Introduction to Resurgence, Trans-Series and Non-Perturbative Physics

Introduction to Resurgence, Trans-series and Non-perturbative Physics

Gerald Dunne

University of Connecticut

ICTS Bangalore January - February, 2018

GD & Mithat Ünsal, reviews: 1511.05977, 1601.03414, 1603.04924

recent KITP Program: Resurgent Asymptotics in Physics and Mathematics, Fall 2017 Physical Motivation

non-perturbative deﬁnition of non-trivial QFT, in the continuum• analytic continuation of path integrals • "sign problem" in ﬁnite density QFT • dynamical & non-equilibrium physics from path integrals (strong• coupling) uncover hidden ‘magic’ in perturbation theory • new understanding of weak-strong coupling dualities • infrared renormalon puzzle in asymptotically free QFT • non-perturbative physics without instantons: physical meaning• of non-BPS saddles exponentially improved asymptotics & resummation • Physical Motivation

sign problem: "complex probability" at ﬁnite baryon density?• 0 A e−SYM [A]+ln det(D/+m+i µγ ) D Z phase transitions and Lee-Yang & Fisher zeroes • Physical Motivation

equilibrium thermodynamics Euclidean path integral • ↔ Kubo-Martin-Schwinger: antiperiodic b.c.’s for fermions •

non-equilibrium physics Minkowski path integral • ↔ Schwinger-Keldysh time contours • quantum transport in strongly-coupled systems • Physical Motivation

what does a Minkowski path integral mean, computationally? i 1 A exp S[A] versus A exp S[A] D D − Z ~ Z ~

1.0

0.5

-6 -4 -2 2 4 6

-0.5

-1.0

− 2 x3/2 e√3 , x + ∞ 2 π x1/4 → ∞ 1 i 1 t3+x t e ( 3 ) dt  2π −∞ ∼  2 3/2 π Z  sin( 3 (−x) + 4 )  √ , x π (−x)1/4 → −∞  massive cancellations  Ai(+5) 10−4 • ⇒ ≈ Physical Motivation

what does a Minkowski path integral mean? • i 1 A exp S[A] versus A exp S[A] D D − Z ~ Z ~ since we need complex analysis and contour deformation to •make sense of oscillatory ordinary integrals, it is natural to expect to require similar tools also for path integrals an obvious idea, but how to make it work ... ? • Mathematical Motivation

Resurgence: ‘new’ idea in mathematics (Écalle, 1980; Stokes, 1850) resurgence = unification of perturbation theory and non-perturbative physics perturbation theory generally divergent series • ⇒ series expansion trans-series expansion • −→ trans-series ‘well-defined under analytic continuation’ • perturbative and non-perturbative physics entwined • applications: ODEs, PDEs, difference equations, fluid mechanics,• QM, Matrix Models, QFT, String Theory, ... philosophical shift: go• beyond the Gaussian approximation and view semiclassical expansions as potentially exact Resurgence, Trans-series and Non-perturbative Physics

1. Lecture 1: Basic Formalism of Trans-series and Resurgence

I asymptotic series in physics; Borel summation

I trans-series completions & resurgence

I examples: linear and nonlinear ODEs 2. Lecture 2: Applications to Quantum Mechanics and QFT

I instanton gas for double-well & periodic potential

I infrared renormalon problem in QFT

I from hyperasymptotics to Picard-Lefschetz thimbles 3. Lecture 3: Resurgence and Large N

I parametric resurgence

I Gross-Witten-Wadia Matrix Model

I Mathieu equation and Nekrasov-Shatashvili limit of = 2 SUSY QFT N Trans-series

an interesting observation by Hardy: • No function has yet presented itself in analysis, the laws of whose increase, in so far as they can be stated at all, cannot be stated, so to say, in logarithmico-exponential terms

G. H. Hardy, Divergent Series, 1949

deep result: “this is all we need” (J. Écalle, 1980) • also as a closed logic system: Dahn and Göring (1980) • Resurgent Trans-Series

Écalle: resurgent functions closed under all operations: • (Borel transform) + (analytic continuation) + (Laplace transform)

basic trans-series expansion in QM & QFT applications: • ∞ ∞ k−1 k l 2 2p c 1 f(g ) = ck,l,p g exp ln −g2 ±g2 p=0 k=0 l=1 X X X perturbative ﬂuctuations k−instantons quasi-zero-modes | {z }− 1 trans-monomial elements: g2, e g2 , ln(| g2), are{z familiar} | {z } • “multi-instanton calculus” in QFT • new: analytic continuation encoded in trans-series • new: trans-series coeﬃcients ck,l,p highly correlated • new: exponentially improved asymptotics • Resurgence

resurgent functions display at each of their singular points a behaviour closely related to their behaviour at the origin. Loosely speaking, these functions resurrect, or surge up - in a slightly diﬀerent guise, as it were - at their singularities J. Écalle, 1980

resurgence = global complex analysis with divergent series Perturbation theory

hard problem = easy problem + “small” correction • perturbation theory generally divergent series • →

∞ n e.g. QM ground state energy: E = n=0 cn (coupling) P n I Zeeman: cn ( 1) (2n)! ∼ − I Stark: cn (2n)! ∼ 1 I cubic oscillator: cn Γ(n + ) ∼ 2 n 1 I quartic oscillator: cn ( 1) Γ(n + ) ∼ − 2 I periodic Sine-Gordon (Mathieu) potential: cn n! ∼ I double-well: cn n! ∼ note generic factorial growth of perturbative coeﬃcients Perturbation theory

but it works ... Perturbation theory works

QED perturbation theory:

g − 2 1 α α 2 α 3 α 4 α 5 = − (0.32848...) + (1.18124...) − 1.9097(20) + 9.16(58) + ... 2 2 π π π π π

1 (g 2) = 0.001 159 652 180 73(28) 2 − exper 1 (g 2) = 0.001 159 652 181 78(77) 2 − theory

QCD: asymptotic freedom

g3 11 4 N (g )= s N F s 16⇥2 3 C 3 2 ⇥ nobelprize.org

The left-hand panel shows a collection of different measurements by S. Bethke from High- Energy International Conference in Quantum Chromodynamics, Montpellier 2002 (hep- ex/0211012). The right-hand panel shows a collection by P. Zerwas, Eur. Phys. J. C34(2004)41. JADE was one of the experiments at PETRA at DESY. NNLO means Next-to- Next-to-Leading Order computation in QCD.

Although there are limits to the kind of calculations that can be performed to compare QCD with experiments, there is still overwhelming evidence that it is the correct theory. Very ingenious ways have been devised to test it and the data obtained, above all at the CERN LEP accelerator, are bounteous. Wherever it can be checked, the agreement is better than 1%, often much better, and the discrepancy is wholly due to the incomplete way in which the calculations can be made.

The Standard Model for Particle Physics

QCD complemented the electro-weak theory in a natural way. This theory already contained the quarks and it was natural to put all three interactions together into one model, a non- abelian gauge field theory with the gauge group SU(3) x SU(2) x U(1). This model has been called ‘The Standard Model for Particle Physics’. The theory explained the SLAC experiments and also contained a possible explanation why quarks could not be seen as free particles (quark confinement). The force between quarks grows with distance because of ‘infrared slavery’, and it is easy to believe that they are permanently bound together. There are many indications in the theory that this is indeed the case, but no definite mathematical proof has so far been advanced.

The Standard Model is also the natural starting point for more general theories that unify the three different interactions into a model with one gauge group. Through spontaneous symmetry breaking of some of the symmetries, the Standard Model can then emerge. Such

12 Asymptotic Series vs Convergent Series

N−1 n f(x) = cn (x x ) + RN (x) − 0 n=0 X convergent series:

RN (x) 0 ,N , x ﬁxed | | → → ∞

asymptotic series:

N RN (x) x x , x x ,N ﬁxed | | | − 0| → 0

“optimal truncation”: −→ truncate just before least term (x dependent!) Asymptotic Series vs Convergent Series

∞ n n 1 1 1 alternating asymptotic series : ( 1) n! x e x E − ∼ x 1 x n=0 X

æ 1.10 æ æ 0.920 æ 1.05 0.918 1.00

æ æ æ 0.95 0.916 æ æ æ æ æ æ æ æ æ æ æ 0.90 æ æ 0.914 æ æ æ 0.85 æ æ æ 0.912 0.80 æ 0.75 æ ææ N æ N 0 5 10 15 20 2 4 6 8 10

(x = 0.1) (x = 0.2)

optimal truncation order depends on x: N 1 opt ≈ x Asymptotic Series vs Convergent Series

∞ n 1 − 1 1 non-alternating asymptotic series : n! x e x E ∼ −x 1 −x n=0 X

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æ N N 5 10 15 20 2 4 6 8 10

x = 0.1 x = 0.2

optimal truncation order depends on x: N 1 opt ≈ x Asymptotic Series vs Convergent Series

contrast with behavior of a convergent series, for which • more terms always improves the answer, independent of x

