and we can read offΓ = − 7 2 + 3 = −1 2

36in agreement withΓ = 2 − Q γ = 1 12[(d − 1) − √ ](d − 25)(d − 1)for d = 0.

37All this was for pure 2D gravity (c = 0). What if we areinterested in adding conformal matter?Consider multi-matrix modelTr 1 2 A2 + 1 2 B2 + gA 3 + gB 3 + cABIsing model (c = 1/2 theory)

38Can also consider∫dt 1 2 (∂ tM) 2 + 1 2 M 2 + gM 3 (c = 1)Also consider adding higher order terms in the potentialV (M) = 1 2 M 2 + g 3 M 3 + g 4 M 4 + ...• Different discretization generically lead to samecontinuum limit (universality)• Fine tuning coupling gives rise to new critical behavior(multi-criticality)

39Double Scaling LimitIf one also compute 1/N corrections,F ∼ ∑ ∑N χ n (Γ−2)χ/2−1 (g/g c ) n ∼ (g c − g) (2−Γ)χ/2 N χχ nso scale g → g c keepingκ = (g c − g) (2−Γ)/2 N = fixedThenF = κ 2 F (0) + F (1) + κ −2 F (2) + ...

40Method of Orthogonal PolynomialsZ =∫dMe −TrV (M) =Because of anti-symmetry⎛∆(λ) = det(λ j−1i ) = det⎜⎝N∏i=1dλ ∆(λ) 2 e − ∑ i V (λ i)1 λ 1 λ 2 1 · · · λ N−111 λ 2 λ 2 2 · · · λ N−12. . . . .1 λ N−1 λ 2 N−1 · · · λN−1 N−1⎞⎟⎠

41Now, define a set of polynomialsP n (λ) = λ n + . . .satisfying orthogonality relation∫ ∞−∞e −V (λ) P n (λ)P m (λ) = h n δ mnThen,∆(λ) = det(λ j−1i ) = det(P j−1 (λ i ))

42andZ = N!N−1∏i=0h i = N!h N 0N−1∏i=1f N−kk, f k = h kh k−1Finding P n (and therefore h n ) is a finite procedure, so Zcan be computed exactlyIf interest is in large N limit,1N 2F = 1 ∑ ( 1 − k )ln f n ∼N Nwhere ξ = k/N.∫ 10dξ(1 − ξ) ln(f(ξ))

43Need f n for large n.This can be gotten from studying the recursion relationλP n (λ) =∑n+1c i P i (λ)but in facti=0λP n (λ) = P n+1 + r n P n−1becausefor i < n − 1.∫λP n (λ)P i (λ)e −V (λ) = 0

44Now,∫e −V (λ) (P n (λ)λ)P n−1 (λ) = r n h n−1∫= e −V (λ) P n (λ)(λP n−1 ) = h nsor n =h nh n−1= f n

45Similarly,∫=∫e −V (P ′ n(λ)λ)P n (λ) =e −V P ′ n(λ)(λP n (λ)) =∫∫= −∫=−e −V (nP n (λ) + . . .)P n (λ) = nh ne −V P n(λ)r ′ n P n−1 (λ)∫P n (λ)(r n e −V P n−1 ) ′r n P n (λ)P n−1 (λ)e −V V ′ (λ)∫P n (λ)r n e −V P n−1′sonh n = r n∫e −V V ′ (λ)P n (λ)P n−1 (λ)

46To apply these structures, consider for simplicity apotential with even terms onlyV = 1)(λ 2 + λ42g N + b λ6N 2ThenInsert this innh n = r n∫gV ′ = λ + 2λ3N+ 3bλ5N 2e −V V ′ (λ)P n (λ)P n−1 (λ)

47P n+1soλ : P n (λ) ↗ ↘r n P n−1gn = r n + 2 N r n(r n−1 + r n + r n+1 ) + 3bn2( 10 rrr terms)In the large n limit,ξ = n N ,r(ξ) = r nN

