### The Resurgence of Instantons in String Theory

Part 2: Applications to Matrix Models and Strings

Ricardo Schiappa

(Instituto Superior T´ecnico)

Taiwan String Theory Workshop 2011

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

Resurgence and Trans–Series: Overview

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

Resurgence and Trans–Series: Overview

### Generalized Trans–Series and ODEs

String theoreticcontexts (minimal strings, matrix models): must include trans–series depending onmultiple parameters.

Rank–n system of non–linear ordinary differential equations, du

dz(z) = F(z, u(z)), may always be written in normal form:

du

dz(z) = −A ⋅ u(z) − 1

zB ⋅ u(z) + G(z, u(z)).

Here{α_{i}}_{i}_{=1⋯n} are eigenvalues of linearized system,
A = [∂F_{i}

∂uj

(∞, 0)]

i ,j=1⋯n

,

with A = diag (α1, . . . , αn) andB = diag (β1, . . . , βn) diagonal
matrices and one further insures thatG(z, u(z)) = O (∥u∥^{2}, z^{−2}u).

Convenient to choose variables such that α_{1}>0.

Resurgence and Trans–Series: Overview

### Formal Trans–Series Solutions

Aformal trans–seriessolution to our system of differential equations is [Costin]

u(z, σ) = u^{(0)}(z) + ∑

n∈N^{n}/{0}

σ^{n}z^{−n⋅β}e^{−n⋅α z}u^{(n)}(z),

whereσ = (σ_{1}, . . . , σ_{n})are trans–series parameters.

Both perturbativecontribution, u^{(0)}(z), as well as instanton and
multi–instantoncontributions,u^{(n)}(z), are formal asymptotic power
series of the form

u^{(n)}(z) =

+∞

∑

g=0

u^{(n)}g

z^{g} .

Resurgence and Trans–Series: Overview

### Resonant and Proper Trans–Series Solutions

Non–resonant trans–series: eigenvalues {αi}_{i}_{=1⋯n} areZ–linearly
independent, in many cases witharg α_{i} =/arg α_{j}.

This will notbe the case in here, as string theoretic systemsresonate,

∃_{n}_{/=n}′ ∣n ⋅ α = n^{′}⋅α.

Proper trans–series: only exponentiallysuppressed contributions
appear ⇒ Eigenvalues α such that, chosen direction in complex
z–plane, all contributions along this direction withσ_{i} =/0 are
exponentially suppressed; Re (n ⋅ α z ) > 0.

If one wishes to addressmulti–instanton series in string theoretic context one will have to allow for non–propertrans–series [Garoufalidis-Its-Kapaev-Mari˜no].

Resurgence and Trans–Series: Overview

### Borel Resumation and Trans–Series Solutions

Asymptotic series need to be Borel resummed in order to extract sensible information from them.

It follows that[Costin]

S_{θ}±u(z, σ_{±}) = S_{θ}±u^{(0)}(z) + ∑

n∈N^{n}/{0}

σ^{n}_{±}z^{−n⋅β}e^{−n⋅α z}S_{θ}±u^{(n)}(z),

is a good solutionto our problem along a properdirection (at least for sufficiently large∣z∣).

Many earlier concepts have straightforward generalization, for instance a simple extension of Stokes’ automorphism

S_{θ}+u(z, σ) = S_{θ}^{−}u(z, σ + S)

for crossing a Stokes line, with Sassociated Stokes constants.

Matrix Model Basics

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

Matrix Model Basics

### Review and Notation

Hermitian one–matrix model, N × N matrix M
Z_{N} =

1

vol(U(N))∫ dM e^{−}^{gs}^{1} ^{TrV}^{(M)},
vol(U(N)) volume factor of gauge group.

Diagonal gauge:

Z_{N} =
1
N!(2π)^{N} ∫

N

∏

i=1

dλ_{i}∆^{2}(λ) e^{−}^{gs}^{1} ^{∑}^{N}^{i}^{=1}^{V}^{(λ}^{i}^{)},

∆(λ) = ∏_{i}_{<j}(λi−λj) familiar Vandermonde determinant.

Free energy has perturbativegenus expansionat large N F =

+∞

∑

g=0

F_{g}(t) g_{s}^{2g}^{−2},
t = gsN is ’t Hooft coupling.

Matrix Model Basics

### Large N One–Cut Solution: Spectral Curve

At large N zero–instanton sector of matrix model characterized by density of eigenvaluesρ(λ).

In one–cut case, density has support on single, connected interval C = [a, b] in complex plane.

