Nonperturbative Effects and the Large-order Behaviour of Matrix Models and Topological Strings Communications in Number Theory and Physics
, Volume 2 Number 2

Marcos Mariño (Department of Physics, CERN, Genève, Switzerland)
Ricardo Schiappa (Department of Physics, CERN, Genève, Switzerland)
Marlene Weiss (ITP, ETH Zürich, Switzerland)

“This work addresses nonperturbative effects in both matrix models andtopological strings, and their relation with the large-order behaviorof the 1/N1/N expansion. We study instanton configurations in genericone-cut matrix models, obtaining explicit results for theone-instanton amplitude at both one and two loops. The holographicdescription of topological strings in terms of matrix models impliesthat our nonperturbative results also apply to topological strings ontoric Calabi–Yau manifolds. This yields very precise predictions forthe large-order behavior of the perturbative genus expansion, both inconventional matrix models and in topological string theory. We test these predictions in detail in various examples, including the quartic matrix model, topological strings on the local curve and the Hurwitz theory. In all these cases, we provide extensive numerical checks which heavily support our nonperturbative analytical results.Moreover, since all these models have a critical point describing two-dimensional gravity, we also obtain in this way the large-order asymptotics of the relevant solution to the Painlevé I equation,including corrections in inverse genus. From a mathematical point of view, our results predict the large-genus asymptotics of simple Hurwitz numbers and of local Gromov–Witten invariants”


It’s indisputable that the laws of physics and the physical reality they constitute develop from the same first principles as the laws of mathematics.

What is needed is serious research into the constraints on how the laws of physics emerge from purely mathematical principles.

a rigorous counting system is, by definition, physics

hence, the natural numbers are a sequence of physical constants

Seemingly esoteric notions of thegeometric Langlands program, arise naturally from the physics

yet more evidence that numerical relationships are physical constants

the natural numbers are a human invention like the formulation F=ma is a human invention

the physical relationships they describe are real

That numbers are physical constants should come as no surprise, for the reasons outlined in this chapter of Trick or Treat, The Mysterious Connection Between Physics and Mathematics published by Springer in 2016, titled ‘Cognitive Science and the Connection Between Physics and Mathematics‘.


“The human mind is endowed with innate primordial perceptions such as space, distance, motion, change, flow of time, matter. The field of cognitive science argues that the abstract concepts of mathematics are not Platonic, but are built in the brain from these primordial perceptions, using what are known as conceptual metaphors. Known cognitive mechanisms give rise to the extremely precise and logical language of mathematics. Thus all of the vastness of mathematics, with its beautiful theorems, is human mathematics. It resides in the mind, and is not ‘out there’. Physics is an experimental science in which results of experiments are described in terms of concrete concepts—these concepts are also built from our primordial perceptions. The goal of theoretical physics is to describe the experimentally observed regularity of the physical world in an unambiguous, precise and logical manner. To do so, the brain resorts to the well-defined abstract concepts which the mind has metaphored from our primordial perceptions. Since both the concrete and the abstract are derived from the primordial, the connection between physics and mathematics is not mysterious, but natural. This connection is established in the human brain, where a small subset of the vast human mathematics is cognitively fitted to describe the regularity of the universe. Theoretical physics should be thought of as a branch of mathematics, whose axioms are motivated by observations of the physical world. We use the example of quantum theory to demonstrate the all too human nature of the physics-mathematics connection: it is at times frail, and imperfect. Our resistance to take this imperfection sufficiently seriously (since no known experiment violates quantum theory) shows the fundamental importance of experiments in physics. This is unlike in mathematics, the goal there being to search for logical and elegant relations amongst abstract concepts which the mind creates”.

The mathematical equation 1a + 2a = 3a describes a mathematical law of quantitative relationships. It also describes a physical absolute, according to the law of conservation of energy. Numbers and numerical relationships are not merely abstract concepts, they are developed from the same fundamental principles as physical laws. This is not insignificant, it suggests that the physical universe developed according to natural constraints which are purely mathematical.

If we expect advanced aliens to know the natural number sequence then humans didn’t invent numbers, we discovered them.

If numbers are a human discovery then so are their mathematical relationships.

If mathematical relationships are discoveries not inventions then they’re natural laws.

1+2=3 is a natural law.

This is not insignificant.