aainstitutetext: School of Natural Sciences, Institute for Advanced Study,
Princeton, NJ, 08540, USAbbinstitutetext: Mathematical Institute, Andrew Wiles
Building, Woodstock Rd, Oxford, UKccinstitutetext: Max-Plank-Institüt fur
Physik, Werner-Heisenberg-Institut, D-80805 München, Germanyddinstitutetext:
Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-
Saclay, IUF, Franceeeinstitutetext: LaCIM, Département de Mathématiques,
Université du Québec à Montréal, Montréal, QC, Canada

#  All Loop Scattering as a Counting Problem

N. Arkani-Hamed b H. Frost c G. Salvatori d P-G. Plamondon e H. Thomas
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>pierre-
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This is the first in a series of papers presenting a new understanding of
scattering amplitudes based on fundamentally combinatorial ideas in the
kinematic space of the scattering data. We study the simplest theory of
colored scalar particles with cubic interactions, at all loop orders and to
all orders in the topological ’t Hooft expansion. We find a novel formula for
loop-integrated amplitudes, with no trace of the conventional sum over Feynman
diagrams, but instead determined by a beautifully simple counting problem
attached to any order of the topological expansion. These results represent a
significant step forward in the decade-long quest to formulate the fundamental
physics of the real world in a radically new language, where the rules of
spacetime and quantum mechanics, as reflected in the principles of locality
and unitarity, are seen to emerge from deeper mathematical structures.

##  1 Introduction and Summary

Scattering amplitudes are perhaps the most basic and important observables in
fundamental physics. The data of a scattering process—the on-shell momenta and
spins of the particles—are specified at asymptotic infinity in Minkowski
space. The conventional textbook formalism for computing amplitudes
“integrates in” auxiliary structures that are not present in the final
amplitude, including the bulk spacetime in which particle trajectories are
imagined to live, and the Hilbert space in which the continuous bulk time
evolution of the wavefunction takes place. These auxiliary structures are
reflected in the usual formalism for computing amplitudes, using Feynman
diagrams, which manifests the rules of spacetime (locality) and quantum
mechanics (unitarity). As has been increasingly appreciated over the past
three decades, this comes at a heavy cost—the introduction of huge
redundancies in the description of physics, from field redefinitions to gauge
and diffeomorphism redundancies, leading to enormous complexities in the
computations, that conceal a stunning hidden simplicity and seemingly
miraculous mathematical structures revealed only in the final result
Parke:1986gb ; Bern_1994 ; Bern_1995 ; Witten_2004 ; Cachazo_2004 ;
Britto_2005 ; Britto_2005W .

This suggests that we should find a radically different formulation for the
physics of scattering amplitudes. The amplitudes should be the answer to
entirely new mathematical questions that make no reference to bulk spacetimes
and Hilbert space, but derive locality and unitarity from something more
fundamental. A number of concrete examples of this have already been found in
special cases. The discovery of deep and simple new structures in
combinatorics and geometry has led to new definitions of certain scattering
amplitudes, without reference to spacetime or quantum mechanics. Notably, the
amplituhedron determines the scattering amplitudes in planar N =4 SYM, and
associahedra and cluster polytopes determine colored scalar amplitudes at
tree-level and one-loop Arkani_Hamed_2014 ; 2017ABHY ; Salvatori_2019 ;
2019AHHST .

Up to now, these results have been limited in how much of the perturbative
expansion they describe—at all loop orders for maximally supersymmetric
theories, but only in the planar limit, and only through to one loop for non-
supersymmetric theories. Furthermore, the connection between combinatorial
geometry and scattering amplitudes at loop level has only been made through
the integrand (pre-loop integration) of the amplitudes, and not the amplitudes
themselves. Both of these limitations must be transcended to understand all
aspects of particle scattering in the real world.

This article is the first in a series reporting on what we believe is major
new progress towards this goal. These ideas set the foundation for a number of
other interrelated threads and results that will appear in various groups of
papers. So we take this opportunity to give a birds-eye view of the nature of
these developments and the new concepts that are driving this progress.

Our departure point is a new formulation of a simple theory,—colored scalar
particles with cubic interactions,—at all loop orders and to all orders in the
topological ’t Hooft expansion, in the form of what we call a _curve
integral_. This approach has no hint of a sum over Feynman diagrams anywhere
in sight and is instead associated with a simple counting problem defined at
any order in the topological expansion. This counting problem defines a
remarkable set of variables, $u_{C}$, associated with every curve, $C$, on a
surface. The $u$-variables non-trivially define _binary geometries_ Arkani-
Hamed:2019mrd by dint of satisfying the remarkable non-linear equations us_us

$u_{C}+\prod_{D}u_{D}^{n(C,D)}=1,$ (1)

where $n(C,D)$ is the intersection number of the curves $C,D$. In the
_positive region_ , where all the $u_{C}$ are non-negative, the $u$-equations
force all the $u_{C}$ to lie between 0 and 1: $0\leq u_{C}\leq 1$. Of
mathematical interest, this positive region is a natural and invariant
compactification of _Teichmüller space_. This algebraic presentation of
Teichmüller space is a counterpart to the famous synthetic compactification of
Teichmüller spaces and surface-type cluster varieties given by Fock-Goncharov
2003math…..11149F ; penner2012decorated (refneed). The new compactifications
defined by the $u_{C}$ variables are immediately relevant for physics, and
lead to the new _curve integral_ formulation of all-loop amplitudes presented
in this article.

The curve integral does more than reformulate the perturbative series in a new
way. It also exposes basic new structures in field theory. For instance, a
striking consequence of our formulation is that amplitudes for large $n$
particles at $L$-loops effectively factorise into a tree and a loop
computation. The full large $n$ amplitudes can be reconstructed from
computations of $n$-point tree amplitudes and low-point $L$-loop amplitudes.
Moreover, our curve integral formulas make manifest that amplitudes satisfy a
natural family of differential equations in kinematic space. The solutions of
these equations give novel and efficient recursion relations for all-loop
amplitudes.

This article focuses on colored scalar amplitudes. However, the results here
have extensions to other theories. New curve integral formulations have been
discovered for theories of colored scalar particles with arbitrary local
interactions, as well as for the amplitudes of pions and non-supersymmetric
Yang-Mills theories. These formulas reveal striking inter-relations between
these theories, together with surprising hidden properties of their amplitudes
that are made manifest by the curve integral formalism.

Our results also have implications for the understanding of strings and UV
completion. The counting problem at the heart of this paper not only defines
QFT amplitudes, it also defines amplitudes for bosonic strings, via the
$u$-variables, $u_{C}$, mentioned above. This gives a combinatorial
formulation of string amplitudes that makes no reference to worldsheet CFTs
and vertex operators. This new approach to string amplitudes differs from the
conventional theory in a very fundamental way. The $u$-variables, which are
derived from a simple counting problem, have a beautiful and direct connection
to the geometry of two-dimensional surfaces. But this connection is via the
_hyperbolic geometry_ of Teichmüller space, and not via the conventional
picture of Riemann surfaces with a complex structure. The new string formulas
are not just an exercise in passing between the complex and the hyperbolic
pictures for Teichmüller space. We find that we can reproduce bosonic strings
at loop level, but other choices are just as consistent, at least insofar as
the field theory limit is concerned. This allows us to deform string
amplitudes into a larger, but still highly constrained, space of interesting
objects. This runs counter to the lore that string theory is an inviolable
structure that cannot be modified without completely breaking it. Our larger
class of string amplitudes transcends the usual strictures on spacetime
dimension, as well as the famous instabilities of non-supersymmetric strings.
Moreover, our new combinatorial-geometric point of view also makes it easier
to recover particle amplitudes from strings in the $\alpha^{\prime}\to 0$
limit. By contrast, recovering field theory from conventional string theory
involves vastly (technically, infinitely!) more baggage than is needed
us_strings .

There are several other related developments, including the discovery of a
remarkable class of polytopes, _surfacehedra_ , whose facet structure
captures, mathematically, the intricate boundary structure of Teichmüller
space, and, physically, the intricate combinatorics of amplitude singularities
at all loop orders, and whose _canonical form_ determines (an appropriate
notion of the) loop integrand at all orders in the topological expansion.

