Resurgence in Quantum Theories Motivation Cancelling Ambiguities Resurgence in Quantum Theories: Resurgence Real Transseries Perturbative Theory and Beyond Airy Quartic MM Summary/Future Directions Inˆ es 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 Perturbation Theory & Asymptotic Series Quantum Theories Motivation Perturbation theory: fundamental in computations of Cancelling Ambiguities Resurgence ◮ ground-state energies in quantum mechanics Real Transseries ◮ beta-functions in quantum field theory Airy ◮ genus expansions of string theory Quartic MM Summary/Future ◮ large N expansion of non-abelian gauge theories Directions · · · BUT... most perturbative expansions are asymptotic, i.e. zero radius of convergence! ◮ Why? existence of singularities in the complex Borel plane, usually related to ◮ instantons ◮ renormalons (21 November) Resurgence in Quantum Theories 2 / 28
Resurgence in Perturbation Theory Quantum Theories Motivation Perturbative expansion of quantity F ( z ) in parameter z ∼ ∞ Cancelling Ambiguities Resurgence F g z − g − 1 , � F ( z ) Asymptotic series: F g ∼ g ! ≃ Real Transseries g ≥ 0 Airy Quartic MM Summary/Future ◮ How to find F ( z )? Directions ◮ Borel transform B [ F ]: ” remove”the factorial growth ◮ Analytically continue B [ F ] to full complex plane ◮ Define resummation S F by the inverse Borel transform ◮ BUT: S F 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 Perturbation Theory Quantum Theories Motivation Perturbative expansion of quantity F ( z ) in parameter z ∼ ∞ Cancelling Ambiguities Resurgence F g z − g − 1 , � F ( z ) Asymptotic series: F g ∼ g ! ≃ Real Transseries g ≥ 0 Airy Quartic MM Summary/Future ◮ How to find F ( z )? Directions ◮ Borel transform B [ F ]: ” remove”the factorial growth ◮ Analytically continue B [ F ] to full complex plane ◮ Define resummation S F by the inverse Borel transform ◮ BUT: S F 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 (21 November) Resurgence in Quantum Theories 3 / 28
Resurgence in Aside: Borel Transform & Resummation Quantum Theories Motivation Cancelling Ambiguities F g z − g − 1 , � Asymptotic series: F ( z ) ≃ with F g ∼ g ! Resurgence g ≥ 0 Real Transseries Airy ∞ F g � Quartic MM ◮ Borel transform : g ! s g B [ F ]( s ) = Summary/Future g =0 Directions � 1 � ( s ) = s α / Γ( α + 1) Rule: B z α +1 ◮ finite radius of convergence - find function B [ F ]( s ) ◮ In general B [ F ]( s ) will have singularities ◮ Borel resummation of F is the Laplace transform ˆ ∞ ds B [ F ]( s ) e − s z S F ( z ) = 0 (21 November) Resurgence in Quantum Theories 4 / 28
Resurgence in Nonperturbative Ambiguity Quantum Theories Motivation Borel resummation of F along direction θ is the Laplace transform Cancelling Ambiguities ˆ e i θ ∞ Resurgence ds B [ F ]( s ) e − s z S θ F ( z ) = Real Transseries 0 Airy Quartic MM ◮ Take B [ F ]( s ) with singularities in direction θ : Summary/Future Directions Nonperturbative ambiguity: ◮ B [ F ]( s ) ∼ 1 s − A in direction θ 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 (21 November) Resurgence in Quantum Theories 5 / 28