Equivalently, may describe largeN solution viaspectral curve y (z) = M(z)

√

(z − a)(z − b),

with singlebranch–cut alongC and M(p)is moment function M(p) =∮

∞

dz 2πi

V^{′}(z)
z − p

1

√

(z − a)(z − b).
Eigenvalue density is ρ(z) =_{2πt}^{1} Im y (z ).

Matrix Model Basics

### Holomorphic Effective Potential

Line integral along spectral curve
V_{h;eff}(λ) =∫

λ

a dz y (z).

Appears at leading order in large N expansion of the matrix integral Z ∼∫

N

∏

i=1

dλ_{i} exp (−1
g_{s}

N

∑

i=1

V_{h;eff}(λ_{i}) + ⋯).

Real part of spectral curve relates to forceexerted on a given
eigenvalue: effective potentialV_{eff}(z) = Re Vh;eff(z)constant inside
the cutC, i.e., inside the cut eigenvalues are free.

Effective potential on eigenvalue

V_{eff}(λ) = V (λ) − 2t∫ dλ^{′}ρ(λ^{′})log ∣λ − λ^{′}∣.

Matrix Model Basics

### Spectral Curve, Free Energies and Correlation Functions

Genus g free energies Fg(t).

Generating functions formulti–trace correlators
W_{h}(z_{1}, . . . , z_{h}) = ⟨Tr 1

z_{1}−M⋯Tr 1
z_{h}−M⟩

(c), with genusexpansion

W_{h}(z1, . . . , z_{h}) =

+∞

∑

g=0

g_{s}^{2g}^{+h−2}W_{g ,h}(z1, . . . , z_{h}; t).

Key Result in Matrix Model Theory

Quantities F_{g}(t) andW_{g ,h}(z_{1}, . . . , z_{h}; t) can beentirely computed in
terms of spectral curve y (z)alone [Ambj¨orn-Chekhov-Kristjansen-Makeenko,
Eynard, Eynard-Orantin].

Matrix Model Basics Multi–Cut Models

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

Matrix Model Basics Multi–Cut Models

### Potential V (z) with s Extrema

Mostgeneral saddle–point solution, at largeN, characterized by density of
eigenvalues supported on disjoint union ofs intervalsC = ⋃^{s}_{I}_{=1}A_{I}, where
A_{I} = [x_{2I}_{−1}, x_{2I}] ares cuts and x_{1}<x_{2}< ⋯ <x_{2s}.

Matrix Model Basics Multi–Cut Models

### Hyperelliptic Geometry of Spectral Curve

Spectral curve y (z) describeshyperelliptic geometry, y (z) = M(z)

¿ Á Á À

2s

∏

k=1

(z − x_{k}).

Still need to specify endpoints ofs cuts, {x_{k}}.
Assume filling fractions

_{I} ≡ N_{I}
N = ∫

A_{I}dλ ρ(λ), I = 1, 2, . . . , s,
fixed ⇒ Regarded as parameters, or moduli, of model.

May also use as modulipartial ’t Hooft couplings t_{I} =t_{I}=g_{s}N_{I}
tI = 1

4πi∮

A^{I}dz y (z),
with ∑^{s}_{I=1}t_{I} =t.

Matrix Model Basics “c= 1” Models

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

Matrix Model Basics “c= 1” Models

### Double–Scaling Limit

Matrix models describespecific string backgrounds.

StrictN → +∞ limit retainsonly genus zero planar diagrams

⇒No perturbative genusexpansion.

Simultaneously approach criticalityt → t_{c} ⇒ Higher–genus
contributionsenhanced& ∃ full perturbative genus expansion.

Matrix models in double–scaling limitdescribeminimal string theory

⇒Characterized bycentral charge

c_{p,q}=1 − 6(p − q)^{2}
pq .

Matrix modelsoff–criticality: topological stringsontoricbackgrounds.

Models in universality class of c = 1string atself–dual radius
F_{DSL}(µ) = F_{c}_{=1}(µ) = 1

2µ^{2}log µ − 1

12log µ +

+∞

∑

g=2

B2g

2g (2g − 2)µ^{2}^{−2g}.

Matrix Model Basics “c= 1” Models

### The Gaussian Model: V

_{G}

### ( z) =

^{1}

_{2}

### z

^{2}

Gaussian partition function

ZG= g

N2

s2

(2π)^{N}^{2}

G2(N + 1).

Gaussian free energy

F_{g}^{G}(t) = B_{2g}

2g (2g − 2)t^{2}^{−2g}, g ≥ 2.

Double–scaling limit,critical point tc =0,
t → 0, g_{s}→0, µ =t − t_{c}

gs

fixed, obtainc = 1 string at self–dual radius behavior.

Spectral geometry

y (z) dz =

√

z^{2}−4t dz.

