Hirotaka Irie (YITP, Kyoto Univ.)

Duality Constraints on String Theory

Hirotaka Irie (YITP, Kyoto Univ.)

September 27th, 2013 @ Taiwan String Seminar in NTU

based on collaborations with

Chuan-Tsung Chan (Tunghai Univ.) and Chi-Hsien Yeh ^(NCTS)

Main Reference

[CIY4], Analytic Study for the String Theory Landscapes via Matrix Models, Phys.Rev. D86 (2012) 126001 [arXiv:1206.2351 [hep-th] ]

[CIY5], Duality Constraints on String Theory I: spectral networks and instantons, arXiv:1308.6603 [hep-th]

THEME: Non-perturbative Definition

Perturbative Def.

See by asymptotic expansion

F(g) = ln Z(g) '

X1 n=0

g^{2n 2}Fⁿ + X

I2Inst

✓_I ⇥ g^1/2 exph 1 g

X1 n=0

gⁿFn^(I)

i

Missing Information

θ: D-instanton fugacity

Non-Pert. Definition : the principle for θ

Vacuum Structure [CIY4]（Meta-stability/true vacuum）

☑ Matrix Model (Gauge Thy) is still not enough

CFT CFT

What is matrix model ?

F(g) = ln Z(g) '

X1 n=0

g^{2n 2}Fⁿ + X

I2Inst

✓_I ⇥ g^1/2 exph 1 g

X1 n=0

gⁿFn^(I)

i

e.g.) one-matrix models

Z = Z

dXe ^{N trV (X)}

CFT CFT

F = ln Z ' X

Feynman Graphs (genus n=0,1,2,··· )

N ^{2 2n}

A model defined for finite g or N, which gives this series as its perturbative expansion

Matrix Model

Large N expansion = Perturbative Strings

(non-perturbative completion)

D-instanton Fugacity Problem

F(g) = ln Z(g) '

X1 n=0

g^{2n 2}Fⁿ + X

I2Inst

✓_I ⇥ g^1/2 exph 1 g

X1 n=0

gⁿFn^(I)

i

Basics of asymptotic expansion

☑ Only leading asymptotic expansion is visible

☑ Sub-Leading terms play Stokes phenomena

（i.e. consider difference of θ : δθ ） Real Coeff. _{✓I = ✓}^(<)

I + i✓_I⁽⁼⁾

(F₀^(I) < 0)

leading

1. For a given Matrix Model, θis fixed [HHIKKMT 04]

2. obtained also from Non-perturbative reconstruction [CIY4 12]

BUT, a number of MMs are possible

= Ambiguity in Non-Pert. Definition itself

sub-leading

D-instanton Fugacity Problem

F(g) = ln Z(g) '

X1 n=0

g^{2n 2}Fⁿ + X

I2Inst

✓_I ⇥ g^1/2 exph 1 g

X1 n=0

gⁿFn^(I)

i

Mathematically

☑ Only leading asymptotic expansion is visible

☑ Sub-Leading terms play Stokes phenomena

（i.e. consider difference of θ : δθ ） Real Coeff. _{✓I = ✓}^(<)

I + i✓_I⁽⁼⁾

(F₀^(I) < 0)

leading

1. For a given Matrix Model, θis fixed [HHIKKMT 04]

2. obtained also from Non-perturbative reconstruction [CIY4 12]

BUT, a number of MMs are possible

= Ambiguity in Non-Pert. Definition itself

sub-leading

Matrix Model (Gauge Thy) itself can define non-perturbatively well-defined theory;

BUT, it does not really know anything more than “perturbative definition (+α)”

i.e. Non-pert. Principle is Necessary

Ambiguity of MM ＝ Contour of MM

Z = Z

dXe ^{N trV (X)}

Several ways to converge integral

Z = Z

dXe ^{N trV (X)} ' Z

dxe ^{N Veff(x)}

V_e↵(x)

Potential in (2,9) minimal string theory

Condensation

of other eigenvalues Instantons

x

Each contour defines different expansions:

F '

X1 n=0

g^{2n 2}Fⁿ + X

I2J^relev

✓_I ⇥ g^1/2 exp⇥ 1 g

X1 n=0

Fn^(I)

