Math 262A: Lecture 15

The $N\to \infty$ Feynman diagram expansion $\int\limits_\mathbb{R} e^{-N S(x)}\frac{\mathrm{d}x}{\sqrt{2\pi N^{-1}}}\sim \sum\limits_\Gamma \frac{\mathrm{Amp}_S\ \Gamma}{|\mathrm{Aut}\ \Gamma|} N^{V(\Gamma)-E(\Gamma)}$

is both fascinating and frustrating: while it is fascinating that there exists a close connection between the asymptotics of integrals and the combinatorics of maps, it is frustrating that this connection is purely combinatorial, and does not “see” the topology of the surface on which the diagram $\Gamma$ is drawn. Wouldn’t it be nice if the exponent of $N$ in the expansion had an extra term, and thus became Euler’s formula $V(\Gamma)-E(\Gamma)+F(\Gamma) = 2-2g(\Gamma),$

relating the alternating sum of vertices, edges, and faces of a graph to the genus of the surface on which it is drawn? Could there be some way to tinker with the integral $\int e^{-NS}$ to bring this about?

Note first that it is impossible to get the missing $F(\Gamma)$ terms by somehow choosing the right action $S$ in the above integral, because the diagrammatic expansion is essentially universal: the only part of it which depends on $S$ is the “amplitude” $\mathrm{Amp}_S\ \Gamma,$ which is a function of $\Gamma$ determined by the Taylor expansion of $S.$ In order to make the jump from the combinatorics of graphs to the topology of maps, we have to alter not the integrand, but the domain of integration itself.

Perhaps the first such alteration one would try is to introduce extra dimensions: instead of integrating over $\mathbb{R},$ we would integrate over $\mathbb{R}^d$ for $d > 1.$ It turns out that there is a Laplace method and a corresponding Feynman diagram expansion for the multidimensional integral $\int\limits_{\mathbb{R}^d} e^{-N S(x)}\mathrm{d}x,$

which you can read about for example in these notes of Pavel Etingof. I was hoping to cover the vector Laplace principle in this course, but unfortunately was not able to do so. However, the answer is more or less that both the principle and the corresponding diagrammatic expansion are essentially the same, and in particular moving from scalars to vectors does not allow one to make contact with the topology of maps by counting faces.

It turns out that what must be done is not to replace numbers with vectors, but rather with their quantum qounterparts (sorry, couldn’t resist): operators. Consider a Hilbert space $\mathrm{V},$ i.e. a complex vector space equipped with a scalar product $\langle \cdot,\cdot \rangle$, which is required to be complete with respect to the norm induced by the scalar product. We can consider the operator algebra $\mathrm{End} \mathrm{V}$ as a noncommutative generalization of $\mathbb{C}$: it actually is $\mathbb{C}$ if $\mathrm{dim} \mathrm{V}=1,$ and in any dimension we can add and multiply operators in a manner analogous to the addition and multiplication of numbers. What subsets of $\mathrm{End} \mathrm{V}$ would be the counterparts of the real line $\mathbb{R} \subset \mathbb{C}$ and the unit circle $\mathbb{T} \subset \mathbb{C}$? Answer: the Hermitian operators $\mathrm{H}(\mathrm{V}) = \{ X \in \mathrm{End}\mathrm{V} \colon X^*=X\}$

are like real numbers, in that they are self-conjugate and consequently have real spectrum, while the unitary operators $\mathrm{U}(\mathrm{V}) = \{U \in \mathrm{End}\mathrm{V} \colon U^*=U^{-1}\}$

are like complex numbers of modulus one, in that conjugation is the same as inversion and consequently the spectrum lies in $\mathbb{T}.$ Thus, a noncommutative analogue of the Laplace method would mean a statement on the $N \to \infty$ asymptotics of the integral $\int\limits_{\mathrm{H}(\mathrm{V})} e^{-NS(X)} \mathrm{d}X,$