Matrix Model Basics “c= 1” Models

### Chern–Simons/Stieltjes–Wigert: V

_{SW}

### ( z) =

^{1}

_{2}

### ( log z)

^{2}

Introduce

q = e^{g}^{s}, [n]_{q}=q^{n}^{2}−q^{−}^{n}^{2}.
Stieltjes–Wigert partition function

ZSW= (gs

2π)

N

2 q^{12}^{N}^{(7N}^{2}^{−1)}

N−1

∏

n=0

[n]_{q}!.

Regard Stieltjes–Wigert model asq–deformation of Gaussian model.

Chern–Simons free energy
F_{g}^{CS}(t) = B2gB2g−2

2g (2g − 2) (2g − 2)!+

B2g

2g (2g − 2)!Li_{3}_{−2g}(e^{−t}), g ≥ 2.

Matrix Model Basics “c= 1” Models

### Topological Strings on the Resolved Conifold

After analytical continuation g_{s}→i¯g_{s},Chern–Simons free energies
coincide with free energies of topological strings onresolved conifold
(identify’t Hooft couplingand K¨ahler parameter) [Gopakumar-Vafa].
Spectral geometry

y (z) dz = 2

z log1 + e^{−t}z +

√

(1 + e^{−t}z)^{2}−4z
2√

z dz.

Coincides with one–formlog Y (Z )^{dZ}_{Z} on mirror curveH(Z , Y ) = 0 of
resolved conifold, written in terms ofC^{∗} variables Z = e^{z},Y = e^{y}!

Approaching Topological String Theory

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

Approaching Topological String Theory

### A–Model ◁Mirror▷ B–Model

MapsΦ ∶ Σ_{g} → X^{3} Calabi–Yauthreefold.

A–model: physical formulation ofGromov–Wittentheory.

B–model: related to theory ofdeformation of complex structures.

Genus expansion of string free energy
F (g_{s}; {t_{i}}) =

+∞

∑

g=0

g_{s}^{2g}^{−2}F_{g}(t_{i}).

Free energy of A–model (large values of K¨ahler parameters) F (Q, gs) = ∑

d ,g

g_{s}^{2g}^{−2}Nd ,gQ^{d} ≡

+∞

∑

g=0

Fg(Q) g_{s}^{2g}^{−2}.
HereQ = exp(−t) with t “size” of CY 3–fold.

Gromov–Witten invariants ofX^{3},N_{d ,g}, count world–sheet instantons
(number of curves of genus g and degree d).

Approaching Topological String Theory

### Nonperturbative Topological Strings?

Free energy computed perturbativelyin two expansion parameters:

Q (world–sheet) andg_{s} (spacetime).

World–sheet: Power series in e^{−t},finiteconvergence radiust_{c},
estimated fromasymptotic behaviorof GW invariants[BCOV]

Ng ,d∼d(γ−2)(1−g )−1e^{dt}^{c}, d → +∞.

Critical value (conifold point) geometric large–radius phase breaks
down ⇒ Non–geometric phase,nonperturbative inα^{′}.

World–sheet: Characterize theory bycritical behaviorat conifold point:

some local CYs in universality class of 2d gravityc = 0(local curve [CGMPS]) most cases inc = 1universality class (resolved conifold).

World–sheet: Considering B–model onmirrorCY, computeF_{g}(Q)
exactly asmodular functionin Q.

Spacetime: There arenon–perturbativecorrections∼e^{−}^{gs}^{1}.

Approaching Topological String Theory

### Topological Strings as Matrix Models

B–model on some local CYs haslarge N dualityto a matrix model [Dijkgraaf-Vafa]. Also true for generic mirrors of toricgeometries [Mari˜no, Bouchard-Klemm-Mari˜no-Pasquetti].

B–model class of target CYgeometries uv = H(X , Y ).

HereH(X , Y )=polynomial inX , Y ∈ C, andu, v ∈ C.

Non–trivial information about this geometry: encoded in Riemann surfaceΣ described byH(X , Y ) = 0.

Spectral curve of matrix model is precisely Riemann surfaceΣ.

Extended totoric CY:mirror geometries as above, butX , Y ∈ C^{∗}.
Matrix model recursive formulation of1/N expansion: all information
encodedin spectral curve ⇒generates topological string amplitudes.

Approaching Topological String Theory

### Gopakumar–Vafa Integral Representations

Schwinger–like integral representation forF_{X}(g_{s}; {t_{i}})

= ∑

{di},r,m

n^{(d}_{r} ^{i}^{)}(X ) ∫

+∞

0

ds

s (2 sins 2)

2r−2

exp (−2πs

g_{s} (d ⋅ t + i m)).

