The Quartic Matrix Model: Transseries, Resurgence and Resummation - - PowerPoint PPT Presentation

Ricardo Vaz

The Quartic Matrix Model:

Transseries, Resurgence and Resummation

Stokes Phenomenon, Resurgence and Physics IRMA, Strasbourg October 14, 2016

Outline

Part I – Transseries and Resurgence

➢ Matrix models at large ➢ 1-cut QMM: transseries solution ➢ 1-cut QMM: large-order resurgent relations and Stokes constants ➢ 2-cut QMM

Part II – Resummation

➢ Resummation of large transseries ➢ Finite (exact) solution ➢ Analytic Continuation: complex ➢ Analytic Continuation II: complex

[Aniceto, Schiappa, Vonk '11] [Schiappa, RV '13] [Couso-Santamaría, Schiappa, RV '15]

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

➢ is a contour integral ➢ determined from boundary condition

Eigenvalue density 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 Partition Function

Recursion coefficients 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

have transseries expansions [Mariño '08] n-instanton, g-loop coefficient Instanton Action Transseries 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 After plugging ansatz into string equation:

• Differential equations when
• Algebraic equations in all other cases
• effects also present (physical interpetation?)
• Sectors with have no exponential weight

Resonance

Two-parameter transseries

[Garoufalidis, Its, Kapaev, Mariño '10] [Aniceto, Schiappa, Vonk '11]

Final twist: logarithmic sectors!

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

Painlevé I Equation

[Mariño, Schiappa, Weiss '07] [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

Cauchy Theorem Bridge Equations Alien derivatives/ Stokes automorphism

Large-order/asymptotic resurgent relations

Schematically: [Écalle] [Seara, Sauzin '04]

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 Asymptotic series Borel-Padé approximant

Tests of Resurgence

Large-order relations for other sectors

Stokes Constants

From large-order relations for Painlevé I transseries [Aniceto, Schiappa, Vonk '11]

➔ Unexpected relations: ➔ QMM Stokes constants are trivially related to P1 Stokes constants

Two-cut QMM

Two-cut solution with -symmetry

[Schiappa, RV '13] Spectral curve: Q: Q: What are nonperturbative effects? All backgrounds contribute at order 1-instanton “Elliptic” component cancelled by -symmetry Need to consider QMM as general three-cut problem

Two-cut QMM

Orthogonal polynomials

Generalization to multi-cut scenarios is not obvious Inspiration from numerics '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

+b +a

• a

while is kept fixed String Equation

Painlevé II

DSL + DSL ansatz for

➢ 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 up to

• BPE resummation to extract numbers

(“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

• f the second kind

has branch cut along Non-trivial 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

above

Fixed , changing Transseries Parameter

Stokes constant of the QMM

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

Other generalizations [Luschny '06]

➢Hadamard Gamma Function ➢ is an entire function ➢ is the unique continuation satisfying

(poles at )

Analytic Continuation II

Example: Gaussian Matrix Model Exactly solvable

➢ original partition function defined for integer ➢ Barnes G-function is an entire function (holomorphic for all )

(Hermite polynomials)

Barnes G-function

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

Current/Future Work

Open problems to be addressed:

➔ Derivation of Stokes constants ➔ Other applications of resummation/interpolation (e.g. Painlevé II) ➔ Transasymptotic resummation of QMM Marcel's talk

sum over with fixed

➔ Technically involved because are functions of ➔ What can we learn from distributions of poles/zeroes?

Current/Future Work

[Bertola, Tovbis '11] [Aniceto, Couso-S., Schiappa, Vaz, Vonk '16/'17] I :

• ne-cut phase

Origin of n.p. effects Transseries structure Large-order relations DSL → Painlevé I Stokes constants I I : two-cut phase Origin of n.p. effects Transseries structure Large-order relations DSL → Painlevé II Stokes constants

Current/Future Work

I I I : three-cut phase (anti-Stokes) I V : trivalent phase Construction of spectral curve Origin of n.p. effects (trivalent) Transseries structure ~ -function

Current/Future Work

Thank you!