The Resurgence of Instantons in String Theory Part 1: The Mathematical Framework

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The Resurgence of Instantons in String Theory

Part 1: The Mathematical Framework

Ricardo Schiappa

(Instituto Superior T´ecnico)

Taiwan String Theory Workshop 2011

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Outline

1 Introduction

Divergence of Perturbation Theory Borel Transform

Instantons and Large–Order Behavior

2 Resurgence from Hyperasymptotics

Hyperasymptotics for Integrals with Saddles Resurgence and Stokes’ Phenomenon A Simple Example

Instantons and Stokes’ Phenomena

3 Borel Analysis, Resurgence and Asymptotics

Asymptotic Series and Borel Transforms Revisited Alien Calculus and the Stokes Automorphism Trans–Series and the Bridge Equations Stokes Constants and Asymptotics

4 The Airy Function and Stokes Discontinuities

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Outline

1 Introduction

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Outline

1 Introduction

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Outline

1 Introduction

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Introduction

Outline

1 Introduction

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Introduction Divergence of Perturbation Theory

Outline

1 Introduction

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Perturbation Theory

Perturbative expansion of some physical quantity (e.g., energy of the ground state, free energyF (g )), with perturbative expansion

parameter g:

F (g ) =

+∞

∑

k=0

Fkg^k.

In many examples the coefficients behave asF_k ∼k!at largek, rendering the perturbative expansion divergent.

In fact [Dyson], physical arguments tell us that many series should have zero radius of convergence.

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Example

Ground State Energy of Anharmonic Oscillator

V = 1

2x²−g x⁴

Wheng > 0 the theory has anunstablevacuum at the origin, which decays via instanton tunneling. This vacuum gets stabilized when g < 0.

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Example

Instantons in Quantum Mechanics

Instantons mediate thedecay of the false vacuum.

Ground state energy will have branch cutalong real, positiveg axis, with purely imaginary discontinuity—associated toinstabilityof potential which becomes unbounded for negative values of coupling.

How to deal with the fact that the perturbative expansion has zero convergence radius?

How do we associate a valueto the divergent sum?

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Introduction Borel Transform

Outline

1 Introduction

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Borel Transform

Introduce Borel transform of the asymptotic series as B[F ](s) =

+∞

∑

k=0

F_k k!s^k,

removing the divergent part of the coefficientsF_k and rendering B[F ](s)with finite convergence radius.

IfF (g ) originally had afiniteradius of convergence, B[F ](s) would be anentire function in the Borel complex s–plane.

In general B[F ](s) will have singularitiesand it is crucial to locate them in the complex plane.

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Inverse Borel Transform

IfB[F ](s)hasnosingularities for s ∈ R⁺ one may analytically continue it and define inverseBorel transform via Laplace transform

SF (g ) =∫

+∞

0 ds B[F ](g s) e^−s.

Function SF (g ) has, by construction, thesame asymptotic expansion as F (g )and provides asolution to original question; it associates a value to the divergent sum.

If the functionB[F ](s)haspolesor branch cuts on therealaxis, things get more subtle: to perform the integral one needs to choose a contour whichavoids such singularities.

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Nonperturbative Ambiguity

Choice of contour introduces nonperturbativeambiguity in reconstruction of function, renderingF (g ) non–Borel summable.

Different integration paths produce functions with the same

asymptotic behavior, but differing byexponentially suppressed terms.

In the presence of a pole singularity at distance Afrom the origin, on the real axis, one may define contoursC_±, either avoiding singularity from above, leading toS₊F (g ), or from below, leading toS₋F (g ).

These two functionsdifferby a nonperturbativeterm S₊F (g ) − S₋F (g ) ∼ i exp (−A

g).

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Example

Instantons in Quantum Mechanics Revisited

Borel transform of ground state energy has singularities on positive real axis, leading to ambiguity of order ∼i e^−1/g.

In the quantum mechanical example the nonperturbative ambiguity has clear physical interpretation: signals the presence—at positive g—ofinstantons mediating decay from unstable to true vacuum, via tunneling under local maximum of the potential.

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Introduction Instantons and Large–Order Behavior

Outline

1 Introduction

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Instantons and Partition Function

Write for full partition function:

Z (g ± i) = Z⁽⁰⁾(g ) ±1

2Disc Z (g ).

DefinesZ⁽⁰⁾ and discontinuity across branch cut, Disc Z (g ) = Z (g + i) − Z (g − i).