Integersn_{r}^{(d}^{i}^{)}(X )are GV invariants ofX;
d_{i} K¨ahler class, r ≥ 0a spin label;

m ∈ Zwinding number (M–theory onX ×S^{1} →IIA onX).

Resolved conifold: toric CY threefold, dim H2(X, Z) = 1,single
non–vanishing integer GV invariant n^{(1)}_{0} =1⇒

F_{X}(g_{s}; t) = 1
4 ∑

m∈Z∫

+∞

0

ds s

1
sin^{2}(^{s}

2)

e^{−}^{2πs}^{gs} ^{(t+i m)}.

Recoverperturbativegenus expansion from integral representation!

Approaching Topological String Theory

### Instantons and String Theory

Perturbative series inzero–instantonsector of closed string theory or its matrix model dual

F^{(0)}(gs) =g_{s}^{2}F^{(0)}(gs) =

+∞

∑

g=0

Fg(t) g_{s}^{2g}.

Heret is’t Hooft couplingt = gsN in context of matrix models, or geometric modulus in string theory.

Multi–instanton path integral yields a series of the form

F^{(`)}(z) = i
z^{`b}^{/2}e^{−}

√`A z

+∞

∑

n=0

F_{n}^{(`)}_{+1}z^{n}^{/2}.
Herez = g_{s}^{2}.

Approaching Topological String Theory

### Large–Order Behavior and String Theory

Ifdispersion relation(from Cauchy integral representation) holds in here

F (g ) = 1 2πi∫

+∞

0 dw Disc F (w ) w − g . Large–order coefficientsin asymptotic series expansion:

F_{g}^{(0)}(t) ∼ 1
2πi

Γ(2g + b)
A(t)^{2g}^{+b}

⎛

⎝

F_{1}^{(1)}(t) +F_{2}^{(1)}(t) A(t)
2g + b − 1 + ⋯

⎞

⎠ .

Explicitly shows(2g )!growth [Shenker]. This is what we shall address:

Compute analytically andcheck numerically!

Instantons in One–Cut Matrix Models

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

Instantons in One–Cut Matrix Models

### Effective Potential and Instanton Sectors

Effective potential constant along cut C = [a, b], where there is local
minimum, and haslocal maximumat pointx_{0}.

Standard (0–instanton) 1/N expansion computed considering

saddle–point configuration with allN eigenvalues supported in cut C. k–instanton configuration corresponds to distinct saddle–point: N − k eigenvalues remaining in C, while k eigenvalues placed at local maximumx0 (still assume k ≪ N) [David].

Instantons in One–Cut Matrix Models

### Geometry of One–Instanton Sector

Integration contour, over nontrivial saddle characterizing one–instantonsector, defined by saddle–point requirement:

V_{h,eff}^{′} (x0) =0 ⇒ y (x0) =0, with x0 located outside the cut.

Explicit form of spectral curve⇒Equivalent conditionM(x_{0}) =0.

Geometrically, spectral curve is genus zero curvepinched atx0. First observed in context of spectral curves for double–scaled matrix models in minimal strings[Seiberg-Shih].

Instantons in One–Cut Matrix Models

### Analytical Results

Tree Level, One and Two Loops

Find thatF^{(1)} has the structure
F^{(1)}=i g

1

s2 exp (−A
g_{s})

+∞

∑

n=0

F_{n}^{(1)}_{+1}g_{s}^{n}.
Instanton action

A = V_{h,eff}(x_{0}) −V_{h,eff}(b) =∫

x0

b dz y (z).

Hasgeometric interpretation as contour integral of one–formy (z) dz,
from endpoint of cutC to singular pointx_{0}.

Computed one–loopfluctuations around one–instanton configuration andtwo–loop fluctuations around one–instanton configuration.

F_{1}^{(1)}and F_{2}^{(1)} dependuniquely on data specified byspectral curve.

Instantons in One–Cut Matrix Models

### Analytical Results

One and Two loops

F_{1}^{(1)}= −ib − a
4

¿ Á Á ÁÁ À

1

2πM^{′}(x0)[(x0−a)(x0−b)]

5 2

,

F_{2}^{(1)}
F_{1}^{(1)}

= 1

4(a − b)

√

(x_{0}−a)(x_{0}−b)
(

(x0−b)M^{′}(a)
M^{2}(a) −

(x0−a)M^{′}(b)
M^{2}(b) ) −

−

√

(x0−a)(x0−b)

12(a − b)^{2} (8(x0−a) + 17(a − b)

