A String Theorist's Journey with Python

Goal

In this post I will try to illustrate how Python is integrated throughout the progress of this project

• to Python programmers and developers who want to know how a math-oriented theoretical physicist use Python, and
• to scientists who are interested in using Python but have difficulty in figuring out where and how to start

as a case study of a collaborative research project using Python.

What is string theory?

So, let me start by telling you what a string theory is.

More commonly known as theoretical particle physics, high-energy theory is a study to understand the fundamental forces of the universe, and you may have heard of the four: electromagnetic force, weak force, strong force, and gravity.

String theory is part of high energy theory. It started as a candidate theory of quantum gravity, and now it’s evolved into a field of study to build a physical and mathematical framework to describe the fundamental forces in a unified way.

Why computational analysis in string theory?

Then why are string theorists interested in computational analysis in tackling questions of mathematical nature?

Indeed string theorists want to find a mathematical and exact result.

However, such a beautiful moment that we have a concise and exact formula for a question rarely happens because many problems in string theory pose significant challenges to a traditional pen-and-paper approach.

We tackle such a problem by using computational method, which gives us hints toward the formulation of a complete answer.

Also it's largely thanks to the recent progress of both hardwares and mathematical software packages that enabled us to physical and mathematical problems by combining analytic and numerical methods together.

String theory and M-theory

Then let me introduce a few concepts from string theory that will appear in this post.

The first one is M-theory, which is sometimes called the mother of all string theories, because various string theories are believed to be different limits of M-theory.

M-theory is a prospective theory of quantum gravity that lives in an 11-dimensional spacetime, that is a ten-dimensional space plus one time direction. 10-dimensional string theories and 11-dimensional supergravity are believed to be different limits of M-theory.

Then we usually wrap this eleven-dimensional theory onto a very small seven-dimensional space, so that we can obtain a four-dimensional spacetime that we live in.

M-branes

M-theory has a higher-dimensional analogue of electromagnetic field.

M-theory has two kinds of extended objects, called M2-branes and M5-branes, that couple to the field electrically and magnetically, respectively. They are the main characters of this story.

An Mp-brane spans a (p+1)-dimensional spacetime. That is, M2-brane spans 2 spatial direction and one time direction, hence the name M2, and M5-brane spans 5 spatial directions and one time direction.

They saturate an inequality called Bogomolny-Prasad-Sommerfield (BPS) bound,

where \(M\) is the mass and \(Z\) is a conserved quantity called a central charge of the object, which is an analogue of electric and magnetic charges. Therefore each M-brane has a mass tied to the central charge.

Supersymmetric gauge theory from M5-branes

What we are going to do with those M-branes is we wrap them on a two-dimensional surface, more precisely a Riemann surface with punctures.

When we wrap \(N\) M5-branes on a punctured Riemann surface \(C\), we obtain a 4d superysymmetric gauge theory, more precisely \(\mathcal{N}=2\) \(\mathrm{SU}(N)\) theory of class \(S\) [1] [2].

In the Coulomb branch of a class \(S\) theory, the M5-branes merge into a single M5-brane wrapping a surface \(\Sigma\),

\begin{align} f(z, x)=0,\ z \in C, \label{eq:SW_curve} \end{align}

which is a multi-sheeted cover \(\{ x_{i = 1, \ldots, N} \}\) over \(C\).

When we wrap an M5-brane on such a surface, more precisely a complex 1-dimensional curve, which is called a Seiberg-Witten curve, the remaining four-dimensional spacetime that comes from a high-level math of six minus two equals 4, becomes populated with a supersymmetric gauge theory, which you can imagine as a mathematical model of the real world particle physics.

A nice thing is that the geometry of the Riemann surface \(\Sigma\) determines the low-energy theory of the four-dimensional physics [3]. This is one example of the interplay between physics and geometry in string theory.

Supersymmetric particles from M2-branes

So we got a theory, but what about particles living in the theory?

A supersymmetric particle comes from an M2-brane whose boundary ends along a closed curve \(\gamma \in H_1(\Sigma; \mathbb{Z})\) on the two-dimensional Riemann surface \(\Sigma\).