∞ ( 1)n convergent series : − xn = PolyLog(2, x) n2 − n=0 X

æ -0.75 æ -0.62

æ æ æ -0.80 -0.64 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ -0.66 æ -0.85 æ -0.68 -0.90 -0.70 -0.72 -0.95 -0.74 æ N æ N 5 10 15 20 5 10 15 20

(x = 0.75) (x = 1) Asymptotic Series: exponential precision ∞ n n 1 1 1 ( 1) n! x e x E − ∼ x 1 x n=0 X optimal truncation: error term is exponentially small

−1/x N −N −N e RN (x) N! x N!N √Ne | |N≈1/x ≈ N≈1/x ≈ ≈ ≈ √x

e.g. alternating exponential integral: • æ æ

æ 0.920 æ 0.90

æ æ 0.918 æ æ 0.85 æ æ 0.916 æ æ æ æ æ æ æ æ æ æ æ æ æ 0.914 æ 0.80 æ æ

0.912 0.75

æ ææ N æ N 0 5 10 15 20 2 4 6 8

(x = 0.1) (x = 0.2) Borel summation: basic idea

alternating factorially divergent series:

∞ ( 1)n n! xn =? − n=0 X ∞ −t n write n! = 0 dt e t R

∞ ∞ 1 ( 1)n n! xn = dt e−t (?) − 1 + x t n=0 0 X Z integral is convergent for all x > 0: “Borel sum” of the series Borel Summation: basic idea

∞ ∞ n n −t 1 1 1 1 ( 1) n! x = dt e = e x E − 1 + x t x 1 x n=0 0 X Z 1.2

1.1

1.0

0.9

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x 0.0 0.1 0.2 0.3 0.4 Borel summation: basic idea

∞ −t n write n! = 0 dt e t non-alternatingR factorially divergent series:

∞ ∞ 1 n! xn = dt e−t (??) 1 x t n=0 0 X Z −

pole on the (real, positive) Borel axis!

i π − 1 non-perturbative imaginary part = e x ⇒ ± x but every term in the series is real !?! Borel Summation: basic idea

∞ ∞ n −t 1 1 − 1 1 Borel e n! x = dt e = e e x E ⇒ R P 1 x t R −x 1 −x "n=0 # 0 X Z − 2.0

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x 0.5 1.0 1.5 2.0 2.5 3.0

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note: E 1 also has an imaginary part = iπ • 1 − x ± 1 − 1 ±i π 1 1 − 1 1 e x E e = e x Ein ln x γ i π −x 1 x −x −x − − ∓ Borel encodes this non-perturbative "connection formula" • Borel summation

Borel transform of series, where cn n! , n ∼ → ∞ ∞ ∞ n cn n f(g) cn g [f](t) = t ∼ −→ B n! n=0 n=0 X X new series typically has a ﬁnite radius of convergence

Borel resummation of original asymptotic series:

1 ∞ f(g) = [f](t)e−t/gdt S g B Z0

note: [f](t) may have singularities in (Borel) t plane B Borel singularities

avoid singularities on R+: directional Borel sums:

eiθ∞ 1 −t/g θf(g) = [f](t)e dt S g B Z0

C+

C-

go above/below the singularity: θ = 0± non-perturbative ambiguity: Im[ f(g)] −→ ± S0 physics challenge: use physical input to resolve ambiguity Resurgence and Analytic Continuation

another view of resurgence:

resurgence can be viewed as a method for making formal asymptotic expansions consistent with global analytic continuation properties

resurgence = global complex analysis for divergent series

(analytic continuation, transforms, monodromy, ...)

“the trans-series really IS the function” ⇒

question: to what extent is this true/useful in physics? Resurgence: canonical example = Airy function

formal large x solution to ODE: "perturbation theory" • ∓ 2 x3/2 ∞ 1 5 2 Ai(x) e 3 Γ n + Γ n + y00 = x y ( 1)n 6 6 ⇒ Bi(x) ∼ √π x1/4 ∓ 2 n 3n/2 n=0 n! 3 x X non-perturbative connection formula: • ∓ 2πi i ∓ πi 1 ∓ πi Ai e 3 x = e 3 Bi (x)+ e 3 Ai (x) ±2 2 Borel sum: cut along negative t axis: t ( , 1] • ∈ −∞ − ∞ n ∞ ( 1) an 4 3/2 − 4 x3/2t 1 5 Z(x) = − | | = x dt e 3 2F1 , , 1; t x3n/2 3 6 6 − n=0 0 X Z discontinuity across cut correct connection formula • ⇒ 2πi − 2πi − 4 x3/2 Z e 3 x Z e 3 x = i e 3 Z (x) − Resurgence: canonical example = Airy function "path integral" allowed z integration contours

∞ 3 1 i x t+ t Ai(x) = dt e 3 2π Z−∞ write x r eiθ, t i√rz: • ≡ ≡ − 3 √r r3/2 eiθ z− z Ai(x) = dz e 3 2πi Zγk saddles at z = eiθ/2 • ± saddle exponent ( "action") = 2 r3/2e3iθ/2 • ≡ ± 3 x > 0 θ = 0 contour through only 1 saddle (z = 1) ⇒ ⇒ − action = 2 r3/2 = 2 x3/2 ⇒ − 3 − 3 x < 0 θ = π contour through 2 saddles (z = i) ⇒ ± ⇒ ± action = i 2 r3/2 = i 2 ( x)3/2 ⇒ ± 3 ± 3 − Resurgence: canonical example = Airy function

3 √r r3/2 eiθ z− z Ai(x) = dz e 3 2πi Zγk saddles at z = eiθ/2 , action = 2 r3/2e3iθ/2 • ± ± 3 real action when θ = 0, 2π : "Stokes lines" • ± 3 imaginary action when θ = π, π : "anti-Stokes lines" • ± 3 ��

Stokes lines in complex x-plane ��-��

x = r ei θ ��-��

moral: keep both saddle

contributions as we analytically ��-�� continue in complex x plane �� Resurgence: canonical example = Airy function

expansions about the two saddles are explicitly related • 5 385 85085 37182145 5391411025 an = 1, , , , , ,... −72 10368 −2239488 644972544 −46438023168 large order/low order relation: • (n 1)! 5 2 385 22 an − 1 + ... ∼ 2n − 72 (n 1) 10368 (n 1)(n 2) − − − −

large order/low order relations are generic ! (see later) • Resurgence: Preserving Analytic Continuation

d Stirling expansion for ψ(x) = dx ln Γ(x) is divergent 1 1 1 1 174611 ψ(1 + z) ln z + + + + ... ∼ 2z − 12z2 120z4 − 252z6 ··· 6600z20 − functional relation: ψ(1 + z) = ψ(z) + 1 • z X reﬂection formula: ψ(1 + z) ψ(1 z) = 1 π cot(π z) • − − z − ∞ 1 π Im ψ(1 + iy) + +π e−2π k y ⇒ ∼ −2y 2 k X=1 formal series only has the two "perturbative" terms • “raw” asymptotics is inconsistent with analytic continuation resurgence: add inﬁnite series of non-perturbative terms • "non-perturbative completion" Resurgence: Preserving Analytic Continuation

∞ 1 π Im ψ(1 + iy) + +π e−2π k y ∼ −2y 2 k X=1 function satisfies infinite order linear ODE • infinitely many exponential terms in trans-series ⇒ Borel representation:

∞ 1 1 ψ(1 + z) ln z = e−zt dt − t − et 1 Z0 − Borel transform: poles at t = 2nπi, n = 1, 2, 3, ... • ± meromorphic (poles, no cuts) no "fluctuation factors" • ⇒ this simple example arises often in QFT: Euler-Heisenberg, finite• temperature QFT, de Sitter, exact S-matrices, Chern-Simons partition functions, matrix models, ... Resurgence in Differential Equations

trans-series from nth order linear ODE has n non-perturbative •exponential terms trans-series from nonlinear ODE has inﬁnitely many non-perturbative• exponential terms

rd e.g.: y1(x) y2(x) satisfies 3 order linear ODE • × nd but y1(x)/y2(x) satisfies 2 order non-linear ODE also generalizes to (some) PDE’s, linear and non-linear • Painlevé = "special functions of nonlinear ODE’s" many• physical applications: fluids, statistical physics, gravity, random matrices, matrix models, optics, QFT, strings, ... resurgent trans-series are the natural language for their asymptotics• Resurgence in Nonlinear ODEs: e.g. Painlevé II

Painlevé II: y00 = x y(x) + 2 y3(x)

I Tracy-Widom law for statistics of max. eigenvalue for Gaussian random matrices

I correlators in polynuclear growth; directed polymers (KPZ)