Z⁽⁰⁾ given by path integral aroundperturbative vacuum (the zero–instanton configuration).

Leading contribution toDisc Z (g ) ∼ Z⁽¹⁾(g ) given by path integral aroundone–instantonconfiguration.

May compute Z⁽¹⁾(g ) ∼ e^−1/g, exponentially suppressed for smallg as compared toZ⁽⁰⁾.

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How do we compute Z

⁽⁰⁾

and Z

⁽¹⁾

?

Path Integral Approach

C+ C-

S1 S2

Complex plane for functional integration: C⁺ andC⁻ are rotated contours for g > 0.

Z⁽⁰⁾ computed as integral oversum of both contours: contribution of saddle–point at origin.

Disc Z (g )computed on difference of rotated contours: contribution of sub–leading saddle–points.

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Instantons and Free Energy

Free energyF = log Z.

Zero–instantonsector has perturbative (asymptotic) expansion F⁽⁰⁾(g ) =

+∞

∑

n=0

F_n⁽⁰⁾gⁿ. Contribution from `–instantonsector has expansion

F^(`)(g ) = i g^`be⁻^`A^g

+∞

∑

n=0

F_n^(`)₊₁gⁿ, A=one–instanton action, b=characteristic exponent, F_n^(`)=n–loop contribution around `–instanton configuration.

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Instantons and Large–Order

Assuming F (g )analytic in Cwith cut along [0, +∞), Cauchy integral representation yields

F (g ) = 1 2πi∫

+∞

0 dw Disc F (w ) w − g − ∮

(∞)

dw 2πi

F (w ) w − g. In certain situations, it is possible to show that last integral above doesnotcontribute.

One thus obtainsa remarkable connection between perturbative and nonperturbative expansions.

Large–order coefficientsin asymptotic series expansion:

F_k⁽⁰⁾= ∫

+∞

0

dz 2πi

F⁽¹⁾(z) z^k⁺¹ ∼

Γ (k + b) 2πA^k^+b

+∞

∑

n=0

Γ (k + b − n)

Γ (k + b) F_n+1⁽¹⁾Aⁿ.

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Instantons Yield Large–Order

F_k⁽⁰⁾≃

Γ (k + b) 2πA^k^+b

⎡

⎢⎢

⎣ F₁⁽¹⁾+

F₂⁽¹⁾A k + b − 1+

F₃⁽¹⁾A²

(k + b − 2)(k + b − 1)+ ⋯

⎤

⎥⎥

⎦ .

Computation of one–loop one–instanton partition function determines leading order ofasymptotic expansion for perturbative coefficientsof zero–instantonpartition function.

Higher loop corrections yield successive ¹_k corrections.

Ideas addressed a long time ago[Bender-Wu]for the quartic

anharmonic oscillator ⇒ Perturbative computation of ground–state energy

E_k ∼ (−1)^k⁺¹

√ 6 π³²

3^kΓ (k + 1 2). Impressivenumerical tests in the early 70s!

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Resurgence from Hyperasymptotics

Outline

1 Introduction

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Resurgence from Hyperasymptotics Hyperasymptotics for Integrals with Saddles

Outline

1 Introduction

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Steepest–Descent Approximation

Usesteepest descent to find asymptotic expansion, as ∣κ∣ → ∞ with κ = ∣κ∣ e^iθ, of “partition function”

Z (κ) =∫

Cdz e^{−κW (z)}, with C a contour we specify below.

Typical calculation:

Saddle points{zk}k=1,2,⋯ such thatW^′(zk) =0.

Chosen reference saddle–pointzn, contour of integrationC deformed to infinite orientedpath of steepest descentthroughzn,Cn(θ), defined as

Im [κ (W (z ) − W (zn))] =0, withκ (W (z) − W (z_n))increasing away fromz_n.

Obtain “partition function”Z_n(κ)evaluated on then–th saddle.

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Steepest–Descent Partition Function

Partition function evaluated on then–th saddle Zn(κ) ≡ 1

√κe^{−κW (z}ⁿ⁾Z_n(κ), Z_n(κ) =√ κ∫

Cn(θ)dz e−κ(W (z)−W (zⁿ)). Z_n(κ)will display Stokes phenomenain the form of discontinuity associated to jump in steepest–descent path whenever it passes through one of other saddles,k /=n.

