Resurgence of Instantons in Resurgence Applications String Theory - - PowerPoint PPT Presentation

Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Resurgence of Instantons in String Theory

Inˆ es Aniceto

(Based on ongoing work with R. Schiappa and M. Vonk, 1106.5922 and 1308.1115)

Bern, 2 September 2013

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Perturbation Theory & Asymptotic Series

Perturbation theory: fundamental in computations of

◮ ground-state energies in quantum mechanics ◮ beta-functions in quantum field theory ◮ genus expansions of string theory ◮ large N expansion of non-abelian gauge theories

· · · BUT... most perturbative expansions are asymptotic, i.e. zero radius of convergence!

◮ Why? existence of singularities in the complex Borel plane,

usually related to

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Perturbation Theory

Perturbative expansion of quantity F(z) in parameter z ∼ ∞ F(z) ≃

• g≥0

Fg z−g , Asymptotic series: Fg ∼ g!

◮ How to find F(z)?

◮ Borel transform B[F]: ”

remove”the factorial growth

◮ Analytically continue B[F] to full complex plane ◮ Define resummation SF by the inverse Borel transform

◮ BUT: SF is just a Laplace transform - needs an integration

contour to be properly defined! What happens when the contour of integration meets a singularity in the complex plane?

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Perturbation Theory

Perturbative expansion of quantity F(z) in parameter z ∼ ∞ F(z) ≃

• g≥0

Fg z−g , Asymptotic series: Fg ∼ g!

◮ How to find F(z)?

◮ Borel transform B[F]: ”

remove”the factorial growth

◮ Analytically continue B[F] to full complex plane ◮ Define resummation SF by the inverse Borel transform

◮ BUT: SF is just a Laplace transform - needs an integration

contour to be properly defined!

◮ If we have a singularity in the complex Borel plane:

Nonperturbative ambiguity: ambiguity in choosing how integration contour will avoid the singularity

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Nonperturbative Ambiguity

Borel resummation of F along direction θ is the Laplace transform SθF(z) = ˆ eiθ∞ ds B[F](s)e−s z

◮ Take B[F](s) with singularities in direction θ:

Nonperturbative ambiguity:

◮ B[F](s) ∼

1 s−A in direction θ

◮ Difference between S±F(z):

S+F(z) − S−F(z) ∼ exp (−z)

◮ around z ∼ ∞ this is non-analytic ◮ Singularities in the Borel plane occur along Stokes lines

Perturbative series is non-Borel resummable along Stokes lines

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Beyond Perturbation Theory?

How can we make sense out of perturbation theory?

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Beyond Perturbation Theory?

Learn from the example of anharmonic potential in QM

[Bender,Wu’73] ◮ Coefficients of perturbative series of ground-state energy obey

Fg ∼ g! A−g , g ≫ 1

◮ Borel plane: singularity in positive real axis, governed by real

instanton action A

◮ Resummation along real axis leads to a nonperturbative ambiguity

BUT: not only the perturbative sector which has an ambiguity!!!

◮ Perturbatively expand around a fixed multi-instanton sector

n − instanton sector: F (n)(z) = e−nAz F (n)

g z−g

Expansion is also asymptotic, with large-order behaviour F (n)

g

∼ g! n A−g , g ≫ 1 Any multi-instanton series suffers from nonperturbative ambiguities!

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Problem or Solution?

◮ Multi-instanton series suffers from nonperturbative ambiguities! ◮ In most cases there is an infinite number of instanton sectors...

Seems to make the problem with perturbation theory even worse!

◮ BUT: for the ground state energy of double-well potential [Bogomolny,Zinn-Justin,’80-83]

◮ ambiguity in 2-instanton sector precisely cancels ambiguity in

perturbative expansion

◮ ambiguity in 3-instanton sector cancels ambiguity in

1-instanton sector

◮ · · ·

Multi-instantonic ambiguities are the solution to our problem!

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Beyond Perturbation Theory!

