Integer partitions, q-series, and conformal field theories.
Olivia Beckwith, PhD
University of Illinois at Urbana-Champaign
Are the answers to the hardest questions in math and physics within our grasp? Is it possible that some of them have been staring us in the face since we learned to count? Could they come down to math as simple as 1+2=3? Imagine that in one hand you have the nature of matter and gravity, and in the other hand, arithmetic. Both are fundamental aspects of our reality, but feel like completely disparate subjects. Yet they may not be as independent of each other as they seem. Integer partitions give us a glimpse of the interconnectedness of arithmetic, modular forms, algebraic topology, and theoretical physics.
It all starts with an elementary question. Let n be a positive integer. How many ways can you write n as a sum of nonincreasing positive integers? We let p(n) denote the answer.
For example, you can write 4 as 4, 3+1, 2+2, 2 + 1 + 1, and 1 + 1 + 1 +1, so p(4) = 5. A pretty simple idea, right?
Number theorists call p(n) the partition function (not to be confused with the partitions functions in statistical mechanics or quantum field theory). The various sums are the partitions of n.
I want to say just a few words about why someone would care to study integer partitions in the first place. From a number theorist's perspective, p(n) is naturally intriguing because it speaks to the additive structure of the integers. Furthermore, the sequence p(1), p(2), p(3), ... is full of the sorts of patterns that number theorists love. For example, in the table below, observe that if the units digit of n is either 4 or 9, then p(n) is divisible by 5. This regularity in divisibility holds for all n and doesn't happen by accident, it is just one example of a large class of patterns for p(n) and other coefficients of modular forms. But I don't want to get carried away - this is perhaps a topic for another post.
You might wonder how you could go about computing p(n), especially for large n. I sometimes like to try to compute partition numbers in my head by counting partitions while I'm out running, to pass the time. However, I can never get very far. Past n=8, it becomes difficult for me to keep track of all the partitions. The following formula helps me get a little bit farther:
Where does this come from, and what exactly comes next on the right hand side? A direct counting argument might feel like the right approach, but good luck! It is easier to work with the generating function for p(n):
This is a common strategy in combinatorics: if you're interested in a sequence, look at its generating function and try to manipulate it algebraically or study its analytic behavior. If you can show that the generating function can be written in a nice way, see what that new formula tells you about your sequence.
In the case of p(n), it turns out that the generating function can be rewritten as a product:
This formula isn't terribly hard to show - you can express each factor on the right as a geometric series and think about what happens when you distribute each factor. It is extremely important, though, because it relates p(n) to the theory of q-series, which are loosely defined as functions involving products such as the right hand side of (2).
Here's an identity that's a little harder to prove:
We won't go into the proof, which involves some very clever reasoning. Multiplying both sides of (3) by the generating function for p(n), you get:
You can rewrite the product as an infinite power series in q, but because the left hand side is 1, all of the coefficients except the constant term have to equal 0. This is how you get the recursive formula (1).
So we see that an identity between two functions, (3), held the key to understanding the recursive formula for p(n) (1). Often in the study of partitions, intricate combinatorial structure is elegantly conveyed by means of q-series. Next we'll see a few more increasingly complex examples of this.
Combinatorial structure beyond p(n)
We defined p(n) to be the number of ways of writing n as a sum of positive integers in nonincreasing order. There is a lot of room to modify this definition to get other interesting functions. One typical modification is to require that all of the summands belong to some set S. If you took S to be the set of odd numbers, you'd be looking at the number of ways of writing n as a sum of odd numbers. In the case of n=4, you'd have 1+1+1+1 and 3+1, so the answer would be 2.
It is also common to put restrictions on multiplicities of the summands. For example, you could require that each of the summands be distinct. In the case of n=3, you'd have 3 and 2+1, so the answer would be 2 (you wouldn't count 1+1+1 because the 1 appears more than once).
Sometimes these functions actually coincide - even if their definitions look completely different! These relations are sometimes easiest to prove using algebraic manipulations. For example,
shows that the number of partitions of a number into odd parts is the same as the number of partitions into distinct parts.
The Rogers-Ramanujan identities are a fascinating example:
Like (3), these identities relate infinite summations to infinite products, but also have a combinatorial interpretation that tells us something about integer partitions. In (4), the left hand side is the generating function for partitions such that the adjacent parts differ by at least 2, and the right hand side is the generating function for the number partitions into parts of the form 5n + 1 and 5n+4. Since the generating functions are equal, these quantities must be the same.
Let's check this for n=6. On the left hand side, the coefficient of q^6 is the number of partitions of 6 for which all of the summands differ by at least 2. If we check each of the 11 partitions of 6, we find that 6, 5+1 and 4+2 are the ones satisfying this property, so the coefficient for q^6 is 3.
To obtain the coefficient of q^6 on the right, you count partitions of 6 for which all of the summands are either 1, 4, and 6. Those partitions are 1+1+1+1+1+1, 4+1+1, and 6, so again you have 3 partitions. Notice that we just counted two different sets of partitions, but the total number of partitions in each set was the same.