48andIn general, ifthengξ = r + 6r 2 + 30br 3 ≡ W (r)V (λ) = 12g a pλ 2pW (r) = a p(2p − 1)!(p − 1)! 2rp

49For generic W (r),gξ = W (r) = g c + 1 2 W ′′ (r c )(r − r c ) 2 + ...Then r − r c ∼ (g c − ξg) −Γ with Γ = −1/2 sor = r c + (g c − ξg) −Γ , so that∫11N 2F =∼∫010dξ(1 − ξ)f(ξ)dξ (1 − ξ)(g c − ξg) −Γ ∼ (g c − g) −Γ+2Agree with Γ for pure gravity computed earlier.

50Multi-criticalityIn generalW (r) = g c + c(r − r c ) 2 → Γ = − 1 2W (r) = g c + c(r − r c ) 3 → Γ = − 1 3.W (r) = g c + c(r − r c ) m → Γ = − 1 mΓ = 2 −Qγ min=2, (p, q) = (2l + 1, 2)1 − p − q

51Now, if one is interested in higher-genus contributions,gξ = W (r) + 2r(ξ)(r(ξ + ɛ) + r(ξ − ɛ) − 2r(ξ))where ɛ = 1/N. Now scaleg c − ξg = a 2 zr − r c = au(z)N = a −5/2g − g c = κ −4/5 a 2⇒ z = u(z) 2 − 1 3 u′′ (z), u(κ −4/5 ) = Z ′′ (κ −4/5 )

52This is the KdV equation.Can be solved perturbativelyu = z 1/2 (1 − ∑ u k z −5/2k )( k= z 1/2 1 − 1 24 z−5/2 − 491152 z−5 − 1225 )6912 z−15/2 + ...and computes the genus expansion of Z

53Summary of Orthogonal PolynomialsZ =∫dMe −TrV (M) =∫N∏dλ ∆(λ)e − ∑ i V (λ i) ∆(λ)i=1|N〉 =N−1∏i=0S = S nm b † nb m ,det(P j−1 (λ i ))Z = 〈N|S|N〉S mn = δ mn h n

54KdV HeirarchyNormalize Π n so that∫dλe −V Π n Π m = δ nmand define Q nm by√hn+1λΠ n√hn−1= Π n + r n Π n−1h n h n= √ r n + 1Π n+1 + √ r n Π n−1 ≡ Q nm Π mQ nm = Q mn

55Along similar lines, define∂∂λ Π n = A nm Π mwhich has [Q, A] by definitnion. No particular symmetry∫0 = dλ ∂∂λ Π nΠ m e −V = (A nm + A mn − V ′ )Π n Π m e −V⇒ A + A T = V ′ (Q)P = A − 1 2 V ′ (Q) = 1 2 (A − AT ) is antisymmetric[P, Q] = 1

56In the double scaling limit, Q nm becomes a differentialoperatorAnticipate scalingThenQ = 2r 1/2cr(ξ) = r c + a 2 u(z)+ a2rc1/2(u + r c κ 2 ∂ 2 z) ∼ d 2 + uP = d 3 + 3 {u, d} Cubic in d4

57(2,3) model:1 = [P, Q] =( 34 u2 + 1 4 u′′ ) ′⇒ KdVP = (Q 3/2 ) +

58Orthogonal Polynomials, Lax Pairs, etc generlizesMulti-matrix modelZ =∫ n∏a=1dM a exp [−TrV a (M a ) + c a M a M a+1 ]are also solvable....

