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 19th 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| ~ 1054 ~ 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:
R24/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)