There is a similar interpretation of (5). And this is just the beginning. Other beautiful identities similar to (4-5) have been discovered by Andrews, Gordon, Bressoud, Slater, Schur, and much more recently Kanade and Russell. Finding new identities remains an area of interest. In addition to being interesting in their own right, one may hope that such identities might fuel the development of algebraic structures useful in theoretical physics, as we will discuss next.
Equation (3) is a special case of an equation known as the Jacobi Triple Product identity:
If you set x = q^3/2 and y = i q^1/4, this simplifies to (3). Formula (6) can be interpreted using Lie algebras.
Lie algebras are mathematical objects defined as vector spaces endowed with a pairing called a Lie bracket. They famously describe symmetries for systems in quantum physics. One attempts to understand all of the representations of a Lie algebra - that is, all the ways the Lie algebra can be written in terms of linear maps on vector spaces. The Weyl Character Formula relates the representations to intrinsic properties of the Lie algebra by writing the characters of representations as ratios where the numerator is a sum and the denominator is a product. The character of the trivial representation is 1, so in that situation the Weyl Character Formula says that a certain product is equal to a sum. In the case of the affine Kac-Moody algebra with root system type A_1, you get (6).
So (3) represents an overlap between the representation theory of Lie algebras and integer partitions. This overlap is not a fluke, but a feature of the connectedness of the two areas. The Macdonald identities are a larger class of identities which include (6) and come from characters of affine Lie algebras.
Affine Lie algebras are famous for their role in the construction of conformal blocks in two-dimensional conformal field theory. The conformal blocks are used to construct correlators, the functions governing predictions about the behavior of particles in the spacetime.
So the algebraic structures giving rise to identities like (3) are used to study conformal field theories. One reason this is fascinating is that conformal field theories play a vital role in cutting edge work in theoretical physics on quantum gravity. Famously, a major goal in theoretical physics is finding a physical theory that is compatible with both quantum physics and general relativity. One approach involves the study of Anti de Sitter space - a spacetime that is a solution to Einstein's field equations which is different from our universe but admits a quantum gravitational theory. There is a duality - the AdS/CFT correspondence - which says that every AdS corresponds to a CFT on a spacetime with one less dimension. This dimension reduction is the ``holographic principle" of string theory, which is discussed in other articles on this site.
Modular forms and K-theory
An interesting feature of (4-5) is the appearance of a quadratic form in the exponent on the summation side. The product side, on the other hand, is a modular form, a function with symmetry under Mobius transformations. You might wonder if it is possible to characterize all of the identities of this type. Nahm's Conjecture connects this question with algebraic K-theory and conformal field theory.
The functions in Nahm's conjecture are easiest described using Q-Pochhammer symbols. We let (q)_n be the denominator on the left hand side of (4), that is:
Nahm's Conjecture involves functions defined by sums over vectors m = (m_1, ... , m_r), where the $m_i$ are integers. Suppose that the q-series
is a modular form, where Q(m) is a quadratic multinomial on the m_i.
When r = 1, it is known (due to work of Zagier) that the series (7) is modular for just seven quadratic forms Q(m) = (A/2) m^2 + Bm + C:
Number theorists would love to see a nice explicit description when r > 1. Obtaining one might require a better understanding of how such q-series come about from elements of groups in K-theory.
Nahm studied the restrictions enforced on the coefficients of Q(m) by the modular symmetry and obtained a system of equations. He demonstrated that the solutions of these systems of equations produce torsion elements in the Bloch group B(C), a group connected to algebraic K-theory and used in the study of three-dimensional hyperbolic manifolds.
On the other hand, every conformal field theory has characters which have modular invariance flowing from the conformal symmetry and q-series of a particular shape. Nahm also studies a map from torsion elements in the Bloch group to central charges of conformal field theories. So we have a 3-way diagram , in which conformal field theories produce Rogers-Ramanujan-like identities, and Rogers-Ramanujan-like identities predict conformal field theories with properties encoded by modular q-expansions.
So we started with a simple addition question, and quickly found ourselves observing ramifications of the interactions of quantum field theory, modular forms, and algebraic K-theory. That elementary combinatorial identities like (3, 4, 5) can have anything to do with high energy physics amazes me.
Some look for the answers to the grand mysteries of the universe by delving deep into the physical laws, the nature of gravity, and the behavior of small particles that are all around us. Some of us seek beauty and meaning in the most elementary of concepts - the numbers we count with. Perhaps we're really looking at 2 sides of the same coin.
Acknowledgements: This article was inspired by many enlightening conversations over the years with Dan Fretwell, Laura Johnson, and Robert Schneider.
The theory of partitions by George Andrews.
Conformal field theory and torsion elements in the Bloch group by Werner Nahm.
Notes on Lie algebras by Hans Samerson.