Ground-state energy = sum over all multi-instanton sectors

◮ usual asymptotic perturbative expansion ◮ all asymptotic expansions around each nonperturbative (instanton)

sector Ambiguities arising in different sectors will conspire to cancel each other The final result is real and free from any nonperturbative amiguities! How to implement this sum? Transseries ansatz! Transseries: formal power series in two or more variables, each a function of the parameter z

F(z, σ) =

• n≥0

σnF (n)(z) , F (n)(z) ≃ e−nAz

g≥1

F (n)

g

z−g

◮ our case has e−Az and z ◮ σ: instanton counting parameter

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Ambiguities along Stokes lines

◮ Nonperturbative ambiguity of F(z) along a Stokes line:

◮ B[F] has singularities along corresponding singular direction θ ◮ Lateral Borel resummations Sθ±F differ

(Sθ+ − Sθ−) F = 0

◮ BUT: these lateral resummations are still related via the Stokes

automorphism Sθ: Sθ+F = Sθ− ◦ SθF

◮ Discontinuity in the direction θ of the Borel transform:

Sθ = 1 − Discθ

◮ Sθ = 1 encodes information on the Stokes transition at θ ◮ Determined up to unknowns called Stokes Constants Sk ◮ How? Via Alien Calculus and Resurgence

Determine the nonperturbative ambiguities using the Stokes automorphism

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Median Resummation

◮ Re-write the lateral Borel resummations as

S± = 1 2 (S+ + S−) ± 1 2 (S+ − S−)

◮ To cancel ambiguities set

S+ − S− ∼ 0 at the level of the transseries

◮ We are left with an unambiguous result given by

Smed ∼ 1 2 (S+ + S−) which is just the median resummation!

◮ In terms of Stokes automorphism: [Delabaere,Pham,’99]

Smed = S+ ◦ S−1/2

θ

= S− ◦ S1/2

θ

Note: For multi-parameter transseries, appearance of extra singularities and Stokes constants severely increases difficulty [IA,Schiappa,’13]

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Constructing a real Transseries F(z, σ)

◮ Physical set-up for coupling z real positive: [IA,Schiappa,’13]

◮ Stokes line in the positive real axis: θ = 0 - singular direction ◮ Coefficients of transseries F (n)

g

real

◮ Nonperturbative ambiguities for each F (n) are imaginary

ImF (n) = 1 2i (S+ − S−) F (n)

◮ Canceling these defines an unambiguous real transseries in the

positive real axis FR(z, σ) = SmedF = S−F(z, σ + 1 2S1) = ReF (0) + σReF (1) +

• σ2 − 1

2S2

1

• ReF (2) + · · ·

where σ ∈ R and S1 is Stokes constant.

All instanton sectors contribute! Even when σ = 0 ...

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Real Transseries: Cancellations

Constructing the real transseries

We want to cancel the ambiguities coming from perturbative expansion:

F0 Fz, Σ 0

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Real Transseries: Cancellations

F0 e F0 i m F0

Constructing the real transseries

Where the nonperturbative ambiguity of F0 is: 2 i m F0

m1

• S1m Fm

S1 e F1 1 2 S13 e F3 S15 e F5 ...

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Real Transseries: Cancellations

F0 e F0 1 2 S1 e F1 1 4 S13 e F3 ...

Constructing the real transseries

1 2 S1 F1 1 2 S1 e F1 i 2 S1 m F1

To cancel the first imaginary term we need to add

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Real Transseries: Cancellations

e F0 1 4 S13 e F3 i 2 S1 m F1 ...

Constructing the real transseries

F0 1 2 S1 F1

||

Nonperturbative ambiguity of F1 is: 2 i m F1Α e F2 Β e F3 ...

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Real Transseries: Cancellations

e F0 1 2 S12 e F2 1 4 S13 e F3 1 2 S14 e F4 ...

Constructing the real transseries

F0 1 2 S1 F1

||

Second imaginary term we need cancel against

Α F2 Α e F2 Α i m F2 Β F3 Β e F3 Β i m F3

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Real Transseries: Cancellations

Constructing the real transseries

Repeating the same procedure iteratively, we finally find the real transseries:

...........

