The Resurgence of Instantons in String Theory Part 2: Applications to Matrix Models and Strings

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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

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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

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Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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Resurgence and Trans–Series: Overview

Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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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⁻²u).

Convenient to choose variables such that α₁>0.

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Formal Trans–Series Solutions

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

u(z, σ) = u⁽⁰⁾(z) + ∑

n∈Nⁿ/{0}

σⁿz^−n⋅βe^{−n⋅α z}u⁽ⁿ⁾(z),

whereσ = (σ₁, . . . , 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 .

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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].

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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∈Nⁿ/{0}

σⁿ_±z^−n⋅βe^{−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.

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Matrix Model Basics

Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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Matrix Model Basics

Review and Notation

Hermitian one–matrix model, N × N matrix M Z_N =

1

vol(U(N))∫ dM e⁻^gs¹ ^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⁻^gs¹ ^∑^Nⁱ⁼¹^V^(λⁱ⁾,

∆(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.

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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) =_2πt¹ Im y (z ).

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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^′∣.

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Matrix Model Basics

Spectral Curve, Free Energies and Correlation Functions

Genus g free energies Fg(t).

Generating functions formulti–trace correlators W_h(z₁, . . . , z_h) = ⟨Tr 1

z₁−M⋯Tr 1 z_h−M⟩

(c), with genusexpansion

W_h(z1, . . . , z_h) =

+∞

∑

g=0

g_s^2g^+h−2W_{g ,h}(z1, . . . , z_h; t).

Key Result in Matrix Model Theory

Quantities F_g(t) andW_{g ,h}(z₁, . . . , z_h; t) can beentirely computed in terms of spectral curve y (z)alone [Ambj¨orn-Chekhov-Kristjansen-Makeenko, Eynard, Eynard-Orantin].

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Matrix Model Basics Multi–Cut Models

Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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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₌₁A_I, where A_I = [x_2I₋₁, x_2I] ares cuts and x₁<x₂< ⋯ <x_2s.

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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_Idλ 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_sN_I tI = 1

4πi∮

A^Idz y (z), with ∑^s_I=1t_I =t.

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Matrix Model Basics “c= 1” Models

Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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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

2µ²log µ − 1

12log µ +

+∞

∑

g=2

B2g

2g (2g − 2)µ²^−2g.

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The Gaussian Model: V

_G

( z) =

¹₂

z

²

Gaussian partition function

ZG= g

N2

s2

(2π)^N²

G2(N + 1).

Gaussian free energy

F_g^G(t) = B_2g

2g (2g − 2)t²^−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 =

√

z²−4t dz.

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Chern–Simons/Stieltjes–Wigert: V

_SW

( z) =

¹₂

( log z)

²

Introduce

q = e^g^s, [n]_q=qⁿ²−q⁻ⁿ². Stieltjes–Wigert partition function

ZSW= (gs

2π)

N

2 q¹²^N^(7N²⁻¹⁾

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)!Li₃_−2g(e^−t), g ≥ 2.

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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^−tz +

√

(1 + e^−tz)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!

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Approaching Topological String Theory

Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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A–Model ◁Mirror▷ B–Model

MapsΦ ∶ Σ_g → X³ 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⁻²F_g(t_i).

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

d ,g

g_s^2g⁻²Nd ,gQ^d ≡

+∞

∑

g=0

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

Gromov–Witten invariants ofX³,N_{d ,g}, count world–sheet instantons (number of curves of genus g and degree d).

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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⁻^gs¹.

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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.

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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^(dⁱ⁾(X )are GV invariants ofX; d_i K¨ahler class, r ≥ 0a spin label;

m ∈ Zwinding number (M–theory onX ×S¹ →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 sin²(^s

2)

e⁻^2πs^gs ^{(t+i m)}.

Recoverperturbativegenus expansion from integral representation!

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Instantons and String Theory

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

F⁽⁰⁾(gs) =g_s²F⁽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^/2e⁻

√`A z

+∞

∑

n=0

F_n^(`)₊₁zⁿ^/2. Herez = g_s².

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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

⎛

⎝

F₁⁽¹⁾(t) +F₂⁽¹⁾(t) A(t) 2g + b − 1 + ⋯

⎞

⎠ .

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

Compute analytically andcheck numerically!

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Instantons in One–Cut Matrix Models

Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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Effective Potential and Instanton Sectors

Effective potential constant along cut C = [a, b], where there is local minimum, and haslocal maximumat pointx₀.