The results of all these parallel threads of investigation will be presented
in various groups of papers. We end this preview of coming attractions by
explaining a quite different sort of motivation for our works that will be
taken up in near-future work. The counting problem that lies at the heart of
this paper has an entirely elementary definition. But the central importance
of this counting problem will doubtless seem mysterious at first sight. It
finds its most fundamental origin in remarkably simple but deep ideas from the
“quiver representation theory” 2017DW ; haupt2012 of (triangulated) surfaces.
Arrows between the nodes of a quiver can be associated with maps between
vector spaces attached to the nodes. Choosing compatible linear maps between
the nodes defines a _quiver representation_. In this context, our counting
problem is equivalent to counting the _sub-representations_ of these quiver
representations. This perspective illuminates the mathematical structure
underlying all of our formulas. But these ideas also hint at a fascinating
prospect. The amplitudes we study are associated with the class of surface-
type quivers, which are dual to triangulated surfaces. Nothing in our formulas
forces this restriction on us: we are free to consider a much wider array of
quivers. All of these quivers can be associated with amplitude-like functions.
This vast new class of functions enjoys an intricate (amplitude-like)
structure of “factorisations” onto simpler functions. This amounts to a
dramatic generalisation of the notion of an “amplitude”, and in a precise
sense also generalises the rules of spacetime and quantum mechanics to a
deeper, more elementary, but more abstract setting.

Having outlined this road map, we return to the central business of this first
paper. We will study the simplest theory of $N^{2}$ colored particles with any
mass $m$, grouped into an $N\times N$ matrix $\Phi^{I}_{J}$ with
$I,J=1,\cdots,N$. The Lagrangian, with minimal cubic coupling, is

${\cal L}={\rm Tr}(\partial\Phi)^{2}+m^{2}{\rm Tr}(\Phi^{2})+g{\rm
Tr}(\Phi^{3}),$ (2)

in any number $D$ of spacetime dimensions. This theory is a simpler cousin of
all theories of colored particles, including Yang-Mills theories, since the
singularities of these amplitudes are the same for all such theories, only the
_numerators_ differ from theory to theory. The singularities of amplitudes are
associated with some of the most fundamental aspects of their conventional
interpretation in terms of spacetime processes respecting unitarity. So
understanding the amplitudes for this simple theory is an important step
towards attacking much more general theories.

We will show that all amplitudes in this theory, for any number $n$ of
external particles, and to all orders in the genus (or $1/N$) expansion 1973H
, are naturally associated with a strikingly simple counting problem. This
counting problem is what allows us to give _curve integral_ formulas for the
amplitudes at all orders. The curve integral makes it easy to perform the loop
integrations and presents the amplitude as a single object.

As an example, consider the single-trace amplitude for $n$-point scattering at
1-loop. Let the particles have momenta $p_{i}^{\mu}$, $i=1,...,n$. The curve
integral for this amplitude (pre-loop integration) is

$\mathcal{A}^{1-{\rm loop}}_{n}=\int d^{D}l\int\limits_{\sum_{i}t_{i}\geq
0}d^{n}t\,{\rm
exp}\left[-\sum_{i=1}^{n}\alpha_{i}(l+p_{1}+\cdots+p_{i})^{2}-\sum_{i,j}\alpha_{i,j}(p_{i}+\cdots+p_{j-1})^{2}\right]$
(3)

where

$\displaystyle\alpha_{i,j}$
$\displaystyle=f_{i,j}+f_{i+1,j+1}-f_{i,j+1}-f_{i+1,j},$ (4)
$\displaystyle\alpha_{i}$ $\displaystyle=\alpha_{i,i+n},$ $\displaystyle
f_{i,j}$ $\displaystyle={\rm
max}(0,t_{j},t_{j}+t_{j-1},\cdots,t_{j}+t_{j-1}+\cdots t_{i+2}).$ (5)

The propagators that arise in the 1-loop Feynman diagrams are either loop
propagators, with momenta $(l+p_{1}+\cdots+p_{i})$, or tree-like propagators,
with momenta $(p_{i}+p_{i+1}+\cdots+p_{j-1})$. The exponential in (3) looks
like a conventional Schwinger parametrisation integral, except that _all_ the
propagators that arise at 1-loop are included in the exponent. Instead of
Schwinger parameters, we have _headlight functions_ : $\alpha_{i}$ (for the
loop propagators) and $\alpha_{i,j}$ (for the tree propagators). The headlight
functions are piecewise linear functions of the $t_{i}$ variables. The magic
is that (3) is a single integral over an $n$-dimensional vector space. Unlike
conventional Schwinger parametrisation, which is done one Feynman diagram at a
time, our formulas make no reference to Feynman diagrams. Amazingly, the
exponent in (3) breaks $t$-space into different cones where the exponent is
linear. Each of these cones can be identified with a particular Feynman
diagram, and the integral in that cone reproduces a Schwinger parameterisation
for that diagram. This miracle is a consequence of the properties of the
headlight functions $\alpha_{i}(t)$ and $\alpha_{i,j}(t)$. These special
functions arise from a simple counting problem associated with the
corresponding propagator.

As in conventional Schwinger parametrisation, the dependence on the loop
momentum variable, $l^{\mu}$, in the curve integral, (3), is Gaussian. We can
perform the loop integration to find the a second curve integral for the
amplitude (post loop integration),

$\mathcal{A}^{1-{\rm loop}}_{n}=\int\limits_{\sum_{i}t_{i}\geq
0}d^{n}t\left(\frac{2\pi}{\mathcal{U}}\right)^{\frac{D}{2}}e^{-\frac{\mathcal{F}}{\mathcal{U}}}.$
(6)

In this formula, the polynomials ${\cal U}$ and ${\cal F}$ are given by

$\mathcal{U}=\sum_{i}\alpha_{i},\qquad\mathcal{F}=\sum_{i,j}\alpha_{i}\alpha_{j}(p_{i}+\cdots
p_{j-1})^{2}-\left(m^{2}\sum_{i}\alpha_{i}+2\sum_{i,j}\alpha_{i,j}\,p_{i}.p_{j}\right)\mathcal{U}.$
(7)

These polynomials are analogs of the familiar Symanzik polynomials, but
whereas the Symanzik polynomials appear in individual Feynman integrals, this
one curve integral above computes the whole amplitude.

These 1-loop curve integrals generalise to all orders in perturbation theory,
at any loop order and genus. In the rest of this introductory section we give
a birds-eye view of the key formulas and results.

###  1.1 Kinematic space

To begin with, we have to define the _kinematic space_ where all the action
will take place. In our theory, each Feynman diagram is what is called a
‘double-line notation diagram’, ‘ribbon graph’ or ‘fatgraph’ in the
literature; we will call them fatgraphs in what follows. Examples of fatgraphs
are shown in Figure 1. Order by order, in the ’t Hooft expansion, these
Feynman diagrams get organised into partial amplitudes, labeled by their
shared _color structure_. Conventionally, when we do a ’t Hooft expansion, we
think of these fat graphs as ‘living on’ or ‘being drawn on’ a surface with
some genus and number of boundary components. We will think of them in a
different way: a single fat graph itself defines a surface. In fact, we will
use a single fat graph to define all the data we need to compute an amplitude!

Figure 1: Fat graphs at tree-level, 1-loop single trace, 1-loop double trace,
and 2-loop single trace, respectively.

Take some fatgraph, $\Gamma$, at any order in the ’t Hooft expansion. Suppose
that it has $n$ external lines and $E$ internal edges. Then this fat graph has
loop order, $L$, with

$E=n+3(L-1).$ (8)

Let the external lines have momenta $p_{1},\ldots,p_{n}$, and introduce $L$
loop variables, $\ell_{1},\ldots,\ell_{L}$. Then, by imposing momentum
conservation at each vertex of $\Gamma$, we can find a consistent assignment
of momenta to all edges of the fat graph in the usual way: if each edge, $e$,
gets a momentum $p_{e}^{\mu}$, then whenever three edges, $e_{1},e_{2},e_{3}$,
meet at a vertex, we have

$p_{e_{1}}^{\mu}+p_{e_{2}}^{\mu}+p_{e_{3}}^{\mu}=0.$ (9)

For example, Figure 2 is an assignment of momenta to the edges of a tree
graph.

Figure 2: A tree fat graph with momenta assigned to all edges.