where $S$ is a scalar-valued function on $\mathrm{H}(\mathrm{V}).$ Similarly, a noncommutative analogue of the imaginary version of the Laplace method, which is called the method of stationary phase and deals with integrals over the circle $\mathbb{T}$ rather than the line $\mathbb{R},$ would be something about the $N \to \infty$ asymptotics of $\int\limits_{\mathrm{U}(\mathrm{V})} e^{-NS(U)} \mathrm{d}U,$

where $S$ is a scalar-valued function on unitary operators. Here there is an immediate issue in that these are ill-defined functional integrals as soon as $\mathrm{V}$ is infinite-dimensional, because of the lack of Lebesgue (or Haar) measure in infinite dimensions. So in order to rigorously study the above integrals, we have to restrict to the case that $\mathbf{V}$ is of finite dimension $M<\infty,$ in which case we can identify $\mathrm{H}(\mathrm{V})$ with the set $\mathrm{H}(M)$ of $M \times M$ Hermitian matrices, and $\mathrm{U}(\mathrm{V})$ with the set of $M \times M$ unitary matrices. The former is a real vector space of dimension $M^2,$ which is isomorphic to the Euclidean space $\mathbb{R}^{M^2}$ when given the scalar product $\langle X,Y \rangle = \mathrm{Tr} XY,$ where $\mathrm{Tr}$ is the usual matrix trace, and the latter is a compact real Lie group of dimension $M^2.$ So, we are led to study the $N\to \infty$ asymptotics of the matrix integrals $\int\limits_{\mathrm{H}(M)} e^{-NS(X)} \mathrm{d}X$

and $\int\limits_{\mathrm{U}(M)} e^{-NS(U)} \mathrm{d}U,$

which are perfectly well-defined objects. Our goal in the remainder of this course is to develop versions of the Laplace principle and the method of stationary phase for these objects, and see that their diagrammatic expansions actually do connect up with the topology of maps.

The first step in this direction is to identify what the counterpart of the the scalar integral $\int\limits_\mathbb{R} x^d e^{-N\frac{x^2}{2}} \mathrm{d}x = \sqrt{\frac{2\pi}{N}} |\mathrm{Inv}(d)| N^{-\frac{d}{2}},$

which computes the moments of the Gaussian measure on $\mathbb{R},$ becomes when we replace the real line with Hermitian matrices. The answer to this was found by physicists long ago, and was rediscovered by mathematicians somewhat later in this fundamental paper of Harer and Zagier.

Theorem 1: For any $\mathrm{d} \in \mathbb{N},$ we have $\int\limits_{\mathrm{H}(N)} \mathrm{Tr} X^d e^{-\frac{N}{2}\mathrm{Tr}X^2}\mathrm{d}X = \mathcal{Z}_N \sum\limits_{g=0}^\infty \frac{\varepsilon_g(d)}{N^{2g}},$

where $\mathcal{Z}_N$ is an explicit number depending only on $N,$ and $\varepsilon_g(d)$ is the number of ways in which the edges of a $d$-gon can be identified in pairs so as to produce a compact orientable surface of genus $g.$ On the right hand side, if we set $N=1,$ we obtain $\sum\limits_{g=0}^\infty \varepsilon_g(d) = |\mathrm{Inv}(d)|,$

since if we forget the stratification by genus encoded by $N^{-2g}$, the (finite) sum is simply counting ways to glue the edges of a $d$-gon in pairs. Note however that setting $N=1$ on the left hand side of Theorem 1 does *not* produce $\int\limits_\mathbb{R} e^{-N\frac{x^2}{2}} \mathrm{d}x,$

but rather $\int\limits_\mathbb{R} e^{-\frac{x^2}{2}} \mathrm{d}x.$

In particular, the fact that we are integrating over $\mathrm{H}(N)$ rather than $\mathrm{H}(M)$ in Theorem 1 is not a typo: to get the connection with surface topology we need $M=N.$ This simple fact causes massive problems on the analytic side, effectively invalidating the fundamental notion underlying Laplace’s method, which is that the main contribution to an integral containing a large parameter comes from a small neighborhood of a single value of the integrand. We will discuss both the combinatorics and the analysis of this “quantum Laplace method” next week.