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].

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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(x₀) =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].

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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⁾₊₁g_sⁿ. Instanton action

A = V_h,eff(x₀) −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 pointx₀.

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

F₁⁽¹⁾and F₂⁽¹⁾ dependuniquely on data specified byspectral curve.

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Analytical Results

One and Two loops

F₁⁽¹⁾= −ib − a 4

¿ Á Á ÁÁ À

1

2πM^′(x0)[(x0−a)(x0−b)]

5 2

,

F₂⁽¹⁾ F₁⁽¹⁾

= 1

4(a − b)

√

(x₀−a)(x₀−b) (

(x0−b)M^′(a) M²(a) −

(x0−a)M^′(b) M²(b) ) −

−

√

(x0−a)(x0−b)

12(a − b)² (8(x0−a) + 17(a − b)

(x0−a)²M(a) +8(x0−b) + 17(b − a) (x0−b)²M(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))³²

+

140 (2x₀− (a + b))²+33(a − b)² 96M^′(x₀) ((x₀−a)(x₀−b))⁵²

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Instantons in One–Cut Matrix Models The Quartic Model

Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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Data and Set–Up

Potential V (z) =¹₂z²+λz⁴.

Single cutC = [a, b] ≡ [−2α, 2α], with α²= ¹

24λ(−1 +

√

1 + 48λt).

Spectral curve y (z) = M(z)

√

z²−4α², with M(z) = 1 + 8λα²+4λz². Non–trivial saddle–pointsx₀²= − ¹

4λ(1 + 8λα²). Instanton action

A = −

√ 3 α² 4 (1 − α²)

√

4 − α⁴−2 log [

√ 3

√

−2 + α²+

√

−2 − α²] + +log 4 (1 − α²).

Computed F_g via orthogonal polynomials up to genusg = 10 [Bessis-Itzykson-Zuber].

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The Effective Potential

-x0 x0

a b

C

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Large–Order Results

Instanton Action

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

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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:

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

Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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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

u²(z) −1

6u^′′(z) = z.

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

F_ds⁽¹⁾(z) = i 8 ⋅ 3³⁴√

π

z⁻⁵⁸ exp (−8√ 3

5 z⁵⁴) {1 − 37 64

√

3z⁻⁵⁴ + ⋯}. Can actually compute full multi–loopcorrections!

ComputeFg numerically up to arbitrary genus.

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Large–Order Results

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

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Multi–Instantons and Multi–Cuts

Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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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

ζ₁^N¹⋯ζ_s^N^sZ (N1, . . . , Ns). Different sectors regarded asinstanton sectors of multi–cut model!

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Instantons in the Multi–Cut Phase

Z (N1, . . . , Ns) partition function in one sector,Z (N₁^′, ⋯, N_s^′) partition function corresponding to different choice of filling fractions ⇒

Z (N₁^′, . . . , N_s^′) Z (N1, . . . , Ns)

∼exp{−1 gs

s

∑

I=1

(N_I−N_I^′)∂F₀(t_I)

∂tI

}.

If we pick set of filling fractions(N₁, . . . , 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}.

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Multi–Instanton Sector in the Two–Cut Case

Reference configuration(N₁, N₂)

Z =

N1

∑

`=−N2

ζ^`Z (N1−`, N2+`) = Z⁽⁰⁾(N1, N2)

N1

∑

`=−N2

ζ^`Z^(`).

HereZ⁽⁰⁾(N1, N2) =Z (N1, N2) andZ^(`)= ^Z^(N¹^−`,N²^+`)

Z⁽⁰⁾(N1,N2) . Change variables t = t₁+t₂,s = ¹₂(t₁−t₂)

Z^(`) =q^`2² exp (−`A gs

) {1 − g_s(` ∂_sF₁+

`³

6 ∂³_sF₀) + O(g_s²)}.

Ais theinstanton action, integral over spectral curve A =∫

x3

x2 dz y (z), andq = exp(∂_s²F0).

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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,ϑ₃(q ∣ z) = ∑_`_∈Zq^`2² 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 − qⁿ) andlog φ(q) gives contribution toF₁ coming frominstanton/anti–instanton interactions in partition function.

