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

Resurgence in String Theory Motivation Cancelling Ambiguities Resurgence of Instantons in Resurgence Applications String Theory Summary/Future Directions Inˆ es Aniceto (Based on ongoing work with R. Schiappa and M. Vonk, 1106.5922 and 1308.1115) Bern, 2 September 2013 (2 September) Resurgence in String Theory 1 / 18

Resurgence in String Perturbation Theory & Asymptotic Series Theory Motivation Perturbation theory: fundamental in computations of Cancelling Ambiguities Resurgence ◮ ground-state energies in quantum mechanics Applications ◮ beta-functions in quantum field theory Summary/Future Directions ◮ 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 ◮ instantons ◮ renormalons (2 September) Resurgence in String Theory 2 / 18

Resurgence in String Perturbation Theory Theory Motivation Perturbative expansion of quantity F ( z ) in parameter z ∼ ∞ Cancelling Ambiguities Resurgence F g z − g , � F ( z ) Asymptotic series: F g ∼ g ! ≃ Applications g ≥ 0 Summary/Future Directions ◮ How to find F ( z )? ◮ 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? (2 September) Resurgence in String Theory 3 / 18

Resurgence in String Perturbation Theory Theory Motivation Perturbative expansion of quantity F ( z ) in parameter z ∼ ∞ Cancelling Ambiguities Resurgence F g z − g , � F ( z ) Asymptotic series: F g ∼ g ! ≃ Applications g ≥ 0 Summary/Future Directions ◮ How to find F ( z )? ◮ 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 (2 September) Resurgence in String Theory 3 / 18

Resurgence in String Nonperturbative Ambiguity Theory 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 ) = Applications 0 Summary/Future Directions ◮ 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 (2 September) Resurgence in String Theory 4 / 18

Resurgence in String Beyond Perturbation Theory? Theory Motivation Cancelling Ambiguities Resurgence Applications Summary/Future Directions How can we make sense out of perturbation theory? (2 September) Resurgence in String Theory 5 / 18

Resurgence in String Beyond Perturbation Theory? Theory Motivation Learn from the example of anharmonic potential in QM Cancelling Ambiguities [Bender,Wu’73] Resurgence ◮ Coefficients of perturbative series of ground-state energy obey Applications F g ∼ g ! A − g , g ≫ 1 Summary/Future Directions ◮ 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 ∼ g ! n A − g , g ≫ 1 F ( n ) g Any multi-instanton series suffers from nonperturbative ambiguities! (2 September) Resurgence in String Theory 5 / 18

Resurgence in String Problem or Solution? Theory Motivation ◮ Multi-instanton series suffers from nonperturbative ambiguities! Cancelling Ambiguities ◮ In most cases there is an infinite number of instanton sectors... Resurgence Applications Summary/Future Directions 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! (2 September) Resurgence in String Theory 6 / 18

Resurgence in String Beyond Perturbation Theory! Theory Motivation Ground-state energy = sum over all multi-instanton sectors Cancelling Ambiguities Resurgence ◮ usual asymptotic perturbative expansion Applications ◮ all asymptotic expansions around each nonperturbative (instanton) Summary/Future Directions 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 ( 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 (2 September) Resurgence in String Theory 7 / 18

Resurgence in String Ambiguities along Stokes lines Theory Motivation Cancelling Ambiguities ◮ Nonperturbative ambiguity of F ( z ) along a Stokes line: Resurgence ◮ B [ F ] has singularities along corresponding singular direction θ Applications ◮ Lateral Borel resummations S θ ± F differ Summary/Future Directions ( S θ + − S θ − ) F � = 0 ◮ 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 (2 September) Resurgence in String Theory 8 / 18

Resurgence in String Median Resummation Theory Motivation ◮ Re-write the lateral Borel resummations as Cancelling Ambiguities S ± = 1 2 ( S + + S − ) ± 1 Resurgence 2 ( S + − S − ) Applications Summary/Future ◮ To cancel ambiguities set Directions S + − S − ∼ 0 at the level of the transseries ◮ We are left with an unambiguous result given by S med ∼ 1 2 ( S + + S − ) which is just the median resummation ! ◮ In terms of Stokes automorphism: [Delabaere,Pham,’99] S med = S + ◦ S − 1 / 2 = S − ◦ S 1 / 2 θ θ Note: For multi-parameter transseries, appearance of extra singularities and Stokes constants severely increases difficulty [IA,Schiappa,’13] (2 September) Resurgence in String Theory 9 / 18

Resurgence in String Constructing a real Transseries F ( z , σ ) Theory Motivation Cancelling Ambiguities ◮ Physical set-up for coupling z real positive: [IA,Schiappa,’13] Resurgence ◮ Stokes line in the positive real axis: θ = 0 - singular direction Applications ◮ Coefficients of transseries F ( n ) real Summary/Future g Directions ◮ Nonperturbative ambiguities for each F ( n ) are imaginary I m F ( n ) = 1 2 i ( S + − S − ) F ( n ) ◮ Canceling these defines an unambiguous real transseries in the positive real axis S med F = S − F ( z , σ + 1 F R ( z , σ ) = 2 S 1 ) � σ 2 − 1 � R e F (0) + σ R e F (1) + R e F (2) + · · · 2 S 2 = 1 where σ ∈ R and S 1 is Stokes constant. All instanton sectors contribute! Even when σ = 0 ... (2 September) Resurgence in String Theory 10 / 18

Resurgence in String Real Transseries: Cancellations Theory Motivation Cancelling Ambiguities Constructing the real transseries Resurgence Applications Summary/Future Directions We want to cancel the ambiguities coming from perturbative expansion: � � F � 0 � � � � F � z , Σ � 0 � (2 September) Resurgence in String Theory 11 / 18

Resurgence in String Real Transseries: Cancellations Theory Motivation Cancelling Ambiguities Constructing the real transseries Resurgence Applications Summary/Future Directions � � F � 0 � � � e F � 0 � � i � m F � 0 � Where the nonperturbative ambiguity of F � 0 � is: � 2 i � m F � 0 � � � S 1 m � � F � m � m � 1 � S 1 � e F � 1 � � 1 2 S 13 � e F � 3 � � S 15 � e F � 5 � � ... (2 September) Resurgence in String Theory 11 / 18

Resurgence in String Real Transseries: Cancellations Theory Motivation Cancelling Ambiguities Constructing the real transseries Resurgence Applications Summary/Future Directions � � F � 0 � � � e F � 0 � � 1 2 S 1 � e F � 1 � � 1 4 S 13 � e F � 3 � � ... To cancel the first imaginary term we need to add 1 2 S 1 � � F � 1 � � 1 2 S 1 � e F � 1 � � i 2 S 1 � m F � 1 � (2 September) Resurgence in String Theory 11 / 18