⇤ Mean-field (effective) potential of a single eigenvalue

Many choices keeping perturbative results

(Not necessary Hermite )

Ambiguity of MM ＝ Contour of MM

Examples [CIY4 12]

V_e↵(x)

Potential in (2,9) minimal string theory

Condensation

of other eigenvalues Instantons

x

In this model, minimal string theory is true vacuum of this system

1

2

3

4

F(g) '

X1 n=0

g^{2n 2}Fⁿ + ✓₁e ^{g1 S1(inst.)} + · · ·

Model 1:

Small instanton corrections

Examples [CIY4 12]

V_e↵(x)

Potential in (2,9) minimal string theory

Condensation

of other eigenvalues Instantons

x

In this model, the string theory is meta-stable

1

2

3

4

Model 2:

F(g) '

X1 n=0

g^{2n 2}Fⁿ + ✓₂e^{+ 1g S2(inst.)} + ✓₄e^{+ g1 S4(inst.)} · · ·

Large instanton corrections

With the ambiguity, even stability can change

Proposal [CIY5]：

Principle of Non-pert. String Duality

idea Consider pairs of matrix models

Mat. Mod. A Mat. Mod. B

Pert. String A Pert. String B

Large N expansion (extract the saddles)

(perturbatively) dual

Non-perturbatively, not necessarily dual [CIY5]

Equivalence of each saddle

Non-perturbative Duality should be imposed in matrix models (Duality Constraints)

Equivalence of combinations of saddles

?

Combinations in A are not necessarily realized by B

Duality in question

Integrate Y X-system

Integrate X

Y-system spectral (p-q) dual

Z =

Z

dXdY e

^{N tr[V1(X)+V2(Y ) XY ]}

Two-matrix model

NOTE）X and Y describe dual spacetimes  Double Field Theory

See it in Large N

Resolvent of X

R(x) =

⌧ 1

N tr 1

x X

Resolvent of Y

R(y) =e

⌧ 1

N tr 1 y Y

X-system Y-system

Spectral Curves

x y

F (x, R) = 0 _{F (y, e} ^e _{R) = 0}

Actually, they are dual

probe eigenvalues of X probe eigenvalues of Y

x y

F (x, R) = 0 _{F (y, e} ^e _{R) = 0}

Actually, they are dual

They are given by the same algebraic equation:

F (x, R) = f x, R V₁⁰(x) = 0 F (y, ee R) = f eR V₂⁰(y), y = 0 That is,

f (x, y) = 0

if x is spacetime

X-system

if y is spacetime

Y-system p-q dual

We can identify

[(x,y) is like phase space]

Duality in spectral curve

x $ y

Symplectic Invariance

Topological Recursion [Eynard-Orantin 07](=Loop Eqs) enables us to evaluate all-order perturbative amplitudes

f (x, y) = 0

Recursions

W₀^(g) = F^g

We can obtain free energy: _compare Wf₀^(g) = eF^g

W_n(x₁, · · · , xⁿ) =

* _n Y

j=1

1

N tr 1 x_j X

+

c

=

X1 g=0

N^{2 2g n}W_n^(g)(x₁, · · · , xⁿ)

Perturbative correlators in X and Y

Wf_n(y₁, · · · , yⁿ) =

* _n Y

j=1

1

N tr 1 y_j Y

+

c

=

X1 g=0

N^{2 2g n}Wf_n^(g)(y₁, · · · , yⁿ)

Duality in spectral curve

x $ y

Symplectic Invariance

f (x, y) = 0

Topological Recursion [Eynard-Orantin 07](=Loop Eqs) enables us to evaluate all-order perturbative amplitudes

Instantons (deform. of spectral curves) are also

X-system Y-system

Perturbative coefficients are the same for all-order

F

^pert

(g) =

X

1 n=0

g

^{2n 2}

F

ⁿ

F

_Inst^(I)

(g) =

X

1 n=0

g

^{n 1}

F

n^(I)

I 2 instantons

Equivalent under All-order perturb. theory

(Including all-order instanton corrections)

World-sheet（Liouville Theory）

Q = b + 1

b , Q = be 1

b b =

r p q S = 1

4⇡

Z

d² p

gh

g^ab@_a @_b + QR + 4⇡µe^2b i + + 1

4⇡

Z

d² p

gh

g^ab@_aX@_bX + i eQRXi

+ S_ghost

p-q duality: p q

_{, b $} ¹

b

q

p

x y

F (x, R) = 0 F (y, e e R) = 0

A basic assumption for 3-pt function（DOZZ）!

