## The Romance Between Maths and Physics

Miranda C. N. Cheng University of Amsterdam

Very happy to be back in NTU indeed!

Question 1:

Why is Nature predictable at all (to some extent)?

Question 2:

Why are the predictions in the form of mathematics?

the unreasonable effectiveness of mathematics in natural sciences.

Eugene Wigner (1960)

First we resorted to gods and spirits to explain the world , and then there were ….. mathematicians?!

### Physicists or Mathematicians?

Until the 19^{th} century, the relation between physical
sciences and mathematics is so close that there was
hardly any distinction made between “physicists”

and “mathematicians”.

Even after the specialisation starts to be made, the two maintain an extremely close relation and

cannot live without one another.

Some of the love declarations …

Dirac (1938)

Our experience up to date justifies us in feeling sure that in Nature is actualized the ideal of mathematical simplicity. It is my conviction that pure mathematical construction enables us to discover the concepts and the laws connecting them, which gives us the key to understanding

nature… In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed. Einstein (1934)

If you want to be a physicist, you must do three things—

first, study mathematics, second, study more mathematics,

and third, do the same. Sommerfeld (1934)

Indeed, the most irresistible reductionistic charm of physics, could not have been possible without mathematics …

## Love or Hate?

## It’s Complicated…

In the era when Physics seemed invincible (think about the standard model), they thought they didn’t need each other anymore.

Dyson (1972)

If all of mathematics disappeared, physics would be set back by

exactly one week.

Richard Feynman Sir Michael Atiyah

That was the week that God created the world.

(An anecdote recounted by Robbert Dijkgraaf.)

But, is physics really so invincible? What about these problems?

heavy and small e.g. black holes

point particles

String Theory: solving quantum gravity

unifies

Quantum Mechanics General Relativityand

conventional field theory

singularities string theory

no distinguished singular point

point particles

String Theory: the ultimate dream of reductionism.

unifies

Quantum Mechanics General Relativityand

world invisible

(microscopic)

world visible

(macroscopic)

world

The invisible determines the physics we see!

(Quantum) Gravity Standard model Beyond SM physics

...

point particles

String Theory: a looming conundrum.

unifies

Quantum Mechanics General Relativityand

world invisible

(microscopic)

world visible

(macroscopic)

world The invisible determines the physics

we see!

Different choices of the

compactification manifolds lead to different “worlds” with different

physics.

String Theory: as a model for all sorts of physics.

world invisible

(microscopic)

world visible

(macroscopic)

world

...

Quantum Gravity

(black holes, holographic principle)

Gauge Theory

Mathematical Physics

(matrix models, integrability, .... ) Condensed Matter Physics

Particle Physics Cosmology

or, turning weakness into strength.

Physics Maths

invisible

(microscopic)

world visible

(macroscopic)

world

black holes gauge theory supersymmetry

String Theory Geometry, Algebra, Number Theory

Mirror Symmetry Gromov-Witten Invariants Donaldson-Thomas Invariants

Knot Invariants

Geometric Langlands Programme Wall-Crossing

SCFT-VOA correspondence

...

String Maths

Geometry, Topology, Algebra, Number Theory, …

In the form of “String Math”, “Physical Mathematics”*, “Physmatics”**, ….

*: from Greg Moore, **: from Eric Sharpe

### I. Moonshine

Finite

Groups Modular

Forms Moonshine

Finite

Groups Modular

Forms Moonshine

functions with special symmetries

Modular Forms

= A holomorphic function on the upper-half plane that transforms “nicely” under SL(2,Z).

-1 -1/2 1/2 1

Example: the J-function

x y

M CHENG

Modular Forms

reflect the symmetry of a torus.

The modular action leaves the torus the same, just a different way to parametrise it.

2. Background/Modular Forms

M CHENG

Modular Forms

are natural products of string theory.

A string moving in time = a cylinder.

time

2. Background/Modular Forms

partition function

The partition functions are computed by identifying the initial and final time.

This turns the cylinder into a torus. As a result the string partition functions are modular forms!

String theory is good at producing functions with symmetries!

All symmetries have to be reflected in suitable partition functions.

More generally, there can be space-time symmetries (such as E-M, S-, T-dualities) as well as world-sheet symmetries (such as SL(2,Z)).

Finite

Groups Modular

Forms Moonshine

Finite Groups

Discrete Symmetries of Objects

Example 1. Symmetry of a square = the dihedral group “Dih4”.

M CHENG

Finite Groups

Discrete Symmetries of Objects

Example 2. Close Packing Lattices: considering the most efficient way to stack up identical balls.

Face-Centered Cubic (fcc) Lattices

e.g. Cu, Ag, Au

2. Background/Finite Groups

M CHENG

Finite Groups

the Representations

A representation V of a finite group G is a space that G acts on. V G

e.g. G = Dih4 =Symmetry group of a square.

x y

0

2-dimensional representation

2. Background/Finite Groups g=

M CHENG

Some Interesting Examples

the “Sporadic Groups”

The only 26 groups that cannot be studied systematically.

They are usually symmetries of very interesting objects.

2. Background/Finite Groups

Example 1. M24= the oldest sporadic “Mathieu Groups”.

