Math 262A: Lecture 11

We continue with the material from Lecture 10. Let $S$ be a smooth function on $\mathbb{R}$ of the form

$S(x) = \frac{x^2}{2} + V(x),$

where $V$ is itself a smooth function with Taylor expansion of the form

$V(x) = \sum_{d=3}^\infty t_d \frac{x^d}{d},$

where $t_3,t_4,t_3,\dots$ are real numbers such that the corresponding path integral

$\mathcal{Z} = \int\limits_{\mathbb{R}} e^{-\frac{1}{\hbar}S(x)} \mathrm{d}x$

converges. The function $V$ is called the “potential,” and we view the action $S(x)$ as a deformation of $S_0(x) = \frac{x^2}{2}$ which corresponds to choosing all $t_i=0$ so that the potential $V=0$ and we get the exact evaluation

$\mathcal{Z}_0 = \int\limits_{\mathbb{R}} e^{-\frac{1}{\hbar}S(x)} \mathrm{d}x = \sqrt{2\pi \hbar}.$

The by the Laplace principle we know that we have an $\hbar \to 0$ asymptotic expansion

$\frac{1}{\mathcal{Z}_0} \int\limits_{\mathbb{R}} e^{-\frac{1}{\hbar}S(x)} \mathrm{d}x \sim \sum\limits_{k=0}^\infty a_k \hbar^k,$

and we want to calculate the coefficients $a_k$ , which are functions of the parameters $t_i$ .

Taking this all back to Lecture 1, if we set $\hbar = 1/N$ and take the action

$S(x) = x-\log(x) = \frac{x^2}{2} - \frac{x^3}{3} + \frac{x^4}{4}-\dots,$

then we are looking at “Stirling’s QFT,” in which the potential $V(x)$ has coefficients $t_d=(-1)^d$ . The result of our computation should then be a some sort of understanding of the subleading corrections $a_k$ in Stirling’s approximation,

$N! \sim \sqrt{2\pi}\frac{N^{N+\frac{1}{2}}}{e^N}(1+\frac{a_1}{N} + \frac{a_2}{N^2} + \dots),$

which are the “quantum corrections” in Stirling’s QFT, whatever that means.

To begin, let’s not yet think about the integral $\mathcal{Z}_N$ , and focus solely on the integrand

$e^{-\frac{1}{\hbar}(\frac{x^2}{2} + V(x))} = e^{-\frac{x^2}{2\hbar}}e^{-\frac{1}{\hbar}\sum_{d \geq 3} t_d \frac{x^d}{d}}.$

Let us focus even more narrowly on the non-Gaussian part $e^{-\frac{1}{\hbar}V(x)}$ of the integrand, and apply the Exponential Formula as in Lecture 3. First, we write

$V(x) = \sum\limits_{d=1}^\infty t_d\frac{x^d}{d} = \sum\limits_{d=1}^\infty t_d(d-1)!\frac{x^d}{d!},$

where $t_1=t_2=0$ . Then we have that the coefficients $u_d$ in the expansion

$e^{-\frac{1}{\hbar}V(x)} = 1 + \sum_{d=1}^\infty u_d \frac{x^d}{d!}$

are given by

$u_d = \sum\limits_{P \in \mathrm{Par}(d)} \prod\limits_{B \in P} (-\frac{1}{\hbar})t_{|B|}(|B|-1)!,$

where the sum is over all partitions $P$ of $\{1,\dots,d\}$ , and the product is over the blocks $B$ of the partition $P$ . Since $(|B|-1)!$ is the number of ways to cyclically order the subset $B \subseteq \{1,\dots,d\}$ , we have that

$u_d = \sum\limits_{\sigma \in \mathrm{S}(d)} \left(-\frac{1}{\hbar} \right)^{c(\sigma)} t_1^{c_1(\sigma)} \dots t_d^{c_d(\sigma)},$

where $c_k(\sigma)$ denotes the number of $k$ -cycles in the disjoint cycle decomposition of the permutation $\sigma \in \mathrm{S}(d)$ , and $c(\sigma)=c_1(\sigma)+\dots+c_d(\sigma)$ is the total number of cycles. Thus the polynomial $u_d=u_d(t_1,\dots,t_d)$ is a generating function for the permutations in $\mathrm{S}(d)$ which keeps track of cycle type; this polynomial is called the “augmented cycle index polynomial” of $\mathrm{S}(d)$ . The actual (i.e. non-augmented) cycle index polynomial is $\frac{1}{d!} u_d$ , and it is the basic tool in Polya’s theory of enumeration under group action. Equivalently, we can write the modified cycle index as

$u_d = \sum\limits_{\mu \vdash d} |C_\mu| \left(-\frac{1}{\hbar} \right)^{\ell(\mu)}\prod\limits_{i=1}^{\ell(\mu)} t_{\mu_i},$

where the sum is over Young diagrams with $d$ cells and $C_\mu \subseteq \mathrm{S}(d)$ is the conjugacy class of permutations of cycle type $\mu$ .