Integral evaluated via steepest–descent method; obtain function ofκ for each saddle,n, given by series in negative powers ofκ

Z_n(κ) ∼

+∞

∑

g=0

ζg(n)

κ^g , ζ_g(n) = Γ(g +1 2) ∮

zn

dz 2πi

1

(W (z) − W (z_n))^g⁺

1 2

.

This series is asymptotic.

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Resurgence from Hyperasymptotics Resurgence and Stokes’ Phenomenon

Outline

1 Introduction

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Resurgence: Preliminaries

Why is the series for Z_n(κ)asymptotic?

Understand this divergence as consequence of the existence of other saddles{z_k_/=n}, through which C_n does not pass [Berry-Howls]. Free to choose the reference saddle n at will ⇒ all possible

asymptotic series related by requirement of mutual consistency, also known as principle of resurgence.

Each divergent series contains, in its late terms, and albeit in coded form due to divergent nature, all terms associated to the asymptotic series from all other saddles.

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Stokes’ Phenomenon: Preliminaries

Asymptotic series Z_n(κ)only holds inwedgeof complex κ–plane.

Varyingθ / steepest–descent contour through saddlez_n ⇒

Discontinuityifθsuch that contour passes throughsecond saddle,zm: θ → −σ_nm= −arg (W (z_m) −W (z_n)).

At this point,exponentially suppressed contributions to Z_n(κ)

“suddenly” appear, eventually become of order one.

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Adjacent Saddles

zm adjacenttozn if it may be reached fromzn through steepest–descent path, i.e.,θ = −σnm and thusarg (W (z) − W (zn)) =σnm. Adjacent contourthrough adjacent saddle: steepest–descent contourC_m(−σ_nm), throughz_m.

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Resurgence

Hyperasymptotics begins withfinite truncation of asymptotic series

Z_n(κ) =

N−1

∑

g=0

ζg(n) κ^g +

1 2πi

1 κ^N ∑

m

∫

∞⋅e^−iσnm

0 dη η^N⁻¹

1 −^η_κ e^−ηW^nmZ_m(η)

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

R^(N)n (κ)

,

summed overalladjacent saddles to z_n,{z_m}, and definedsingulant for every adjacent saddleW_nm≡W (zm) −W (zn) ≡ ∣W_nm∣e^iσ^nm. Hyperasymptotic optimaltruncation: choose N^∗ such that

approximation error reduced from polynomially to exponentiallysmall.

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Resurgent Relations

Resurgent expression with N = 0[Berry-Howls],

Z_n(κ) = 1 2πi∑

m

∫

∞⋅e^−iσnm

0 dηe^−ηW^nm η −^η_κ²

Z_m(η).

Follows formal resurgent relation:

ζ_g(n) = 1 2πi∑

m +∞

∑

h=0

(g − h − 1)!

W_nm^g^−h

ζ_h(m).

Expresses late terms of asymptotic series atgiven saddle as sumover early terms of corresponding asymptotic series atadjacent saddles.

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Crossing the Stokes’ Line

The resurgence formula precisely incorporates Stokes phenomenon: appearance of suppressed exponentialterms as steepest–descent contour C_n(θ)sweeps through one of the adjacent saddles, m, i.e., asθcrosses theStokes lineC_n(−σnm).

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Stokes’ Phenomenon

Discontinuityacross the Stokes line Disc Zn(κ)∣

θ=−σnm

≡ Z_n(∣κ∣ eⁱ^(−σ^nm⁺⁰⁺⁾) − Z_n(∣κ∣ eⁱ^(−σ^nm⁺⁰⁻⁾) /=0.

Use resurgence formula to obtain Disc Zn(κ)∣

θ=−σnm

=e^−κW^nmZ_m(κ).

Discontinuityexponentially smallas, on Stokes line, κWnm∈R⁺. Notice that this is not discontinuity of the functionZ (κ) but rather discontinuity of asymptotic approximation to Z (κ).

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Resurgence from Hyperasymptotics A Simple Example

Outline

1 Introduction

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The Gamma Function

Gamma function via Euler’s integral Γ(κ) =∫

+∞

0 dw w^κ⁻¹e^−w, Re (κ) > 0, and

log Γ(κ) = (κ −1

2)log κ − κ +1

2log 2π + Ω(κ).

Ω(κ)meromorphic with simple poles at κ = −n,n ∈ N0. Stirlingseries

Ω(κ) ∼

+∞

∑

g=1

B2g

2g (2g − 1) 1 κ^2g⁻¹, valid as ∣κ∣ → +∞, in the sector∣arg(κ)∣ < π.