The amplitude itself depends on momenta only through Lorentz invariant
combinations. So we want to define a collection of Lorentz invariant kinematic
variables. Consider a curve, $C$, drawn on the fatgraph $\Gamma$ that starts
at an external line, passes through the graph and exits at another external
line. For example, the curve in Figure 3 starts at $p_{2}$, and exits at
$p_{5}$. Every such curve can be assigned a unique momentum. It is given by
the momentum of the first edge plus the sum of all momenta on the graph
entering the curve ‘from the left’. For example, in Figure 3, the curve starts
with momentum $p_{2}$, and then takes two right turns. At the first right
turn, momentum $p_{3}$ enters from the left. At the second right turn,
momentum $p_{4}$ enters from the left. The total momentum of the curve is then
given by

$p_{C}^{\mu}=p_{2}^{\mu}+p_{3}^{\mu}+p_{4}^{\mu}.$ (10)

Notice that if we had gone in the opposite direction (starting at $p_{5}$), we
would have got

$-p_{C}^{\mu}=p_{5}^{\mu}+p_{1}^{\mu}.$ (11)

But by total momentum conservation ($p_{1}+\ldots+p_{n}=0$), it does not
matter which direction we take.

For a general curve, $C$, on any fatgraph, this rule can be written as:

$P^{\mu}_{C}=p^{\mu}_{{\rm start}}+\sum_{{\rm right\,turns}}p^{\mu}_{{\rm
from\,left}}.$ (12)

This rule assigns to every curve $C$ on our fatgraph $\Gamma$ some momentum,
$P_{C}^{\mu}$. Each $P_{C}^{\mu}$ is a linear combination of external momenta,
$p_{i}$, and loop variables, $\ell_{a}$. Each curve, $C$, then also defines a
Lorentz invariant kinematic variable

$X_{C}=P_{C}^{2}+m^{2}.$ (13)

The collection of variables $X_{C}$, for _all_ curves $C$ on the fatgraph,
defines a complete set of kinematic variables in our kinematic space. Modulo a
small detail about how to deal with internal color loops, this completes the
description of our kinematic space.

Figure 3: Every curve, $C$, drawn on the fat graph inherits a momentum,
$P_{C}^{\mu}$, from the momenta assigned to the fat graph itself.

It is significant in our story that we can use the momenta of a _single_ fat
graph (or Feynman diagram) to define a complete set of kinematic variables
$X_{C}$. As we will see in more detail in Section 6, this basic idea ends up
solving the long-standing problem of defining a good notion of loop integrand
beyond the planar limit!

###  1.2 The First Miracle: Discovering Feynman diagrams

We now look for a question whose answer produces scattering amplitudes. We
just saw how we can define all our kinematics using a single fatgraph. So with
this starting point, what would make us consider _all_ possible Feynman
diagrams (i.e. all spacetime processes)? And why should these be added
together with equal weights (as demanded by quantum mechanics)? Amazingly, the
answer to both of these fundamental questions is found right under our noses,
once we think about how to systematically describe all the curves on our
fatgraph.

How can we describe a curve on our fat graph without drawing it? We can do
this by labeling all the edges, or “roads”, on the fatgraph. Any curve passes
through a series of these roads. Moreover, at each vertex, we demand that the
curve must turn either left or right: we do not allow our curves to do a ‘U
turn’. It follows that a curve is fully described by the order of the roads
and turns it takes as it passes through the graph. For example, the curve in
Figure 4 enters through edge ‘$1$’, takes a left turn, goes down ‘$x$’, takes
a left turn, goes down ‘$y$’, takes a right turn, and then exits via ‘$4$’.

We can represent this information graphically as a _mountainscape_ , where
left turns are represented by upward slopes, and right turns are represented
by downward slopes. The mountainscape for the curve in Figure 4 is shown in
the Figure.

Figure 4: Describing a curve on a fatgraph (left) using a mountainscape
diagram (right).

Once again, let our fatgraph have $E$ internal edges. To every curve $C$, we
will associate a vector ${\bf g}_{C}$ in _curve space_. As a basis for this
vector space, take $E$ vectors ${\bf e}_{i}$, associated to each internal
edge. Then ${\bf g}_{C}$ can be read off from the mountainscape for $C$ using
the following rule:

${\bf g}_{X}=\sum_{{\rm peaks\,p}}{\bf e}_{{\rm p}}-\sum_{{\rm
valleys\,v}}{\bf e}_{{\rm v}}.$ (14)

For example, the curve in Figure 4 has a peak at ‘$y$’ and no valleys. So the
$g$-vector for this curve is

${\bf g}_{C}={\bf e}_{y}.$ (15)

Figure 5: The five (non boundary) curves that we can draw on a 5-point tree
fatgraph.

Now consider _every_ curve that we can draw on the fatgraph in Figure 4. There
are 10 possible curves. 5 of these are ‘boundaries’, and their g-vectors end
up vanishing (because their mountainscapes have no peaks or valleys). The
remaining 5 curves are drawn in Figure 5. If we label the external lines, each
curve can be given a name $C_{ij}$ ($i,j=1,2,3,4,5$), where $C_{ij}$ is the
curve connecting $i$ and $j$. Their g-vectors are

${\bf g}_{13}={\bf e}_{x},\leavevmode\nobreak\ \leavevmode\nobreak\ {\bf
g}_{14}={\bf e}_{y},\leavevmode\nobreak\ \leavevmode\nobreak\ {\bf
g}_{24}=-{\bf e}_{x}+{\bf e}_{y},\leavevmode\nobreak\ \leavevmode\nobreak\
{\bf g}_{25}=-{\bf e}_{x},\leavevmode\nobreak\ \leavevmode\nobreak\ {\bf
g}_{35}=-{\bf e}_{y}.$ (16)

If we draw these five g-vectors, we get Figure 6. This has revealed a
wonderful surprise! Our g-vectors have divided curve space into five regions
or _cones_. These cones are spanned by the g-vectors for the following pairs
of curves:

$(C_{13},C_{14}),\leavevmode\nobreak\ (C_{14},C_{24}),\leavevmode\nobreak\
(C_{24},C_{25}),\leavevmode\nobreak\ (C_{25},C_{35}),\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ (C_{35},C_{13}).$ (17)

These pairs of curves precisely correspond to all the five Feynman diagrams of
the 5-point tree amplitude!

Figure 6: The collection of ${\bf g}$-vectors for the fat graph in Figure 4
cuts the 2-dimensional vector space into five regions.

This is a general phenomenon. The collection of g-vectors for all the curves
$C$ on a fatgraph is called _the g-vector fan_ 2005math…..10312F ;
fomin2006cluster ; Fomin_2018 , or _the Feynman fan_ , associated to that
fatgraph. Each top-dimensional cone of the fan is spanned by an $E-$tuple of
curves, $C_{a_{1}},\cdots,C_{a_{E}}$, and these $E-$tuples of curves are
precisely the propagators of Feynman diagrams. Moreover, the cones are non-
overlapping, and together they densely cover the entire vector space! The
g-vector fan is telling us that all the Feynman diagrams for the amplitude are
combined in curve space.

Even better, each of the cones in the g-vector fan have the same size. It is
natural to measure the size of a cone, bounded by some g-vectors ${\bf
g}_{1},\cdots,{\bf g}_{E}$, using the determinant of these vectors:
$\langle{\bf g}_{1}\cdots{\bf g}_{E}\rangle$. Remarkably, the cones of the
g-vector fan all satisfy: $\langle{\bf g}_{1}\cdots{\bf g}_{E}\rangle=\pm 1$.

To summarise, starting with a _single_ fatgraph at any order in perturbation
theory, simply recording the data of the curves on the fatgraph, via their
g-vectors, brings all the Feynman diagrams to life. Furthermore, we see why
they are all naturally combined together into one object, since they
collectively cover the entire curve space! This represents a very vivid and
direct sense in which the most basic aspects of spacetime processes and the
sum-over-histories of quantum mechanics arise as the answer to an incredibly
simple combinatorial question.

###  1.3 An infinity of diagrams and the spectre of Gravity

An important novelty appears with the first non-planar amplitudes. Consider
the double-trace one-loop amplitude at 2-points. A fatgraph for this amplitude
is given in Figure 7. There are now infinitely many curves that we can draw on
this fat graph: they differ from one another only in how many times they wind
around the graph.

Figure 7: A double-trace 1-loop fat graph, which has infinitely many possible
curves.

The g-vector fan for this infinity of curves is shown in Figure 8. These
g-vectors break curve space up into infinitely many cones. Each of these cones
is bounded by a pair of g-vectors, $g_{C_{m}}$ and $g_{C_{m+1}}$, where
$C_{m}$ and $C_{m+1}$ are two curves that differ by exactly one _winding_. If
we use our rule for the momenta of curves, (12), the momenta of these curves
are

$P_{C_{m}}^{\mu}=mk^{\mu}+\ell^{\mu},\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ P_{C_{m+1}}^{\mu}=(m+1)k^{\mu}+\ell^{\mu}.$
(18)

So the momenta associated to each cone are related to each other by a
translation in the loop variable, $\ell^{\mu}\mapsto\ell^{\mu}+k^{\mu}$. It
follows that _every_ cone in Figure 8 corresponds to a copy of the _same_
Feynman diagram.