Let’s integrate. We write

$\frac{\mathcal{Z}}{\mathcal{Z}_0} = \frac{1}{\mathcal{Z}_0} \int\limits_{\mathbb{R}} e^{-\frac{1}{\hbar}(\frac{x^2}{2} + V(x))} \mathrm{d}x = 1+\sum\limits_{d=1}^\infty \frac{u_d}{d!} \int\limits_{\mathbb{R}} x^d e^{-\frac{x^2}{2\hbar}} \frac{\mathrm{d}x}{\mathcal{Z}_0}.$

We know how to compute moments of the Gaussian distribution: we recall from Lecture 4 that

$\int\limits_{\mathbb{R}} x^d e^{-\frac{x^2}{2\hbar}} \frac{\mathrm{d}x}{\mathcal{Z}_0} = \hbar^{\frac{d}{2}} (d-1)!!,$

where $(d-1)!!$ is the number of fixed point free involutions in $\mathrm{S}(d)$ , i.e. permutations which factor into disjoint cycles of order $2$ (so $(d-1)!!$ is zero if $d$ is odd). We thus have the series

$\frac{\mathcal{Z}}{\mathcal{Z}_0} = 1 + \sum\limits_{d=1}^\infty \frac{u_d}{d!} \hbar^{\frac{d}{2}} (d-1)!! = 1 + \sum\limits_{d=1}^\infty \frac{1}{d!} \left( \sum\limits_{\mu \vdash d} (-1)^{\ell(\mu)}(d-1)!!|C_\mu| \hbar^{\frac{d}{2}-\ell(\mu)} \right)t_\mu,$

where $t_\mu = \prod_{i=1}^{\ell(\mu)} t_{\mu_i}$ , and the number $(d-1)!!|C_\mu|$ is equal to the number of pairs $(\alpha,\varphi)$ in which $\alpha \in \mathrm{S}(d)$ is a fixed point free involution, and $\varphi \in C_\mu$ is a permutation in $\mathrm{S}(d)$ of cycle type $\mu$ . Such pairs of permutations are special objects, and they have a special name.

Definition 1: A pair of permutations $(\alpha,\varphi) \in \mathrm{S}(d) \times \mathrm{S}(d)$ such that $\alpha$ is a fixed point free involution and all cycles of $\varphi$ have length at least $3$ 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 $\mathbf{X}$ together with a set $P$ of distinct points on $\mathbf{X}$ and a set $C$ of curves on $\mathbf{X}$ , each of which joins a point of $P$ to a point of $P$ in such a way that the curves in $C$ intersect only at points of $P$ , and moreover if $P$ and $C$ are deleted from $\mathbf{X}$ 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 $\mathrm{G}=(V,E)$ we can try to embed it on a surface $\mathbf{X}$ by choosing a set of points $P$ on $\mathbf{X}$ in bijection with $V$ , and a set of curves $C$ on $\mathbf{X}$ in bijection with $E$ , such that the requirements to be a topological map are satisfied (this may not be possible if the connectivity of $\mathrm{G}$ is high and the genus of $X$ is low, and the minimal genus of $\mathbf{X}$ such that this can be done is called the genus of $\mathrm{G}$ .

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 $\varphi$ 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 $\alpha$ , 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 $(\alpha,\varphi)$ a closed oriented surface $\mathbf{X},$ together with a collection of points on the surface connected by curves meeting only at those points — a topological map. The points $P$ 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 $d=4$ , i.e. pairs $(\alpha,\varphi) \in \mathrm{S}(4) \times \mathrm{S}(4)$ such that $\alpha = (**)(**)$ is a product of $2$ -cyles and $\varphi=(****)$ is a $4$ -cycle. There are thus $(4-1)!! = 3 \cdot 1$ possibilities for $\alpha$ and $(4-1)!=3\cdot 2 \cdot 1$ possibilities for $\varphi$ , for a total of $18$ combinatorial maps of degree $4$ .

We consider those combinatorial maps as above with $\varphi=(1\ 2\ 3\ 4)$ , the full forward cycle in $\mathrm{S}(4)$ . These are the following:

$A = \left[(1\ 2)(3\ 4), (1\ 2\ 3\ 4) \right],\ B=\left[(1\ 3)(2\ 4), (1\ 2\ 3\ 4) \right],\ C=\left[(1\ 4)(2\ 3), (1\ 2\ 3\ 4) \right].$

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 $A$ and $C$ 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 $\frac{\mathcal{Z}}{\mathcal{Z}_0}$ is a generating function for topological maps, with face degrees determined by the Maclaurin series of the potential $V$ . 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.