Resurgence Perturbative Theory and Beyond

Ines Aniceto
(Based on ongoing work with R. Schiappa and M. Vonk, 1106.5922 and 1308.1115)
University of Minnesota, 21 November 2013
(21 November) Resurgence in Quantum Theories 1 / 28
Resurgence in Quantum Theories
I large N expansion of non-abelian gauge theories
· · ·
BUT... most perturbative expansions are asymptotic, i.e. zero radius of convergence!
I Why? existence of singularities in the complex Borel plane, usually related to
I instantons I renormalons
F (z) ' ∑ g≥0
I How to find F (z)?
I Borel transform B[F ]: ”remove” the factorial growth I Analytically continue B[F ] to full complex plane I Define resummation SF by the inverse Borel transform
I 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?
F (z) ' ∑ g≥0
I How to find F (z)?
I Borel transform B[F ]: ”remove” the factorial growth I Analytically continue B[F ] to full complex plane I Define resummation SF by the inverse Borel transform
I BUT: SF is just a Laplace transform - needs an integration contour to be properly defined!
I If we have a singularity in the complex Borel plane:
Nonperturbative ambiguity: ambiguity in choosing how integration contour will avoid the singularity
Fg z −g−1 , with Fg ∼ g !
I Borel transform: B[F ](s) = ∞∑ g=0
Fg
I finite radius of convergence - find function B[F ](s)
I In general B[F ](s) will have singularities
I Borel resummation of F is the Laplace transform
SF (z) =
ˆ ∞ 0
Quartic MM
Summary/Future Directions
Nonperturbative Ambiguity
Borel resummation of F along direction θ is the Laplace transform
SθF (z) =
ˆ eiθ∞
I Take B[F ](s) with singularities in direction θ:
Nonperturbative ambiguity:
in direction θ
I around z ∼ ∞ this is non-analytic
I Singularities in the Borel plane occur along Stokes lines
Perturbative series is non-Borel resummable along Stokes lines
Airy differential equation Z”(κ)− κZ(κ) = 0
Objective: finding a real solution in the whole real line κ ∈ R
I First look at κ > 0:
ZAi (κ) = 1
I ZAi (κ) is Borel resummable at κ > 0
I Can we analitically continue the solution to κ < 0?
No! Φ−1/2(κ) has a pole in Borel plane before κ < 0!
Airy differential equation Z”(κ)− κZ(κ) = 0
Objective: finding a real solution in the whole real line κ ∈ R
I We need to understand how to reach κ < 0
ZAi(κ = eiθ)
θ = 2π
I Analytically continue until arg κ = 2π
3
I ZAi is asymptotically same before and after
I We need to understand how to jump the dicontinuous direction
Beyond Perturbation Theory?
Learn from the example of anharmonic potential in QM [Vainshtein’64, Bender,Wu’73]
I Coefficients of perturbative series of ground-state energy obey
Fg ∼ g !A−g , g 1
I Borel plane: singularity in positive real axis, governed by real instanton action A
I Resummation along real axis leads to a nonperturbative ambiguity
BUT: not only the perturbative sector which has an ambiguity!!!
I 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!
I Multi-instanton series suffers from nonperturbative ambiguities!
I In most cases there is an infinite number of instanton sectors...
Seems to make the problem with perturbation theory even worse!
I BUT: for the ground state energy of double-well potential [Bogomolny,Zinn-Justin,’80-83]
I ambiguity in 2-instanton sector precisely cancels ambiguity in perturbative expansion
I ambiguity in 3-instanton sector cancels ambiguity in 1-instanton sector
I · · ·
I usual asymptotic perturbative expansion I 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
I our case has e−Az and z I σ: instanton counting parameter
I Nonperturbative ambiguity of F (z) along a Stokes line:
I B[F ] has singularities along corresponding singular direction θ I Lateral Borel resummations Sθ±F differ
(Sθ+ − Sθ−)F 6= 0
I BUT: these lateral resummations are still related via the Stokes automorphism Sθ:
Sθ+F = Sθ− SθF
I Discontinuity in the direction θ of the Borel transform: Sθ = 1−Discθ
I Sθ 6= 1 encodes information on the Stokes transition at θ
I Determined up to unknowns called Stokes Constants Sk
I How? Via Alien Calculus and Resurgence
Determine the nonperturbative ambiguities using the Stokes automorphism
The cancelation of nonperturbative ambiguities in multi-instanton sectors is but an example of a larger structure behind perturbation theory!
Resurgence analysis and Transseries
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 (`) n in terms of F
(`′) n where `′ is close to `
How does it work?
-A A 2 A 3 A 4 A 5 A ...