(x0−a)^{2}M(a) +8(x0−b) + 17(b − a)
(x0−b)^{2}M(b) ) +
+5 (M^{′′}(x0))^{2}−3M^{′}(x0)M^{(3)}(x0)

24 (M^{′}(x0))^{3}

√

(x0−a)(x0−b)

+ 35 (2x0− (a + b)) M^{′′}(x0)
48 (M^{′}(x0))^{2}((x0−a)(x0−b))^{3}^{2}

+

+

140 (2x_{0}− (a + b))^{2}+33(a − b)^{2}
96M^{′}(x_{0}) ((x_{0}−a)(x_{0}−b))^{5}^{2}

Instantons in One–Cut Matrix Models The Quartic Model

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

Instantons in One–Cut Matrix Models The Quartic Model

### Data and Set–Up

Potential V (z) =^{1}_{2}z^{2}+λz^{4}.

Single cutC = [a, b] ≡ [−2α, 2α], with α^{2}= ^{1}

24λ(−1 +

√

1 + 48λt).

Spectral curve y (z) = M(z)

√

z^{2}−4α^{2}, with M(z) = 1 + 8λα^{2}+4λz^{2}.
Non–trivial saddle–pointsx_{0}^{2}= − ^{1}

4λ(1 + 8λα^{2}).
Instanton action

A = −

√
3 α^{2}
4 (1 − α^{2})

√

4 − α^{4}−2 log [

√ 3

√

−2 + α^{2}+

√

−2 − α^{2}] +
+log 4 (1 − α^{2}).

Computed F_{g} via orthogonal polynomials up to genusg = 10
[Bessis-Itzykson-Zuber].

Instantons in One–Cut Matrix Models The Quartic Model

### The Effective Potential

-x0 x0

a b

C

Instantons in One–Cut Matrix Models The Quartic Model

### Large–Order Results

Instanton Action

Numerical asymptotics forinstanton action, along with matrix model prediction, at λ = −0.1:

Instantons in One–Cut Matrix Models The Quartic Model

### Large–Order Results

One and Two–Loop Corrections

Leading asymptotics divided by one–loop matrix model prediction, and subleading asymptotics divided by two–loopmatrix model prediction:

Instantons in One–Cut Matrix Models The Painlev´e I Equation

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

Instantons in One–Cut Matrix Models The Painlev´e I Equation

### Double–Scaling Limit

c = 0(2d gravity) double–scaling limitof previous model.

Perturbative amplitudes governed by Painlev´e I equation, fulfilled by
specific heatu(z) = F_{ds}^{′′}(z)as

u^{2}(z) −1

6u^{′′}(z) = z.

Action[Shenker], one[David]and two loops corrections:

F_{ds}^{(1)}(z) = i
8 ⋅ 3^{3}^{4}√

π

z^{−}^{5}^{8} exp (−8√
3

5 z^{5}^{4}) {1 − 37
64

√

3z^{−}^{5}^{4} + ⋯}.
Can actually compute full multi–loopcorrections!

ComputeFg numerically up to arbitrary genus.

Instantons in One–Cut Matrix Models The Painlev´e I Equation

### Large–Order Results

Convergence towards one–loop matrix model prediction, and convergence towards two–loopmatrix model prediction:

Multi–Instantons and Multi–Cuts

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

Multi–Instantons and Multi–Cuts

### Partition Function and the Multi–Cut Phase

s cuts: “topological sectors”characterized by fillings N1, . . . , Ns. General partition function as sum over allpossible arrangements of eigenvalues across cuts[Bonnet-David-Eynard, Eynard-Mari˜no]

Z (ζ1, . . . , ζs) = ∑

N1+⋯+Ns=N

ζ_{1}^{N}^{1}⋯ζ_{s}^{N}^{s}Z (N1, . . . , Ns).
Different sectors regarded asinstanton sectors of multi–cut model!

Multi–Instantons and Multi–Cuts

### Instantons in the Multi–Cut Phase

Z (N1, . . . , Ns) partition function in one sector,Z (N_{1}^{′}, ⋯, N_{s}^{′}) partition
function corresponding to different choice of filling fractions ⇒

Z (N_{1}^{′}, . . . , N_{s}^{′})
Z (N1, . . . , Ns)

∼exp{−1 gs

s

∑

I=1

(N_{I}−N_{I}^{′})∂F_{0}(t_{I})

∂tI

}.

If we pick set of filling fractions(N_{1}, . . . , N_{s})asreference background,
all other sectors willnot be seenin gs perturbation theory.

Regard them as different instanton sectors of the matrix model.

However: depending on value of real part of exponent, sectors with
filling fractions {N_{I}^{′}} will be either exponentially suppressedor

exponentially enhanced, with respect to reference configuration {N_{I}}.