Figure 8: The g-vector fan for the 2-point double-trace 1-loop fat graph,
which has infinitely many regions.

What has gone wrong? The g-vector fan is telling us to include infinitely many
copies of one Feynman diagram. This is a consequence of the _mapping class
group_ of the fat graph in Figure 7. The mapping class group of this fatgraph
acts by increasing the winding of curves drawn on the fatgraph. In fact, this
infinity of windings is the heart of the well-known difficulty in defining a
loop integrand for non-planar amplitudes. Fortunately, as we will see, it is
easy to _mod out_ by the action of the mapping class group, which we will do
using what we call the _Mirzakhani trick_ mirzakhani2007 . Getting rid of
these infinities using the Mirzakhani trick is the final ingredient we need in
order to define amplitudes directly from the combinatorics of a single
fatgraph.

As an aside, note that the infinite collection of cones in Figure 8 does not
quite cover the entire vector space! The g-vectors asymptotically approach the
direction $(-1,1)$, but never reach it. This is the beginning of fascinating
story: it turns out that the vector $(-1,1)$ is the g-vector for the closed
curve that loops once around the fat graph. Nothing in our story asks us to
consider these closed curves, but the g-vector fan forces them on us.
Physically, these new closed curves are associated with the appearance of a
new uncoloured particle, $\sigma$. These missing parts of the fan are then
seen to have a life of their own: they tell us about a theory with uncoloured
self-interactions, $\sigma^{3}$, that is minimally coupled to our colored
particle by an interaction $\sigma$ Tr $(\Phi)$. The appearance of $\sigma$ is
a scalar avatar of how the graviton is forced on us in string theory even if
we begin only with open strings. From our perspective, however, this has
absolutely nothing to do with the worldsheet of string theory; it emerges
directly from the combinatorics defined by a fatgraph.

###  1.4 The Amplitudes

The g-vector fan gives a beautiful unified picture of all Feynman diagrams
living in an $E$-dimensional vector space, _curve space_. This result suggests
a natural formula for the full amplitude in the form of an integral over curve
space. To find this formula, we need one extra ingredient. For every curve,
$C$, we will define a piecewise-linear _headlight function_ , $\alpha_{C}({\bf
t})$. We will define the headlight function $\alpha_{C}$ so that it “lights
up” curve space in the direction ${\bf g}_{C}$, and vanishes in all other
g-vector directions:

$\alpha_{C}({\bf g}_{D})=\delta_{C,D}$ (19)

This definition means that $\alpha_{C}$ vanishes everywhere, except in those
cones that involve ${\bf g}_{C}$. Moreover, $\alpha_{C}$ is _linear_ inside
any given cone of the Feynman fan.

Using linear algebra, we can give an explicit expression for $\alpha_{C}$ in
any cone where it is non-vanishing. Suppose that the g-vectors of such a cone
are $({\bf g}_{C},{\bf g}_{D_{1}},\cdots,{\bf g}_{D_{E-1}})$. The unique
linear function of ${\bf t}$ which evaluates to $1$ on ${\bf g}_{C}$ and 0 on
all the other g-vectors is

$\alpha_{C}=\frac{\langle{\bf t}\,{\bf g}_{D_{1}}\cdots{\bf
g}_{D_{E-1}}\rangle}{\langle{\bf g}_{C}{\bf g}_{D_{1}}\cdots{\bf
g}_{D_{E-1}}\rangle}.$ (20)

In what follows, imagine that we already know these functions,
$\alpha_{C}({\bf t})$.

We now define an _action_ , $S$, given by a sum over all curves on a fatgraph:

$S({\bf t})=\sum_{C}\alpha_{C}({\bf
t})X_{C},\qquad\text{with}\leavevmode\nobreak\ X_{C}=P_{C}^{2}+m^{2}.$ (21)

Recall that $P_{C}^{\mu}$ is the momentum we associate to a curve $C$. If we
restrict $S({\bf t})$ to a single cone, bounded by some g-vectors, ${\bf
g}_{C_{1}},\ldots,{\bf g}_{C_{E}}$, then the only $\alpha$’s that are non-zero
in this cone are precisely $\alpha_{C_{1}},\ldots,\alpha_{C_{E}}$. Moreover,
$S({\bf t})$ is linear in this cone. It is natural to parametrise the region
inside this cone by ${\bf t}=\rho_{1}{\bf g}_{C_{1}}+\cdots\rho_{E}{\bf
g}_{C_{E}}$, with $\rho_{i}\geq 0$ positive. Then we can integrate $\exp(-S)$
in this cone. The result is identical to the result of a standard Schwinger
parametrisation for a single Feynman diagram:

$\int\limits_{\text{cone}}d^{E}t\,e^{-S}=\int\limits_{0}^{\infty}d^{E}\rho\,|\langle
g_{C_{1}}\cdots
g_{C_{E}}\rangle|\prod_{i=1}^{E}e^{-\rho_{i}X_{C_{i}}}=\prod_{i=1}^{E}\frac{1}{P_{C_{i}}^{2}+m^{2}}.$
(22)

The factor $|\langle g_{X_{1}}\cdots g_{X_{E}}\rangle|$ is the Jacobian of the
change of variables from $(t_{1},\cdots,t_{E})$ to
$(\rho_{1},\cdots,\rho_{E})$. As we have remarked, the cones are _unimodular_
and these Jacobian factors are all equal to 1!

In order to get the full amplitude, all we have to do now is integrate
$\exp(S)$ over the whole vector space, instead of restricting it to just a
single cone. However, to account for the infinity resulting from the _mapping
class group_ , we also need to factor out this MCG action in our integral,
which we denote by writing the measure as

$\frac{d^{E}t}{{\rm MCG}}.$ (23)

Before doing the loop integrations, the full amplitude is then given by a
_curve integral_ :

${\cal A}=\int d^{D}\ell_{1}\cdots d^{D}\ell_{L}\int\frac{d^{E}t}{{\rm
MCG}}\,{\rm exp}\left(-\sum_{X}\alpha_{X}({\bf t})(p_{X}^{2}+m^{2})\right).$
(24)

The dependence on loop momenta in this formula is Gaussian. When we integrate
the loop momenta, we find the final amplitude is given by a curve integral

${\cal A}=\int\frac{d^{E}t}{{\rm MCG}}\,\left(\frac{\pi^{L}}{{\cal
U}(\alpha)}\right)^{\frac{D}{2}}{\rm exp}\left(\frac{{\cal F}(\alpha)}{{\cal
U}(\alpha)}\right).$ (25)

${\cal U}(\alpha)$ and ${\cal F}(\alpha)$ are homogeneous polynomials in the
headlight functions. They are analogous to Symanzik polynomials, but are not
associated with any particular Feynman diagram. We give simple formulas for
${\cal A}$ and ${\cal F}$ in Section 7.

The key to using these curve integral formulas lies in how we mod out by the
MCG. One way of doing this would be to find a _fundamental domain_ in ${\bf
t}$-space that would single out one copy of each Feynman diagram. However, in
practice this is no easier than enumerating Feynman diagrams. Instead, we will
use an elegant way of modding out that we call _the Mirzakhani trick_ , which
is analogous to the Fadeev-Popov trick familiar from field theory. As we will
see, any MCG invariant function, $f$, can be integrated as,

$\int\frac{d^{E}t}{{\rm MCG}}f=\int d^{E}t\,{\cal K}(\alpha)f,$ (26)

where the _Mirzakhani kernel_ ${\cal K}(\alpha)$ is a simple rational function
of the $\alpha_{C}$’s.111The restriction on the integration region
$\sum_{i}t_{i}\geq 0$ in equation (6) for 1-loop amplitudes can be thought of
as the smallest example of a Mirzakhani kernel. In this formula, we are
modding out by a discrete $Z_{2}$ symmetry, described more in Section 4.5. We
will describe several formulas for these kernels. In all cases, ${\cal K}$ has
support on a finite region of the fan, so that only a small number of the
$\alpha_{C}$’s is ever needed to compute the amplitude. We will also show how
some of our methods for producing ${\cal K}$ give new systematic recursive
methods for computing amplitudes.

###  1.5 The Second Miracle: The Counting Problem

We have given a formula, (25), for partial amplitudes at any order in the ‘t
Hooft expansion of our theory. However, the meat of this formula is in the
headlight functions, $\alpha_{C}$. The problem is that headlight functions
are, naively, hard to compute!