Instanton Instanton action singularities
-A A 2 A 3 A 4 A 5 A ...
Perturbative series:
Instanton series:
¥
-A A 2 A 3 A 4 A 5 A ...
Large-order behaviour (g >>1): Use Cauchy's Theorem for each FHnLHzL
FHzL = 1
...
FH1L FH2L FH3L FH4L FH5L
Fg H0L ~ S1 â
2 â n>0
...
Equivalently: Perturbative series for large g ENCODES all other sectors
FH1L FH2L FH3L FH4L FH5L
Fg H0L ~ S1 F1
determine F1 H1L
...
Equivalently: Perturbative series for large g ENCODES all other sectors
FH2L FH3L FH4L FH5L
Fg H0L - S1 â
2 F1 H2L
determines F1 H2L
...
FH2L FH3L
2 â n>0
...
FH3L FH4L FH5L
H3L + 2-g S1 2 âbnHgL Fn
H4L + H-1Lg f IS-Γ M âcnHgL Fn H1L + ...
FH1L
Large - order behaviour - r - instanton series for large g
Fg HrL
H3L + H-1Lg f1IS-Γ M c1HgL F1
Hr-1L + ...
3 Ak
2 Ak
Ak A1
2 A1
3 A1
We are now ready to see how non-perturbative ambiguities cancel!
I Nonperturbative ambiguity of F (z) along a Stokes line:
I B[F ] has singularities along corresponding singular direction θ
I Lateral Borel resummations Sθ±F differ
(Sθ+ − Sθ−)F 6= 0
I Lateral resummations are related via the Stokes automorphism Sθ
Determine the nonperturbative ambiguities using the Stokes automorphism
S± = 1
S+ − S− ∼ 0 at the level of the transseries
I We are left with an unambiguous result given by
Smed ∼ 1
I In terms of Stokes automorphism: [Delabaere,Pham,’99]
Smed = S+ S−1/2 θ = S− S1/2
θ
I Physical set-up for coupling z real positive: [IA,Schiappa,’13]
I Stokes line in the positive real axis: θ = 0 - singular direction
I Coefficients of transseries F (n) g real
I Nonperturbative ambiguities for each F (n) are imaginary
ImF (n) = 1
coming from perturbative expansion:
Constructing the real transseries
¥
2 S1 Re FH1L +
Nonperturbative
ambiguity of FH1L is: 2 i Im FH1L ~Α Re FH2L + Β Re FH3L + ...
Α S- FH2L = Α Re FH2L - Α i Im FH2L
Β S- FH3L = Β Re FH3L - Β i Im FH3L
we finally find the real transseries:
...........
¥ S1
n
2 S1O
I Physical set-up for coupling z real positive: [IA,Schiappa,’13]
I Stokes line in the positive real axis: θ = 0 - singular direction I Coefficients of transseries F
(n) g real
ImF (n) = 1
2i (S+ − S−)F (n)
I Canceling these defines an unambiguous real transseries in the positive real axis
FR(z , σ) = SmedF = S−F (z , σ + 1
2 S1)
Quartic MM
Summary/Future Directions
Airy Function - Asymptotic Solutions Airy differential equation Z”(κ)− κZ(κ) = 0
Two independent solutions
ZAi (κ) = 1
I Asymptotic expansions: Φ±1/2(κ) ' ∑ n≥0
(1)n anκ − 3
Transseries solution: Z(κ, σ1, σ2) = σ1ZAi (κ) + σ2ZBi (κ)
ZΓ(λ) =
3 x3
I Γ passes through saddle point x∗Ai = −eiκ/2 (x∗Bi = eiκ/2)
I infinite oriented path of steepest descent
Im (V (x)− V (x∗)) = 0
I Changing κ = eiθ leads to different paths: choose Γ s.t.
Re (V (x)− V (x∗)) > 0 for x large
Moving in κ space: Stokes phenomena
Π
6
Π
3
Transseries solution: Z(z = k3/2, σ1, σ2) = σ1ZAi (z) + σ2ZBi (z)
Two Stokes transitions in singular directions arg z = 0, π
S0Z(z , σ1, σ2) = Z (z , σ1 − iσ2, σ2)
SπZ(z , σ1, σ2) = Z (z , σ1, σ2 − iσ1)
In the ”physical” variable κ, Stokes lines unfold:
Stokes line at arg z = 0 ⇒ Stokes lines at arg κ = 0, 4π
3 ,
2π
3
Stokes line at arg z = π ⇒ Stokes lines at arg κ = 2π
3 , 0,
Airy Function- ”Physical”vs ”Working
Stokes line at arg z = 0 ⇒ Stokes lines at arg κ = 0, 4π
3 ,
2π
3
Stokes line at arg z = π ⇒ Stokes lines at arg κ = 2π
3 , 0,
4π
3
S0
SΠ
S0
SΠ
S0
SΠ
Objective: Real solution in whole real line, i.e. for arg κ = 0, π (21 November) Resurgence in Quantum Theories 22 / 28
Airy Function- Real Transseries
Objective: Real solution in whole real line, i.e. for arg κ = 0, π
I At arg κ = 0:
I Stokes line!