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Rewrite the Gamma Function

Change variables

Γ(κ) = κ^κ∫

+∞

−∞ dz e^{−κW (z)}, with potential W (z) = e^z−z andsaddlesz_k =2πik.

Selectreference saddlez0=0, defineΓ0(κ) ≡

√

2π κ^κ⁻¹²e^−κG₀(κ),

G₀(κ) =

√ κ 2π∫

C0(θ)dz e−κ(W (z)−1), Re (κ) > 0, with log G₀(κ) = Ω(κ).

Identify all saddles{z_m}_m_/=0 asadjacent saddles to z₀: singulantsare W_0m = −2πim.

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Stokes’ Phenomena in the Gamma Function

Discontinuities evaluated on Stokes lines,κ = ±i∣κ∣, always exponentially suppressed.

Disc Ω(κ)∣

θ=±^π₂ =

+∞

∑

m=1

e^±2πiκm

m .

Understand these terms as instanton contributions,“instanton action”S_inst^(m)=W (zm) −W (z0) = W_0m = −2πim.

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Stokes’ Phenomena as Instanton Contributions

Instanton(s) with least action, S⁽⁻¹⁾=2πi,S⁽¹⁾= −2πi, control leading large–order behavior of perturbative expansion "

Identify Stokes discontinuities with instanton contributions?

Multi–instanton action S^(m)=mS⁽¹⁾ ⇒Disc Ω(κ)∣

θ=±^π₂ exact:

includes all multi–instanton corrections, to all loop orders.

Fully reconstruct perturbative coefficientsΩ⁽⁰⁾_g out of complete multi–instanton series"

Identify Stokes discontinuities with instanton contributions!

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Resurgence from Hyperasymptotics Instantons and Stokes’ Phenomena

Outline

1 Introduction

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Instantons as Stokes’ Phenomena in Matrix Models

Exact free energy for Gaussian matrix modelis FG= 1

2N²log gs−1

2N log 2π + log G2(N + 1), with G2(z) Barnes function,G2(z + 1) = Γ(z)G2(z).

Integral representation forlog G2(N + 1)

= 1

2N log 2π −1

2N(N − 1) + N log Γ(N) −∫

N

0 dn log Γ(n).

Stokes structure of Barnesin termsof Stokes structure of Gamma!

Disc log G₂(N + 1)∣

θ=±^π₂ =

+∞

∑

m=1

( N m∓

1

2πim²)e^±2πiNm.

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Stokes’ Phenomena and Quantum Geometry

At Sokes’ lines N = ±i∣N∣obtain Disc F_G=

i 2π ¯g_s

+∞

∑

m=1

( 2πt

m +

¯ g_s

m²)e⁻^{2πt m}^gs^¯ , used t = gsN and restricted to the Stokes line θ = +^π₂.

Matrix models and topological strings: regardinstantonsas Stokes phenomena of the 1/N expansion ⇒ Nonperturbative ambiguity artifact of semiclassical, largeN analysis.

Only insemiclassical limit notion oftarget spacein holographically dual theory emerges ⇒ Considering exact free energies as

nonperturbative definitions, “exact quantum” target spaces very different from semiclassical ones; at nonperturbative level notion of target space as smooth geometryis lost.

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Borel Analysis, Resurgence and Asymptotics

Outline

1 Introduction

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Borel Analysis, Resurgence and Asymptotics Asymptotic Series and Borel Transforms Revisited

Outline

1 Introduction

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The String Theoretic Asymptotic Series

String theory may be defined perturbatively, as topological genus expansion, with two couplings,α^′,gs,

F (g_s; {t_i}) =

+∞

∑

g=0

g_s^2g⁻²F_g(t_i),

whereF = log Z is string free energy andZ partition function. At fixed genus: free energiesFg(ti) perturbatively expanded in α^′. Topological string theory: {t_i} moduli areK¨ahler parametersin A–model andcomplex structure parameters in B–model.

Theα^′ expansion is the milder one, withfinite convergence radius.

Topological genus expansion: one is faced with string theoretic large–order behaviorFg ∼ (2g )! rendering the topological expansion as an asymptotic expansion[Shenker].

How can one go beyond perturbation theory in g_s and define nonperturbative string theory in general?