The issue can already been seen at tree level. For $n$-points at tree level,
the number of possible curves, $C$, is $\sim n^{2}$, whereas the number of
Feynman diagrams (or cones) grows exponentially as $\sim 4^{n}$. Each
$\alpha_{C}$ restricts to a different linear function on each of the $\sim
4^{n}$ cones. So we would expect that it takes an exponentially-growing amount
to work to compute all of the $\alpha_{C}$,—about as much work as it would
take us to just enumerate all the Feynman diagrams to begin with! So, is there
an easier way to compute $\alpha_{C}$?

This is where a second miracle occurs. It turns out that headlight functions
can be computed efficiently by matrix multiplication. In fact, the calculation
is completely _local to the curve_ , in the sense that we only need to know
the path taken by $C$, and nothing else about the fatgraph it lives in. There
are always many fewer curves than there are Feynman diagrams. This means that
the amount of work to compute the $\alpha_{C}$’s should grow slower than the
amount of work it takes to enumerate all Feynman diagrams.

This way of computing $\alpha_{C}$ is based on a simple combinatorial problem.
For a curve, $C$, draw its _mountainscape_. We are going to record all the
ways in which we can pick a subset of letters of $C$, subject to a special
rule: if we pick a letter $y$, we also have to pick any letters _downhill_ of
$y$. We will then define an _F polynomial_ for the curve, $F(C)$, which
records the valid subsets.

(a)

(b)

(c)

Figure 9: Three mountainscapes.

For example, for the mountainscape in Figure 9(a), we get

$F=1+a+c+ac+abc.$ (27)

This is because we can choose the following subsets: no-one (“1”); just $a$;
just $c$; $a$ and $c$ together; or finally we can pick $b$, but if we do, we
must also pick $a$ and $c$, which are both downhill of $b$. In Figure 9(b), we
get

$F=1+b+ab+bc+abc,$ (28)

because in this example we can choose: no-one; just $b$; we can pick $a$, but
if we do we must also pick $b$; we can pick $c$, but we must then also pick
$b$; and finally we can can pick both $a$ and $c$, but then we must also pick
$b$. Finally, we leave Figure 9(c) as an exercise. The result is

$F=1+a+d+ad+ab+abd+abcd.$ (29)

In general, there is a fast method for computing $F(C)$ by reading the
mountainscape for $C$ from left to right. Say the leftmost letter is $Y$, and
call the next letter $y$. Then write $F(C)=F_{{\rm no}}+F_{{\rm yes}}$, where
we group the terms in $F(C)$ according to whether they include $Y$
($F_{\text{yes}}$) or not ($F_{\text{no}}$). Similarly write $f_{{\rm
no}},f_{{\rm yes}}$ for what we would get starting instead from $y$. Suppose
that in our mountainscape we move “up” from $Y$ to $y$. Then if we do not pick
$Y$, then we cannot pick $y$ either, since if we choose $y$ we must choose
$Y$. On the other hand if we do choose $Y$, we can either pick or not pick
$y$. Thus, in this case, we have

$F_{{\rm no}}=f_{{\rm no}},\qquad F_{{\rm yes}}=Y(f_{{\rm no}}+f_{{\rm
yes}}).$ (30)

Similarly if, in our mountainscape, we move down from $Y$ to $y$, we find that

$F_{{\rm no}}=f_{{\rm no}}+f_{{\rm yes}},\qquad F_{{\rm yes}}=Yf_{{\rm yes}}.$
(31)

In matrix form, we find that

$\left(\begin{array}[]{c}F_{{\rm no}}\\\ F_{{\rm
yes}}\end{array}\right)=M_{L,R}(Y)\left(\begin{array}[]{c}f_{{\rm no}}\\\
f_{{\rm yes}}\end{array}\right),$ (32)

where $M_{L}$ and $M_{R}$ are the matrices

$M_{L}(Y)=\left(\begin{array}[]{cc}1&0\\\ Y&Y\end{array}\right),\qquad
M_{R}(Y)=\left(\begin{array}[]{cc}1&1\\\ 0&Y\end{array}\right).$ (33)

Now suppose that the curve $C$ is given explicitly by the following series of
edges and turns:

$(y_{1},{\rm turn}_{1},y_{2},{\rm turn}_{2},\cdots,y_{m-1},{\rm
turn}_{m-1},y_{m}),$ (34)

where ${\rm turn}_{i}$ is either a left or right turn, immediately following
$y_{i}$. Given (32), we find

$\left(\begin{array}[]{c}F_{{\rm no}}\\\ F_{{\rm
yes}}\end{array}\right)=M\left(\begin{array}[]{c}1\\\
y_{{m}}\end{array}\right),$ (35)

where

$M(C)=M_{{\rm turn}_{1}}(y_{1})M_{{\rm turn}_{2}}(y_{2})\cdots M_{{\rm
turn}_{m-1}}(y_{m-1}).$ (36)

So our counting problem is easily solved simply by multiplying a series of
$2\times 2$ matrices (equation 33) associated with the left and right turns
taken by the curve $C$.

Suppose that the initial edge of $C$, $y_{1}$, and the final edge, $y_{m}$,
are external lines of the fatgraph. It is natural to write $F(C)$ as a sum
over four terms:

$F(C)=F_{{\rm no},\,{\rm no}}+F_{{\rm no},\,{\rm yes}}+F_{{\rm yes},\,{\rm
no}}+F_{{\rm yes},\,{\rm yes}},$ (37)

where we group terms in $F(C)$ according to whether they do or do not include
the first and last edges: $y_{1}$ and/or $y_{m}$. Indeed, these terms are also
the entries of the matrix $M(C)$,

$M(C)=\left(\begin{array}[]{cc}F_{{\rm no},{\rm no}}&F_{{\rm no},{\rm yes}}\\\
F_{{\rm yes},{\rm no}}&F_{{\rm yes},{\rm yes}}\end{array}\right),$ (38)

if we now set $y_{m}=1$. In fact, we will also set $y=1$ for every external
line of the fatgraph, and will reserve $y$-variables for internal edges of the
fatgraph.

Notice that $\det M_{L}(y)=\det M_{R}(y)=y$, so that

$\det M(C)=\prod_{i=2}^{m-1}y_{i}.$ (39)

In other words, we have the identity

$F_{{\rm no},{\rm no}}F_{{\rm yes},{\rm yes}}=F_{{\rm no},{\rm yes}}F_{{\rm
yes},{\rm no}}+\prod_{i}y_{i}.$ (40)

Motivated in part by this identity, we will define $u$-variables for every
curve,

$u_{C}=\frac{F(C)_{{\rm no},{\rm yes}}\,F(C)_{{\rm yes},{\rm no}}}{F(C)_{{\rm
no},{\rm no}}\,F(C)_{{\rm yes},{\rm
yes}}}=\frac{M(C)_{12}M(C)_{21}}{M(C)_{11}M(C)_{22}}.$ (41)

These $u_{C}$ variables are most interesting to us in the region $y_{i}\geq
0$. Equation (40) implies that $0\leq u_{C}\leq 1$ in this region. They vastly
generalise the $u$-variables defined and studied in brown2009 ;
Arkani_Hamed_2021 .

We now define the headlight functions. We define them to capture the
asymptotic behaviour of the $u$-variables when thought of as functions of the
${\bf y}$ variables. We define

$\alpha_{C}=-{\rm Trop\ }u_{C}.$ (42)

where ${\rm Trop\ }u_{C}$ is the so-called _tropicalization_ of $u_{C}$.

The idea of tropicalization is to look at how functions behave asymptotically
in ${\bf y}$-space. To see how this works, parameterise the $y_{i}\geq 0$
region by writing $y_{i}=\exp t_{i}$, where the $t_{i}$ are real variables.
Then, as the $t_{i}$ become large, ${\rm Trop\ }u_{C}$ is defined such that

$u_{C}({\bf t})\rightarrow\exp\left({\rm Trop\ }u_{C}\right).$ (43)

For example, consider a simple polynomial,
$P(y_{1},y_{2})=1+y_{2}+y_{1}y_{2}=1+e^{t_{2}}+e^{t_{1}+t_{2}}$. As we go to
infinity in ${\bf t}=(t_{1},t_{2})$ in different directions, different
monomials in $P$ will dominate. In fact, we can write, as we go to infinity in
${\bf t}$,

$P\rightarrow\exp\max(0,t_{2},t_{1}+t_{2}),$ (44)

and so ${\rm Trop\ }(P)=\max(0,t_{2},t_{1}+t_{2})$. If we have a product of
polynomials, $F=\prod_{a}P_{a}^{c_{a}}$, then as we go to infinity in ${\bf
t}$ we have $F\to e^{{\rm Trop(F)}}$, where ${\rm Trop\ }F=\sum c_{a}{\rm
Trop\ }(P_{a})$.