Then the mass of the particle is given by integrating a differential 1-form \(\lambda = x\, \mathrm{d} z\), called Seiberg-Witten differential, along \(\gamma\),

and its charges are determined again from the geometry [4] [5].

Spectral network

Thanks to supersymmetry, the boundary of such an M2-brane satisfies a differential equation,

\begin{align} \frac{\partial}{\partial t} \left( \lambda_{j} - \lambda_{k} \right)= \left( x_j(z) - x_k(z) \right) \frac{\partial z}{\partial t} = e^{i \theta},\label{eq:class_S_diff_eq} \end{align}

where \(t\) is a real parameter along \(\gamma\), \(\lambda_i = x_i\, \mathrm{d}z\) is the value of \(\lambda\) on the \(i\)-th sheet, and \(\theta= \arg(Z)\). [6]

Then we can reverse the question, such that we first find the solutions of the differential equation on the Riemann surface and then find particles out of the geometric flows. This is the idea of spectral network.

Extending the previous construction, we consider a set of curves on \(C\) that are the solutions of (\(\ref{eq:class_S_diff_eq}\)) for a fixed \(\theta\), which is called a spectral network of the class \(\mathcal{S}\) theory determined by \(f(z, x)\). [7]

Spectroscopy we learned at high school

So what can we do by finding particles of a theory? We are going to do supersymmetric spectroscopy!

First let me remind you what we learned in a high-school science class about spectroscopy. When we put a small amout of chemical compound into a torch we see a flame, and its color tells us about an atomic element in the compound. This is flame test, a qualitative method to identify an element.

More quantitatively we can use a diffraction grating to the light of the flame to obtain an emission spectrum to identify an atomic element.

Supersymmetric spectroscopy via spectral network

The idea is the same: we obtain a spectrum of supersymmetric particles, then using the spectrum we try to identify what the theory that has such a spectrum is.

Using spectral networks, we can find the particle spectrum of a given supersymmetric theory. More precisely, we find two-way streets of finite \(\mathcal{S}\)-walls corresponding to BPS states, thereby obtaining the BPS spectrum of a 4d \(\mathcal{N} = 2\) theory of class \(\mathcal{S}\) on the Coulomb branch.

We use the spectrum to identify what the theory is, which is useful when the theory is strongly coupled and we lack any perturbative method to understand it.

Examples of spectral networks from loom

In practice, we generate a familiy of colorful networks called spectral networks, stare it until we spot a particle, do some taxonomy of them until we claim a victory. For example, in the following figure, we have a family of spectral networks, and between #24 and #25 we find one particle, and another around #74. By calculating their masses and charges, we can conjecture what the underlying theory is.

loom, a Python program to generate spectral networks

Now all the groundworks are laid down so let me tell you how loom is actually built.

Previously there were a few codes for generating spectral networks that are written in Mathematica, which is a favourite mathematics package among string theorists thanks to its user-friendly manipulation of symbolic math.

However there have been several issues when using Mathematica. A Mathematica code is not particularly convenient during collaborative development of a program and therefore each of us maintained separate codes, which was an overlapping investment of time and effort; there is the issue of the number of licenses available because Mathematica is (quite a pricey) commerical program; and in practice the numerical accuracy of Mathematica is not good and its warnings and errors are hard to understand.

So I launched an open-source project loom on GitHub and started a collaboration with my friend string theorists working on spectral networks, hoping that we can converge our effort into a common project. The name came from the fact that drawing a pretty figure of a spectral network is like weaving a tapestry with a loom.

Why I chose Python to write loom

First of all, my mother tongue is C, and at first I was thinking of writing a C module for Mathematica.

However, after some experiments I decided to use Python.

It was because Python is easy enough for my colleagues without programming experience to learn, but at the same time is a full-fledged standard programming language.

All the Python libraries that I needed and used were open-source, meaning I can look into the box if I want and don’t have to second-guess an incomprehensible error message.

Of course an active and helpful Python community was a big factor.

Diverse and powerful libraries, including SciPy stack, was another crucial factor.

And when I needed to use something outside Python libraries, including SageMath and C/C++ libraries, it was relatively painless to glue them into a Python program.

Python in the core module of loom

Let me go through the gut of loom to give you the detail of what is used in which way.

Python in the front-end of loom