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Asymptotic Series Set–Up: Perturbative Contribution

Consider asymptotic perturbative expansion (do perturbation theory aroundz ∼ ∞, rather thangs∼0 as usual)

F (z) ≃

+∞

∑

g=0

Fg

z^g⁺¹. Coefficients behave asFg∼g !;

Nonperturbative instanton corrections∼e^−nAz; Ainstanton action,ninstanton number[Zinn–Justin].

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Asymptotic Series Set–Up: Multi–Instanton Contribution

Perturbative expansion around (nonperturbative) contribution at given fixed instanton number

F⁽ⁿ⁾(z) ≃ z^−nβe^−nAz

+∞

∑

g=1

Fg⁽ⁿ⁾

z^g . β characteristic exponent;

Fg⁽ⁿ⁾ theg–loop contribution aroundn–instanton configuration;

In many examplesalsocoefficientsF_g⁽ⁿ⁾∼g !, rendering all multi–instanton contributionsasymptotic.

Possible to understand asymptotics, ing, of multi–instanton

contributions Fg⁽ⁿ⁾, in terms of coefficientsFg⁽ⁿ^′⁾, with n^′ closeto n.

This means that the asymptotic expansions are resurgent! [´Ecalle]

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Revisiting the Borel Transform

If we do not know the exact function, F (z), but only its asymptotic series expansion, how do we associate a valueto the divergent sum?

Borel transform

B [ 1

z^α⁺¹] (s) = s^α Γ(α + 1). Borel transform of asymptotic series

B[F ](s) =

+∞

∑

g=0

Fg

g ! s^g. Borel resummation ofF (z)alongθ

S_θF (z) =∫

e^iθ∞

0 ds B[F ](s) e^−zs.

S_θF (z)has by construction same asymptotic expansion as F (z)and may provide asolution to our original question.

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Lateral Borel Resummations

Lateral Borel resummations along θ,S_θ±F (z):

Θ

Choice of contour introduces nonperturbativeambiguity S_θ+F (z) − Sθ⁻F (z) ∝∮

(A)ds e^−zs

s − A ∝e^−Az.

Ambiguities arise reconstructing function along different directions ⇒ Different integration paths yield functions with sameasymptotic behavior, differing by exponentially suppressed terms.

To be fully precise about these, need to dwell into resurgence!

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Borel Analysis, Resurgence and Asymptotics Alien Calculus and the Stokes Automorphism

Outline

1 Introduction

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Simple Resurgent Functions

Asymptotic expansion is said a simple resurgent function if its Borel transform, B[F ](s), only has simple polesor logarithmic branch cuts as singularities; near each singular pointω

B[F ](s) = α

2πi (s − ω)+Ψ(s − ω)log (s − ω)

2πi +Φ(s − ω), whereα ∈ C andΨ,Φare analyticaround the origin.

Simple resurgent functions allow for resummationof formal power series along any direction in complex s–plane

⇒Family of sectorialanalytic functions{S_θF (z)}.

Up to nonperturbative ambiguities, in different sectors one obtains different resummations ⇒ Need to fully understand Borelsingularities in order to “connect”sectorial solutions together.

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Alien Derivative

Alien calculus based uponalien derivative: operator which, when acting onresurgent functions, behaves as a derivative∆_ω [´Ecalle]. For simpleresurgent functions:

ω notasingularpoint ⇒∆_ωF (z) = 0.

ω asingularpoint with Borel transform

B[F ](ω + s) = α

2πi s + B[G ](s)log s

2πi +holomorphic

⇒ S∆ωF (z) = α + Sarg ωG (z).

Alien derivatives encode the whole singular behavior of the Borel transform—how muchB[F ](s)“jumps” at a singularity—allowing for the“connection” of sectorial solutions.

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The Stokes’ Automorphism

Singular direction θ: direction along which there are singularities in Borel complex plane.

In original complex z–plane such direction is known as Stokes line.

Connecting distinct sectorial solutions on both sides of such direction entails understanding their “jump”, accomplished via theStokes’

automorphism,S_θ,

S_θ+= S_θ−○S_θ≡ S_θ−○ (1 − Discθ⁻).

Actionof S_θ on resurgent functions translates into the required connection of distinct sectorial solutions, across asingular directionθ.

In particular,

S_θ+− S_θ− = −S_θ−○Disc_θ⁻,

Disc_θ encodes full discontinuityof resurgent function acrossθ!

