Introduction to Matrix Models and Non-Critical String Theory Akikazu Hashimoto Taipei, July 2005

Introduction to Matrix Models and Non-Critical String Theory

Akikazu Hashimoto

Taipei, July 2005

These lectures are meant to provide pedagogical

introduction to continuum and matrix model formulation of non-critical string theory

They typically describe strings in 1+0 or 1+1 dimensions with 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

The subject is reviewed extensively in the literature

• Brezin, Wadia (Collection of Reprints)

• Di Francesco, Ginsparg, Zinn-Justin

• Dijkgraaf

• Ginsparg and Moore

• Klebanov

• Morozov

• Seiberg, Shih

Warning: “These theories are exactly solvable. That does not necessarily mean we understand them.”

Plan

1) Review of Liouville Theory and Matrix Model 2) Methods of Orthogonal Polynomials and KdV heirarchy

3) Branes and open/closed string duality

Non-critical string theory

Bosonic string in d dimensions Z =

Z Dg DX

Vol(diff)e^{−4πα01 R d2ξ}

√gg^µν∂_µXⁱ∂_νXⁱ

, i = 0...d − 1 Measure:

Z

Dge^−12|δg2| = 1, |δg²| = Z

d²ξ√

g(g^acg^bd − 2g^abg^cd)δg_abδg_cd Z

DXe^−|δX2| = 1, |δX²| = Z

d²ξ√

gδXⁱδXⁱ

Action and measure are diffeomorphism invariant

Gauge fix by chosing conformal gauge g = e^ϕg(τ )ˆ

by introducing ghosts Z

DbD¯bDcD¯c e^−Sgh S_gh = Z

d²ξ√

gb_zz∇_z¯c^z−b_{z ¯¯z}∇_zc^z¯ Under g → e^σg, the action is invariant, but the measure transforms

D_gXD_g(gh) → D_e^σ_gXD_e^σ_g(gh) = e^{24πα0d−26 SL(σ,g)}D_gXD_g(gh)

S_L = Z

dξ²√

g 1

2g^ab∂_aσ∂_bσ + Rσ + µe^σ

 This is the conformal anomaly which is cancelled by chosing d = 26

Alternatively, make ϕ dynamical Z

Dϕe^{−4πα01 R d2ξ}

√gˆ(^25−d₁₂ ^{gˆab∂aϕ∂}bϕ+^25−d₆ Rϕˆ )

Then δˆg = (ξ)ˆg, δϕ = −(ξ) is a symmetry

Rescale

q25−d

12 ϕ → ϕ Z

Dϕe^{−4πα01 R d2ξ}

√gˆ(^{gˆab∂aϕ∂}bϕ+Q ˆRϕ)

where

Q =

r25 − d 3 The stress tensor

T = −1

2∂ϕ∂ϕ + Q

2 ∂²ϕ

T (z)T (w) ∼ 1 2

c

(z − w)⁴ + ..., c = 1 + 3Q²

Finally, the µe^γϕ term adjusted so that the operator has dimension

[e^γϕ] = −1

2γ(γ − Q) = 1 So the bottom line

Z

Dϕe^{−4πα01 R d2ξ}

√gˆ(^{gˆab∂aϕ∂}bϕ+Q ˆRϕ+µe^γϕ)

γ = 1

√12(√

25 − d ∓ √

1 − d) = Q

2 ± 1 2

pQ² − 8

pick branch γ = 1

√12(√

25 − d − √

1 − d) = Q

2 − 1 2

pQ² − 8 so that in d → −∞ limit, γ → 0, so that

Q = 2/γ − γ → 2/γ, giving rise to “classical” Liouville theory

S = Z

d²ξp ˆ g

 ˆ

g^ab∂_aϕ∂_bϕ + Q ˆRϕ + µ

γ²e^γϕ



which is Weyl invariant under ˆg → e^2ρg, ϕ → ϕ −ˆ _γ²ρ

These can be coupled to d free bosons (or CFT with central charge c) so that

c_matter + c_liouv − 26 = 0

c ≤ 1 and c ≥ 25 picked out because γ is real. At c = 25 the signature of ϕ coordinate flips. We will focus on

c ≤ 1 for which ϕ coordinate is Eucledean.

