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
Resurgence in Quantum Theories
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?
(21 November) Resurgence in Quantum Theories 3 / 28
Resurgence in Quantum Theories
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
(21 November) Resurgence in Quantum Theories 3 / 28
Resurgence in Quantum Theories
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
(21 November) Resurgence in Quantum Theories 4 / 28
Resurgence in Quantum Theories
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
(21 November) Resurgence in Quantum Theories 5 / 28
Resurgence in Quantum Theories
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!
(21 November) Resurgence in Quantum Theories 6 / 28
Resurgence in Quantum Theories
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
(21 November) Resurgence in Quantum Theories 6 / 28
Resurgence in Quantum Theories
(21 November) Resurgence in Quantum Theories 7 / 28
Resurgence in Quantum Theories
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!
(21 November) Resurgence in Quantum Theories 7 / 28
Resurgence in Quantum Theories
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 · · ·
Resurgence in Quantum Theories
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
(21 November) Resurgence in Quantum Theories 9 / 28
Resurgence in Quantum Theories
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
(21 November) Resurgence in Quantum Theories 10 / 28
Resurgence in Quantum Theories
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?
Resurgence in Quantum Theories
-A A 2 A 3 A 4 A 5 A ...
Instanton Instanton action singularities
(21 November) Resurgence in Quantum Theories 12 / 28
Resurgence in Quantum Theories
-A A 2 A 3 A 4 A 5 A ...
Perturbative series:
Instanton series:
¥
Resurgence in Quantum Theories
-A A 2 A 3 A 4 A 5 A ...
Large-order behaviour (g >>1): Use Cauchy's Theorem for each
FHnLHzL
FHzL = 1
Resurgence in Quantum Theories
...
FH1L FH2L FH3L FH4L FH5L
Fg H0L ~ S1 â
2 â n>0
(21 November) Resurgence in Quantum Theories 12 / 28
Resurgence in Quantum Theories
...
Equivalently: Perturbative series for large g ENCODES all other
sectors
FH1L FH2L FH3L FH4L FH5L
Fg H0L ~ S1 F1
determine F1 H1L
Resurgence in Quantum Theories
...
Equivalently: Perturbative series for large g ENCODES all other
sectors
FH2L FH3L FH4L FH5L
Fg H0L - S1 â
2 F1 H2L
determines F1 H2L
Resurgence in Quantum Theories
...
FH2L FH3L
2 â n>0
Resurgence in Quantum Theories
...
FH3L FH4L FH5L
H3L + 2-g S1 2 âbnHgL Fn
H4L + H-1Lg f IS-Γ M âcnHgL Fn H1L + ...
FH1L
Resurgence in Quantum Theories
Large - order behaviour - r - instanton series for large g
Fg HrL
H3L + H-1Lg f1IS-Γ M c1HgL F1
Hr-1L + ...
Resurgence in Quantum Theories
3 Ak
2 Ak
Ak A1
2 A1
3 A1
(21 November) Resurgence in Quantum Theories 12 / 28
Resurgence in Quantum Theories
We are now ready to see how non-perturbative ambiguities
cancel!
(21 November) Resurgence in Quantum Theories 13 / 28
Resurgence in Quantum Theories
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
(21 November) Resurgence in Quantum Theories 14 / 28
Resurgence in Quantum Theories
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
(21 November) Resurgence in Quantum Theories 15 / 28
Resurgence in Quantum Theories
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
(21 November) Resurgence in Quantum Theories 16 / 28
Resurgence in Quantum Theories
coming from perturbative expansion:
(21 November) Resurgence in Quantum Theories 17 / 28
Resurgence in Quantum Theories
Constructing the real transseries
¥
(21 November) Resurgence in Quantum Theories 17 / 28
Resurgence in Quantum Theories
2 S1 Re FH1L +
Resurgence in Quantum Theories
Nonperturbative
ambiguity of FH1L is: 2 i Im FH1L ~Α Re FH2L + Β Re FH3L +
...
(21 November) Resurgence in Quantum Theories 17 / 28
Resurgence in Quantum Theories
Α S- FH2L = Α Re FH2L - Α i Im FH2L
Β S- FH3L = Β Re FH3L - Β i Im FH3L
(21 November) Resurgence in Quantum Theories 17 / 28
Resurgence in Quantum Theories
we finally find the real transseries:
...........
¥ S1
n
2 S1O
Resurgence in Quantum Theories
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)
Resurgence in Quantum Theories
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)
(21 November) Resurgence in Quantum Theories 19 / 28
Resurgence in Quantum Theories
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
Resurgence in Quantum Theories
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,
Resurgence in Quantum Theories
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
Resurgence in Quantum Theories
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)
(21 November) Resurgence in Quantum Theories 23 / 28
Resurgence in Quantum Theories
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)
(21 November) Resurgence in Quantum Theories 23 / 28
Resurgence in Quantum Theories
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
(21 November) Resurgence in Quantum Theories 24 / 28
Resurgence in Quantum Theories
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!
(21 November) Resurgence in Quantum Theories 25 / 28
Resurgence in Quantum Theories
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
(21 November) Resurgence in Quantum Theories 26 / 28
Resurgence in Quantum Theories
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
(21 November) Resurgence in Quantum Theories 27 / 28
Resurgence in Quantum Theories
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?
(21 November) Resurgence in Quantum Theories 28 / 28