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The Stokes’ Automorphism: Geometrical Description

Geometrically, Discθ⁻ is the sum over allHankel contours which encircle each singular point in θ–direction, on the left, and part off to infinity, on the right:

Θ

=

Θ

Left Borel resummation = right Borel resummation plus discontinuity of singular direction (the sum over Hankel contours aroundsingular points).

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Stokes’ Automorphism and Alien Derivative

The main interesting point is that S_θ=exp⎛

⎝

∑

{ωθ}

e^−ω^θ^z∆_ω_θ⎞

⎠ ,

where{ωθ}denote all singular points along theθ–direction.

For singularities along the θ–direction inordered sequence, S_θ+F (z) = S_θ−F (z) +

+ ∑

r≥1;{ni≥1}

1

r !e^−(ωⁿ¹^+ωⁿ²^+⋯ω^nr^)zS_θ−(∆_ω_n1∆_ω_n2⋯∆_ω_nrF (z)) .

Given all possiblealien derivatives, this result provides the necessary connection, thus allowing for full construction of exact

nonperturbative solutionalongside with its Riemann surface domain.

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Stokes’ Automorphism and Multi–Instantons

Positive real axis,θ = 0, with Borel singularities located at multi–instantonpointsnA with n ∈ N^∗,

S₀=exp (

+∞

∑

n=1

e^−nAz∆_nA) =1 + e^−Az∆_A+e^−2Az(∆_2A+1

2∆²_A) + ⋯. Question that remains: how tocomputealien derivatives in a simple and systematic fashion? ... Answer arises in the bridge equations, constructing a “bridge” between ordinary and alien calculus.

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Borel Analysis, Resurgence and Asymptotics Trans–Series and the Bridge Equations

Outline

1 Introduction

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Trans–Series Set–Up

Considertrans–series ans¨atz for a resurgent function, F (z, σ) =

+∞

∑

n=0

σⁿF⁽ⁿ⁾(z).

Formal asymptotic power series F⁽⁰⁾(z) ≃ ∑^+∞_g₌₀^F

g(0)

z^g+1. n–instanton contributions areF⁽ⁿ⁾(z) ≃ z^−nβe^−nAz ∑^+∞_g₌₁

Fg⁽ⁿ⁾

z^g . DefineF⁽ⁿ⁾(z) ≡ e^−nAzΦn(z), withΦn(z)further formal asymptotic power series assumed in the following assimple resurgent functions.

σ is nonperturbative ambiguity or trans–seriesparameter, with trans–series an expansion in C[[z⁻¹, σ e^−Az]].

Resurgence framework: most general solutionto non–linear system!

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Trans–Series Framework

Use trans–series innon–linear equation satisfied byF (z).

Yield backs non–linear equation forF⁽⁰⁾(z), which is now to be solvedperturbatively.

Yieldslinearandhomogeneous equation for F⁽¹⁾(z).

Yieldslinearbut inhomogeneousequations forF⁽ⁿ⁾(z),n ≥ 2.

Feasible to solve forallmembers of this hierarchy of equations and fully compute the trans–series ans¨atz.

Asymptoticsof trans–series coefficientsFg⁽ⁿ⁾ exactly determined in terms ofneighboring coefficients Fg⁽ⁿ^′⁾, with n^′ close to n, and in terms of finite number of Stokes constants.

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Bridge Equations

Introducepointed alien derivative

∆˙_ω ≡e^−ωz∆_ω, commuting with usual derivative [ ˙∆ω,_dz^d] =0.

Immediately implies that linear differential equation which

∆˙_`AF (z, σ)satisfies (assumed to be first order for simplicity) is exact same differential equation as ^∂F_∂σ(z, σ)satisfies!

Then it must be the case that

∆˙`AF (z, σ) = S`(σ)∂F

∂σ(z, σ);

Ecalle’s´ bridge equation, relating alien derivatives to familiar ones!

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Bridge Equations and Analytic Invariants

Introducedegree such thatdeg (σⁿe^mAz) =n + m. Using bridge equations, deg F (z, σ) = 0 immediately yieldsdeg S_`(σ) = 1 − `, i.e.,

S_`(σ) = S_`σ¹^−`, ` ≤ 1.

Can rewrite bridge equationsas

∆_`AΦ_n=

⎧⎪

⎪

⎨

⎪⎪

⎩

0, ` > 1,

S_`(n + `) Φ_n_+`, ` ≤ 1.