• c < 1: 1 linear dilaton dimension

• c = 1: 1 linear dilaton dimension, one flat direction

linear dilaton: c.f. NS5-brane

X

φ

µe^γϕ marginal tachyon condensate no X for c < 1, c = 1 is a barrier.

Other interesting example: (p, q) Minimal CFT











O₁₁ · · · O_1(p−1)

O₂₁ O_2(p−1)

... ...

O_(q−1)1 · · · O_{(q−1)(p−1)}











c = 1 − 6(p − q)² pq

• (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

[O_r,s] = (pr − qs)² − (p − q)² 4pq

• (3,2): 0 0

• (4,3):

 ₁

2

1

16 0 0 ₁₆^{1 1}₂



• (5,2): 0 −¹₅ −¹₅ 0

Gravitational dressing

[e^αϕO] = 1 [O] = ∆₀, [e^αϕ] = −1

2α(α − Q), Q =

r25 − d so 3

α = 1

√12

h√

25 − d − p

1 − d + 24∆₀ i α < Q

2 , Seiberg Bound

Some simple observables: String Susceptibility Γ Let

Z(A) = Z

DϕDX e^−Sδ

Z

d²ξp ˆ

ge^γϕ − A

 which for large A scales as

Z(A) ∼ A^{(Γ−2)χ/2−1}

String suceptibility is this scaling exponent.

Simple scaling argument: Dϕ invariant under ϕ → ϕ + ρ/γ

Under this transformation Q

8π Z

d²ξp ˆ

g ˆRϕ → Q 8π

Z

d²ξp ˆ

g ˆRϕ + Q 8π

ρ

γd²ξp ˆ g ˆR so

Z[A] =

Z

DϕDXe^−Se^Qρχ2γ δ

Z

d²ξp ˆ

ge^γϕ+ρ − A



=

Z

DϕDXe^−Se^{Qρχ2γ −ρ}δ

Z

d²ξp ˆ

ge^γϕ − e^−ρA



= e^{Qρχ2γ −ρ}Z[e^−ρA] = A^{−2γ−1Qχ} Z[1], and so Γ = 2 − Q

γ = 1 12

h

(d − 1) − p

(d − 25)(d − 1)i

Along similar lines, one can define scaling dimension ∆_r,s

1 Z[A]

Z

DϕDXe^−Sδ

Z

d²ξp ˆ

ge^γϕ − A

 Z

d²ξp ˆ

ge^αr,sϕO_r,s

For ∆₀ = ^(pr−qs)_4pq^2−(p−q)2 and c = 1 − ^6(p−q)_pq ², α

γ = 1 − ∆_rs = p − q − |pr − qs|

2q

Physical observables are correlation functions of BRST cohomology

For L ⊗ M (p, q),

H = {O_ne^αnϕ}, O_n : matter, ghosts, Liouville α_n

γ = p + q − n

2q , n > 1, 6= 0 mod q

BRST cohomology has infinite element even though number of primary matter fields were finite.

Matrix Model

Theory of random matrces: (applications in disordered systems and quantum chaos)

Z = Z

dM e^{−TrV (M )} M is hermitan N × N matrix. Measure

DM = Y

i

dM_ii Y

i<j

dM_ijdM_ij^∗ invariant under M → U M U^†.

Parametrizing

M = U^†ΛU, dM = dU Y

i

dλ_i∆(λ)² where

Vandermonde : ∆(λ) = Y

i<j

(λ_i − λ_j)

One way to see this. Let dU = idT U . Line element:

TrdM ² = X

i

dM_ii² + X

i<j

dM_ij^∗ dM_ij

= Tr U^†(dΛ + i[Λ, dT ])U )
²

= X

i

dλ²_i + X

i<j

(λ_i − λ_j)²|dT_ij|² analgoue of dx² + dy² = dr² + r²dΩ².

Simplest Matrix Model: Gaussian ensumble Z =

Z

dM e^−TrM2 = Z

dm_i Y

i<j

(m_i − m_j)²e^{−Pi m2i}

=

Z

dm_ie^{− Pi m2i+2 log(mi−mj)}

Ensumble of N particles in harmonic oscillator potential with Logarighmic repulsion

Large N limit described in terms of the eigenvalue density ρ(m).