Multi–Instantons and Multi–Cuts

### Multi–Instanton Sector in the Two–Cut Case

Reference configuration(N_{1}, N_{2})

Z =

N1

∑

`=−N2

ζ^{`}Z (N1−`, N2+`) = Z^{(0)}(N1, N2)

N1

∑

`=−N2

ζ^{`}Z^{(`)}.

HereZ^{(0)}(N1, N2) =Z (N1, N2) andZ^{(`)}= ^{Z}^{(N}^{1}^{−`,N}^{2}^{+`)}

Z^{(0)}(N1,N2) .
Change variables t = t_{1}+t_{2},s = ^{1}_{2}(t_{1}−t_{2})

Z^{(`)} =q^{`2}^{2} exp (−`A
gs

) {1 − g_{s}(` ∂_{s}F_{1}+

`^{3}

6 ∂^{3}_{s}F_{0}) + O(g_{s}^{2})}.

Ais theinstanton action, integral over spectral curve A =∫

x3

x2 dz y (z),
andq = exp(∂_{s}^{2}F0).

Multi–Instantons and Multi–Cuts

### Multi–Cuts, Multi–Instantons and Theta Functions

At large N1 andN2 can extend sum from −∞ to+∞, exchange sum
over`with expansion in g_{s}, and writeZ in terms of aJacobi theta
function,ϑ_{3}(q ∣ z) = ∑_{`}_{∈Z}q^{`2}^{2} z^{`}.

Express free energy F = log Z in terms of infinite series which formally has structure of instanton/anti–instanton expansion

F = F^{(0)}(t1, t2) +log φ(q) + ∑

`/=0

(−1)^{`}

`(q^{`}^{2}−q^{−}^{`}^{2})

ζ^{`}e^{−`A/g}^{s}(1 + O(gs)).

φ(q) = ∏^{∞}_{n}_{=1}(1 − q^{n}) andlog φ(q) gives contribution toF_{1} coming
frominstanton/anti–instanton interactions in partition function.

Multi–Instantons and Multi–Cuts

### Multi–Instantons in the One–Cut Phase

Multi–Instantons and Multi–Cuts

### Multi–Instantons in the One–Cut Model

Degeneration limitt_{2}→0(singular⇒need to regularizeby removing
Gaussian contribution of second cut ∝log t2).

Two–loops`–instanton amplitude in arbitrary one–cut matrix model:

F^{(1)} = g_{s}^{1/2}

√

2πˆq^{1}^{/2}{1 − g_{s}(∂_{s}F̂_{1}+1

6∂_{s}^{3}F̂_{0}) + O(g_{s}^{2})}

F^{(`)} =

(−1)^{`}^{−1}g_{s}^{`}^{/2}

` (2π)^{`}^{2}
ˆ

q^{`}^{/2}{1 − ` g_{s}(∂_{s}F̂_{1}+
1

6∂_{s}^{3}F̂_{0}+ˆq) + O(g_{s}^{2})}

Evaluated att1=t andt2=0.

Multi–Instantons and Multi–Cuts The Quartic Model Revisited: Two Cuts

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

Multi–Instantons and Multi–Cuts The Quartic Model Revisited: Two Cuts

### Spectral Curve Geometry: One–Instanton Sector

Multi–Instantons and Multi–Cuts The Quartic Model Revisited: Two Cuts

### Large–Order Results: Instanton Action

Multi–Instantons and Multi–Cuts The Quartic Model Revisited: Two Cuts

### Orthogonal Polynomials: The Good and the Bad

Nonperturbative Structure of Topological Strings

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

Nonperturbative Structure of Topological Strings The Local Curve

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

Nonperturbative Structure of Topological Strings The Local Curve

### Data and Set–Up

ConsiderA–model topological strings ontoric CY (local curve)
Xp= O(p − 2) ⊕ O(−p) → P^{1}.

Free energies at genus g depend on single complexified K¨ahler
parameter t, associated to complexified areaofP^{1}.

Spectral curve (description ofmirror B–model)[Mari˜no]

y (λ) = 2

λ(tanh^{−1}[

√

(λ − a)(λ − b)

λ −^{a}^{+b}_{2} ] −p tanh^{−1}[

√

(λ − a)(λ − b) λ +

√ ab

]).