Yieldsall alien derivatives, in terms of (possibly) infinite sequence of analytic invariants S`∈C,` ∈ {1, −1, −2, ⋯}.

Analytic invariants allow for full nonperturbative reconstruction of original functionF (z)! Generically analytic invariants are

transcendentalfunctions of initial data and quite hardto compute.

Resurgence: alien derivatives, encodingsingular behavior of Borel transform,find back original asymptotic power series one started off with (multiplied by suitableStokes’ constants).

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Bridge Equations and Borel Transform

Bridge/resurgence equations may be translated back to the structure of Borel transform, at least near eachsingularity `A.

For asimpleresurgent function with

Φ_n(z) =

+∞

∑

g=1

F_g⁽ⁿ⁾

z^g^+nβ and B[Φ_n](s) =

+∞

∑

g=1

F_g⁽ⁿ⁾

Γ(g + nβ)s^g^+nβ−1, it follows

B [Φn] (s + `A) = S_`(n + `) B [Φ_n+`] (s)log s

2πi, ` ≤ 1.

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Bridge Equations and Stokes’ Automorphism: θ = 0

Can use bridge equations to explicitly compute Stokes’ automorphism.

Action of S₀ onF (z, σ) entirelyencoded by action ofS₀ on several Φ_n(z), which vanishes whenever` > 1.

Stokes’ automorphism forpositive real axis:

S₀=exp (e^−Az∆A) =1 + e^−Az∆A+1

2e^−2Az∆²_A+ 1

3!e^−3Az∆³_A+ ⋯. Using ∆^N_AΦn= (S1)^N⋅ ∏^N_i₌₁(n + i ) ⋅ Φn+N it follows

S₀Φ_n=

+∞

∑

k=0

(n + k

n )S₁^ke^−kAzΦ_n_+k. Standard multi–instantonflavor!

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Bridge Equations and Stokes’ Automorphism: θ = π

Stokes’ automorphism fornegative real axis:

S_π=exp (

+∞

∑

`=1

e^Àz∆_−À) =1 + eÂz∆_−A+e^2Az(∆_−2A+1

2∆²_−A) + ⋯. Need∏^N_i₌₁∆_−`_(N+1−i)_AΦ_n. Orderingof alien derivatives important;

when computed at differentsingular points they do notcommute.

Alien derivatives vanishas soon as one considers ∆_−nAΦn=0, always get afinite sum. For example,

S_πΦ₀ = Φ₀, S_πΦ₁ = Φ₁,

S_πΦ2 = Φ2+S₋₁e^AzΦ1,

S_πΦ3 = Φ3+2S₋₁e^AzΦ2+ (S₋₂+S₋₁² )e^2AzΦ1.

Notquite standard multi–instantonflavor...

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Bridge Equations and Stokes’ Phenomena

Applying bridge equations directly toStokes’ automorphism, when ω = 1,arg ω = 0, find

S₊F (z, σ) = S₋F (z, σ + S₁).

S₁ acts as aStokes constant for the trans–series expression!

In original complex z–plane the Borel–plane singular–direction corresponds toStokes line ⇒Expression above describes Stokes phenomena of classical asymptotics—fully and naturally incorporated in the resurgence analysis.

The“connection” expression yields relation between coefficient(s) in trans–series solution, in different partsof its domain, or, on different sides of theStokes line.

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Borel Analysis, Resurgence and Asymptotics Stokes Constants and Asymptotics

Outline

1 Introduction

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Large–Order Dispersion Relation

Resurgence allows to understand fullasymptotic behavior of all multi–instanton sectors.

Large–order dispersion relation fromCauchy’s theorem[Zinn–Justin]

F (z) = 1 2πi∫

e^iθ⋅∞

0 dw Disc_θF (w ) w − z − ∮

(∞)

dw 2πi

F (w ) w − z. Function F (z)hasbranch-cutalong some direction (Stokes’

direction),θ, inC, and analytic elsewhere.

In certain situations possible to show byscaling argumentsthat integral around infinity doesnotcontribute [Bender–Wu]. In such cases Cauchy’s theorem providesremarkable connectionbetween perturbative and nonperturbative expansions.

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Asymptotics of Perturbative Series

Perturbative expansionF⁽⁰⁾(z) ≃ ∑^+∞_g₌₀^F

(0) g

z^g+1 =Φ0(z).

