Nonperturbative effects and nonperturbative definitions in matrix models and topological strings

By Marcos Mariño. 2008

207 citations recorded by the source corpus.

We develop techniques to compute multi-instanton corrections to the 1/N expansion in matrix models described by orthogonal polynomials. These techniques are based on finding trans-series solutions, i.e. formal solutions with exponentially small corrections, to the recursion relations characterizing the free energy. We illustrate this method in the Hermitian, quartic matrix model, and we provide a detailed description of the instanton corrections in the Gross-Witten-Wadia (GWW) unitary matrix model. Moreover, we use Borel resummation techniques and results from the theory of resurgent functions to relate the formal multi-instanton series to the nonperturbative definition of the matrix model. We study this relation in the case of the GWW model and its double-scaling limit, providing in this way a nice illustration of various mechanisms connecting the resummation of perturbative series to nonperturbative results, like the cancellation of nonperturbative ambiguities. Finally, we argue that trans-series solutions are also relevant in the context of topological string theory. In particular, we point out that in topological string models with both a matrix model and a large N gauge theory description, the nonperturbative, holographic definition involves a sum over the multi-instanton sectors of the matrix model

Subfields: Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c89b99d12204001728ac4ffef7589eea9311ecd7

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. M. Unsal
2. R. Schiappa
3. G. Dunne
4. Inês Aniceto
5. Daniele Dorigoni
6. Phil Saad
7. G. Basar
8. M. Vonk

In Nuclear and High Energy Physics

1. Marcos Mariño
2. Pavel Putrov
3. N. Drukker
4. M. Unsal
5. F. Benini
6. G. Dunne
7. D. Gaiotto
8. Daniele Dorigoni

In Statistical and Nonlinear Physics

1. Marcos Mariño
2. N. Drukker
3. Pavel Putrov
4. M. Unsal
5. K. Hosomichi
6. Daniele Dorigoni
7. P. Argyres
8. G. Dunne

Citations in the corpus

• Localization techniques in quantum field theories (2016, methodology): “We have included an overall sign in Zgauge, missing in [13,14], whose necessity was pointed out in [55,86].”
• Nonperturbative aspects of ABJM theory (2011, background): “The non-perturbative partition function of Chern-Simons requires summing over all partitions (N1, N2) of the total numbers of eigenvalues N between the two cuts [33, 46].”
• The Resurgence of Instantons in String Theory (2011, methodology): “, [15, 12, 42, 13] for discussions in the context of matrix models, and topological and minimal strings) F (z) ' z−nβ e−nAz +∞ ∑”
• Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity (2019, methodology): “The rules for including D-brane boundaries do not have a clear origin in gravity; we must rely upon the matrix integral description of JT gravity to understand how to properly (31)These branes are also closely related to branes in topological string theories described by matrix models [64, 65, 66, 6”
• JT gravity as a matrix integral (2019, unknown): “[130].”
• A primer on resurgent transseries and their asymptotics (2018, methodology): “In fact, a similar albeit more involved analysis was followed for the Painlevé I equation, describing 2d gravity, in [180, 181, 130, 16] and for the Painlevé II equation, describing 2d supergravity, in [176, 210].”
• Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles (2014, background): “3.1 Perturbation theory on R2 and IR-renormalons 22”
• Resurgence theory, ghost-instantons, and analytic continuation of path integrals (2013, methodology): “We call the set of all saddles the generalized instanton sector, following the works in d = 0 dimensional matrix models [11–19].”
• The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects (2012, methodology): “We note that similar expressions have appeared in the context of matrix models and topological string theory [38, 67] and by using the bridge equations in the context of resurgence theory in [39].”
• Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c = 1 Matrix Models (2009, background): “3.1 The Gaussian Matrix Model 18 3.2 The Penner Matrix Model 20 3.3 The Chern–Simons Matrix Model 21 3.4 Double–Scaling Limit and c = 1 Behavior 25”