Example 2. “The Monster” = the largest sporadic groups.

|M24| = the number of elements in M24 = 244,823,040. cf. |Dih4| = 8

|M| ~ 10^{54} ~ the number of atoms in the solar
system.

The smallest non-trivial representation has 196,883 dimensions.

[Mathieu 1860]

[Fischer, Griess 1970-80]

Finite

Groups Modular

Objects Moonshine

[McKay late 70’s]

Moonshine

relates modular forms and finite groups.

Monstrous Moonshine Conjecture

??

dim irreps of Monster

[Conway–Norton ’79]

The Largest

Sporadic Group The Most Natural

Modular Function

String Theory

explains Monstrous Moonshine.

Q: What’s the most efficient way to stack up 24-dimensional identical balls?

A: It’s given by the Leech lattice ΛLeech. [Leech 1967]

ΛLeech has very interesting sporadic symmetries.

Q: But it’s 24-dimensional! What can we do with it?

A: Just the right number of dimensions for string theory!

Strings in the Leech lattice background:

R^{24}/ΛLeech

The Partition Function = J(τ)

ΛLeech

Co1 ^{Z/2} Monster

Monstrous Moonshine

Sporadic Symmetry Modular Symmetry

String Theory

explains Monstrous Moonshine.

To prove the Monstrous Moonshine Conjecture

• Use ideas shared with string theory

(orbifold conformal field theory, no-ghost theorem, ...)

• Led to important developments in algebra and representation theory

(vertex operator algebra, Borcherds–Kac–Moody algebra, ...)

Monstrous Moonshine

A Meeting Place of Different Subjects

[Frenkel–Lepowsky–Meurman, 80’s]

[Borcherds, 80-90’s]

### Umbral Moonshine

(‘13 w. J. Duncan, J. Harvey)

“Niemeier”

Finite Groups

“Mock”

Modular Forms

Umbral Moonshine

Physical Origin:

superstring theory on K3 surface.

(unique non-trivial CY 2-fold, with great mathematical importance)

### II. Topological Invariants, and Quantum Modular Forms

Topology

Vertex Alg.

Q: How to distinguish one knot from another?

A: Topological Invariants.

A closely related question:

Q: How to distinguish one 3-manifold from another?

A: Topological Invariants.

Knot Complement:= 3-manifold – (the nbhd of a) knot

Can string theory help?

A: Yes. Via compactification.

M-theory: non-pert formulation of string theory.

M5 brane: a 6-dim object in M-theory.

From the point of view of the 3d SQFT:

Modularity from the boundary theory.

/

1. Physics gives a more powerful top. inv.

than Chern-Simons.

2. The mathematical (quantum) modular properties helps to get new information on physics .

The mathematics involved in string theory … in subtlety and sophistication vastly exceeds previous uses of mathematics in physical theories. … String

theory has led to a whole host of amazing results in mathematics in areas that seem far removed from physics.

2005, Sir Michael Atiyah

world invisible

(microscopic)

world visible

(macroscopic)

world The invisible determines the physics

we see!

Different choices of the

compactification manifolds lead to different “worlds” with different

physics.

But, how about physics?

Which internal manifold is ours? ? ? How many options to choose from?

?!?!?

String Theory Landscape

not your universe not your

universe not your

universe

your universe!

Why? LIVE WHERE YOU CAN (?).

String Theory Landscape:

a blessing that provides the ultimate answer?

Living in the multiverse, 2015, S. Weinberg

Makes sense! But at the same time it relies on a probabilistic view on fundamental physics and is distinctively nontheistic and non-anthropocentric.

Not feeling comfortable? You are not alone.

The invisible determines the physics we see!

Q: Can we at least find the exact string theory background that gives our universe?

A: A daunting task.

Involving a series of NP-hard and undecidable tasks.

The invisible determines the physics we see!

Q: Can we at least find the exact string theory background that gives our universe?

A: A daunting task.

Involving a series of NP-hard and undecidable tasks.

In recent years, artificial intelligence has been employed to look for a string universe like ours.

at the moment, it’s especially good for solving things approximately and probabilistically.

AI as the new physicist:

are we out of job?

AI as a new colleague.

So far, AI has been lots of help in astro-, particle, and material physics, among other things.

AI as a new helper.

So far, AI has been lots of help in astro-, particle, and cm physics, among other things.

Some argue that the purpose of physics is to make reliable predictions about the future, given data about the present and the past.

If AI can produce approximately correct prediction almost all the time (say within 1% error 99.78% of the time), has it then learned the physics?

AI as the new physicist:

learn math first*.

Dogs and babies understand that objects fall instead up flying upwards (they act surprised or frightened if

objects behave abnormally). In this sense they too can predict the future motion of a moving object and

hence “understand physics”. However, I won’t say it is physics unless they show me

*: work in progress.

To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature ...

If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.

Richard Feynman (1967)

We can eavesdrop on nature not only by paying attention to experiments but also by trying to understand how their results can be explained with the deepest mathematics. You could say that the universe speaks to us in numbers.

Nima Arkani-Hamed (2019)

To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature ...

If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.

Richard Feynman (1967)

We can eavesdrop on nature not only by paying attention to experiments but also by trying to understand how their results can be explained with the deepest mathematics. You could say that the universe speaks to us in numbers.

Nima Arkani-Hamed (2019)