S = − Z

dm m²ρ(m)+

Z

dm dm⁰ log(m−m⁰)ρ(m)ρ(m⁰) Equation of motion

δS = −m² + 2 Z

dm⁰ log(m − m⁰)ρ(m⁰) = 0 and differentiating with respect to m,

m = Z

dm⁰ 1

m − m⁰ρ(m⁰)

Solution:

ρ(m) = 1 π

p2N − m² Wigner semi-circule distribution

ρ

m

One can also consider more general potential (BIPZ) e.g. V (M ) = 1

2M² + gM⁴ so that equation of motion becomes

1

2λ + 2gλ³ = Z

dλ⁰ 1

λ − λ⁰ρ(λ⁰) solved by

ρ(λ) = 1 π

 1

2 + 4ga² + 2gλ² N



p4a²N − λ², 12ga⁴ + a² − 1 = 0

Such generalization is interesting because the matrix action

Z

dM e^{−TrM 22 +gM 33!}

can be represented in Feynman expansion

↔ triangulation of 2D surface

1/N expansion ↔ genus expansion g → g_c ↔ continuum limit

Resolvent: useful computational tool Z =

Z

dM e^{−N V (M )}

Change integration variable M → M + _{M −z}¹ Under this transformation

δdM = −Tr

 1

M − z



Tr

 1

M − z



dM δe^{N V (M )} = −N V ⁰(M )

M − z

So we arrive at an identity hTr

 1

M − z



Tr

 1

M − z



i + hN V ⁰(M )

M − z i = 0 In the large N limit, one can factorize

hTr

 1

M − z



ihTr

 1

M − z



i + hN V ⁰(M )

M − z i = 0 Schwinger-Dyson equation. A little rewriting:

hTr

 1 M − z



ihTr

 1 M − z



i + N V ⁰(z)h 1

M − zi = −hN V ⁰(M ) − N V ⁰(z)

M − z i

= N²f (z), Polynomial order p − 2

Let’s denote

R(z) = 1

N Tr 1 M − z Then

R(z)² + R(z)V ⁰(z) = f (z) For the simple case of V (M ) = M ²/2,

R² + zR(z) = c or

R(z) = −z + √

4c + z²

2 = −z + √

z² − 4 2

• Constant c fixed by requirement that R(z) = 1/z + O(z⁻²)

• R(z) imaginary for −2 < z < 2.

ρ(z) = 1 2π

p4 − z²

ρ

m

For the interacting cubic theory V = M²

2 + gM³ one has

R(z) = −V ⁰(z) + pV ⁰(z)² + 4f 2

f = cz + d, and the descriminant is quartic in z. Arrange f so that descriminiant has two real roots and one

double root.

With this ansatz, one has

R(z) = −z + 3gz²

2 + (1 + 3g(a + b) + 3gz)p(z − 2a)(z − 2b) 2

where

3g(a − b)²+ 2(a + b)(1 + 3g(a + b)) = 0, (b − a)²(1 + 6g(a + b)) = 4 so that

R(z) = 1

z + O(z⁻²)

This determines the eigenvalue distribution ρ(z) and it solves the equation of motion.

F⁽⁰⁾ =

Z 2b 2a

dλρ(λ)  1

2λ² + gλ³



− 1 2

Z

dλdλ⁰ρ(λ)ρ(λ⁰) ln(λ − λ⁰)

= −1 3

σ(3σ² + 6σ + 2)

(1 + σ)(1 + 2σ)² + 1

2 ln(1 + 2σ) where

σ = 3g(a + b) is a solution of

18g² − σ(1 + σ)(1 + 2σ) with a bit more massaging

F⁽⁰⁾ = − X

k

1 2

(72g²)^k (k + 2)!

Γ(3k/2)

Γ(k/2 + 1) ≈

r 2

3πk⁷(108√

3g²)^2k

Now, series

X k^Γ−3  g g_c

^2k

= (g_c − g)^2−Γ The area scales like

A = g_∂g^∂ F⁽⁰⁾

F⁽⁰⁾ ∼ 1 g_c − g so

Z ∼ A^Γ−2 and we can read off

Γ = −7

2 + 3 = −1 2

in agreement with Γ = 2 − Q

γ = 1 12

h

(d − 1) − p

(d − 25)(d − 1) i

for d = 0.

All this was for pure 2D gravity (c = 0). What if we are interested in adding conformal matter?

