The Quartic Matrix Model: Transseries, Resurgence and Resummation Stokes Phenomenon, Resurgence and Physics IRMA, Strasbourg October 14, 2016 Ricardo Vaz
Outline Part I – Transseries and Resurgence [Aniceto, Schiappa, Vonk '11] ➢ Matrix models at large ➢ 1-cut QMM : transseries solution ➢ 1-cut QMM : large-order resurgent relations and Stokes constants [Schiappa, RV '13] ➢ 2-cut QMM Part II – Resummation [Couso-Santamaría, Schiappa, RV '15] ➢ Resummation of large transseries ➢ Finite (exact) solution ➢ Analytic Continuation: complex ➢ Analytic Continuation II: complex Summary / Conclusions / Current work
Matrix Models Why Matrix Models (Random Matrices)? Toy models of QFT Very tractable, ideal to investigate nonperturbative phenomena Related to richer theories via dualities/localization Well-studied critical points
Matrix Models at Large N Partition Function (after gauge-fixing) Vandermonde determinant Quartic Matrix Model (QMM) 't Hooft large N limit: large, small, fixed ➢ Eigenvalues condense around critical points of ➢ Physical quantities have topological expansions
Matrix Models at Large N Simplest case: one-cut solution eigenvalues condense in Spectral curve Eigenvalue density ➢ is a contour integral ➢ determined from boundary condition Our example: QMM
Matrix Models at Large N Q: What are nonperturbative effects? Take two different eigenvalue distributions Different backgrounds are instanton sectors ➢ Consider 1-cut solution as “ reference background” ➢ Study fluctuations around it All objects determined from spectral curve
Matrix Models at Large N [Bessis, Itzykson, Zuber '80] Orthogonal Polynomials Recursion coefficients Partition Function O.P. satisfy recursion relations String Equation
Matrix Models at Large N String Equation 't Hooft limit becomes continuous variable promoted to a function Compute the Free Energy using ~Euler-Maclaurin formula Normalized free energy
Transseries Solutions [Mariño '08] have transseries expansions n -instanton, g -loop coefficient Transseries Instanton Action parameter ➢ Expansion in two “coupling constants”: ➢ Perturbative sector is ➢ Each instanton sector is itself an asymptotic series Not the whole story: need two-parameter transseries
Two-parameter transseries [Garoufalidis, Its, Kapev, Mariño '10] [Aniceto, Schiappa, Vonk '11] Non-zero starting genus ● effects also present (physical interpetation?) ● Sectors with have no exponential weight After plugging ansatz into string equation : ● Differential equations when Resonance ● Algebraic equations in all other cases
Two-parameter transseries [Garoufalidis, Its, Kapaev, Mariño '10] Final twist: logarithmic sectors ! [Aniceto, Schiappa, Vonk '11] Better variable General structure of results: Polynomial of degree in
Two-parameter transseries Log and non-log terms are related: we can perform the sum over The QMM Free Energy ● Similar transseries structure inherited ● General pattern for coefficients ● Log/non-log relations and summation also hold
Double-scaling limit The one-cut QMM has a very interesting double-scaling limit with kept fixed Double-scaling ansatz [Mariño, Schiappa, Weiss '07] Painlevé I Equation [Aniceto, Schiappa, Vonk '11] (Marcel's talk) ➢ small adjustments needed to match P1 two-parameter transseries ● Reparametrization invariance of the transseries ● String equation not affected
Tests of Resurgence ➢ Transseries coefficients are not a random, infinite collection of functions/numbers ➢ Formalism of resurgence produces a web of (large-order) relations [Écalle] Schematically: [Seara, Sauzin '04] Bridge Equations Alien derivatives/ Cauchy Theorem Stokes automorphism Large-order/asymptotic resurgent relations
Tests of Resurgence Large-order behaviour of the perturbative sector ● Leading behaviour ● Exponentially suppressed contributions from higher instantons Leading order: Stokes constant
Tests of Resurgence Beyond 1-instanton sector: contributions Borel-Padé Asymptotic series approximant
Tests of Resurgence Large-order relations for other sectors
Stokes Constants [Aniceto, Schiappa, Vonk '11] From large-order relations for Painlevé I transseries ➔ Unexpected relations: ➔ QMM Stokes constants are trivially related to P1 Stokes constants
Two-cut QMM [Schiappa, RV '13] Two-cut solution with -symmetry Spectral curve: Q: What are nonperturbative effects? Q: Need to consider QMM as general three-cut problem All backgrounds contribute at order 1-instanton “Elliptic” component cancelled by -symmetry
Two-cut QMM Orthogonal polynomials Inspiration from numerics Generalization to multi-cut scenarios is not obvious 't Hooft limit of string equation should be split: ➢ Introduce (two-parameter) transseries expansions for ➢ Plug into string equation(s) and solve order by order ➢ Compute free-energy via large N limit of
Two-cut QMM ➢ Results for qualitatively similar to one-cut case ➢ Connection between log and non-log sectors ➢ Tests of large-order/resurgence (instanton action)
Two-cut QMM Very interesting double-scaling limit -b -a +a +b while is kept fixed DSL + DSL ansatz for Painlevé II String Equation ➢ More computational power, tests of higher-order resurgence relations ➢ Extraction of Stokes constants (related to off-critical Stokes constants) “Unexpected” relations also occur, e.g.