Returning to headlight functions, our definition can also be written as

$\alpha_{C}={\rm Trop\ }(M(C)_{11})+{\rm Trop\ }(M(C)_{22})-{\rm Trop\
}(M(C)_{12})-{\rm Trop\ }(M(C)_{21}).$ (45)

For example, consider again the $n=5$ tree amplitude. Take the curve $C$ from
Figure 4 (left). This curve has path $(1,L,x,R,y,R,4)$. So it has a matrix
(with $y_{23},y_{15}\equiv 1$)

$M(C)=M_{L}(1)M_{R}(x)M_{R}(y)=\left(\begin{array}[]{cc}1&1+y\\\
1&1+y+xy\end{array}\right).$ (46)

Using this matrix, we find that its $u$-variable is

$u_{C}=\frac{1+y}{1+y+xy},$ (47)

and so its headlight function is

$\alpha_{C}={\rm max}(0,t_{y},t_{x}+t_{y})-{\rm max}(0,t_{y}).$ (48)

Amazingly, this function satisfies the key property of the headlight
functions: $\alpha_{C}$ vanishes on every g-vector, except for its own
g-vector, ${\bf g}_{C}=(1,0)$.

###  1.6 Back to the Amplitude!

We have now formulated how to compute all-order amplitudes in Tr$\Phi^{3}$
theory as a counting problem. The final expression for the integrated
amplitude at any order of the topological expansion associated with a surface
${\cal S}$ is given as

$\displaystyle{\cal A}=\int d^{E}t\,{\cal K}(\alpha)\left(\frac{\pi^{L}}{{\cal
U}(\alpha)}\right)^{\frac{D}{2}}{\rm exp}\left(\frac{{\cal F}(\alpha)}{{\cal
U}(\alpha)}\right),$ (49)

where ${\cal F}(\alpha),{\cal U}(\alpha)$ are homogeneous polynomials in the
$\alpha_{C}$’s, ${\cal K}(\alpha)$ is the Mirzakhani kernel that mods out by
the mapping-class-group, and crucially, each $\alpha_{C}$ is determined
entirely by the path of its curve, using a simple counting problem on the
curve. The presence of ${\cal K}$ ensures that only a finite number of
$\alpha_{C}$’s ever appear in our computations, which makes the formula easy
to apply. There is no trace of the idea of ‘summing over all spacetime
processes’ in this formula. Instead, small combinatorial problems attached to
the curves on a fatgraph, treated completely independently of each other,
magically combine to produce local and unitary physics, pulled out of the
platonic thin air of combinatorial geometry.

Our goal in the rest of this paper is to describe these ideas systematically.
Our focus in here will exclusively be on simply presenting the formulas for
the amplitudes. This presentation will be fully self-contained, so that the
interested reader will be fully equipped to find the curve integrals for the
$\text{Tr}\phi^{3}$ theory at any order in the topological expansion. The
methods can be applied at any order in the topological expansion, but there
are a number of novelties that need to be digested. We illustrate these
subtleties one at a time, as we progress from tree level examples through to
one and two loops, after which no new phenomena occur. We begin at tree-level
to illustrate the basic ideas. At one-loop single-trace, we show how to deal
with _spiralling_ curves. Then, as we have seen above, double-trace amplitudes
at 1-loop expose the first example of the infinities associated with the
mapping class group. Finally, we study the leading $1/N$ correction to single-
trace at 2-loops—the genus one amplitude—to show how to deal with a non-
abelian mapping class group. This non-abelian example illustrates the
generality and usefulness of the Mirzakhani trick.

In all cases discussed in this paper we will use use the smallest example
amplitudes possible to illustrate the new conceptual points as they arise. The
next paper in this series will give a similarly detailed set of formulae for
amplitudes for any number of particles, $n$. In this sense this first pair of
papers can be thought of as a “user guide” for the formalism. A systematic
accounting of the conceptual framework underlying these formulae, together
with an exposition of the panoply of related developments, will be given in
the later papers of this series.

##  2 The partial amplitude expansion

Consider a single massive scalar field with two indices in the fundamental and
anti-fundamental representations of $\mathrm{SU}(N)$,
$\phi=\phi^{I}_{J}\,t_{I}t^{J}$, and with a single cubic interaction,

$\displaystyle\mathcal{L}_{int}=g\mathrm{Tr}\left[\phi^{3}\right]=g\,\phi_{I}^{J}\phi_{J}^{K}\phi_{K}^{I}.$
(50)

The trace of the identity is $\mathrm{Tr}(1)=\delta_{I}^{I}=N$. The propagator
for the field $\phi$ can be drawn as a double line and the Feynman diagrams
are _fatgraphs_ with cubic vertices. The Feynman rules follow from (50). To
compute the $n$ point amplitude, $\mathcal{A}_{n}$, fix $n$ external particles
with momenta $k_{i}^{\mu}$ and colour polarisations $t_{i}^{IJ}$. A fatgraph
$\Gamma$ with $V$ cubic vertices contributes to the amplitude as

$\displaystyle(ig)^{V}\,C_{\Gamma}\,\mathrm{Val}(\Gamma),$ (51)

where $C_{\Gamma}$ is the tensorial contraction of the polarisations
$t_{i}^{IJ}$ according to $\Gamma$. The kinematical part is given by an
integral of the form

$\displaystyle\mathrm{Val}(\Gamma)=\int\prod_{i=1}^{L}d^{D}\ell_{i}\prod_{\mathrm{edges}\leavevmode\nobreak\
e}\frac{1}{P_{e}^{2}+m^{2}},$ (52)

for some assignment of loop momenta to the graph. Each momentum $P_{e}^{\mu}$
is linear in the external momenta $k_{i}^{\mu}$ and in the loop momentum
variables $\ell_{i}^{\mu}$. To find $P_{e}^{\mu}$, the edges of $\Gamma$ need
to be oriented, so that momentum conservation can be imposed at each cubic
vertex.

The colour factors $C_{\Gamma}$ organise the amplitude $\mathcal{A}_{n}$ into
partial amplitudes. This is because $C_{\Gamma}$ depends only on the topology
of $\Gamma$ regarded as a surface, and forgets about the graph. Write
$S(\Gamma)$ for the surface obtained from the fatgraph $\Gamma$ by
‘forgetting’ the graph. Two fatgraphs $\Gamma_{1},\Gamma_{2}$ share the same
colour factor, $C_{\Sigma}$, if they correspond to the same marked surface,
$\Sigma=S(\Gamma_{1})=S(\Gamma_{2})$. The amplitude can therefore be expressed
as

$\displaystyle\mathcal{A}_{n}=\sum_{L=0}^{\infty}(ig)^{n-2+2L}\sum_{\begin{subarray}{c}\Sigma\leavevmode\nobreak\
\mathrm{s.t.}\\\ h+2g=L+1\end{subarray}}C_{\Sigma}{\cal A}_{\Sigma},$ (53)

where we sum over marked bordered surfaces $\Sigma$ having $n$ marked points
on the boundary. At loop order $L$, this second sum is over all surfaces
$\Sigma$ with $h$ boundary components and genus $g$, subject to the Euler
characteristic constraint: $h+2g=L+1$. The partial amplitudes appearing in
(53) are

$\displaystyle{\cal A}_{\Sigma}=\sum_{\begin{subarray}{c}\Gamma\\\
S(\Gamma)=\Sigma\end{subarray}}\mathrm{Val}(\Gamma).$ (54)

Examples of some ribbon graphs $\Gamma$ and their corresponding surfaces are
shown in Figure 10.

Figure 10: Feynman graphs $\Gamma$ and the surfaces $S(\Gamma)$ that label
their colour factors.