59Key identity: Itzykson-Zuber integral∫∫dAe TrV (A)+cAB ∆(a)= da ∑ i∆(b) e− i V (a i)+ca i b ithenZ =∫ n∏a=1dM a exp [−TrV a (M a ) + c a M a M a+1 ]=∫ n∏a=1dλ a ∆(λ 1 )e −S(λ a) ∆(λ n )

60Define biorthogonal polynomials∫ n∏a=1Π i (λ 1 )˜Π j (λ n )e −V a(λ a )+c a λ a λ a+1= δ nmfrom which one derivesQ = d q + {v q−2 (z), d q−2 } + {v q−4 (z), d q−4 } + . . . v 0 (z)and adjust V ’s such thatP = (Q) p/q+

61P = (Q p/q ) + , [P, Q] = 1defines differential equation for u(z)One can also turn on “coupling” t nP → P + 1 ∑nt n (Q n/q−1 ) +qGeneralized KdV flow equationn∂∂t nQ = [(Q n/q ) + , Q]

62Solve for u(z, t i ) = F ′′ (z, t i )τ = Z = e −Fis called the τ-function: compute correlatorsExpectation value of generic single trace operatorTrf(M)computes insertion of integrated lowest dimensionoperator. Fine tune for higher dimension operators

63∂∂t nQ = [(Q n/q ) + , Q]α n ∼ p + q − nqγ ∼ α p−1,q−1 = 2⇒ α nγ = p + q − n2qcompare with BRST cohomology

64Alternative Matrix formulation of KdV flow∫ ]dM exp[−Tr 1 2 ΛM 2 + i M 3definee F(Λ) =Expand in small t i∫ [dM exp −Tr12ΛM 2]t i (Λ) = −(2i − 1)!!TrΛ −2i−1ln τ = F = t3 06 + t 124 + t3 0t 16 + 1 24 t 0t 2 + t2 148 + ...6

65So we haveDouble Scaled Matrix Model (gauge theory)↕Non-critical string theory (gravity theory)↕Kontsevich Matrix Model (gauge theory)Can they be thought of as analogues of AdS/CFT in anyway?

66Think about D-branes• Matrix point of view1M TrΦM

67In double scaling limit1M ΦM = 1 M (2r c + a2√rcQ) MscaleM = 2r cla 2Then1M ΦM = 1 l elQ ⇐ 1 L TreLΦLaplace transform∫dL e −xL 1 L TreLΦ = Tr log(x − Φ)

68Differentiate wrt x1R(x) = Trx − Φis the resolventInterpret as insertion of boundary cosmologial constant∫d 2 ξ √ ĝ(ĝ ab ∂ a ϕ∂ b ϕ + Q ˆRϕ) ∫+ µe γϕ + Kϕ+µ B e γϕ/2These are branes considered by FZZ/T∂Σ

69What does the resolvent measure?1R(x) = TrΦ − x = ∑ i1λ i − xforce due to log interaction with other Eigenvaluesy ≡ V ′ (x) + 2R(x) = Effective Force∫ydx = Effective PotentialR(x) = −V ′ (x) + √ V ′ (x) 2 + 4f2

70For example for cubic potential theoryy 2 = (1 double root and pair of single root)In the continuum limit g 2 → 1/108 √ 3 the double rootapproach the cut....

71In this limit, one obtainsT 2 (y) = 2y 2 − 1 = x(4x 2 − 3) = T 3 (x)for the (2,3) model. Along limilar lines, (2, 2l + 1) modelgives rise to a cut and l stationary pointsT 2 (y) = T 2l+1 (x)....

72One arrives at a following global picture of 1-pointfunction of FZZT branexx

73CFT side: one has the Liouville Boundary StateΓ(1 + 2iP b)Γ(1 + 2iP/b) cos(2πσP )|µ B 〉 = µ −iP b |P 〉2 1/4 (−2iπP )µ B√ = cosh πbσ, b 2 = q µ pThese branes are semi-localizedΨ(ϕ) = 〈ϕ|µ B 〉 = e −µ Be bϕ φ=−(1/b)log( µB)

74CFT and Matrix Model agree e.g. annulus∫Z = dτZ ghost Z Lioville Z matterZ Liouville (τ) = 〈l 1 |e −τ(L 0+¯L 0 ) |l 2 〉Z(l 1 , l 2 ) ∼ ∑ k=1k sin(πk/q)Kkq(l 1 )Ikq(l 2 )Small l 2 limit: loops = ∑ BRST cohomology

75FZZT expectation value probe target space (as functionof µ B )xφ=−(1/b)log(µ B )....