Fz, Σ 0

n0 S1n

2n Fn Fz, 1 2 S1

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Resurgence [´

Ecalle’81]

The cancelation of nonperturbative ambiguities in multi-instanton sectors is but an example of a larger structure behind perturbation theory! Resurgence analysis and Transseries Given a transseries:

F(z, σ) =

• n≥0

σnF (n) , F (n)(z) ≃ e−nAz

g≥0

F (n)

g z−g

defines a resurgent function if it relates the asymptotics of multi- instanton contributions F (ℓ)

n

in terms of F (ℓ′)

n

where ℓ′ is close to ℓ How does it work?

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Resurgence at play[IA,Schiappa,Vonk’11]

A A 2 A 3 A 4 A 5 A ... Instanton

Instanton action singularities Multi-instanton singularities

Singularities in the Borel Plane

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Resurgence at play[IA,Schiappa,Vonk’11]

A A 2 A 3 A 4 A 5 A

...

Perturbative series: Instanton series:

Multi-instanton asymptotic series

Fnz enA z

g1

• Fg

n zg

F0z

g0

• Fg

0 zg1

F1 F2 F3 F4 F5 F1 Fz

n0

• Σn Fnz

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Resurgence at play[IA,Schiappa,Vonk’11]

A A 2 A 3 A 4 A 5 A

...

Large-order behaviour (g >>1): Use Cauchy's Theorem for each Fnz Fz 1 2 Πi

Disc0 FΩ

Ω z Ω 1 2 Πi

DiscΠ FΩ

Ω z Ω

F1 F2 F3 F4 F5

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Resurgence at play[IA,Schiappa,Vonk’11]

...

Large-order behaviour - Perturbative series for large g

F1 F2 F3 F4 F5

Fg

0S1 n0

ang Fn

1 2g S1 2 n0

bng Fn

2 ...

Fg

S1 Fn

1

S1

2

2g Fn

2

S1

3

3g Fn

3

S1

4

4g Fn

4

S1

k

kg Fn

k

...

All multi-instanton sectors contribute to the large-order behavior of coefficients Fg

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Resurgence at play[IA,Schiappa,Vonk’11]

...

Equivalently: Perturbative series for large g ENCODES all other sectors

F1 F2 F3 F4 F5

Fg

0S1 F1 1

A g 1 F2

1 ... O2g

Fg

S1 F1

1 ...

From the leading large g behaviour of Fg

0:

determine F1

1, F2 1, ...

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Resurgence at play[IA,Schiappa,Vonk’11]

...

Equivalently: Perturbative series for large g ENCODES all other sectors

F2 F3 F4 F5

Fg

0 S1 n0

ang Fn

12g S1 2 F1 2 2 A

g 1 F2

2 ... O3g

Fg

S1

2

2g F1

2 ...

Re-summing results for F1: next leading behaviour: determines F1

2, F2 2, ...

F1

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Resurgence at play[IA,Schiappa,Vonk’11]

...

Large-order behaviour - 1-instanton series for large g

F2 F3 F4 F5

Fg

1 S1 Fn

2

S1

2

2g Fn

3

S1

3

3g Fn

4

S1

k

kg Fn

k1

...

Fg

1S1 n0

ang Fn

2 2g S1 2 n0

bng Fn

3 ...

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Resurgence at play[IA,Schiappa,Vonk’11]

...

Large-order behaviour - 2-instanton series for large g

F3 F4 F5 fSΓ 1g Fn

1

S1 Fn

3

S1

2

2g Fn

4

S1

k

kg Fn

k2

...

Fg

2S1 ang Fn 3 2g S1 2 bng Fn 4 1g fSΓ cng Fn 1 ...

F1

Fg

2

...

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Resurgence at play[IA,Schiappa,Vonk’11]

Large order behaviour r instanton series for large g

Fg

r f1SΓ 1g Fn

r1

fkSΓ kg Fn

rk

...

S1

2

2g Fn

r2

... ... ...

S1 Fn

r1

S1

k

kg Fn

kr

Fg

rS1 a1g F1 3 1g f1SΓ c1g F1 r1 ...

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Resurgence at play[IA,Schiappa,Vonk’11]

3 Ak 2 Ak Ak A1 2 A1 3 A1

...

A1 2 A2 3 A3

...

