Direct Integration and Non-Perturbative Effects in Matrix Models

Albrecht Klemm; Marco Rauch; Marcos Marino

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Direct Integration and Non-Perturbative Effects in Matrix Models

Albrecht Klemm, Marcos Marino, Marco Rauch

High Energy Physics - Theory · Submitted 22 Feb 2010

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Finally, we argue that a full understanding of the large genus asymptotics in the multi-cut case requires a new class of non-perturbative sectors in the matrix model.

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We show how direct integration can be used to solve the closed amplitudes of multi-cut matrix models with polynomial potentials. In the case of the cubic matrix model, we give explicit expressions for the ring of non-holomorphic modular objects that are needed to express all closed matrix model amplitudes. This allows us to integrate the holomorphic anomaly equation up to holomorphic modular terms that we fix by the gap condition up to genus four. There is an one-dimensional submanifold of the moduli space in which the spectral curve becomes the Seiberg--Witten curve and the ring reduces to the non-holomorphic modular ring of the group $\Gamma(2)$. On that submanifold, the gap conditions completely fix the holomorphic ambiguity and the model can be solved explicitly to very high genus. We use these results to make precision tests of the connection between the large order behavior of the 1/N expansion and non-perturbative effects due to instantons. Finally, we argue that a full understanding of the large genus asymptotics in the multi-cut case requires a new class of non-perturbative sectors in the matrix model.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We show how direct integration can be used to solve the closed amplitudes of multi-cut matrix models with polynomial potentials.	contribution 0.90	0.94	0.39	asserted 1.00
s1	In the case of the cubic matrix model, we give explicit expressions for the ring of non-holomorphic modular objects that are needed to express all closed matrix model amplitudes.	contribution 0.50	0.95	0.11	asserted 1.00
s2	This allows us to integrate the holomorphic anomaly equation up to holomorphic modular terms that we fix by the gap condition up to genus four.	method 0.53	0.92	0.11	asserted 1.00
s3	There is an one-dimensional submanifold of the moduli space in which the spectral curve becomes the Seiberg--Witten curve and the ring reduces to the non-holomorphic modular ring of the group $\Gamma(2)$.	result 0.97	0.81	0.09	asserted 1.00
s4	On that submanifold, the gap conditions completely fix the holomorphic ambiguity and the model can be solved explicitly to very high genus.	result 0.95	0.83	0.14	asserted 1.00
s5	We use these results to make precision tests of the connection between the large order behavior of the 1/N expansion and non-perturbative effects due to instantons.	contribution 0.70	0.91	0.12	asserted 0.99
s6	Finally, we argue that a full understanding of the large genus asymptotics in the multi-cut case requires a new class of non-perturbative sectors in the matrix model.	future work 0.53	0.70	0.44	hedged possibility 0.45

What kind of paper the classifier reports this is: theoretical result 0.79 · evidence from proof or derivation 1.00.

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Original abstract

We show how direct integration can be used to solve the closed amplitudes of multi-cut matrix models with polynomial potentials. In the case of the cubic matrix model, we give explicit expressions for the ring of non-holomorphic modular objects that are needed to express all closed matrix model amplitudes. This allows us to integrate the holomorphic anomaly equation up to holomorphic modular terms that we fix by the gap condition up to genus four. There is an one-dimensional submanifold of the moduli space in which the spectral curve becomes the Seiberg--Witten curve and the ring reduces to the non-holomorphic modular ring of the group $\Gamma(2)$. On that submanifold, the gap conditions completely fix the holomorphic ambiguity and the model can be solved explicitly to very high genus. We use these results to make precision tests of the connection between the large order behavior of the 1/N expansion and non-perturbative effects due to instantons. Finally, we argue that a full understanding of the large genus asymptotics in the multi-cut case requires a new class of non-perturbative sectors in the matrix model.

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Cited by 14 Pith papers

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How the literature uses it

20 arXiv papers cite this in 55 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 38 sentences without a confident label

Builds on it 1 paper

In Section 3 we extend previous results on instantons in matrix models and related systems [ref] .