I double-scaling limit in unitary matrix models

I double-scaling limit in 2d Yang-Mills

I double-scaling limit in 2d supergravity

I non-intersecting Brownian motions

I longest increasing subsequence in random permutations

I ... universal ! Resurgence in Nonlinear ODEs: e.g. Painlevé II

y00 = x y(x) + 2 y3(x)

x + asymptotics: y00 x y(x) + ... • → ∞ ≈ y 0 as x + y(1)(x) σ Ai(x) + ... → → ∞ ⇒ + ∼ + trans-series solution generated from ODE: • k− ∞ − 2 x3/2 2 1 e 3 (k) y+(x) σ+ y (x) ∼ 2√π x1/4 + k ! X=1 infinite number of non-perturbative terms • fluctuations factorially divergent & alternating • σ = real trans-series parameter (for real solution) • + large-order/low-order relations for fluc. coefficients • Resurgence in Nonlinear ODEs: e.g. Painlevé II

y00 = x y(x) + 2 y3(x) x asymptotics: 0 x y(x) + 2 y3(x) • → −∞ ≈ x 1 smoothness y(0)(x) − 1 + O ⇒ − ∼ 2 ( x)3/2 r − different (!) trans-series solution generated from ODE: • √ k ∞ − 2 2 (−x)3/2 x e 3 (k) y−(x) − σ− y (x) ∼ 2 2√π ( x)1/4 − r k=0 ! X − fluctuations: • ∞ (k) k an y( )(x) − ∼ ( x)3n/2 n=0 X − fluctuations factorially divergent & non-alternating • σ− = pure imaginary trans-series parameter (for real solution);• fixed by resurgent cancellations Resurgence in Nonlinear ODEs: e.g. Painlevé II

y00 = x y(x) + 2 y3(x) , y(x) σ Ai(x) , x + ∼ + → ∞ trans-series structurally • diﬀerent as x note diﬀerent→ exponents! ±∞ • − 2 x3/2 e 3 x + → ∞ ⇒ 2√π x1/4 √ − 2 2 (−x)3/2 e 3 x → −∞ ⇒ 2√π ( x)1/4 − Hastings-McLeod: σ = 1 (σ− = i) unique real solution on R • + connection formula for σ < 1:(d2 π−1 ln 1 σ2 ) • + ≡ − − + −1/4 2 3/2 3 2 y−(x) σ x sin x d ln x θ ∼ +| | 3| | − 4 | | − 0 intricate "condensation of instantons" across transition • Resurgence, Trans-series and Non-perturbative Physics

1. Lecture 1: Basic Formalism of Trans-series and Resurgence

I asymptotic series in physics; Borel summation

I trans-series completions & resurgence

I examples: linear and nonlinear ODEs 2. Lecture 2: Applications to Quantum Mechanics and QFT

I instanton gas for double-well & periodic potential

I infrared renormalon problem in QFT

I from hyperasymptotics to Picard-Lefschetz thimbles 3. Lecture 3: Resurgence and Large N

I parametric resurgence

I Gross-Witten-Wadia Matrix Model

I Mathieu equation and Nekrasov-Shatashvili limit of = 2 SUSY QFT N Borel Summation and Dispersion Relations: QM examples

2 3 cubic oscillator: V = x + λ x A. Vainshtein, 1964

2 z=

1 E(z) z E(z ) = dz . o 0 2πi C z z0 R I − 1 R Im E(z) = dz π z z Z0 − 0 ∞ C 1 R Im E(z) = zn dz 0 π zn+1 n=0 0 X Z WKB Im E(z) √a e−b/z , z 0 n ⇒ ∼ z → ↔ → ∞ ∞ −b/z 1 a e a Γ(n + 2 ) cn dz = X ⇒ ∼ π zn+3/2 π bn+1/2 Z0 Instability and Divergence of Perturbation Theory

x2 x4 quartic AHO: V (x) = 4 + λ 4 Bender/Wu, 1969 Instability and Divergence of Perturbation Theory

an important part of the story ...

The majority of nontrivial theories are seemingly unstable at some phase of the coupling constant, which leads to the asymptotic nature of the perturbative series

A. Vainshtein (1964) Borel summation in practice ∞ n n f(g) cn g , cn β Γ(γn + δ) ∼ ∼ n=0 X alternating series: real Borel sum • 1 ∞ dt 1 t δ/γ t 1/γ f(g) exp ∼ γ t 1 + t βg − βg Z0 " # nonalternating series: ambiguous imaginary part • 1 ∞ dt 1 t δ/γ t 1/γ Re f( g) exp − ∼ γ P t 1 t βg − βg Z0 − " # π 1 δ/γ 1 1/γ Im f( g) exp − ∼ ±γ βg − βg " # γ determines power of coupling in the exponent • β and γ determine coeﬃcient in the exponent • β, γ and δ determine the prefactor • recall: divergence of perturbation theory in QM

∞ n e.g. ground state energy: E = n=0 cn (coupling) P n Zeeman: cn ( 1) (2n)! • ∼ −

Stark: cn (2n)! • ∼ n 1 quartic oscillator: cn ( 1) Γ(n + ) • ∼ − 2 1 cubic oscillator: cn Γ(n + ) • ∼ 2

periodic Sine-Gordon potential: cn n! • ∼

double-well: cn n! • ∼ recall: divergence of perturbation theory in QM

∞ n e.g. ground state energy: E = n=0 cn (coupling) P n Zeeman: cn ( 1) (2n)! stable • ∼ − X

Stark: cn (2n)! unstable • ∼ X n 1 quartic oscillator: cn ( 1) Γ(n + ) stable • ∼ − 2 X 1 cubic oscillator: cn Γ(n + ) unstable • ∼ 2 X

periodic Sine-Gordon potential: cn n! stable ??? • ∼

double-well: cn n! stable ??? • ∼ Bogomolny/Zinn-Justin mechanism in QM

......

degenerate vacua: double-well, Sine-Gordon, ... • − S level splitting = real one-instanton eﬀect: ∆E e g2 • ∼ n! surprise: pert. theory non-Borel summable: cn n ∼ (2S) I stable systems

I ambiguous imaginary part

2S − 2 I i e g , a two-instanton eﬀect ± Bogomolny/Zinn-Justin mechanism in QM

......

degenerate vacua: double-well, Sine-Gordon, ... • 1. perturbation theory non-Borel summable: ill-deﬁned/incomplete 2. instanton gas picture ill-deﬁned/incomplete: and ¯ attract I I regularize both by analytic continuation of coupling • ambiguous, imaginary non-perturbative terms cancel ! ⇒ "tip of the (resurgence) iceberg" Bogomolny/Zinn-Justin mechanism in QM

e.g., double-well: V (x) = x2(1 g x)2 −

2n E0 cn g ∼ n perturbation theory: X • 1 n − 2 cn 3 n!: Borel Im E π e 3g ∼ − ⇒ 0 ∼ ∓ 1 non-perturbative analysis: instanton: g x0(t) = 1+e−t • 1 classical Eucidean action: S0 = 6g2 • non-perturbative instanton gas: • − 1 −2 1 ∆E e 6g2 , Im E π e 6g2 0 ∼ 0 ∼ ± BZJ cancellation E is real and unambiguous • ⇒ 0 “resurgence” cancellation to all orders ⇒ Decoding a Resurgent Trans-series

∞ ∞ k−1 k q 2 2n S 1 f(g ) = cn,k,q g exp ln −g2 −g2 n=0 k q=0 X X=0 X

non-perturbative

perturbative

quasi-zero-mode

expansions in diﬀerent directions are quantitatively related Decoding a Resurgent Trans-series

∞ ∞ k−1 k q 2 2n S 1 f(g ) = cn,k,q g exp ln −g2 −g2 n=0 k q=0 X X=0 X ∞ 2n perturbative fluctuations about vacuum: n=0 cn,0,0 g • n! divergent (non-Borel-summable): cn,0,0 α (2S)n • ∼P 2 ambiguous imaginary non-pert energy i π α e−2S/g ⇒ ∼ ± but c , , = α: BZJ cancellation ! • 0 2 1 − −S/g2 2 4 pert flucs about instanton: e 1 + a1g + a2g + ... divergent: n! −3S/g2 1 an n (a ln n + b) i π e a ln 2 + b ∼ (2S) ⇒ ± g 2 2 3-instanton: e−3S/g a ln 1 + b ln 1 + c • 2 − g2 − g2 resurgence: ad infinitum, also sub-leading large-order terms Towards Resurgence in QFT

basic divergence due to combinatoric growth of diagrams • new features arise in QFT due to renormalization • asymptotically free QFT: “renormalons” • Dyson’s argument (QED)

F. J. Dyson (1952): •physical argument for divergence of QED perturbation theory

2 2 4 F (e ) = c0 + c2e + c4e + ...

Thus [for e2 < 0] every physical state is unstable against the spontaneous creation of large numbers of particles. Further, a system once in a pathological state will not remain steady; there will be a rapid creation of more and more particles, an explosive disintegration of the vacuum by spontaneous polarization.

suggests perturbative expansion cannot be convergent • Euler-Heisenberg Eﬀective Action (1935) review: hep-th/0406216

. . .