Our aim is to evaluate ${\cal A}_{\Sigma}$. It is conventional to compute
$\mathrm{Val}(\Gamma)$ using _Schwinger parameters_. Schwinger parameters are
introduced via the identity

$\displaystyle\frac{1}{P^{2}+m^{2}}=\int_{0}^{\infty}d\alpha\,e^{-\alpha(P^{2}+m^{2})}.$
(55)

The integration in $\ell_{i}^{\mu}$ loop variables then becomes a Gaussian
integral, and the result can be written as

$\displaystyle\mathrm{Val}(\Gamma)=\int\limits_{\alpha_{i}\geq
0}d^{E}\alpha\,\left(\frac{2\pi}{{\cal
U}_{\Gamma}}\right)^{\frac{D}{2}}\exp\left(\frac{\mathcal{F}_{\Gamma}}{\mathcal{U}_{\Gamma}}-m^{2}\sum_{i}\alpha_{i}\right),$
(56)

where $\mathcal{U}_{\Gamma}$ and $\mathcal{F}_{\Gamma}$ are the Symanzik
polynomials of $\Gamma$. The Symanzik polynomials depend on $\Gamma$ regarded
as a graph (i.e. forgetting that it is a surface). The first Symanzik
polynomial is given by

$\displaystyle\mathcal{U}_{\Gamma}=\sum_{T}\prod_{e\not\in T}\alpha_{e},$ (57)

where the sum is over all spanning trees, $T$, of $\Gamma$. The second
Symanzik polynomial is given by a sum over all spanning 2-forests,
$(T_{1},T_{2})$, which cut $\Gamma$ into two tree graphs:

$\displaystyle\mathcal{F}_{\Gamma}=-\sum_{(T_{1},T_{2})}\left(\prod_{e\not\in
T_{1}\cup T_{2}}\alpha_{e}\right)\left(\sum_{e\not\in T_{1}\cup
T_{2}}P_{e}\right)^{2},$ (58)

where $P_{e}^{\mu}$ is the momentum of the edge $e$. It can be shown that
$\mathcal{F}_{\Gamma}$ depends only on the external momenta, and not on the
loop momentum variables.

The partial amplitudes ${\cal A}_{\Sigma}$ are given by sums over integrals of
this form, as in (54). But it is the purpose of this paper to show how ${\cal
A}_{\Sigma}$ can be written more compactly as a _single_ Symanzik-like
integral. It does not work to naively sum the integrands of
$\mathrm{Val}(\Gamma)$ for different Feynman diagrams $\Gamma$. One problem is
that there is no conventional way to relate the loop momentum variables for
different Feynman graphs. We will see how this is solved by basic facts from
surface geometry. Moreover, a simple counting problem associated to surfaces
will allow us to define tropical functions we call _headlight functions_.
These simple functions allow us to evaluate the full partial amplitude without
enumerating the Feynman diagrams.

##  3 Momenta and curves

Curves on fatgraphs are the key building block for our formulation of
amplitudes. In this section we show how a fatgraph can be used to assign
momenta to its curves. This momentum assignment solves the problem of finding
a consistent choice of momentum variables for all Feynman diagrams
contributing to an amplitude. This generalizes the _dual momentum variables_
that can be used for planar amplitudes.

###  3.1 Mountainscapes

A _curve_ is a path on the fatgraph that enters from an external line, passes
through the fatgraph without self-intersections, and exits on an external
line. It is sometimes useful to separately consider _closed curves_ , which
are paths on the fatgraph that form a closed loop.

Curves are important because they define _triangulations_ of fatgraphs. A
triangulation is a maximal collection of pairwise non-intersecting curves. The
key point is that each triangulation of $\Gamma$ corresponds, by graph
duality, to some fatgraph $\Gamma^{\prime}$. These fatgraphs $\Gamma^{\prime}$
all have the same colour factor and so contribute, as Feynman diagrams, to the
same amplitude.222There is also a duality between triangulations of a fatgraph
$\Gamma$, and triangulations of the surface $S(\Gamma)$. Defining this
requires some care and is not needed for the results here. The methods in this
paper can be used to automatically find all the triangulations of $\Gamma$
without having to list them, using only the data of the curves on $\Gamma$.

Figure 11: A triangulation of a fatgraph is a maximal set of curves that cuts
the fatgraph into cubic vertices.

A curve $C$ on $\Gamma$ is completely specified by reading off the order in
which $C$ passes through the edges of $\Gamma$. It is also helpful to record
the _left and right turns_ made by the curve. We present this information
using _mountainscape diagrams_. The vertices of a mountainscape are labelled
by the edges of $\Gamma$. Each left turn made by $C$ is recorded with a left
arrow (and a step up), while each right turn is written with a right arrow
(and a step down):

$i$$j$ |  |  $i$$k$
---|---|---
_Turn left from $i$ to $j$._ |  | _Turn right from $i$ to $k$._

For example, the curve in Figure 12(a) passes through the edges
$1,x,w,z,y,w,4$. Its mountainscape is shown in Figure 12(b). If we traverse
$C$ in the opposite direction we obtain the left-right reflection of this
mountainscape. We regard these as being the _same_ mountainscape. For brevity,
it is convenient to write mountainscapes as a _word_ , writing ‘L’ for a left
turn, and ‘R’ for a right turn. The mountainscape in Figure 12(b) is given by
the word

$C=1LxRwRzRyLwL4.$ (59)

Figure 12: A curve on a fatgraph (left) and its mountainscape diagram (right).

###  3.2 Intersections

Mountainscape diagrams encode the intersections of curves. In fact, it is not
necessary to know the whole fatgraph in order to determine if two curves
intersect: their mountainscapes alone have all the data needed.

For example, consider Figure 13. The two curves in Figure 13(a) are

$\displaystyle C=x_{2}RyLx_{4}\qquad\text{and}\qquad
C^{\prime}=x_{1}LyRx_{3}.$ (60)

These two mountainscapes _overlap_ on the edge $y$, which they share in
common. For $C$, $y$ is a _peak_ , whereas for $C^{\prime}$, $y$ is a
_valley_. This is equivalent to the information that $C$ and $C^{\prime}$
_intersect_ at $y$. By contrast, the two curves in Figure 13(b) are

$\displaystyle C=x_{1}LyLx_{4}\qquad\text{and}\qquad
C^{\prime}=x_{2}RyRx_{3}.$ (61)

These curves also overlap on the edge $y$. But $y$ does not appear in these
curves as a peak or valley. This is equivalent to the information that $C$ and
$C^{\prime}$ do not intersect.

Figure 13: A pair of intersecting curves (left), and a pair of non-
intersecting curves (right).

In general, if two curves $C$ and $C^{\prime}$ intersect, their paths must
overlap near the intersection. So suppose that $C$ and $C^{\prime}$ share some
sub-path, $W$, in common. Then $C$ and $C^{\prime}$ _intersect along $W$_ only
if $W$ is a peak for one and a valley for the other. In other words, $C$ and
$C^{\prime}$ intersect at $W$ if they have the form

$\displaystyle C=W_{1}RWLW_{2}\qquad\text{and}\qquad
C^{\prime}=W_{3}LWRW_{4},$ (62)

or

$\displaystyle C=W_{1}LWRW_{2}\qquad\text{and}\qquad
C^{\prime}=W_{3}RWLW_{4},$ (63)

for some sub-paths $W_{1},W_{2},W_{3},W_{4}$. The left/right turns are very
important. If the two curves have the form, say,

$\displaystyle C=W_{1}RWRW_{2}\qquad\text{and}\qquad
C^{\prime}=W_{3}LWLW_{4},$ (64)

then they _do not intersect_ at $W$.

Using this general rule, we can find triangulations of fatgraphs using only
the data of the curves.

For every fatgraph $\Gamma$, there are two special triangulations. Suppose
that $\Gamma$ has edges $e_{i}$, $i=1,\ldots,E$. Let $C_{i}$ be the curve
that, starting from $e_{i}$, turns right in both directions away from $e_{i}$.
Then

$\displaystyle C_{i}=\cdots
LeLe^{\prime}Le_{i}Re^{\prime\prime}Re^{\prime\prime\prime}R\cdots.$ (65)

$C_{i}$ has exactly one peak, which is at $e_{i}$. The intersection rule,
(62), shows that no pair of such curves $C_{i},C_{j}$ ($i\neq j$) intersect.
So the $C_{i}$ give $E$ nonintersecting curves, and these form a
triangulation, $T$. We can also consider the curves

$\displaystyle\tilde{C}_{i}=\cdots
ReRe^{\prime}Re_{i}Le^{\prime\prime}Le^{\prime\prime\prime}L\cdots,$ (66)

that turn left going in both directions away from $e_{i}$. These
$\tilde{C}_{i}$ each have exactly one valley, at $e_{i}$, and so they are
mutually nonintersecting. Together, they give another triangulation of the
fatgraph, $\tilde{T}$. An example of these special triangulations is given in
Figure 14.