Ak 2 Ak 3 Ak

... ...

Fg

ri

... ... Several instanton Actions : Ak Ak ei Θk

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Applications

◮ Exact quantisation conditions in QM:[Voros,Zinn-Justin,...]

◮ Orginally derived via WKB and Bohr-Sommerfeld; ◮ New developments using resurgent analysis

[Basar,Dunne,¨ Unsal,’13] ◮ In quantum field theory:[Dunne,¨ Unsal et al,’11-13]

◮ Semi-classical description of renormalons found; ◮ Problematic as their singularities are dominant compared to

instantons;

◮ In Principal Chiral Model other semi-classical descriptions

have also been found: unitons and fractons.

◮ String theory and Large-N duality:[Mari˜ no;IA,Schiappa et al,’10-13]

◮ Study of large order behaviour in string theory and large N

random matrix models

◮ Topological strings and Holomorphic Anomaly ◮ Large−N duality: phase diagram of matrix models and the

genus expansion

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Resurgence & large-N duality:Quartic MM

One-matrix model partition function (N × N matrix M) Z(N, gs) ∝ ˆ dM exp

• − 1

gs TrV (M)

• ,

V (z) = 1 2z2 − 1 24λz4 Free energy has perturbative genus expansion (t = gsN) F ≃

• g≥0

Fg(t) g 2g−2

s

◮ Full nonperturbative solution include all backgrounds:

Z(σ1, σ2, σ3) =

• N1+N2+N3=N

σN1

1 σN2 2 σN3 3 Z(N1, N2, N3)

◮ Expand around background (instanton sector) {N∗

i } in gs

◮ Start: 1-cut background [David’91]

◮ Perturbative configuration: all N eigenvalues in cut C ◮ instanton corrections: eigenvalues in other saddles

◮ Resurgence: transseries with σ1, σ2, reach other sectors in gs plane

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Phase Diagram & Large-N Duality

P1 P2 I II 2 1 1 2 1 1 2

[Bertola’07, Bertola, Tovbis’11] [IA,Schiappa,Vonk’13] ◮ light red: Stokes regions, standard

large N expansion

◮ I: 1-cut solution is dominant ◮ II: 2-cut sym solution dominant

◮ blue: anti-Stokes region,dominated

by 3-cuts solution, oscillatory behaviour; no genus expansion

◮ gray: 1-cut unstable configuration ◮ Re line in I and II: Stokes lines, exponentially supressed saddles are

as suppressed as possible

◮ Im line in blue region: anti-Stokes lines, 3 cuts with same size ◮ P1 (P2): DSL described by Painlev´

e I - 2d gravity (II - 2d sugra)

Ongoing work: how to jump across these regions? Resurgence!

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Summary and Future directions

Perturbation theory in a QM setting:

◮ All semi-classical sectors had a non-perturbative ambiguity

associated to the corresponding asymptotic series

◮ These ambiguities cancel between each other ◮ A proper way to define perturbation theory is to consider a sum of

all semi-classical sectors in a multi-parameter transseries

◮ The non-ambiguous result is given by the median resummation

Resurgence:

◮ Knowing all semi-classical sectors allows us to define the

non-ambigous result mentioned above via resurgence

◮ Equivalently resurgence tells us that all the other sectors are

encoded in the perturbative series

◮ In many cases we only know the perturbative expansion

◮ We can determine the full multi-instantonic sectors from the

large order behaviour of the perturbative series

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Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions

Summary and Future directions

More Beyond Quantum Mechanics?

The analysis presented works in a general setting, as long as the singularities in the Borel plane have a semi-classical description much like instantons do

QTFs and asymptotically free gauge theories

◮ Description of all singularities as semiclassical data & discovery of

• ther semi-classical objects: unitons and fractons

◮ Towards a nonperturbative definition, starting from perturbative

data, and augmenting them into transseries involving all types of semi-classical data

◮ multi-renormalons, -instantons, -unitons, -fractons, · · ·

String theory and integrable models:

◮ Large-N duality from phase diagram of matrix models ◮ Generalisation to gauge theories via localisation techniques ◮ Other semi-classical constructions: Integrable models?

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