1-loop QED effective action in uniform emag field • the birth of effective field theory • ~ 2 ~ 2 2 2 E B α 1 ~ 2 ~ 2 ~ ~ L = − + 2 E B + 7 E B + ... 2 90π Ec − · encodes nonlinear properties of QED/QCD vacuum • QFT Application: Euler-Heisenberg

Borel transform of a (doubly) asymptotic series • resurgent trans-series: analytic continuation B E • ←→ EH effective action Barnes function ln Γ(x) • ∼ ∼ R Euler-Heisenberg Effective Action: e.g., constant B field

B2 ∞ ds 1 s m2s S = coth s exp −8π2 s2 − s − 3 − B Z0 perturbative (weak field) expansion: • 2 ∞ 2n+2 B n 2B S B2 +4 ∼ −2π2 (2n + 4)(2n + 3)(2n + 2) m2 n=0 X characteristic factorial divergence • ∞ ( 1)n+1 Γ(2n + 2) cn = − 8 (k π)2n+4 k X=1 instructive exercise: reconstruct correct Borel transform • ∞ s 1 1 s = coth s k2π2(s2 + k2π2) −2s2 − s − 3 k X=1 Euler-Heisenberg Effective Action and Schwinger Effect

B field: QFT analogue of Zeeman effect E field: QFT analogue of Stark effect B2 E2: series becomes non-alternating → − 2 2 2 Borel summation Im S = e E ∞ 1 exp k m π ⇒ 8π3 k=1 k2 − eE 328 TheP European Physicalh Journali D Schwinger effect: recent~ technological advances in laser science, and also in 2eE 2mc2 theoreticalmc ∼ refinements of the Heisenberg-Euler computation, suggest that lasers such as those planned for ELI may be⇒ able to reach this elusive nonperturbative regime. 2 3 Thism c has the potential16 to open up an entirely new domain Ec 10 V/cm Fig. 1. PairWKB production tunneling as the from separation Dirac sea of a virtual vacuum ∼of experiments,e~ ≈ with the prospect of fundamental discov- dipole pairIm underS thephysical influence pair of anproduction external electric rate field. eries and practical applications, as are described in many → talks in this conference. Euler-Heisenberg series must be divergent • + asymptotic e e− pairs if they gain the binding energy of 2mc2 from the external field, as depicted in Figure 1.This is a non-perturbative process, and the leading exponential 2TheQEDeffective action part of the probability, assuming a constant electric field, was computed by Heisenberg and Euler [2,3]: In quantum field theory, the key object that encodes vacuum polarization corrections to classical Maxwell electro- π m2 c3 dynamics is the “effective action” Γ [A], which is a func- PHE exp , (3) tional of the applied classical gauge field A (x)[22–24]. ∼ − e ! µ & E ' The effective action is the relativistic quantum field the- building on earlier work of Sauter [18]. This result sets a ory analogue of the grand potential of statistical physics, basic scale of a critical field strength and intensity near in the sense that it contains a wealth of information about which we expect to observe such nonperturbative effects: the quantum system: here, thenonlinearpropertiesofthe quantum vacuum. For example, the polarization tensor 2 3 δ2Γ m c 16 Πµν = contains the electric permittivity %ij and c = 10 V/cm δAµδAν E e ! ≈ the magnetic permeability µij of the quantum vacuum, c 2 29 2 and is obtained by varying the effective action Γ [A]with Ic = c 4 10 W/cm . (4) 8π E ≈ × respect to the external probe Aµ(x). The general formal- As a useful guiding analogy, recall Oppenheimer’s compu- ism for the QED effective action was developed in a se- tation [19]oftheprobabilityofionizationofanatomof ries of papers by Schwinger in the 1950’s [22,23]. Γ [A]is binding energy E in such a uniform electric field: defined [23]intermsofthevacuum-vacuumpersistence b amplitude 3/2 4 √2mEb Pionization exp . (5) i 3 e 0out 0in =exp Re(Γ )+i Im(Γ ) . (8) ∼ (− ! ) & | ' { } E &! ' Taking as a representative atomic energy scale the binding Note that Γ [A]hasarealpartthatdescribesdispersiveef- me4 energy of hydrogen, Eb = 2 13.6eV,wefind fects such as vacuum birefringence, and an imaginary part 2! ≈ that describes absorptive effects, such as vacuum pair pro- 2 m2 e5 P hydrogen exp . (6) duction. Dispersive effects are discussed in detail in Gies’s ∼ −3 !4 contribution to this volume [25]. The imaginary part en- & E ' codes the probability of vacuum pair production as This result sets a basic scale of field strength and inten- 2 sity near which we expect to observe such nonperturbative Pproduction =1 0out 0in ionization effects in atomic systems: − |& | '| 2 =1 exp Im Γ m2e5 − −! ionization = = α3 4 109 V/cm & ' c 4 c 2 E ! E ≈ × Im Γ (9) Iionization = α6I 6 1016 W/cm2. (7) ≈ ! c c ≈ × These, indeed, are the familiar scales of atomic ioniza- here, in the last (approximate) step we use the fact that ionization Im(Γ )/ is typically very small. The expression (9)canbe tion experiments. Note that c differs from c ! 3 7E E viewed as the relativistic quantum field theoretic analogue by a factor of α 4 10− .Thesesimpleestimates explain why vacuum∼ pair× production has not yet been of the well-known quantum mechanical fact that the ion- observed – it is an astonishingly weak effect with con- ization probability is determined by the imaginary part ventional lasers [20,21]. This is because it is primarily a of the energy of an atomic electron in an applied electric non-perturbative effect, that depends exponentially on the field. (inverse) electric field strength, and there is a factor of From a computational perspective, the effective action 107 difference between the critical field scales in the atomic∼ is defined as [22–24] regime and in the vacuum pair production regime. Thus, Γ [A]=! ln det [iD/ m] with standard lasers that can routinely probe ionization, − there is no hope to see vacuum pair production. However, = ! tr ln [iD/ m] . (10) − Euler-Heisenberg and Matrix Models, Large N, Strings, ...

scalar QED EH in self-dual background (F = F˜): • ± F 2 ∞ dt 1 1 1 S = e−t/F + 16π2 t sinh2(t) − t2 3 Z0 Gaussian matrix model: λ = g N • 1 ∞ dt 1 1 1 = e−2λ t/g + F −4 t sinh2(t) − t2 3 Z0 c = 1 String: λ = g N • 1 ∞ dt 1 1 1 = e−2λ t/g F 4 t sin2(t) − t2 − 3 Z0 Chern-Simons matrix model: • 1 ∞ dt 1 1 1 = e−2(λ+2π i m) t/g + F −4 t 2 − t2 3 m∈Z 0 sinh (t) X Z de Sitter/ anti de Sitter eﬀective actions (Das & GD, hep-th/0607168)

explicit expressions (multiple gamma functions) • 2 d/2 n m (AdSd) K AdSd (K) an 2 L ∼ 4π n m X 2 d/2 n m (dSd) K dSd (K) an 2 L ∼ 4π n m X (AdSd) n (dSd) changing sign of curvature: an = ( 1) an • − odd dimensions: convergent • even dimensions: divergent • n d Γ(2n + d 1) a(AdSd) B2 + 2( 1)n − n ∼ n(2n + d) ∼ − (2π)2n+d

pair production in dSd with d even • Towards Resurgence in Asymptotically Free QFT

QM: divergence of perturbation theory due to factorial growth of number of Feynman diagrams

n n! cn ( 1) ∼ ± (2S)n QFT: new physical eﬀects occur, due to running of couplings with momentum faster source of divergence: “renormalons” • n n β0 n! n n! cn ( 1) n = ( 1) n ∼ ± (2S) ± (2S/β0) both positive and negative Borel poles • IR Renormalon Puzzle in Asymptotically Free QFT

2S − 2 perturbation theory: i e β0 g −→ ± − 2S instantons on R2 or R4: i e g2 −→ ±

instanton/anti-instanton poles

UV renormalon poles IR renormalon poles

appears that BZJ cancellation cannot occur asymptotically free theories remain perturbatively inconsistent

’t Hooft, 1980; David, 1981 IR Renormalon Puzzle in Asymptotically Free QFT

resolution: there is another problem with the non-perturbative instanton gas analysis (Argyres, Ünsal 1206.1890; GD, Ünsal, 1210.2423) scale modulus of instantons • spatial compactiﬁcation and principle of continuity • 2 dim. CPN−1 model: • instanton/anti-instanton poles

