Topological differential forms

Related material

Article	JHEP 05 (2025) 120
Preprint	math-ph/2503.09558
Program	Mathematica notebook to compute alpha
Slides	Amplitude Workshop Oxford, December 2024
Slides	PI Math Physics seminar, May 2025

Efficient computation of a parametric integrand

The start of this project is a particular type of Feynman integral, defined in 2403.13049. These integrals control deformations in topological field theories. We are concretely interested in the integrand $\alpha_\Gamma$ of that integral in Schwinger parameters. Here, one assigns one Schwinger parameter $a_e$ to each edge $e\in E_\Gamma$ of a Feynman graph $\Gamma$ , and $\alpha_\Gamma$ is a differential form, to be integrated over all $\text{d} a_e$ . The integrand $\alpha_\Gamma$ represents a single topological direction, it is defined with the help of auxiliary edge variables $s_e:= a_e^{-\frac 12} (x_+(e)- x_-(e) )$ , where $x_\pm(e)\in \mathbb R$ are the (1-dimensional) coordinates of the start vertex and end vertex of the edge $e$ . The parametric Feynman integrand is then

$\begin{align*} \alpha_\Gamma := \int \limits_{\mathbb R^{\abs{V}-1}} \bigwedge_{e\in E_\Gamma} e^{-s_e^2} \; \text{d}s_e. \end{align*}$

The definition of $\alpha_\Gamma$ is impressively simple in notation, but it is not particularly convenient to actually compute $\alpha_\Gamma$ for a given graph $\Gamma$ . A main finding of our preprint 2408.03192 (together with Davide Gaiotto) is a relatively simple explicit formula for $\alpha_\Gamma$ which does not involve any integrals any more. The derivation of that formula is relatively long and technical, but the fundamental steps are straightforward.

Firstly, one realizes that the integral in the definition of $\alpha_\Gamma$ extends over vertex coordinates $\text{d}x_v$ . The differential $\text d s_e$ contains two types of terms: One summand that is proportional to $\text d a_e$ and one that is proportional to $\text d x_+(e) - \text d x_-(e)$ . We need exactly $|V|-1$ factors of $\text d x_v$ , which are necessarily all distinct (because wedge products of identical differentials vanish). This means that exactly $|V|-1$ of the edges contribute their $\text d x_v$ -part, and all other edges contribute $\text d a_e$ . A closer examination shows that these $|V|-1$ edges must form a spanning tree $T$ in $\Gamma$ . Consequently, $\alpha_\Gamma$ is a sum of terms, each of which contributes to one spanning tree in $\Gamma$ .

A second important step is to rewrite the various sums and products in terms of graph matrices. Let $\vec s$ be a vector with $|E_\Gamma|$ components which contains all the $s_e$ . Similarly, collect $|V_\Gamma\-1$ vertex coordinates into a vector $\vec x$ . Finally, let $\mathbb D$ be a diagonal matrix with entries $a_e$ . Then, $\mathbb D^{-\frac 12}$ is a diagonal matrix with entries $a_e^{-\frac 12}$ . If $\mathbb I$ is the edge-vertex incidence matrix of $\Gamma$ , then $\vec s = \mathbb D^{-\frac 12} \mathbb I \vec x$ . The exponent in $\alpha_\Gamma$ is simply

$\begin{align*} -\sum_e s_e^2 &= -\vec s \cdot \vec s =- (\mathbb D^{-\frac 12} \mathbb I \vec x)^T \mathbb D^{-\frac 12} \mathbb I \vec x\\ &=- \vec x^T \mathbb L \vec x. \end{align*}$

Here, we have identified the (vertex-)Laplacian matrix $\mathbb L := \mathbb I^T \mathbb D^{-1} \mathbb I$ . This is a well-studied matrix with many interesting applications and properties.

Having identified the exponential as $e^{-\vec x^T \mathbb L \vec x$ , we see that the integral is Gaussian. However, the integrand is not trivial because the factors $\text d a_e$ in $\text d s_e$ also give rise to factors $(\mathbb I \vec x)_e$ , that is, the integrand is a polynomial in the variables $x_v$ whose degree is the loop number of the graph. We see that $\alpha_\Gamma$ is zero for graphs with odd loop number. Finding a closed-form solution for the integral requires a more systematic examination of the terms, but, of course, the result is a polynomial in the entries of $\mathbb L^{-1}$ (this is a standard fact for Gaussian integrals where the exponent involves a non-trivial matrix).