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Multi–Instantons in the One–Cut Phase

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Multi–Instantons in the One–Cut Model

Degeneration limitt₂→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

√

2πˆq¹^/2{1 − g_s(∂_sF̂₁+1

6∂_s³F̂₀) + O(g_s²)}

F^(`) =

(−1)^`⁻¹g_s^`^/2

` (2π)^`² ˆ

q^`^/2{1 − ` g_s(∂_sF̂₁+ 1

6∂_s³F̂₀+ˆq) + O(g_s²)}

Evaluated att1=t andt2=0.

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

Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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Spectral Curve Geometry: One–Instanton Sector

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Large–Order Results: Instanton Action

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Orthogonal Polynomials: The Good and the Bad

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Nonperturbative Structure of Topological Strings

Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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Nonperturbative Structure of Topological Strings The Local Curve

Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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Data and Set–Up

ConsiderA–model topological strings ontoric CY (local curve) Xp= O(p − 2) ⊕ O(−p) → P¹.

Free energies at genus g depend on single complexified K¨ahler parameter t, associated to complexified areaofP¹.

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

y (λ) = 2

λ(tanh⁻¹[

√

(λ − a)(λ − b)

λ −^a^+b₂ ] −p tanh⁻¹[

√

(λ − a)(λ − b) λ +

√ ab

]).

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Instanton Action: A = V

_h,eff

( x

₀

) − V

_h,eff

( a)

x₀= 4ab (√

a−√

b)², p= 3, x₀= 2√

√a−ab√ b, p= 4,

V_h,eff(x) = − log (f1(x))(log (f1(x)) − 2 log(1 + 2f₁(x) (√

a−√

b)²) + log(1 + 2f₁(x) (√

a+√ b)²))−

−2Li2(− 2f₁(x) (√

a−√

b)²) − 2Li2(− 2f₁(x) (√

a+√

b)²) − log(a − b)² 4 log x−

−p log (f2(x))(log (f2(x)) + 2 log(1 −f₂(x) 2√

ab) − log(1 − 2f₂(x) (√

a+√ b)²))−

−2pLi2(−f₂(x) 2√

ab) + 2pLi2( 2f₂(x) (√

a+√ b)²) +p

2(log x)²+ p log(√ a+√

b)²log x,

f₁(x) = √

(x − a)(x − b) + x −a+ b 2 , f₂(x) = √

(x − a)(x − b) + x +√ ab.

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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 "

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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

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One and Two Loop Coefficients: Numerical Results

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

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Nonperturbative Structure of Topological Strings Hurwitz Theory

Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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Data and Set–Up

Hurwitz theory countsbranched covers of Riemann surfaces (restrict to coverings of sphereP¹ (“target”) by surfaces of genusg

(“worldsheets”)). Number of disconnected coverings of degreed counted by simple Hurwitz numberH_{g ,d}^P¹ (1^d).

Total partition function ⇒ Generating functional

Z^H(t_H, g_H) = ∑

g≥0

g_H^2g⁻²∑

d≥0

H_{g ,d}^P¹ (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}^P¹ (1^d)^●.

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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.

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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

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One and Two Loop Coefficients: Numerical Results

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

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

Outline

“c = 1” Models

Hurwitz Theory

7 Future Directions

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Borel Analysis Example

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

IntroduceF²=¹

4FµνF^µν; natural dimensionless parameter γ = ^2e_m^F2 . Scalar particle, charge e, mass m, admits weak coupling expansion

L ∼ − m⁴ 16π²

+∞

∑

n=1

B2n+2

2n (2n + 2)(2eF m² )

2n+2

. Computing Boreltransform, inverse Boreltransform is

S₀L = e²F² 16π² ∫

+∞

0

ds s (

1 sinh²s −

1 s² +

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.

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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 = em²F 32π³

+∞

∑

n=1

(2π n + γ

n²)e⁻^2πn^γ .

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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

S₀F_G(g_s) = − 1 4∫

+∞

0

ds s

⎛

⎝ 1

sinh²(^g_2t^ss)− ( 2t g_s)

2 1 s² +

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).

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c = 1 Strings

Consider instead imaginarycouplingg¯_s=ig_s ⇒One–loop effective Lagrangian corresponding to electric background[Gross-Klebanov]. Borel integral representation

S₀Fc=1(g¯s) = 1 4∫

+∞

0

dσ σ ( 1

sin²σ − 1 σ² −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

n²)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^¯ .

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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

sinh²(_2(t+2πim)^g^s s)

− (

2 (t + 2πim) g_s )

2 1 s²+

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.