UV renormalon poles IR renormalon poles

neutral bion poles

cancellation occurs ! (GD, Ünsal, 1210.2423, 1210.3646) Topological Molecules in Spatially Compactiﬁed Theories 2 1 1 ℝ → SL x ℝ CPN−1: regulate scale modulus problem with (spatial) compactiﬁcation: R2 S1 R1 → L ×

x2 x2

x1 x1

Euclidean time

Sinst Sinst ZN twist: instantons fractionalize: Sinst = −→ N β0 Perturbative Analysis

weak-coupling semi-classical analysis • perturbative eﬀective QM problem • → perturbation theory diverges & non-Borel summable • perturbative sector: directional Borel summation •

2 8π 2 1 −t/g 2 16 − 2 B± (g ) = dt B (t) e = Re B (g ) iπ e g N E g2 E E ∓ g2 N ZC± compare with non-perturbative instanton gas analysis: • 2 8π 8π g N 16 − 2 16 − 2 i ¯i = ln γ e g N iπ e g N I I ± 8π − g2 N ± g2 N exact ("BZJ") cancellation ! explicit application of resurgence to nontrivial QFT Non-perturbative Physics Without Instantons Dabrowski, GD, 1306.0921,

Cherman, Dorigoni, GD, Ünsal, 1308.0127, 1403.1277, GD, Ünsal, 1505.07803

2d O(N) & principal chiral model have no instantons ! • but they have ﬁnite action non-BPS saddles • Yang-Mills, CPN−1, O(N), principal chiral model, ... all have •non-BPS solutions with ﬁnite action

(Din & Zakrzewski, 1980; Uhlenbeck 1985; Sibner, Sibner, Uhlenbeck, 1989) “unstable”: negative modes of ﬂuctuation operator • what do these mean physically ? • resurgence: ambiguous imaginary non-perturbative terms should cancel ambiguous imaginary terms coming from directional Borel sums of perturbation theory

− 1 S[A] − 1 S[A ] A e g2 = e g2 saddle (ﬂuctuations) (qzm) D × × Z allX saddles The Bigger Picture: Decoding the Path Integral

what is the origin of resurgent behavior in QM and QFT ?

to what extent are (all?) multi-instanton eﬀects encoded in perturbation theory? And if so, why?

QM & QFT: basic property of all-orders steepest descents integrals• Lefschetz thimbles: analytic continuation of path integrals• Towards Analytic Continuation of Path Integrals

The shortest path between two truths in the real domain passes through the complex domain

Jacques Hadamard, 1865 - 1963 All-Orders Steepest Descents: Darboux Theorem

all-orders steepest descents for contour integrals: •

hyperasymptotics (Berry/Howls 1991, Howls 1992)

1 1 − f(z) 1 − fn I(n)(g2) = dz e g2 = e g2 T (n)(g2) C 1/g2 Z n p T (n)(g2): beyond the usual Gaussian approximation • asymptotic expansion of ﬂuctuations about the saddle n: • ∞ T (n)(g2) T (n) g2r ∼ r r=0 X All-Orders Steepest Descents: Darboux Theorem

Berry/Howls: exact resurgent relation between ﬂuctuations • about nth saddle and about neighboring saddles m

∞ −v (n) 2 1 γnm dv e (m) Fnm T (g ) = ( 1) 2 T 2π i m − 0 v 1 g v/(Fnm) v X Z − proof is based on contour deformation • universal factorial divergence of ﬂuctuations (Darboux) •

γnm 2 (n) (r 1)! ( 1) (m) Fnm (m) (Fnm) (m) Tr = − − r T0 + T1 + T2 + ... 2π i m (Fnm) " (r 1) (r 1)(r 2) # X − − −

ﬂuctuations about diﬀerent saddles are explicitly related ! All-Orders Steepest Descents: Darboux Theorem

d = 0 partition function for periodic potential V (z) = sin2(z)

π − 1 sin2(z) I(g2) = dz e g2 Z0 this is a Bessel function • two saddle points: z = 0 and z = π . • 0 1 2

min. saddle min.

vacuum vacuum II¯ All-Orders Steepest Descents: Darboux Theorem

large order behavior about saddle z : • 0 1 2 (0) Γ r + 2 Tr = √πΓ(r + 1) (r 1)! 1 9 75 − 1 4 + 32 128 + ... ∼ √π − (r 1) (r 1)(r 2) − (r 1)(r 2)(r 3) − − − − − − !

low order coeﬃcients about saddle z : • 1 1 9 75 T (1)(g2) i √π 1 g2 + g4 g6 + ... ∼ − 4 32 − 128 ﬂuctuations about the two saddles are explicitly related • simple example of a generic resurgent large-order/low-order perturbative/non-perturbative• relation Resurgence in Path Integrals: “Functional Darboux Theorem”

could something like this work for path integrals?

“functional Darboux theorem” ?

multi-dimensional case is already non-trivial and interesting • Pham (1965); Arnold (1970); Delabaere/Howls (2002); Kontsevich (2016-) Picard-Lefschetz theory •

do a computation to see what happens ... • Resurgence in (Inﬁnite Dim.) Path Integrals (GD, Ünsal, 1401.5202)

periodic potential: V (x) = 1 sin2(g x) • g2 vacuum saddle point • 5 1 13 1 cn n! 1 ... ∼ − 2 · n − 8 · n(n 1) − − instanton/anti-instanton saddle point: • −2 1 5 13 Im E π e 2g2 1 g2 g4 ... ∼ − 2 · − 8 · − double-well potential: V (x) = x2(1 gx)2 • − vacuum saddle point • n 53 1 1 1277 1 1 cn 3 n! 1 ... ∼ − 6 · 3 · n − 72 · 32 · n(n 1) − − instanton/anti-instanton saddle point: • −2 1 53 1277 Im E π e 6g2 1 g2 g4 ... ∼ − 6 · − 72 · − Resurgence in Quantum Mechanics

in fact, the resurgent structure is much deeper than this ... Uniform WKB & Resurgent Trans-Series

Alvarez/Casares (2000, 2003), GD/Unsal (1306.4405, 1401.5202) d2 V (g x) d2 ψ + ψ = E ψ g4 ψ(y) + V (y)ψ(y) = g2 E ψ(y) −dx2 g2 →− dy2

weak coupling: degenerate harmonic classical vacua • 2 c non-perturbative eﬀects: g ~ exp 2 • ↔ ⇒ − g approximately harmonic • uniform WKB with parabolic cylinder functions ⇒ 1 Dν u(y) ansatz (with parameter ν): ψ(y) = g • √u0(y) “similar looking equations have similar looking solutions” Uniform WKB & Resurgent Trans-Series

perturbative expansion for E and u(y): • ∞ 2 2k E = E(ν, g ) = g Ek(ν) k X=0 ν = N: usual perturbation theory (not Borel summable) • global analysis boundary conditions: • ⇒

y -1 - 1 y 2 - 3 Π -Π - Π Π Π 3 Π 2 2 2 2

midpoint 1 ; non-Borel summability g2 e±i g2 • ∼ g ⇒ → trans-series encodes analytic properties of Dν • generic and universal ⇒ Uniform WKB & Resurgent Trans-Series

ν −z2/4 ±iπν √2π −1−ν z2/4 Dν(z) z e (1 + ... ) + e z e (1 + ... ) ∼ Γ( ν) − exact quantization condition −→ −ν 2 1 e±iπ 2 e−S/g = (ν, g2) Γ( ν) g2 π g2 P −

p −S/g2 ν is only exponentially close to N (here ξ e ): ⇒ ≡ √π g2 N 2 2 g2 (N, g ) ν = N + P ξ N! 2N 2 ±iπ g2 ∂ e 2 P + ln ψ(N + 1) 2 ξ2 + O(ξ3) −(N!)2 P ∂N g2 − P 2 ∞ 2k insert: E = E(ν, g ) = g Ek(ν) trans-series • k=0 ⇒ P Connecting Perturbative and Non-Perturbative Sector this proves the Zinn-Justin/Jentschura conjecture: generate entire trans-series from just two functions:

(i) perturbative expansion E = Epert(~,N) (ii) single-instanton ﬂuctuation function inst(~,N) (iii) rule connecting neighbouring vacua (parity,P Bloch, ...)