76This picture is strictly perturbativeNice geometrical picture(Seiberg-Shih)• tachyon backgrounds deforming the Reimann sufracepreserving the singularity• adding ZZ-brane opens the root into a cutIgnores non-perturbative effect such as tunneling ofeigenvalues (ZZ-branes)

77Non-perturbativelyW (x) = log(Φ − x)〈e W (x) 〉 = det(Φ − x)Equivalent to adding fundamental matter∫d¯χdχe¯χ(Φ−x)χ

78To be concrete, pick a simple model: Gaussian potential∫〈e W (x) 〉 = dΦd¯χdχ e − 2g 1 Φ2 +¯χ(x−Φ)χ

79〈e W (x) 〉 ==∫∫= 12πg= 12πg( g=2dΦd¯χdχ e − 12g Φ2 +¯χ(x−Φ)χd¯χdχe x¯χχ−g 2 (¯χχ)2∫∫d¯χdχds e x¯χχ− 12g s2 +is¯χχds (x + is) N e − 12g s2) N/2HN (x/ √ 2g)

80Hermite polynomial is an orthogonal polynomial forGaussian measuredet(Φ − x) = det ij(λ i−1j )∆(λ)= det ij(P i−1 (λ j ))∆(λ)where i = 1..(N + 1), λ N = x So∫ ∏〈det(Φ − x)〉 = dλ detij (λi−1 j )∆(λ)e − 2g 1 λ2 = P N (x)FZZT is probing the wavefunction of fermion at the topof fermi-surface

81Recursion relationλP n (λ) = √ r n+1 P n+1 (λ) + √ r n P n−1asymptotes toQψ(z, λ) =( ∂2∂z 2 − z )ψ(z, λ) = λψ(z, λ)in the double scaling limit.Baker-Akheizer function

82Go back to∫1ds (x + is) N e − 2g 1 s2 = 12πg2πgand scale∫ds e − 12g s2 +N log(x+is)g = ɛ 3 , N = ɛ −3 , s = i + ɛ˜s, x = 2 + ɛ 2˜x12π∫d˜s e −i (˜s 3 3 +˜s˜x )• This is the famous Airy function= Ai(˜x)• This is the famous Kontsevich 1 × 1 matrix model

83Multi-FZZT amplitude generalizes this to the matrix Airyintegral〈n∏a=1det(Φ − x a )〉 =∫dSe −iTr (S 33 +SX )Gaussian matrix model corresponds to (p, q) = (2, 1).No conformal content: topological gravity. c = −2

84• Gaiotto and Rastelli: Open SFT of FZZT intopological (2,1) theory is the Kontsevich matrix integralRank n of OSFT is precisely the number of FZZT branes(not N)• Kontsevich: This integral computes topological closedstring amplitudesThis is AdS/CFT correspondenceOSFT is simple because the theory was topological(much like the duality of Gopakumar-Vafa)

85• Airy function: Non-perturbative FZZT amplitude• The function is entire• Multi-sheeted structure of FZZT moduli-space is lostat the non-perturbative levelHow did this happen?

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”

88The branch cut is lost behind the Stoke’s lineMatrix model is powerful enough to address thesenon-perturbative issues

89• Perturbatively, many possible vacua• non-perturbative FZZT calculation is blind to this,except• # of stationary point must be even for wave functionto decay properly• (2, 2l + 1) model is well defined non-perturbatively onlyfor l even.

90What are the analogues of all these ideas for c = 1 orĉ = 1.What are the analogues of all these ideas for (p, q)

91Open SFTS =∫Ψ ∗ QΨ + Ψ ∗ Ψ ∗ Ψ+Chern-Simions∫AdA + 2 3 A3

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