Liouville theory

(p,q) minimal CFT

( T-duality)

b radius of MCFT

p-q duality and T-duality: References

•

[Fukuma-Kawai-Nakayama 92]

[P,Q]=1 <=> [Q,-P]=1

•

[Kharchev-Marshakov 92]

Kontsevich MM: from (p,1) to (p,q)

•

[Bertola-Eynard-Harnad 01-04]

Two-matrix models, saddle point analysis

•

[Asatani-Kuroki-Okawa-Sugino-Yoneya 96]

Kramers-Wannier duality in Random Surfacs

•

[Kuroki-Sugino 07]

D-instanton fugacity

Integrate Y X-system

Integrate X

Y-system spectral (p-q) dual

Z =

Z

dXdY e

^{N tr[V1(X)+V2(Y ) XY ]}

Two-matrix model

Nonperturbative comparison

For simplicity, we choose as gaussian ^V

₂

^{(Y )}

Integrate Y X-system

Integrate X

Y-system

Z =

Z

dXdY e

^{N tr[V1(X)+V2(Y ) XY ]}

Two-matrix model

Nonperturbative completion

gaussian

Z = Z

dXe ^{N trV (X)} one-matrix model

（We know well）

We can perform！

Non-Pert. in one-matrix

Z = Z

dXe ^{N trV (X)} Mean-field Potential [David 91]

Z ' Z

dxe ^{N Veff(x)} = Z

dx ⌦

det(x X)²↵

e ^{N V (x)}

Look at a single eigenvalue x

⌦det(x X)²↵

= D

e2tr ln(x X)E

' e^2N

R _x

dx⁰D

1

N 1

x0 X

E

is resolvent in Large N： ^2N

Z

dxR(x)

V_e↵(x)

Potential in (2,9) Minimal String

Condensation

of other Eigenvalues Instantons

it is given by spectral curve!

Z = Z

dXe ^{N trV (X)} ' Z

dxe ^{N Veff(x)}

V_e↵(x)

Potential in (2,9) minimal string theory

Condensation

of other eigenvalues Instantons

x

See the expansions of free-energy

F '

X1 n=0

g^{2n 2}Fⁿ + X

I2J^relev

✓_I ⇥ g^1/2 exp⇥ 1 g

X1 n=0

Fn^(I)

⇤ Mean-field (effective) potential of a single eigenvalue

Many choices keeping perturbative results

(Not necessary Hermite )

Non-Pert. in one-matrix

Examples [CIY4 12]

V_e↵(x)

Potential in (2,9) minimal string theory

Condensation

of other eigenvalues Instantons

x

In this model, minimal string theory is true vacuum of this system

1

2

3

4

F(g) '

X1 n=0

g^{2n 2}Fⁿ + ✓₁e ^{g1 S1(inst.)} + · · ·

Model 1:

Small instanton corrections

Examples [CIY4 12]

V_e↵(x)

Potential in (2,9) minimal string theory

Condensation

of other eigenvalues Instantons

x

In this model, the string theory is meta-stable

1

2

3

4

Model 2:

F(g) '

X1 n=0

g^{2n 2}Fⁿ + ✓₂e^{+ 1g S2(inst.)} + ✓₄e^{+ g1 S4(inst.)} · · ·

Large instanton corrections

V_e↵(x)

Potential in (2,9) Minimal String

Condensation of Eigenvalues

instantons

x Examples [CIY4 12]

1

2

3

4

Model 3:

path with up-side-down potential

R(x) x

Simply, contradict with one-cut boundary condition (No matrix-model realization)

F(g) '

X1 n=0

g^{2n 2}Fⁿ + ✓₁e ^{1g S1} + ✓₂e ^{g1 S2} + · · ·

small instantons

up-side-down potential

Integrate Y X-system

Integrate X

Y-system

Z =

Z

dXdY e

^{N tr[V1(X)+V2(Y ) XY ]}

Two-matrix model

gaussian

Z = Z

dXe ^{N trV (X)} one-matrix model

We can perform！

Next !