Finite N from Resurgent Large N ['t Hooft '74] “[…] the 1/N expansion may be a reasonable perturbation expansion, in spite of the fact that N is not very big.” ➢ This is indeed true ➢ An appropriate resummation of asymptotic expansions is needed ➢ Nonperturbative contributions are important (transseries) Borel-Padé-Écalle Resummation
Finite N from Resurgent Large N [Couso-Santamaría, Schiappa, RV '15] One-parameter transseries for (one-cut solution) up to ● BPE resummation to extract numbers up to (“how to associate a number to a divergent sum?”) ● Compare to exact results at finite N ● Interpolation/analytic continuation explore (monodromy, “strength” of n.p. sectors) Interpolate, extend to Other examples: ➢ 3d ABJ(M) partition functions [Codesido, Grassi, Mariño '14] ➢ Cusp anomalous dimension [Aniceto '15] [Dorigoni, Hatsuda '16] (Inês' talk) ➢ Hydrodynamics (Müller-Israel-Stewart) [Heller, Spaliński '15]+[Aniceto, Spaliński '16]
BPE Resummation A : Borel-Padé-Écalle Resummation Borel Resummation ➢ Borel transform cannot be computed exactly ➢ Only a finite # of terms available Borel-Padé Resummation ➢ BP approximant is a rational function ➢ Resummation can be performed BPE Resummation ➢ Independent of ➢ to be determined ➢ Finite N predictions
Exact Results Partition function can be computed exactly for finite (small) ➢ Exact results to be compared to resummed transseries Define moments : Confluent Hypergeometric of the second kind Non-trivial has branch cut along monodromy
Exact Results For fixed we have to go around the complex plane twice to close the curve
Exact vs. Resummation (exact) (resummed transseries) Perturbative 1-instanton 2-instanton 3-instanton Transseries Parameter above Stokes constant of the QMM Fixed , changing
Analytic Continuation Fix , fix and move Stokes Lines (Transseries parameter jumps) Anti-Stokes Lines (All sectors of same order)
Analytic Continuation Free Energy
Analytic Continuation Free Energy
Analytic Continuation Recursion coefficient
Analytic Continuation Recursion coefficient
Analytic Continuation II Example : Factorials ( Euler ) Gamma Function ➢ defined for all ➢ not an entire function (poles at ) Other generalizations [Luschny '06] ➢ Hadamard Gamma Function ➢ is an entire function ➢ is the unique continuation satisfying
Analytic Continuation II Example : Gaussian Matrix Model Exactly solvable (Hermite polynomials) Barnes G-function ➢ original partition function defined for integer ➢ Barnes G-function is an entire function (holomorphic for all )
Analytic Continuation II Claim : Resummed transseries are the unique analytic continuation of the QMM into complex (satisfying the string equation) Entire function? ➢ Possibly... (more data needed!)
Summary ● QMM is the ideal testing ground for ideas of resurgence ● Geometrical/physical interpretation of nonperturbative effects ● Transseries construction via orthogonal polynomials (string equation) ● Generate large amounts of data investigate structure/properties verify asymptotic relations compute Stokes constants (numerically) ● Computational power and knowledge enhanced at critical points ● Transseries can be used to reach finite from large ● QMM partition function can be extended to any complex
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