Resurgence in Example: Reality of the Airy Function Quantum Theories Motivation Cancelling Ambiguities Airy differential equation Z ”( κ ) − κ Z ( κ ) = 0 Resurgence Real Transseries Airy Objective: finding a real solution in the whole real line κ ∈ R Quartic MM Summary/Future Directions ◮ First look at κ > 0: 1 2 √ πκ 1 / 4 e − 1 2 A κ 3 / 2 Φ − 1 / 2 ( κ ) Z Ai ( κ ) = ◮ Φ − 1 / 2 is an asymptotic expansion ◮ Z Ai ( κ ) is Borel resummable at κ > 0 ◮ 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 Example: Reality of the Airy Function Quantum Theories Motivation Cancelling Ambiguities Airy differential equation Z ”( κ ) − κ Z ( κ ) = 0 Resurgence Real Transseries Airy Objective: finding a real solution in the whole real line κ ∈ R Quartic MM Summary/Future Directions ◮ We need to understand how to reach κ < 0 ◮ We start with Z Ai for κ > 0 Z Ai ( κ = e i θ ) ◮ Analytically continue until arg κ = 2 π 3 ◮ Pole in the Borel plane! ◮ Z Ai is asymptotically same ? θ = 2 π before and after 3 ◮ We need to understand how to jump the dicontinuous direction (21 November) Resurgence in Quantum Theories 6 / 28
Resurgence in Beyond Perturbation Theory? Quantum Theories Motivation Cancelling Ambiguities Resurgence Real Transseries Airy Quartic MM Summary/Future Directions How can we make sense out of perturbation theory? (21 November) Resurgence in Quantum Theories 7 / 28
Resurgence in Beyond Perturbation Theory? Quantum Theories Motivation Learn from the example of anharmonic potential in QM Cancelling Ambiguities [Vainshtein’64, Bender,Wu’73] Resurgence ◮ Coefficients of perturbative series of ground-state energy obey Real Transseries F g ∼ g ! A − g , g ≫ 1 Airy Quartic MM ◮ Borel plane: singularity in positive real axis, governed by real Summary/Future Directions 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 ∼ g ! n A − g , g ≫ 1 F ( n ) g Any multi-instanton series suffers from nonperturbative ambiguities! (21 November) Resurgence in Quantum Theories 7 / 28
Resurgence in Problem or Solution? Quantum Theories Motivation ◮ Multi-instanton series suffers from nonperturbative ambiguities! Cancelling Ambiguities ◮ In most cases there is an infinite number of instanton sectors... Resurgence Real Transseries Airy Seems to make the problem with perturbation theory even worse ! Quartic MM Summary/Future Directions ◮ 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! (21 November) Resurgence in Quantum Theories 8 / 28
Resurgence in Beyond Perturbation Theory! Quantum Theories Motivation Ground-state energy = sum over all multi-instanton sectors Cancelling Ambiguities Resurgence ◮ usual asymptotic perturbative expansion Real Transseries ◮ all asymptotic expansions around each nonperturbative (instanton) Airy Quartic MM sector Summary/Future Directions 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 ( n ) � σ n F ( n ) ( z ) , F ( n ) ( z ) ≃ e − nAz � z − g F ( z , σ ) = g n ≥ 0 g ≥ 1 ◮ our case has e − Az and z ◮ σ : instanton counting parameter (21 November) Resurgence in Quantum Theories 9 / 28
Resurgence in Ambiguities along Stokes lines Quantum Theories Motivation Cancelling Ambiguities ◮ Nonperturbative ambiguity of F ( z ) along a Stokes line: Resurgence ◮ B [ F ] has singularities along corresponding singular direction θ Real Transseries ◮ Lateral Borel resummations S θ ± F differ Airy Quartic MM ( S θ + − S θ − ) F � = 0 Summary/Future Directions ◮ BUT : these lateral resummations are still related via the Stokes automorphism S θ : S θ + F = S θ − ◦ S θ F ◮ Discontinuity in the direction θ of the Borel transform: S θ = 1 − Disc θ ◮ S θ � = 1 encodes information on the Stokes transition at θ ◮ Determined up to unknowns called Stokes Constants S k ◮ How? Via Alien Calculus and Resurgence Determine the nonperturbative ambiguities using the Stokes automorphism (21 November) Resurgence in Quantum Theories 10 / 28
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