Consider multi-matrix model Tr1

2A² + 1

2B² + gA³ + gB³ + cAB

Ising model (c = 1/2 theory)

Can also consider Z

dt 1

2(∂_tM )² + 1

2M² + gM³ (c = 1)

Also consider adding higher order terms in the potential V (M ) = 1

2M² + g₃M³ + g₄M⁴ + ...

• Different discretization generically lead to same continuum limit (universality)

• Fine tuning coupling gives rise to new critical behavior (multi-criticality)

Double Scaling Limit

If one also compute 1/N corrections, F ∼ X

χ

X

n

N^χn^{(Γ−2)χ/2−1}(g/g_c)ⁿ ∼ (g_c − g)^(2−Γ)χ/2N^χ so scale g → g_c keeping

κ = (g_c − g)^(2−Γ)/2N = fixed Then

F = κ²F⁽⁰⁾ + F⁽¹⁾ + κ⁻²F⁽²⁾ + ...

Method of Orthogonal Polynomials

Z = Z

dM e^{−TrV (M )} =

N

Y

i=1

dλ ∆(λ)²e^{− Pi V (λi)} Because of anti-symmetry

∆(λ) = det(λ^j−1_i ) = det











1 λ₁ λ²₁ · · · λ^{N −1}₁ 1 λ₂ λ²₂ · · · λ^{N −1}₂

... ... ... ... ...

1 λ_{N −1} λ²_{N −1} · · · λ^{N −1}_{N −1}











Now, define a set of polynomials

P_n(λ) = λⁿ + . . . satisfying orthogonality relation

Z ∞

−∞

e^{−V (λ)}P_n(λ)P_m(λ) = h_nδ_mn Then,

∆(λ) = det(λ^j−1_i ) = det(P_j−1(λ_i))

and

Z = N !

N −1

Y

i=0

h_i = N !h^N₀

N −1

Y

i=1

f_k^{N −k}, f_k = h_k h_k−1

Finding P_n (and therefore h_n) is a finite procedure, so Z can be computed exactly

If interest is in large N limit, 1

N²F = 1 N

X



1 − k N



ln f_n ∼

Z 1 0

dξ(1 − ξ) ln(f (ξ)) where ξ = k/N .

Need f_n for large n.

This can be gotten from studying the recursion relation

λP_n(λ) =

n+1

X

i=0

c_iP_i(λ) but in fact

λP_n(λ) = P_n+1 + r_nP_n−1 because

Z

λP_n(λ)P_i(λ)e^{−V (λ)} = 0 for i < n − 1.

Now,

Z

e^{−V (λ)}(P_n(λ)λ)P_n−1(λ) = r_nh_n−1

= Z

e^{−V (λ)}P_n(λ)(λP_n−1) = h_n so

r_n = h_n

h_n−1 = f_n

Similarly,

Z

e^−V (P_n⁰(λ)λ)P_n(λ) =

Z

e^−V (nP_n(λ) + . . .)P_n(λ) = nh_n

= Z

e^−V P_n⁰(λ)(λP_n(λ)) =

Z

e^−V P_n⁰(λ)r_nP_n−1(λ)

= − Z

P_n(λ)(r_ne^−V P_n−1)⁰

=

Z

r_nP_n(λ)P_n−1(λ)e^−V V ⁰(λ)

− Z

P_n(λ)r_ne^−V P_n−1⁰ so

nh_n = r_n Z

e^−V V ⁰(λ)P_n(λ)P_n−1(λ)

To apply these structures, consider for simplicity a potential with even terms only

V = 1 2g



λ² + λ⁴

N + b λ⁶ N²

 Then

gV ⁰ = λ + 2λ³

N + 3b λ⁵ N² Insert this in

nh_n = r_n Z

e^−V V ⁰(λ)P_n(λ)P_n−1(λ)

λ : P_n(λ) %

&

P_n+1 r_nP_n−1 so

gn = r_n + 2

N r_n(r_n−1 + r_n + r_n+1) + 3b

n²( 10 rrr terms) In the large n limit,

ξ = n

N , r(ξ) = r_n N

and

gξ = r + 6r² + 30br³ ≡ W (r) In general, if

V (λ) = 1

2ga_pλ^2p then

W (r) = a_p(2p − 1)!

(p − 1)!² r^p

For generic W (r),

gξ = W (r) = g_c + 1

2W ⁰⁰(r_c)(r − r_c)² + ...

