We continue with the material from Lecture 10. Let be a smooth function on of the form
where is itself a smooth function with Taylor expansion of the form
where are real numbers such that the corresponding path integral
converges. The function is called the “potential,” and we view the action as a deformation of which corresponds to choosing all so that the potential and we get the exact evaluation
The by the Laplace principle we know that we have an asymptotic expansion
and we want to calculate the coefficients , which are functions of the parameters .
Taking this all back to Lecture 1, if we set and take the action
then we are looking at “Stirling’s QFT,” in which the potential has coefficients . The result of our computation should then be a some sort of understanding of the subleading corrections in Stirling’s approximation,
which are the “quantum corrections” in Stirling’s QFT, whatever that means.
To begin, let’s not yet think about the integral , and focus solely on the integrand
Let us focus even more narrowly on the non-Gaussian part of the integrand, and apply the Exponential Formula as in Lecture 3. First, we write
where . Then we have that the coefficients in the expansion
are given by
where the sum is over all partitions of , and the product is over the blocks of the partition . Since is the number of ways to cyclically order the subset , we have that
where denotes the number of -cycles in the disjoint cycle decomposition of the permutation , and is the total number of cycles. Thus the polynomial is a generating function for the permutations in which keeps track of cycle type; this polynomial is called the “augmented cycle index polynomial” of . The actual (i.e. non-augmented) cycle index polynomial is , and it is the basic tool in Polya’s theory of enumeration under group action. Equivalently, we can write the modified cycle index as
where the sum is over Young diagrams with cells and is the conjugacy class of permutations of cycle type .
Let’s integrate. We write
We know how to compute moments of the Gaussian distribution: we recall from Lecture 4 that
where is the number of fixed point free involutions in , i.e. permutations which factor into disjoint cycles of order (so is zero if is odd). We thus have the series
where , and the number is equal to the number of pairs in which is a fixed point free involution, and is a permutation in of cycle type . Such pairs of permutations are special objects, and they have a special name.
Definition 1: A pair of permutations such that is a fixed point free involution and all cycles of have length at least is called a combinatorial map.
The use of the word “map” in Definition 1 may seem strange, since you are probably more accustomed to seeing it appear in the following definition.
Definition 2: A topological map consists of a closed oriented surface together with a set of distinct points on and a set of curves on , each of which joins a point of to a point of in such a way that the curves in intersect only at points of , and moreover if and are deleted from then what remains are disjoint open sets, called “faces,” each of which is homeomorphic to a disc.
Topological maps are basic objects in graph theory: for example, the famous four color theorem asserts that in order to color the faces of a map on the sphere in such a way that faces which are bounded by a common curve have different colors, it is necessary and sufficient to have four colors. Indeed there is a clear connection between graphs and topological maps, in that starting from a graph we can try to embed it on a surface by choosing a set of points on in bijection with , and a set of curves on in bijection with , such that the requirements to be a topological map are satisfied (this may not be possible if the connectivity of is high and the genus of is low, and the minimal genus of such that this can be done is called the genus of .
The connection between combinatorial maps and topological maps is not quite as obvious, but it is still fairly straightforward. The construction is quite simple and natural. First, draw every cycle of as a polygon, with the elements permuted by the cycle being taken as labeling the edges of the polygon, and the edges being directed counterclockwise around the polygon. Now, identify the edges of these polygons in pairs according to the cycles of the fixed point free involution , ensuring that when we identify two edges they are glued with opposition orientations, i.e. forming a two-way street. This procedure produces from the pair a closed oriented surface together with a collection of points on the surface connected by curves meeting only at those points — a topological map. The points are images of the former vertices of the polygons after having been identified, and its curves are former edges of the polygons after having been identified; the faces of the map are the interiors of the polygons.
As a small example of the above construction, let us consider combinatorial maps of degree , i.e. pairs such that is a product of -cyles and is a -cycle. There are thus possibilities for and possibilities for , for a total of combinatorial maps of degree .
We consider those combinatorial maps as above with , the full forward cycle in . These are the following:
Now we depict the corresponding gluing construction as described above:
So, the three different combinatorial maps described above gave us only two different topological maps: the combinatorial maps and produced the same topological map. Thus, the correspondence between combinatorial maps and topological maps is many-to-one (as is the correspondence between graphs and topological maps). But the general picture that is emerging is that the asymptotic expansion of is a generating function for topological maps, with face degrees determined by the Maclaurin series of the potential . That is, topological maps are the Feynman diagrams of zero-dimensional scalar-valued quantum field theory. We will continue to develop this connection next week.