Nonperturbative completion

Non-Pert. in Y-system

Mean-field Potential（[Kazakov-Kostov 04]）

is evaluated in Large N as Z =

Z

dXdY e ^{N tr[V1(X)+ Y 22 XY ]} '

Z

dxdy hdet(x X) det(y Y )i e ^{N [V1(x)+V2(y) xy]}

' e

^{N [ 1(x)+ 2(y) xy]}

Integrate X:

Z

dxe ^{N [ 1(x) xy]} : like Airy func.

•

Perturbatively, one can just solve saddle point eqn.

   （→A contribution from each saddle）

•

Non-Pert. requires Stokes phenomena [CIY5]

integral of resolvents

'^(j,l)(y) = Z

dy[R^(j)(y) R^(l)(y)]

R^(j)(y) = R(e ^{2⇡i(j 1)}y) F (R^(j)(y), y) = 0

e ^{N eVeff(y)} = X

j,l

(⇤)^j,l ⇥ e^'(j,l)(y)

Generally:

Stokes Phenomena (Result)

(5,2) (2,5) Models

[CIY5]

Stokes Coefficients

<2,5><4,3>

<4,3><1,5>

<1,5>

<4,5>

<2,5><4,3>

<5,3><2,1>

<5,1>

<5,4>

<5,4><3,1>

<5,3><2,1>

[5,3][2,1] [2,5][4,3]

steepest descent

y y

Sol to saddle Eqn

(5,2) (2,5) Models

And we can add other paths

<2,5><4,3>

<4,3><1,5>

<1,5>

<4,5>

<2,5><4,3>

<5,3><2,1>

<5,1>

<5,4>

<5,4><3,1>

<5,3><2,1>

[5,3][2,1] [2,5][4,3]

steepest descent

y y

e ^{N eVeff(y)} = X

j,l

(⇤)^j,l ⇥ e^'(j,l)(y)

<35><12>

<52><34>

'^(i,j)(y)

Roughly speaking,

it is a superposition of two different effective potentials

Integrate Y X-system

Integrate X

Y-system

Z =

Z

dXdY e

^{N tr[V1(X)+V2(Y ) XY ]}

Two-matrix model

Are they Equivalent?

gaussian

Z = Z

dXe ^{N trV (X)} one-matrix model

We can perform！

See large instanton modes

V_e↵(x)

Potential of (2,5) Minimal String

x

D-instanton

large D-instanton

Large instantons → observed on both sides

This cannot be realized in MM !

F ' F^pert + e^{+ g1 SI} + · · ·

y

<35><12>

<52><34>

<42>

'^(i,j)(y)

picked up by vertical contour

<42>

[CIY5]

One-cut Boundary condition

<2,5><4,3>

<4,3><1,5>

<1,5>

<4,5>

<2,5><4,3>

<5,3><2,1>

<5,1>

<5,4>

<5,4><3,1>

<5,3><2,1>

[5,3][2,1] [2,5][4,3]

<42>

<42>

This completion cannot satisfy One-cut BC

R(y)e y

x Therefore,

RULED OUT!

Conjecture [CIY5]

(p,q) minimal string theory consistent with non-perturbative spectral p-q duality does not

include large instantons in the background i.e.

(p,q) minimal string theory is the true vacuum in this non-perturbative string theory

For general (p,q) cases:

Summary

•

Duality in perturbation theory (Large N) v.s. Duality in Non-perturb. Completion

•

Since matrix models only know perturbative definition, non-perturbative [contour] ambiguity appears.

•

Therefore, matrix model itself is not enough for non- perturbative formulation of string theory.

•

Our proposal is to require string duality acts non- perturbatively, then it provides a constraint on non- perturbative ambiguity of string theory

This is the first quantitative observation

on non-perturbative principle of string theory,

Thank you for your attention!