1 N+ 2 ~ 1 32 −S/~ E(~,N) = Epert(~,N) e inst(~,N) + ... ± √2π N! P ~ in fact ... there is much more structure hiding here: • instanton ﬂuctuation factor: • ~ 1 2 ∂Epert d~ ∂Epert(~,N) N + 2 ~ inst(~,N) = exp S ~ + P ∂N 3 ∂N − S " Z0 ~ !#

perturbation theory Epert(~,N) encodes everything ! ⇒ Resurgence at work ﬂuctuations about (or ¯) saddle are determined by those •about the vacuum saddle,I toI all ﬂuctuation orders

1 1 1 − 8 48 16

a1 b 11 b 21

1 1 − "QFT computation": 3-loop ﬂuctuation about 24 12

b12 b 22 • 1 for double-well and Sine-Gordon: 8

b I 23 1 − Escobar-Ruiz/Shuryak/Turbiner 1501.03993, 1505.05115 8

b 24

1 1 1 − S 16 16 8 − 0 71 2 DW : e ~ 1 ~ 0.607535 ~ ... 72 d e f − − − 1 1 − 16 48 g h S0 1 − 2 Figure 2: Diagrams contributing to the coeﬃcient B2. The signs of contributions and symmetry resurgence ~ : e 1 + ~ 102N 174N 71factors are indicated. 72 − − − 1 2 4 3 2 + ~ 10404N + 17496N 2112N 14172N 6299 + ... 10368 − − − known for all N and to essentially any loop order, directly11 from• perturbation theory ! diagramatically mysterious ... • Deconstructing Zero: P/NP Resurgence for SUSY QM

GD & Ünsal: 1609.05770

perturbative SUSY: E (~) = 0 to all orders ! • ground state how can it encode non-perturbative effects ? • nonpert. β −S/~ broken SUSY: Eg.s. (~,N) ~ e fluc(~,N) > 0 • ∼ P ∂Epert ~ d~ ∂Epert(~,N) N ~2 fluc(~,N) = exp S ~ + P ∂N 3 ∂N − S Z0 ~ pert ∂Epert note that E = 0, but ∂N = 0 • N=0 N=0 6 non−pert. h i unbroken SUSY: Eg.s. (~) = 0, due to cancellations •between two saddles resurgence explains SUSY breaking or non-breaking ⇒ Connecting Perturbative and Non-Perturbative Sector

all orders of multi-instanton trans-series are encoded in perturbation theory of ﬂuctuations about perturbative vacuum

why ? turn to path integrals again ... look for a semiclassical explanation Analytic Continuation of Path Integrals: Lefschetz Thimbles

1 i 1 − 2 S[A] − 2 Simag[Ak] − 2 Sreal[A] A e g = k e g A e g D N D k Γk Z thimblesX Z Lefschetz thimble = “functional steepest descents contour” remaining path integral has real measure: (i) Monte Carlo (ii) semiclassical expansion (iii) exact resurgent analysis

resurgence: asymptotic expansions about diﬀerent saddles are closely related requires a deeper understanding of complex conﬁgurations and analytic continuation of path integrals ...

Stokes phenomenon: intersection numbers k can change with phase of parameters N Thimbles from Gradient Flow

gradient ﬂow to generate steepest descent thimble:

∂ δS A(x; τ) = ∂τ −δA(x; τ) keeps Im[S] constant, and Re[S] is monotonic • ∂ S S¯ 1 δS ∂A δS ∂A − = = 0 ∂τ 2i −2i δA ∂τ − δA ∂τ Z ∂ S + S¯ δS 2 = ∂τ 2 − δA Z

Chern-Simons theory (Witten 2010) • comparison with complex Langevin (Aarts 2013, ...) • lattice (Aurora, 2013; Tokyo/RIKEN): Bose-gas • X generalized thimble method: (Alexandru, Başar, Bedaque et al, 2016) • Complex Saddles in Path Integrals

(Behtash, GD, Schäfer, Sulejmanpasic, Ünsal 1510.00978, 1510.03435)

puzzle 1: how do approximate bion solutions yield exact SUSY• answers? puzzle 2: how to explain SUSY breaking for DW semiclassically?• puzzle 3: how to explain SUSY non-breaking for SG semiclassically?• Complex Saddles in Path Integrals

complex classical equations of motion • d2z ∂V d2x ∂V = or equivalently = + r dt2 ∂z dt2 ∂x d2y ∂V = r dt2 − ∂y very diﬀerent from 2d motion ! • Complex Saddles in Path Integrals

complex classical saddles from eﬀective potential •

V = W 0 2 g W 00 eﬀ ± arises from integrating out the fermions •

Complex Bion Real Bion

Complex bion Complex turning point Bounce

Bounce

Real turning point Necessity of Complex Saddles

SUSY QM: g = 1 x˙ 2 + 1 (W 0)2 g W 00 L 2 2 ± 2 complex saddles have complex action: •

S 2SI + iπ complex bion ∼

W = cos x double Sine-Gordon • 2 → E 0 2 e−2SI 2 e−iπe−2SI = 0 ground state ∼ − − X

W = 1 x3 x tilted double-well • 3 − → E 0 2 e−iπe−2SI > 0 ground state ∼ − X

semiclassics complex saddles required for SUSY algebra ⇒ Resurgence, Trans-series and Non-perturbative Physics

1. Lecture 1: Basic Formalism of Trans-series and Resurgence

I asymptotic series in physics; Borel summation

I trans-series completions & resurgence

I examples: linear and nonlinear ODEs 2. Lecture 2: Applications to Quantum Mechanics and QFT

I instanton gas for double-well & periodic potential

I infrared renormalon problem in QFT

I from hyperasymptotics to Picard-Lefschetz thimbles 3. Lecture 3: Resurgence and Large N

I parametric resurgence

I Gross-Witten-Wadia Matrix Model

I Mathieu equation and Nekrasov-Shatashvili limit of = 2 SUSY QFT N Connecting weak and strong coupling

physics question: does weak coupling analysis contain enough information to extrapolate to strong coupling ?

. . . even if the degrees of freedom re-organize themselves in a very non-trivial way?

classical asymptotics is clearly not enough: is resurgent asymptotics (= resurgent semiclassics) enough? "Parametric Resurgence": Both N and g2

trans-series expansion is a double-expansion: can be organized •in diﬀerent ways

− S 2 2n (0) g2 2n (1) F (N, g ) g pn (N)+e g pn (N) + ... ∼ n n X 1 X ck(N) + ??? ∼ g2k k X 1 1 f (0)(N g2)+e−SN f (1)(N g2) + ... ∼ N 2h−2 h N 2h−2 h h h X X how does a divergent trans-series at weak coupling turn into a •convergent series at strong-coupling? what happens to the resurgent structure? • what about the ’t Hooft limit? N ; g2 0; Ng2 = t • → ∞ → separated by a phase transition: “instantons condense” • ∗ Resurgence in = 2 and = 2 Theories (Başar, GD, 1501.05671) N N ~2 d2ψ + cos(x) ψ = u ψ − 2 dx2 u(ℏ) electric sector ←− 2.5 (convergent)

2.0 1.5 magnetic sector ←− 1.0

0.5

ℏ 0.5 1.0 1.5 -0.5 dyonic sector ←− -1.0 (divergent)

energy: u = u(N, ~); ’t Hooft coupling: λ N ~ • ≡ very diﬀerent physics for λ 1, λ 1, λ 1 • ∼ Resurgence of = 2 SUSY SU(2): Mathieu Eqn Spectrum N moduli parameter: u = tr Φ2 • h i electric: u 1; magnetic: u 1 ; dyonic: u 1 • ∼ ∼ − ∂W a = scalar , aD = dual scalar , aD = • h i h i ∂a Nekrasov twisted superpotential (a, ~, Λ): • W Mathieu equation: (Mironov/Morozov) • ~2 d2ψ N~ + Λ2 cos(x) ψ = u ψ , a − 2 dx2 ≡ 2

(quantum) Matone relation: • iπ ∂ (a, ~, Λ) ~2 u(a, ~) = Λ W 2 ∂Λ − 48

= 2∗ Lamé equation •N ↔ ∗ Resurgence in = 2 and = 2 Theories (Başar, GD, 1501.05671) N N ~2 d2ψ + cos(x) ψ = u ψ − 2 dx2 u(ℏ) electric sector ←− 2.5 (convergent)

2.0 1.5 magnetic sector ←− 1.0

0.5

ℏ 0.5 1.0 1.5 -0.5 dyonic sector ←− -1.0 (divergent)

energy: u = u(N, ~); ’t Hooft coupling: λ N ~ • ≡ very diﬀerent physics for λ 1, λ 1, λ 1 • ∼ Mathieu Equation Spectrum

~2 d2ψ + cos(x) ψ = u ψ − 2 dx2 small ~: divergent, non-Borel-summable trans-series • → 2 1 ~2 1 1 u(N, ~) 1 + ~ N + N + + ∼ − 2 − 16 2 4 " # 3 ~3 1 3 1 N + + N + ... −162 2 4 2 − " # large ~: convergent expansion: ?? trans-series ?? • −→ 2 4 2 8 ~ 2 1 2 5N + 7 2 u(N, ~) N + + ∼ 8 2(N 2 1) ~ 32(N 2 1)3(N 2 4) ~ − − − 9N 4 + 58N 2 + 29 2 12 + + ... 64(N 2 1)5(N 2 4)(N 2 9) ~ − − − ! note: poles in coeﬃcients • Mathieu Equation Spectrum: far above the barrier

~2 d2ψ + cos(x) ψ = u ψ − 2 dx2 narrow gaps high in the spectrum: complex instantons • Dyhkne: same formula for band/gap splittings •