Then r − r_c ∼ (g_c − ξg)^−Γ with Γ = −1/2 so r = r_c + (g_c − ξg)^−Γ, so that

1

N²F =

Z 1 0

dξ(1 − ξ)f (ξ)

∼

Z 1 0

dξ (1 − ξ)(g_c − ξg)^−Γ ∼ (g_c − g)^−Γ+2 Agree with Γ for pure gravity computed earlier.

Multi-criticality In general

W (r) = g_c + c(r − r_c)² → Γ = −1 2 W (r) = g_c + c(r − r_c)³ → Γ = −1

... 3

W (r) = g_c + c(r − r_c)^m → Γ = − 1 m Γ = 2 − Q

γ_min = 2

1 − p − q, (p, q) = (2l + 1, 2)

Now, if one is interested in higher-genus contributions, gξ = W (r) + 2r(ξ)(r(ξ + ) + r(ξ − ) − 2r(ξ)) where  = 1/N . Now scale

g_c − ξg = a²z r − r_c = au(z)

N = a^−5/2 g − g_c = κ^−4/5a²

⇒ z = u(z)² − 1

3u⁰⁰(z), u(κ^−4/5) = Z⁰⁰(κ^−4/5)

This is the KdV equation.

Can be solved perturbatively u = z^1/2(1 − X

k

u_kz^−5/2k)

= z^1/2



1 − 1

24z^−5/2 − 49

1152z⁻⁵ − 1225

6912z^−15/2 + ...



and computes the genus expansion of Z

Summary of Orthogonal Polynomials

Z = Z

dM e^{−TrV (M )} =

Z ^N Y

i=1

dλ ∆(λ)e^{− Pi V (λi)}∆(λ)

|N i =

N −1

Y

i=0

det(P_j−1(λ_i)) Z = hN |S|N i

S = S_nmb^†_nb_m, S_mn = δ_mnh_n

KdV Heirarchy

Normalize Π_n so that Z

dλe^−V Π_nΠ_m = δ_nm and define Q_nm by

λΠ_n = rh_n+1

h_n Π_n + r_nrh_n−1

h_n Π_n−1

= √

r_n + 1Π_n+1 + √

r_nΠ_n−1 ≡ Q_nmΠ_m Q_nm = Q_mn

Along similar lines, define

∂

∂λΠ_n = A_nmΠ_m

which has [Q, A] by definitnion. No particular symmetry 0 =

Z

dλ ∂

∂λΠ_nΠ_me^−V = (A_nm + A_mn − V ⁰)Π_nΠ_me^−V

⇒ A + A^T = V ⁰(Q) P = A − 1

2V ⁰(Q) = 1

2(A − A^T) is antisymmetric [P, Q] = 1

In the double scaling limit, Q_nm becomes a differential operator

Anticipate scaling

r(ξ) = r_c + a²u(z) Then

Q = 2r_c^1/2 + a² r_c^1/2

(u + r_cκ²∂_z²) ∼ d² + u P = d³ + 3

4{u, d} Cubic in d

1 = [P, Q] = 3

4u² + 1 4u⁰⁰

⁰

⇒ KdV (2,3) model:

P = (Q^3/2)₊

Orthogonal Polynomials, Lax Pairs, etc generlizes Multi-matrix model

Z =

Z ⁿ Y

a=1

dM_a exp [−TrV_a(M_a) + c_aM_aM_a+1]

...

are also solvable.

Key identity: Itzykson-Zuber integral Z

dAeTrV (A)+cAB

= Z

da_i∆(a)

∆(b)e^{− Pi V (ai)+caibi} then

Z =

Z ⁿ Y

a=1

dM_a exp [−TrV_a(M_a) + c_aM_aM_a+1]

=

Z ⁿ Y

a=1

dλ_a∆(λ₁)e^−S(λa)∆(λ_n)

Define biorthogonal polynomials Z ⁿ

Y

a=1

Π_i(λ₁) ˜Π_j(λ_n)e^{−Va(λa)+caλaλa+1} = δ_nm from which one derives

Q = d^q + {v_q−2(z), d^q−2} + {v_q−4(z), d^q−4} + . . . v₀(z) and adjust V ’s such that