Via bridge equations, F⁽⁰⁾(z)hasdiscontinuities Disc₀Φ₀ = −

+∞

∑

k=1

S₁^ke^−kAzΦ_k, Disc_πΦ₀ = 0.

From perturbative expansion and dispersion relation above F_g⁽⁰⁾≃

+∞

∑

k=1

S₁^k 2πi

Γ (g − kβ) (kA)^g^−kβ

+∞

∑

h=1

Γ (g − kβ − h + 1)

Γ (g − kβ) F_h^(k)(kA)^h⁻¹, here used asymptotic expansions for multi–instantoncontributions.

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Explicit Asymptotics of Perturbative Series

Instructive toexplicitlywrite down first terms in double–series, Fg⁽⁰⁾ ≃ S1

2πi

Γ (g − β)

A^g^−β (F₁⁽¹⁾+ A

g − β − 1F₂⁽¹⁾+ ⋯) + +S₁²

2πi

Γ (g − 2β) (2A)^g^−2β

(F₁⁽²⁾+ 2A

g − 2β − 1F₂⁽²⁾+ ⋯) + ⋯.

Multi–instanton generalization of known result: relates coefficients of perturbative expansion@ zero–instanton sector with sum over coefficients of perturbative expansions @ all multi–instantonsectors.

One–loop one–instanton partition function determinesleading order of asymptotic expansion forperturbativecoefficients of

zero–instanton partition function, up to Stokes factor S1. Higher–loop contributions yield successive _g¹ corrections.

n–instanton contributions yield successiven^−g corrections.

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Asymptotics of Multi–Instanton Series

n–instanton contributionsF⁽ⁿ⁾(z) ≃ z^−nβe^−nAz ∑^+∞_g₌₁

F_g⁽ⁿ⁾ z^g . Bridge equations yield discontinuities Disc₀Φ_n andDisc_πΦ_n. From perturbative expansion and dispersion relation

Fg⁽ⁿ⁾ ≃

+∞

∑

k=1

(n + k n )

S₁^k 2πi

Γ (g − kβ) (kA)^g^−kβ

+∞

∑

h=1

Γ (g − kβ − h)

Γ (g − kβ) F_h^(n+k)(kA)^h+ +S₋₁

2πi(n − 1)Γ (g + β) (−A)^g+β

+∞

∑

h=1

Γ (g + β − h)

Γ (g + β) F_h⁽ⁿ⁻¹⁾(−A)^h+ ⋯ Relates coefficients ofperturbative expansion@ n–instanton sector with sums over coefficients of perturbative expansions @ all other multi–instantonsectors.

AllStokes factors now needed ⇒ Analysis essentially boiled down asymptotic problem tocomputing these Stokes factors!

Leading contribution to this asymptotics[Garoufalidis–Mari˜no].

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The Airy Function and Stokes Discontinuities

Outline

1 Introduction

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The Airy Function

Airy function,Z(x ), is a solution to Airysecond–order(linear) differential equation and is many times written as

Z(x ) = 1 2πi∫

Γdz e^{−V (z)}, whereV (z) = ¹₃z³−xz and Γa contour to specify.

Not every contour is admissible; there are only three directions going off to infinity with Re (¹₃z³−xz) > 0as z → ∞, leading to two homologically independentadmissible integration paths.

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The Airy Function and Steepest Descent

Saddle points: z_±^∗= ±

√x, with V (z_±^∗) = ∓²

3x³².

Saddle z₋^∗, formalasymptotic power series expansion [Delabaere]

Z_Ai(x ) = 1 2√

π x¹⁴ e⁻²³^x

3 2 +∞

∑

n=0

a_n x³ⁿ²

≡ 1

2√ π x¹⁴

e⁻¹²^Ax

3 2Φ₋¹

2

(x ),

A = ⁴₃ instanton action, and a_n= _2π¹ (−³₄)^{n Γ}⁽ⁿ⁺

5 6)Γ(n+¹₆)

n! .

Saddle z₊^∗

Z_Bi(x ) = 1 2√

π x¹⁴ e²³^x

3 2 +∞

∑

n=0

(−1)ⁿ a_n x³ⁿ²

≡ 1 2√

π x¹⁴ e⁺¹²^Ax

3 2Φ₊1

2

(x ).

Trans–seriessolution to Airy integral

Z(x , σ) = ZAi(x ) + σ ZBi(x ).

Simple as solution tolinearequation!