Nonperturbative aspects of ABJM theory · Introduction · 0.91

Uses its method or tool 5 papers

We will closely follow [ref] .

Large N instantons from topological strings · The cubic matrix model and its 1/N expansion · 0.97
The genus one free energy $F_1$ can be computed using the Akemann formula for a two-cut matrix model [ref] F_1 = -1/24∑_i=1^4 logM_i^(1) - 1/2logK(k^2) -1/12logΔ+ 1/8log(x_1 -x_3)^2 + 1/8(x_2 -x_4)^2 + const. , where $Δ$ is the discriminant [equation] and the moments $M_i^(1)$ are all $1$ in the cubic matrix model.

Argyres-Douglas theories, Painlev\'e II and quantum mechanics · The two cut phase of the cubic model · 1.00
Following [ref] we can parametrize the branch points in terms of a pair of “B-model” variables ($z_1$, $z_2$) as

The uses of the refined matrix model recursion · Two-cut examples · 0.96

Uses its data or values 1 paper

The Yukawa coupling was obtained in terms of modular forms in [ref] and it reads,

Holomorphic Anomaly and Quantum Mechanics · The double well and the quartic oscillator · 0.97

Compares with it 2 papers

One hint that the topological recursion cannot lead straightforwardly to the WKB result is that the $F_g$'s produced by the topological recursion, as applied for example to the curves studied in this paper, are simply not the functions $F_n$ produced by the WKB method[In the case of the double-well, this is easily seen by comparing the results in [ref] to the results in section 3.2 above.].

Holomorphic Anomaly and Quantum Mechanics · The double well and the quartic oscillator · 0.90
Notice that the modular ring appearing here and parametrizing the holomorphic ambiguity is different from the one appearing in Seiberg–Witten theory [ref] or in the cubic matrix model [ref] .

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Cites it as support 2 papers

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This was verified in [ref] for the two-cut, cubic matrix model.

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In this subsection we review some results on instantons in matrix models, following the work of [ref] , which contains much more details and references.

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This matrix model was studied in great details in [ref] , which we quickly review in section [ref], as it describes topological string theory on some particular Dijkgraaf-Vafa geometry.

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Non-perturbative aspects in matrix models have also been studied extensively [ref] .

Stokes Phenomena and Non-perturbative Completion in the Multi-cut Two-matrix Models · Introduction · 0.98

What it cites · 103 sentences without a confident label

Uses the method or tool of 5 papers

Now we are prepared to apply the method of directly integrating the holomorphic anomaly equations according to [ref] .

Polynomial Structure of the (Open) Topological String Partition Function · Direct integration and higher genus amplitudes · 1.00
Now we are prepared to apply the method of directly integrating the holomorphic anomaly equations according to [ref] .

Direct Integration of the Topological String · Direct integration and higher genus amplitudes · 1.00
For this we change variables to $y=x-12(x_1+x_2)$ and the integral becomes [equation] Expanding the second square root for $z_1$ small, each term in the series can be computed explicitly and it is most easily given in terms of a generating function [ref] ,

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These results agree with the low-order results obtained in [ref] .

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These results agree with the low-order results obtained in [ref] .

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Cites as support 1 paper

The existence of these sectors is also suggested by a recent analysis of the asymptotic behavior of the instanton solutions of the Painlevé I equation [ref] .

Asymptotics of the instantons of Painleve I · Introduction · 1.00

Cites as background 14 papers

Non-perturbative effects in multi-cut matrix models have been studied in [ref] .

Multi-Instantons and Multi-Cuts · Non-perturbative effects in the cubic matrix model · 0.93
Non-perturbative effects in multi-cut matrix models have been studied in [ref] .

Breakdown of universality in multi-cut matrix models · Non-perturbative effects in the cubic matrix model · 0.93
In the double-scaling limit of the matrix model (see [ref] ), the total free energy of the matrix model becomes a function of a double-scaled variable $z$,

2D Gravity and Random Matrices · Non-perturbative effects in the one-cut matrix model · 0.88

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