One obtains an explicit, but somewhat messy, sum over products of $(\mathbb L^{-1})_{i,j}$ . These entries of the inverse Laplacian are (by definition) cofactors, i.e. determinants of minors, of the Laplacian. One introduces the expanded Laplacian

$\begin{align*} \mathbb M := \begin{pmatrix} \mathbb D & \mathbb I \\ -\mathbb I^T &0 \end{pmatrix} .\end{align*}$

Deleting Edges and rows in $\mathbb M$ gives rise to minors. The determinants of these minors are called Dodgson polynomials. As always with determinants, they satisfy an almost incomprehensible number of relations and identities, many of which also have nice interpretations in terms of the underlying graph (e.g. sums over trees that include or avoid certain edges or vertices). For our application, we note that the entries $(\mathbb L^{-1})_{i,j}$ coincide with the vertex-indexed Dodgson polynomials. Moreover, these terms occur in particular combinations, and one can show that they combine to form edge-indexed Dodgson polynomials. This sounds like a weird and irrelevant technicality, but it leads to dramatic simplifications: An entire sum of terms collapses to a single product of edge-indexed Dodgson polynomials. Eventually, for each spanning tree $T$ , our formula for $\alpha_\Gamma$ involves only a single summation, namely over permutations of the edges not in $T$ . The number of these edges is the loop number (which, in particular, is much less than the total number of edges in the graph), therefore, this sum has only moderately many terms unless the graph is huge. We provide a Mathematica program that computes the differential form $\alpha_\Gamma$ symbolically for a given graph.

Besides reducing the number of terms, our formula for $\alpha_\Gamma$ has the nice property that the quantities in it, the Dodgson polynomials, are determinants of (rather simple) matrices. This means that if one wants to evaluate the expression for numerical values of the Schwinger parameters, one can do that very efficiently by numerically computing a determinant of a numerical matrix, which is much faster than replacing the variables in a large symbolic polynomial.

A non-renormalization theorem

A second consequence of our formula for $\alpha_\Gamma$ is, that we can study its algebraic properties. With another slightly non-trivial combinatorial construction, we prove that $\alpha_\Gamma \wedge \alpha_\Gamma=0$ for all $\Gamma$ which are not trees. This is a “vanishing lemma” for a parametric Feynman integral. in fact, it is a parametric-integral version of the famous vanishing of Kontsevich integrals, which is crucial for associativity of the star product in deformation quantization (the “formality theorem”).

An unexpected relation with the Pfaffian

I the follow-up project 2503.09558 together with Simone Hu, we discovered another formula for the topological form: Up to constants,

$\begin{align*} \alpha = \operatorname{Pf} \left(\operatorname{d} \Lambda \Lambda^{-1} \operatorname{d} \Lambda \right), \end{align*}$

where $\Lambda$ is the cycle Laplacian matrix of the graph. This matrix is analogous to the ordinary (vertex-) Laplacian, but instead of an edge-vertex incidence matrix, one starts from an edge-cycle incidence matrix. Our paper includes a detailed review of these constructions and their relations. The Pfaffian “Pf” is a map from matrices to numbers, similar to a determinant.

The new formula immediately has nice consequences. In particular, the non-renormalization theorem now becomes a fairly simple exercise in linear algebra, and one no longer needs to analyze Dodgson polynomial identities to prove it, see the slides.

But the formula is also interesting from another perspective: Under the name “Pfaffian form”, the same object has recently appeared in cohomology computations in the odd graph complex. The question there, simplified, is the following: Consider a certain class of graphs together with combinatorial data (the orientation of edges and the ordering of vertices). Define a differential “d” which maps one graph to a sum of graphs by shrinking one edge in turn. With suitable details, this differential gives rise to a chain complex. The task is now: Find the cohomology classes of this complex. That is, find the (linear combinations of) graphs $G$ such that d $G$ =0 and there is no linear combination $F$ of graphs such that d $F=G$ . The first can be checked by an explicit computation, but the second is in general hard. This is where a dual description, and a pairing with certain integrals, enters the game: Loosely speaking, one can set up integrals, where the integration variables are identified with the edges of the graph and the structure of the integrand is determined by the structure of the graph, such that the differential in the chain complex of graphs gets identified with the differential that relates integrals to their boundary terms. Then, a clever application of Stokes theorem shows that the cohomology classes in the graph complex can be detected by whether or not these integrals vanish. The “Pfaffian” differential form is not the most general one, it only finds one cohomology class, that of dipole/multiedge graphs. But at the same time, this class is the only one that contributes to quantum corrections of the BRST differential in a 1-dimensional topological QFT.

We are currently working on a generalization beyond the purely topological case.

Research

Efficient computation of a parametric integrand

A non-renormalization theorem

An unexpected relation with the Pfaffian