2 ∂u − 2π Im aD ∆u e ~ 0 ∼ π ∂N

2 2N 4 gap ~ 1 2 2 ∆uN 2 1 + O ∼ 4 (2N−1(N 1)!) ~ " ~ !# − N ~2 e 2N ,N 1 ∼ 2π N ~ Keldysh Approach in QED Brézin/Itzykson, 1970; Popov, 1971

Schwinger eﬀect in E(t) = cos(ωt) • E adiabaticity parameter: γ m ω • ≡ E 2 WKB P exp π m g(γ) • ⇒ QED ∼ − ~ E 2 h i exp π m , γ 1 (non-perturbative) − ~ E P  h i QED ∼  4m/ ω  E ~ , γ 1 (perturbative) ω m  semi-classical instanton (saddle) interpolates between •non-perturbative ‘tunneling pair-production” and perturbative “multi-photon pair production” exact mapping physical interpretation of diﬀerent non-pert expressions• ⇒ 2 4ω m 2 ~ ; N ; u = 1 + 2γ ↔ ↔ ω E Beyond Large N: Multi-instantons at strong coupling

2 4 2 8 ~ 2 1 2 5N + 7 2 u(N, ~) N + + + ... ∼ 8 2(N 2 1) ~ 32(N 2 1)3(N 2 4) ~ − − − ! re-organize as a multi-instanton expansion • 2 2 N−1 2 2N N−1 (±) ~ N αn(N) ~ 1 2 βn(N) u (~) = + ... N 8 4n ± 8 (2N−1(N 1)!)2 4n n=0 ~ ~ n=0 ~ X − X ﬂuctuation series are polynomials ! • 1-instanton gap splitting: (Basar, GD, Unsal, 2014) •

1 ∂u A(N,~) ∂A 4 ∂u ∆uN e = ≡ (2N−1(N 1)!)2 ∂N ⇒ ∂~2 −~4 ∂N − 1-inst. ﬂucts. determined by pert. expansion • resurgent multi-instanton structure in convergent region • Resurgence in Matrix Models: Mariño: 0805.3033, Ahmed & GD: 1710.01812 Gross-Witten-Wadia Unitary Matrix Model

resurgent Borel-Écalle analysis of partition functions, Wilson •loops, etc ... in matrix models

2 1 † Z(g ,N) = DU exp 2 tr U + U U(N) g Z matrix model for 2d lattice Yang-Mills • two variables: g2 and N (’t Hooft coupling: t g2N/2) • ≡ "parametric resurgence" • 3rd order phase transition at N = , t = 1 (universal!) • ∞ double-scaling limit: Painlevé II • 3rd order phase transition: condensation of instantons • similar in 2d Yang-Mills on sphere and disc • GWW Phase Transition in 2d Gauge Theory

"... one can attempt to expand the partition function Z() of two dimensional Yang-Mills in powers of the gauge coupling constant . In doing so (in a suitable topological sector), one finds a remarkable result: the perturbation series in stops after finitely many terms, yet Z() is not a polynomial. Z() contains exponentially small terms which can be identified as contributions of unstable classical solutions to the functional integral.”

E. Witten (Two Dimensional Gauge Theories Revisited, 1992)

... resurgence approach to non-perturbative eﬀects in large N Gross-Witten-Wadia Model: Trans-series Structure

2 1 † Z(g ,N) = DU exp 2 tr U + U U(N) g Z transseries structure: ln Z(g2,N), as fn of both g2 & N • "parametric resurgence" • Resurgence in Gross-Witten-Wadia Model

partition function = N N Toeplitz determinant • × 2 2 Z(g ,N) = det (Ij−k(x)) , x j,k=1,...N ≡ g2 so Z(g2,N) satisﬁes (N + 1)th order linear ODE, N • ∀ weak-coupling resurgent trans-series "guaranteed" ⇒ ∞ (0) N−1 ∞ (1) an (N) (4x) an (N) Z(x, N) Z (x, N) + i e−2x + ∼ 0 xn Γ(N) xn "n=0 n=0 X X ∞ (N) G(N + 1) an (N) ... + e−2Nx N−1 xn i=0 Γ(N i) n=0 # − X but strong-couplingQ expansion is convergent! • N+1 2 2 2 (x/2) 1 (N + 1) x Z(x, N) ex /4 1 1 + ... + ... ∼ − (N + 1)! − 2 (N + 2)2 " # Resurgence in Gross-Witten-Wadia Model Ahmed & GD: 1710.01812

idea: map it to a Painlevé function (Painlevé III) • det [Ij−k+1 (x)]j,k ,...,N ∆(x, N) det U = =1 ≡ h i det [Ij−k (x)]j,k=1,...,N for any N, ∆(x, N) satisﬁes a PIII-type equation: • 1 ∆ N 2 ∆00 + ∆0 + ∆ 1 ∆2 + ∆0 2 = 0 x − (1 ∆2) − x2 − generate trans-series solutions: weak- & strong-coupling ⇒ N is a parameter ! large N limit by rescaling • ⇒ direct relation to the partition function: • Z(x, N 1) Z(x, N + 1) ∆2(x, N) = 1 − − Z2(x, N) 1 x Z(x, N) = exp x dx 1 ∆2(x, N) (1 + ∆(x, N 1)∆(x, N + 1)) 2 − − Z0 Resurgence in Gross-Witten-Wadia Model

weak-coupling expansion is a divergent series: • trans-series non-perturbative completion → strong-coupling expansion is a convergent series: • but it still has a non-perturbative completion ! ∆ small linearize Bessel equation • ⇒ → 1 ∆ N 2 ∆00 + ∆0 +∆ 1 ∆2 + ∆0 2 = 0 x − (1 ∆2) − x2 − ∆(x, N) σ JN (x) ⇒ strong ≈ strong-coupling expansion (x 2 ) is clearly convergent • ≡ g2 but full solution is a non-perturbative trans-series: • ∞ k ∆(x, N) = (σstrong) ∆(k)(x, N) k , , ,... =1X3 5 all higher terms are Bessel kernels with lower terms • Resurgence in Gross-Witten-Wadia Model

strong-coupling trans-series (convergent !!!): • ∞ k ∆(x, N) = (σstrong) ∆(k)(x, N) k , , ,... =1X3 5

Δ(x,5)

0.6

0.4

0.2

x 2 4 6 8 10

-0.2

blue: exact, red: ∆(1) = J5(x) , black: includes ∆(3) Resurgence in GWW: ’t Hooft limit and phase transition

Gross-Witten-Wadia N = phase transition: • ∞

N→∞ 0 , t 1 (strong coupling) ∆(t, N) ≥ −−−−→ (√1 t , t 1 (weak coupling) − ≤

Δ(t,N)

1.0

N Ng2 t x 2 0.8 ≡ ≡ black lines: 0.6 N = 5, 25, 50,... 150 0.4 red dashed line: 0.2 ∆ = √1 t −

t 0.5 1.0 1.5 2.0 Resurgence in GWW: ’t Hooft limit and phase transition

rescaled PIII equation: t Ng2/2 N • ≡ ≡ x N 2∆ ∆ t2∆00 + t∆0 + 1 ∆2 = N 2 t2 ∆0 2 t2 − 1 ∆2 − − GWW N = phase transition: • ∞

N→∞ 0 , t 1 (strong coupling) ∆(t, N) ≥ −−−−→ (√1 t , t 1 (weak coupling) − ≤ large N at weak coupling: • ∆ ∆ 1 ∆2 = 1 ∆2 = t t2 − 1 ∆2 ⇒ − − Resurgence in GWW: ’t Hooft limit and phase transition

rescaled PIII equation: t Ng2/2 N • ≡ ≡ x N 2∆ ∆ t2∆00 + t∆0 + 1 ∆2 = N 2 t2 ∆0 2 t2 − 1 ∆2 − − GWW N = phase transition: • ∞

full large N trans-series at weak-coupling: • ∞ (0) −NS (t) ∞ (1) dn (t) i t e weak dn (t) ∆(t, N) √1 t σ +... ∼ − N 2n − √ weak (1 t)1/4 N n n=0 2 2πN n=0 X − X large N weak-coupling action • 2√1 t Sweak(t) = − 2 arctanh √1 t t − − conﬁrm (parametric!) resurgence relations, for allt: • ∞ (1) 2 dn (t) (3t 12t 8) 1 = 1 + − − + ... N n 96(1 t)3/2 N n=0 X − large-order growth of perturbative coeﬃcients ( t < 1): • ∀ 1 Γ(2n 5 ) (3t2 12t 8) S (t) d(0)(t) − − 2 1 + − − weak + ... n 3/4 3/2 2n− 5 3/2 7 ∼ √2(1 t) π (S (t)) 2 " 96(1 t) (2n ) # − weak − − 2 Resurgence in GWW: ’t Hooft limit and phase transition