Nonperturbative Structure of Topological Strings The Local Curve

### Instanton Action: A = V

_{h,eff}

### ( x

_{0}

### ) − V

_{h,eff}

### ( a)

x_{0}= 4ab
(√

a−√

b)^{2}, p= 3, x_{0}= 2√

√a−ab√ b, p= 4,

V_{h,eff}(x) = − log (f1(x))(log (f1(x)) − 2 log(1 + 2f_{1}(x)
(√

a−√

b)^{2}) + log(1 + 2f_{1}(x)
(√

a+√
b)^{2}))−

−2Li2(− 2f_{1}(x)
(√

a−√

b)^{2}) − 2Li2(− 2f_{1}(x)
(√

a+√

b)^{2}) − log(a − b)^{2}
4 log x−

−p log (f2(x))(log (f2(x)) + 2 log(1 −f_{2}(x)
2√

ab) − log(1 − 2f_{2}(x)
(√

a+√
b)^{2}))−

−2pLi2(−f_{2}(x)
2√

ab) + 2pLi2( 2f_{2}(x)
(√

a+√
b)^{2}) +p

2(log x)^{2}+ p log(√
a+√

b)^{2}log x,

f_{1}(x) = √

(x − a)(x − b) + x −a+ b
2 ,
f_{2}(x) = √

(x − a)(x − b) + x +√ ab.

Nonperturbative Structure of Topological Strings The Local Curve

### Large–Order Results

Use topological vertex [Aganagic-Klemm-Mari˜no-Vafa] to computeF_{g}
up to genus8 (genus6) for p = 3 (p = 4).

Instanton action,one andtwo loops "

Nonperturbative Structure of Topological Strings The Local Curve

### Instanton Action: Numerical Results

Numerical asymptotics forinstanton action, along with matrix model prediction, at ζ = 0.24and p = 3:

1 2 3 4 5 6 7 8 *g*

0.0004 0.0006 0.0008 0.0010

Nonperturbative Structure of Topological Strings The Local Curve

### One and Two Loop Coefficients: Numerical Results

Leading asymptotics divided by one–loop matrix model prediction, and subleading asymptotics divided by two–loopmatrix model prediction:

0.05 0.10 0.15 0.20

0.96 0.98 1.00 1.02 1.04 1.06

0.05 0.10 0.15 0.20 0.25

0.95 1.00 1.05 1.10

Nonperturbative Structure of Topological Strings Hurwitz Theory

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

Nonperturbative Structure of Topological Strings Hurwitz Theory

### Data and Set–Up

Hurwitz theory countsbranched covers of Riemann surfaces (restrict
to coverings of sphereP^{1} (“target”) by surfaces of genusg

(“worldsheets”)). Number of disconnected coverings of degreed
counted by simple Hurwitz numberH_{g ,d}^{P}^{1} (1^{d}).

Total partition function ⇒ Generating functional

Z^{H}(t_{H}, g_{H}) = ∑

g≥0

g_{H}^{2g}^{−2}∑

d≥0

H_{g ,d}^{P}^{1} (1^{d})
(2g − 2 + 2d )!Q^{d},
Q = e^{−t}^{H} andg_{H} parameters keeping track ofdegree andgenus.

Free energy describes connected, simple Hurwitz numbersH_{g ,d}^{P}^{1} (1^{d})^{●}.

Nonperturbative Structure of Topological Strings Hurwitz Theory

### Hurwitz Theory as a Topological String Theory

Hurwitz theory is topological string theory in disguise: Can be realized as special limit of A–model on localcurves [CGMPS]

p → ∞, t → ∞, g_{s} →0.

Computed Fg up to genus16, finding again spectacularagreement.

Nonperturbative Structure of Topological Strings Hurwitz Theory

### Instanton Action: Numerical Results

Numerical asymptotics forinstanton action, along with matrix model prediction, at χ = 0.5:

5 10 15 *g 1*

0.45 0.50 0.55 0.60 0.65 0.70

Nonperturbative Structure of Topological Strings Hurwitz Theory

### One and Two Loop Coefficients: Numerical Results

Leading asymptotics divided by one–loop matrix model prediction, and subleading asymptotics divided by two–loopmatrix model prediction:

0.2 0.4 0.6 0.8 1.0Χ

0.998 1.000 1.002 1.004

0.0 0.2 0.4 0.6 0.8 1.0

0.995 1.000 1.005 1.010

Nonperturbative Structure of Topological Strings c= 1 Strings and the Resolved Conifold

### Outline

1 Resurgence and Trans–Series: Overview

2 Matrix Model Basics Multi–Cut Models

“c = 1” Models

3 Approaching Topological String Theory

4 Instantons in One–Cut Matrix Models The Quartic Model

The Painlev´e I Equation

5 Multi–Instantons and Multi–Cuts

The Quartic Model Revisited: Two Cuts

6 Nonperturbative Structure of Topological Strings The Local Curve

Hurwitz Theory

c = 1 Strings and the Resolved Conifold

7 Future Directions

Nonperturbative Structure of Topological Strings c= 1 Strings and the Resolved Conifold