P = (Q)^p/q₊

P = (Q^p/q)₊, [P, Q] = 1 defines differential equation for u(z)

One can also turn on “coupling” t_n P → P + 1

q

X

n

nt_n(Q^n/q−1)₊ Generalized KdV flow equation

∂

∂t_nQ = [(Q^n/q)₊, Q]

Solve for u(z, t_i) = F⁰⁰(z, t_i)

τ = Z = e^−F

is called the τ -function: compute correlators

Expectation value of generic single trace operator Trf (M )

computes insertion of integrated lowest dimension operator. Fine tune for higher dimension operators

∂

∂t_nQ = [(Q^n/q)₊, Q]

α_n ∼ p + q − n q

γ ∼ α_p−1,q−1 = 2

⇒ α_n

γ = p + q − n 2q

compare with BRST cohomology

Alternative Matrix formulation of KdV flow e^{F (Λ)} =

R dM exp h

−Tr¹₂ΛM² + i^M₆³ i R dM exp −Tr¹₂ΛM²

define

t_i(Λ) = −(2i − 1)!!TrΛ⁻²ⁱ⁻¹ Expand in small t_i

ln τ = F = t³₀

6 + t₁

24 + t³₀t₁

6 + 1

24t₀t₂ + t²₁

48 + ...

So we have

Double Scaled Matrix Model (gauge theory) l

Non-critical string theory (gravity theory) l

Kontsevich Matrix Model (gauge theory)

Can they be thought of as analogues of AdS/CFT in any way?

Think about D-branes

• Matrix point of view

1

M TrΦ^M

In double scaling limit 1

M Φ^M = 1

M (2r_c + a²

√r_cQ)^M scale

M = 2r_c` a²

Then 1

M Φ^M = 1

`e<sup>`Q</sup> ⇐ 1

LTre^LΦ Laplace transform

Z

dL e^−xL 1

LTre^LΦ = Tr log(x − Φ)

Differentiate wrt x

R(x) = Tr 1 x − Φ is the resolvent

Interpret as insertion of boundary cosmologial constant Z

d²ξp ˆ g

 ˆ

g^ab∂_aϕ∂_bϕ + Q ˆRϕ + µe^γϕ

 +

Z

∂Σ

Kϕ+µ_Be^γϕ/2 These are branes considered by FZZ/T

What does the resolvent measure?

R(x) = Tr 1

Φ − x = X

i

1 λ_i − x

force due to log interaction with other Eigenvalues y ≡ V ⁰(x) + 2R(x) = Effective Force

Z

ydx = Effective Potential R(x) = −V ⁰(x) + pV ⁰(x)² + 4f

2

For example for cubic potential theory

y² = (1 double root and pair of single root)

In the continuum limit g² → 1/108√

3 the double root approach the cut

....

In this limit, one obtains

T₂(y) = 2y² − 1 = x(4x² − 3) = T₃(x)

for the (2,3) model. Along limilar lines, (2, 2l + 1) model gives rise to a cut and l stationary points

T₂(y) = T_2l+1(x)

....

One arrives at a following global picture of 1-point function of FZZT brane

x x

CFT side: one has the Liouville Boundary State

|µ_Bi = Γ(1 + 2iP b)Γ(1 + 2iP/b) cos(2πσP )

2^1/4(−2iπP ) µ^{−iP b}|P i µ_B

√µ = cosh πbσ, b² = q p These branes are semi-localized

Ψ(ϕ) = hϕ|µ_Bi = e^−µBebϕ

µB φ=−(1/b)log( )

CFT and Matrix Model agree e.g. annulus Z =

Z

dτ Z_ghostZ_LiovilleZ_matter Z_Liouville(τ ) = hl₁|e^{−τ (L0+ ¯L0)}|l₂i

Z(l₁, l₂) ∼ X

k=1

k sin(πk/q)Kk

q(l₁)Ik

q(l₂) Small l₂ limit: loops = P

BRST cohomology

FZZT expectation value probe target space (as function of µ_B)

=−(1/b)log( )

....

x

µB φ

This picture is strictly perturbative

Nice geometrical picture (Seiberg-Shih)

• tachyon backgrounds deforming the Reimann sufrace preserving the singularity

• adding ZZ-brane opens the root into a cut

Ignores non-perturbative effect such as tunneling of eigenvalues (ZZ-branes)