N large N transseries at strong-coupling: ∆(t, N) σJN • ≈ t ∞ k ∆(t, N) = (σstrong) ∆(k)(t, N) k , , ,... =1X3 5 Debye expansion for Bessel function: JN (N/t) • −NSstrong(t) ∞ √t e Un (t) ∆(t, N) ∼ √2πN (t2 1)1/4 N n − n=0 X 3 −NS (t) ∞ (1) 1 √te strong Un (t) + + ... 4(t2 1) √ 2 1/4 N n 2πN (t 1) ! n=0 − − X large N strong-coupling action: • S (t) = arccosh(t) 1 1/t2 strong − − large-order/low-order (parametric)p resurgence relations: • n 2 ( 1) (n 1)! (2Sstrong(t)) (2Sstrong(t)) Un (t) − − 1 + U (t) + U (t) + ... ∼ 2π(2S (t))n 1 (n 1) 2 (n 1)(n 2) strong − − − Resurgence in GWW: ’t Hooft limit and phase transition

Debye expansion has unphysical divergence at t = 1 • uniform asymptotic expansion: • 1 2 2/3 2/3 4 3 3 Ai N 3 Sstrong(t) N 4 2 Sstrong(t) 2 JN 2 1 t ∼ 1 1/t 3 − ! N

J_5(5/t)

0.4

0.2

t 0.5 1.0 1.5 2.0

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-0.4 nonlinear analogue of uniform WKB (coalescing saddles) • Resurgence in GWW: ’t Hooft limit and phase transition

Wilson loop: 1 ∂ ln Z • W ≡ N ∂x 1 (t, N) = 1 ∆2(t, N) (1 + ∆(t, N 1)∆(t, N + 1)) W 2t − − uniform large N approximation at strong-coupling: • strong 1 2 (t, N) 1 J (N/t) (1 + JN− (N/t)JN (N/t)) W ≈ 2t − N 1 +1

W_1(t,5) 1.0 blue: exact 0.8 red: uniform large N 0.6 dashed: usual large N

0.4

0.2 uniform resummation of instantons & ﬂuctuations t 0.5 1.0 1.5 2.0 Resurgence in GWW: ’t Hooft limit and phase transition

uniform asymptotic expansion: • 1 2 2/3 2/3 4 3 3 Ai N 3 Sstrong(t) 4 2 Sstrong(t) 2 ∆strong(N, t) 2 1 ∼ 1 1/t 3 − ! N physical meaning of "uniform large-N instantons"? • nonlinear anaolgue of "uniform WKB" • technically: coalescence of two saddles "bion" • −→ expect similar phenomena in QFT • Resurgence in GWW: double-scaling limit = Painlevé II

reduction cascade of Painlevé equations • "zoom in" on vicinity of phase transition: • t1/3 κ N 2/3(t 1) ; ∆(t, N) = y(κ) ≡ − N 1/3 N with κ ﬁxed: • → ∞ d2y ∆ PIII equation = 2 y3(κ) + 2κ y(κ)(PII) −→ dκ2 e.g. on strong-coupling side: • 1/3 1/3 2 1/3 lim JN (N N κ) = Ai 2 κ N→∞ − N integral equation form of PII: • ∞ y(χ) = σ Ai(χ) + 2π Ai(χ)Bi(χ0) Ai(χ0)Bi(χ) y3(χ0) dχ0 − Zχ Resurgence in GWW: double-scaling limit = Painlevé II

"zoom in" on vicinity of phase transition: • integral equation form of PII: • ∞ y(χ) = σ Ai(χ) + 2π Ai(χ)Bi(χ0) Ai(χ0)Bi(χ) y3(χ0) dχ0 − Zχ W(χ) 0.7 iterate resummed −→ 0.6 trans-series instanton expansion 0.5 blue: exact 0.4 red: leading uniform

0.3 large N dashed: sub-leading 0.2 uniform large N 0.1 green dashed: usual

χ large N -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Conclusions

Resurgence systematically uniﬁes perturbative and non-perturbative• analysis, via trans-series trans-series ‘encode’ analytic continuation information • expansions about diﬀerent saddles are intimately related • there is extra un-tapped ‘magic’ in perturbation theory • QM, matrix models, large N, strings, SUSY QFT • IR renormalon puzzle in asymptotically free QFT • multi-instanton physics from perturbation theory • = 2 and = 2∗ SUSY gauge theory •N N appliactions to sign problem and non-equil. path integrals •

moral: go complex and consider all saddles, not just minima • A Few References: books

I J.C. Le Guillou and J. Zinn-Justin (Eds.), Large-Order Behaviour of Perturbation Theory

I C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers

I R. B. Dingle, Asymptotic expansions: their derivation and interpretation

I O. Costin, Asymptotics and Borel Summability

I R. B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes Integrals

I E. Delabaere, “Introduction to the Ecalle theory”, In Computer Algebra and Diﬀerential Equations 193, 59 (1994), London Math. Soc. Lecture Note Series

I M. Mariño, Instantons and Large N : An Introduction to Non-Perturbative Methods in Quantum Field Theory, (Cambridge University Press, 2015). A Few References: some papers

I C.M. Bender and T.T. Wu, “Anharmonic oscillator”, Phys. Rev. 184, 1231 (1969); “Large-order behavior of perturbation theory”, Phys. Rev. Lett. 27, 461 (1971).

I E. B. Bogomolnyi, “Calculation of instanton–anti-instanton contributions in quantum mechanics”, Phys. Lett. B 91, 431 (1980).

I M. V. Berry and C. J. Howls, “Hyperasymptotics for integrals with saddles”, Proc. R. Soc. A 434, 657 (1991)

I J. Zinn-Justin & U. D. Jentschura, “Multi-instantons and exact results I: Conjectures, WKB expansions, and instanton interactions,” Annals Phys. 313, 197 (2004), quant-ph/0501136, “Multi-instantons and exact results II” Annals Phys. 313, 269 (2004), quant-ph/0501137

I E. Delabaere and F. Pham, “Resurgent methods in semi-classical asymptotics”, Ann. Inst. H. Poincaré 71, 1 (1999)

I E. Witten, “Analytic Continuation Of Chern-Simons Theory,” arXiv:1001.2933

I E. Witten, “A New Look At The Path Integral Of Quantum Mechanics,” arXiv:1009.6032 A Few References: some papers

I M. Mariño, R. Schiappa and M. Weiss, “Nonperturbative Eﬀects and the Large-Order Behavior of Matrix Models and Topological Strings,” Commun. Num. Theor. Phys. 2, 349 (2008) arXiv:0711.1954

I I. Aniceto, R. Schiappa and M. Vonk, “The Resurgence of Instantons in String Theory,” Commun. Num. Theor. Phys. 6, 339 (2012), arXiv:1106.5922

I M. Mariño, “Lectures on non-perturbative eﬀects in large N gauge theories, matrix models and strings,” arXiv:1206.6272

I I. Aniceto and R. Schiappa, “Nonperturbative Ambiguities and the Reality of Resurgent Transseries”, Commun. Math. Phys. 335, 183 (2015), arXiv:1308.1115

I I. Aniceto, J. G. Russo and R. Schiappa, “Resurgent Analysis of Localizable Observables in Supersymmetric Gauge Theories”, arXiv:1410.5834

I D. Dorigoni, “An Introduction to Resurgence, Trans-Series and Alien Calculus,” arXiv:1411.3585.

I S. Demulder, D. Dorigoni and D. C. Thompson, “Resurgence in η-deformed Principal Chiral Models,” JHEP 1607, 088 (2016), arXiv:1604.07851 A Few References: some papers

I G. V. Dunne & M. Unsal, “New Methods in QFT and QCD: From Large-N Orbifold Equivalence to Bions and Resurgence,” Annual Rev. Nucl. Part. Science 2016, arXiv:1601.03414

I A. Behtash, G. V. Dunne, T. Schaefer, T. Sulejmanpasic & M. Unsal, “Toward Picard-Lefschetz Theory of Path Integrals, Complex Saddles and Resurgence,” Annals of Mathematical Sciences and Applications 2016, arXiv:1510.03435

I G. V. Dunne & M. Ünsal, “Resurgence and Trans-series in Quantum Field Theory: The CP(N-1) Model,” JHEP 1211, 170 (2012), and arXiv:1210.2423

I G. V. Dunne & M. Ünsal, “Generating Non-perturbative Physics from Perturbation Theory,” arXiv:1306.4405; “Uniform WKB, Multi-instantons, and Resurgent Trans-Series,” arXiv:1401.5202.

I G. Basar & G. V. Dunne, “Resurgence and the Nekrasov-Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lamé Systems”, JHEP 1502, 160 (2015), arXiv:1501.05671.

I A. Ahmed and G. V. Dunne, “Transmutation of a Trans-series: The Gross-Witten-Wadia Phase Transition,” JHEP 1711, 054 (2017), arXiv:1710.01812.