I Cancel the non-perturbative ambiguity to obtain real solution
I Type of transition S0Z(z , σ1, σ2) = Z (z , σ1 − iσ2, σ2)
I Median resummation (Smed 0 ≡ S0+ S−1/2
0 )
ZR+ (z , σ1, σ2) = Smed 0 Z (z , σ1, σ2) = S0+Z
( z , σ1 +
I Choose σ1 = 1, σ2 = 0: usual Airy function solution
ZR+ (z) ≡ S0+Z (z , 1, 0) = S0+Z (z , 1, 0) = S0+ZAi (z)
Airy Function- Real Transseries
Objective: Real solution in whole real line, i.e. for arg κ = 0, π
I At arg κ = 0: ZR+ (κ) ≡ S0+Z (κ, 1, 0) = S0+ZAi (κ)
I At arg κ = π:
I No Stokes line...
I ... but we cross one at arg κ = 2π/3!
I type of transition: SπZ(z , σ1, σ2) = Z (z , σ1, σ2 − iσ1)
S0
SΠ
S0
SΠ
S0
SΠ
= ∑
αn cos (Θ + (−1)nπ/4)
Solution is real in the whole real line, defined by ZR±(κ)
Resurgence & large-N duality:Quartic MM One-matrix model partition function (N × N matrix M)
Z(N, gs) ∝ ˆ
F ' ∑ g≥0
I Full nonperturbative solution include all backgrounds:
Z(σ1, σ2, σ3) = ∑
I Expand around background (instanton sector) {N∗ i } in gs
I Start: 1-cut background [David’91]
I Perturbative configuration: all N eigenvalues in cut C I instanton corrections: eigenvalues in other saddles
I Resurgence: transseries with σ1, σ2, reach other sectors in gs plane
I light red: Stokes regions, standard large N expansion
I I: 1-cut solution is dominant I II: 2-cut sym solution dominant
I blue: anti-Stokes region,dominated by 3-cuts solution, oscillatory behaviour; no genus expansion
I gray: 1-cut unstable configuration
I Re line in I and II: Stokes lines, exponentially supressed saddles are as suppressed as possible
I Im line in blue region: anti-Stokes lines, 3 cuts with same size
I P1 (P2): DSL described by Painleve I - 2d gravity (II - 2d sugra)
Ongoing work: how to jump across these regions? Resurgence!
I Exact quantisation conditions in QM:[Voros,Zinn-Justin,...]
I Orginally derived via WKB and Bohr-Sommerfeld; I New developments using resurgent analysis
[Basar,Dunne,Unsal,’13]
I Semi-classical description of renormalons found; I Problematic as their singularities are dominant compared to
instantons; I In Principal Chiral Model other semi-classical descriptions
have also been found: unitons and fractons.
I String theory and Large-N duality:[Marino;IA,Schiappa et al,’10-13]
I Study of large order behaviour in string theory and large N random matrix models
I Topological strings and Holomorphic Anomaly
I Large−N duality: phase diagram of matrix models and the genus expansion
Perturbation theory in a QM setting:
I All semi-classical sectors had a non-perturbative ambiguity associated to the corresponding asymptotic series
I These ambiguities cancel between each other
I A proper way to define perturbation theory is to consider a sum of all semi-classical sectors in a multi-parameter transseries
I The non-ambiguous result is given by the median resummation
Resurgence:
I Knowing all semi-classical sectors allows us to define the non-ambigous result mentioned above via resurgence
I Equivalently resurgence tells us that all the other sectors are encoded in the perturbative series
I In many cases we only know the perturbative expansion
I We can determine the full multi-instantonic sectors from the large order behaviour of the perturbative series
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
I Description of all singularities as semiclassical data & discovery of other semi-classical objects: unitons and fractons
I Towards a nonperturbative definition, starting from perturbative data, and augmenting them into transseries involving all types of semi-classical data
I multi-renormalons, -instantons, -unitons, -fractons, · · ·
String theory and integrable models:
I Large-N duality from phase diagram of matrix models
I Generalisation to gauge theories via localisation techniques I Other semi-classical constructions: Integrable models?

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Resurgence Perturbative Theory and Beyond