### Borel Analysis Example

Charged Scalar Particle in Constant (Euclidean) Self–Dual Background Fµν= ⋆Fµν

IntroduceF^{2}=^{1}

4FµνF^{µν}; natural dimensionless parameter γ = ^{2e}_{m}^{F}2 .
Scalar particle, charge e, mass m, admits weak coupling expansion

L ∼ − m^{4}
16π^{2}

+∞

∑

n=1

B2n+2

2n (2n + 2)(2eF
m^{2} )

2n+2

. Computing Boreltransform, inverse Boreltransform is

S_{0}L =
e^{2}F^{2}
16π^{2} ∫

+∞

0

ds s (

1
sinh^{2}s −

1
s^{2} +

1
3)e^{−}^{2s}^{γ}.
There are two possible self–dual backgrounds:

Magnetic–likebackground: F ∈R, series isalternating⇒ Borel poles onpositive imaginary axis.

Electric–likebackground: F ∈iR, series isnot–alternating⇒ Borel poles on thepositive real axis.

Nonperturbative Structure of Topological Strings c= 1 Strings and the Resolved Conifold

### Borel Analysis Example

Charged Scalar Particle in Constant Self–Dual Background: The Schwinger Effect

Nonperturbativeambiguity: to perform integration on real axis still need to specifyprescriptionin order to avoid poles.

Unitarity dictates contour: integral picks up contributions of all poles as if real axis is approached from above≡ a+iprescription (unitary +i prescription guaranteesprobability of Schwingerpair–production ratein scalar electrodynamics is positive number between 0and 1).

Lagrangian develops animaginary part (sum over residues).

Non–alternating (electric) series & unitarity prescription:

Im L = em^{2}F
32π^{3}

+∞

∑

n=1

(2π n + γ

n^{2})e^{−}^{2πn}^{γ} .

Nonperturbative Structure of Topological Strings c= 1 Strings and the Resolved Conifold

### Gaussian Matrix Model

Borel transform

B[F_{G}](ξ) =

+∞

∑

g=2

F_{g}^{G}(t)

(2g − 3)!ξ^{2g}^{−2}.

No poles on the positive real axis ⇒ inverse Borel transform

S_{0}F_{G}(g_{s}) = −
1
4∫

+∞

0

ds s

⎛

⎝ 1

sinh^{2}(^{g}_{2t}^{s}s)− (
2t
g_{s})

2 1
s^{2} +

1 3

⎞

⎠
e^{−s}.

Coincides with one–loop effective Lagrangian for charged scalar particle in constant self–dual electromagnetic field of magnetictype (here γ = 1/N).

Nonperturbative Structure of Topological Strings c= 1 Strings and the Resolved Conifold

### c = 1 Strings

Consider instead imaginarycouplingg¯_{s}=ig_{s} ⇒One–loop effective
Lagrangian corresponding to electric background[Gross-Klebanov].
Borel integral representation

S_{0}Fc=1(g¯s) = 1
4∫

+∞

0

dσ σ ( 1

sin^{2}σ − 1
σ^{2} −1

3)e^{−}^{2tσ}^{gs}^{¯}
has an integrand withpoleson the positive real axis.

Unitarity prescription Disc Fc=1(¯gs) = − i 2π ¯gs

+∞

∑

n=1

(2πt n + g¯s

n^{2})e^{−}^{2πt n}^{¯}^{gs} =F^{(1)}(g¯s) +F^{(2)}(g¯s) + ⋯.
Relate to large–order and reconstructfull perturbative expansion"

F^{(1)}(g¯_{s}) = − i

¯ gs

(t + g¯_{s}

2π)e^{−}^{2πt}^{gs}^{¯} .

Nonperturbative Structure of Topological Strings c= 1 Strings and the Resolved Conifold

### Chern–Simons Matrix Model

Borel transform

B[FCS](ξ) =

+∞

∑

g=2

F_{g}^{CS}(t)

(2g − 3)!ξ^{2g}^{−2}.

No poles on the positive real axis ⇒ inverse Borel transform

S0FCS= − 1 4 ∑

m∈Z

∫

+∞

0

ds s

⎛

⎜

⎝

1

sinh^{2}(_{2(t+2πim)}^{g}^{s} s)

− (

2 (t + 2πim)
g_{s} )

2 1
s^{2}+

1 3

⎞

⎟

⎠
e^{−s}.

Sum over all integerm corresponding tosum over infinite number of one–loop effective Lagrangian for charged scalar particle in constant self–dual electromagnetic field ofmagnetictype.