Non-perturbatively

W (x) = log(Φ − x) he^{W (x)}i = det(Φ − x) Equivalent to adding fundamental matter

Z

d ¯χdχe^{χ(Φ−x)χ¯}

To be concrete, pick a simple model: Gaussian potential he^{W (x)}i =

Z

dΦd ¯χdχ e^{−2g1 Φ2+ ¯χ(x−Φ)χ}

he^{W (x)}i =

Z

dΦd ¯χdχ e^{−2g1 Φ2+ ¯χ(x−Φ)χ}

=

Z

d ¯χdχe^{x ¯χχ−g2( ¯χχ)2}

= 1

2πg Z

d ¯χdχds e^{x ¯χχ−2g1 s2+is ¯χχ}

= 1

2πg Z

ds (x + is)^Ne^{−2g1 s2}

= g 2

N/2

H_N(x/p

2g)

Hermite polynomial is an orthogonal polynomial for Gaussian measure

det(Φ − x) = det_ij(λⁱ⁻¹_j )

∆(λ) = det_ij(P_i−1(λ_j))

∆(λ) where i = 1..(N + 1), λ_N = x So

hdet(Φ − x)i =

Z Y

dλ det

ij (λⁱ⁻¹_j )∆(λ)e^{−2g1 λ2} = P_N(x) FZZT is probing the wavefunction of fermion at the top of fermi-surface

Recursion relation

λP_n(λ) = √

r_n+1P_n+1(λ) + √

r_nP_n−1 asymptotes to

Qψ(z, λ) =  ∂²

∂z² − z



ψ(z, λ) = λψ(z, λ) in the double scaling limit.

Baker-Akheizer function

Go back to 1

2πg Z

ds (x + is)^Ne^{−2g1 s2} = 1 2πg

Z

ds e^{−2g1 s2}+N log(x+is)

and scale

g = ³, N = ⁻³, s = i + ˜s, x = 2 + ²x˜ 1

2π Z

d˜s e⁻ⁱ



˜ s3

3 +˜s˜x



= Ai(˜x)

• This is the famous Airy function

• This is the famous Kontsevich 1 × 1 matrix model

Multi-FZZT amplitude generalizes this to the matrix Airy integral

h

n

Y

a=1

det(Φ − x_a)i = Z

dSe^−iTr



S3

3 +SX



Gaussian matrix model corresponds to (p, q) = (2, 1).

No conformal content: topological gravity. c = −2

• Gaiotto and Rastelli: Open SFT of FZZT in

topological (2,1) theory is the Kontsevich matrix integral Rank n of OSFT is precisely the number of FZZT branes (not N )

• Kontsevich: This integral computes topological closed string amplitudes

This is AdS/CFT correspondence

OSFT is simple because the theory was topological (much like the duality of Gopakumar-Vafa)

• Airy function: Non-perturbative FZZT amplitude

• The function is entire

• Multi-sheeted structure of FZZT moduli-space is lost at the non-perturbative level

How did this happen?

Stoke’s phenomenon

Ai(˜x) = 1 2π

Z

d˜s e⁻ⁱ



˜ s3

3 +˜s˜x



solution of

 ∂²

∂z² − z



f (z) = 0

Two 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 for positive real z)

Airy integral has three saddle points

Steepest descent contour hits only one of the saddles

0pi/12 7pi/12 9pi/12 11pi/12 12pi/12

3

1 1

0

4 2

3 3 2

4 0 1

2

3 4

0 2

1 0

4 0

4 3

2 1

But as z is taken off axis, different saddle points appear and disappear (along the steepest descent contour)

The locus on parameter space where contributing saddles re-arrange themselves is called “stoke’s line”

The branch cut is lost behind the Stoke’s line

Matrix model is powerful enough to address these non-perturbative issues

• Perturbatively, many possible vacua

• non-perturbative FZZT calculation is blind to this, except

• # of stationary point must be even for wave function to decay properly

• (2, 2l + 1) model is well defined non-perturbatively only for l even.

What are the analogues of all these ideas for c = 1 or ˆ

c = 1.

What are the analogues of all these ideas for (p, q)

Open SFT

S = Z

Ψ ∗ QΨ + Ψ ∗ Ψ ∗ Ψ

+

Chern-Simions

Z

AdA + 2

3A³

s

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

               

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

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