Figure 14: The two special triangulations of a fatgraph, $T$ and $\tilde{T}$,
are defined by curves with exactly one peak (left) and curves with exactly one
valley (right).

###  3.3 Momentum Assignments

The edges of a fatgraph $\Gamma$ are naturally decorated with momenta, induced
by the _external momenta_ of the graph. Let $\Gamma$ have $n$ external momenta
$p_{1}^{\mu},\ldots,p_{n}^{\mu}$, directed _into_ the graph (say). By imposing
momentum conservation at each cubic vertex, we obtain a momentum $p_{e}^{\mu}$
for every edge. If $\Gamma$ has loops (i.e. $E>n-3$), then there is a freedom
in the definition of the $p_{e}^{\mu}$ that we parametrise by some $L$ _loop
momentum variables_ , $\ell_{1}^{\mu},\ldots,\ell_{L}^{\mu}$. This is the
standard rule for assigning momenta to a fatgraph, $\Gamma$.

To go further, we now introduce a way to also assign a momentum to every
_curve_ on $\Gamma$. For a curve with an orientation, $\overrightarrow{C}$,
will assign a momentum $P_{\overrightarrow{C}}^{\mu}$. This momentum
assignment should satisfy two basic rules. If $\overleftarrow{C}$ is the curve
with reversed orientation (Figure 15(a)), then

$\displaystyle P_{\overleftarrow{C}}^{\mu}=-P_{\overrightarrow{C}}^{\mu}.$
(67)

And if three curves,
$\overrightarrow{C}_{1},\overrightarrow{C}_{2},\overrightarrow{C}_{3}$, cut
out a cubic vertex (Figure 15(a)), then we impose momentum conservation at
that vertex:

$\displaystyle
P_{\overrightarrow{C}_{1}}^{\mu}+P_{\overrightarrow{C}_{2}}^{\mu}+P_{\overrightarrow{C}_{3}}^{\mu}=0.$
(68)

(a)

(b)

Figure 15: (a) Reversing a curve reverses its momentum assignment. (b) The
momenta of three curves that cut out a cubic vertex satisfy momentum
conservation.

The solution to satisfying both (67) and (68) is very simple, if we start with
the momenta $p_{e}^{\mu}$ assigned to the edges of $\Gamma$. Suppose
$\overrightarrow{C}$ enters $\Gamma$ via the external line $i$. Then assign
this curve

$P_{\overrightarrow{C}}^{\mu}=p_{i}^{\mu}+\sum_{\text{right
turns}}p_{\text{left}}^{\mu},$ (69)

where $p_{\text{left}}^{\mu}$ is the momentum of the edge incident on $C$ from
the left, at the vertex where $\overrightarrow{C}$ makes a right turn. The
momentum assignment, (69), can easily be checked to satisfy (67) and (68).

For example, take the fatgraph in Figure 16. The assignment of momenta to the
edges of the graph is shown in the Figure. The curve $C_{0}$ in Figure 16
enters the graph with momentum $p^{\mu}$. Then it turns left, traverses an
edge, and then turns right. At the right turn, the momentum incident on the
curve from the left is $-p-\ell^{\mu}$. So the momentum assignment of this
curve is

$P_{\overrightarrow{C}_{0}}^{\mu}=-\ell^{\mu}.$ (70)

The curve $C_{1}$ in Figure 16 has two right turns. At its first right turn,
it gains momentum $p^{\mu}$. At its second right turn, it gains momentum
$-p^{\mu}-\ell^{\mu}$. So the momentum assignment of this curve is

$P_{\overrightarrow{C}_{1}}^{\mu}=p^{\mu}-\ell^{\mu}.$ (71)

Figure 16: An assignment of momenta to the edges of a fatgraph (left) induces
as assignment of momenta to curves on the fatgraph (right).

For _any_ triangulation, $T$, the above rules assign a momentum to every curve
in the triangulation. By construction, these momenta satisfy momentum
conservation at each of the cubic vertices cut out by $T$. The upshot of this
is that we can _re-use_ the same loop momentum variables,
$\ell_{1},...,\ell_{L}$, when assigning momenta to _any_ triangulation of
$\Gamma$. This simple idea makes it possible to do the loop integrations for
all diagrams at once, instead of one Feynman diagram at a time, which is a key
step towards our formulas for amplitudes. This idea also makes it possible to
compute well-defined _loop integrands_ , even beyond the planar limit.

####  3.3.1 Aside on Homology

There is a more formal way to understand the assignment of momenta to curves:
these momentum assignments are an avatar of the homology of the fatgraph. Let
$H_{1}(\Gamma,\Gamma_{\infty})$ be the homology of $\Gamma$ (regarded as a
surface) relative to the _ends_ of the external edges of the fatgraph,
$\Gamma_{\infty}$. An oriented curve $\overrightarrow{C}$ represents a class
$[\overrightarrow{C}]\in H_{1}(\Gamma,\Gamma_{0})$, and

$\displaystyle[\overrightarrow{C}]+[\overleftarrow{C}]=0$ (72)

in homology. Moreover, if three curves cut out a cubic vertex, their classes
satisfy

$\displaystyle[\overrightarrow{C}_{1}]+[\overrightarrow{C}_{2}]+[\overrightarrow{C}_{3}]=0$
(73)

in homology. This means that a momentum assignment to curves satisfying (67)
and (68) defines a linear map

$\displaystyle P:H_{1}(\Gamma,\Gamma_{\infty})\rightarrow\mathbb{R}^{1,D-1},$
(74)

from $H_{1}(\Gamma,\Gamma_{\infty})$ to Minkowski space.

###  3.4 Spirals

The colour factor $C_{\Gamma}$ is a product of trace factors
$\text{tr}(t_{1}...t_{k})$ formed from the colour polarisations
${t_{i}}_{I}^{J}$. If $\Gamma$ has a closed colour loop, this boundary
contributes $\text{tr}(1)=N$ to the colour factor. For such a fatgraph, there
are curves that infinitely spiral around this closed loop. These spiral curves
can be treated just the same as all the other curves. In fact, the momentum
assignment for spiral curves follows again from the same rule above, (69).

Suppose that $\Gamma$ has a closed colour loop, $\beta$. Suppose that there
are some $m\geq 1$ edges incident on the loop, as in Figure 17. By momentum
conservation, the momenta of these edges, $p_{1},\ldots,p_{m}$, must sum up to
zero: $\sum_{i=1}^{m}p_{i}=0$. This ensures that (69) assigns a well-defined
momentum to a curve that spirals around this boundary, because the
contributions from the $p_{i}^{\mu}$ vanish after every complete revolution.

Figure 17: The momenta incident on a closed loop in a fatgraph sum to zero.
This ensures that the assignment of momentum to a spiral curve is well
defined.

##  4 The Feynman Fan

For a fatgraph $\Gamma$ with $E$ edges ($e_{1},\ldots,e_{E}$), consider the
$E$-dimensional vector space, $V$, generated by some vectors, ${\bf
e}_{1},\ldots,{\bf e}_{E}$. To every curve $C$ on the fatgraph, we can assign
a _$g$ -vector_, ${\bf g}_{C}\in V$. These simple integer vectors contain all
the key information about the curves on $\Gamma$. Moreover, the $g$-vectors
define a _fan_ in $V$ that we can use to rediscover the Feynman diagram
expansion for the amplitude.

To define the $g$-vector of a curve, $C$, consider the _peaks_ and _valleys_
of its mountainscape. $C$ has a _peak at $e_{i}$_ if it contains

$\displaystyle\cdots Le_{i}R\cdots.$ (75)

$C$ has a _valley at $i$_ if it contains

$\displaystyle\cdots Re_{i}L\cdots.$ (76)

Let $a^{i}_{C}$ be the number of times that $C$ has a peak at $e_{i}$, and let
$b^{i}_{C}$ be the number of times that $C$ has a valley at $e_{i}$. This
information about the peaks and valleys is recorded by the _$g$ -vector of
$C$_,

$\displaystyle{\bf g}_{C}\equiv\sum_{i=1}^{E}g_{C}^{i}\,{\bf
e}_{i},\qquad\text{where}\leavevmode\nobreak\ g_{C}^{i}=a^{i}_{C}-b^{i}_{C}.$
(77)

Each curve has a distinct $g$-vector. The converse is even more surprising: a
curve is completely specified by its $g$-vector.

For example, consider the curve, $C_{i}$, in the triangulation $T_{\Gamma}$,
which has only one peak, at $e_{i}$. The $g$-vector for $C_{i}$ is then

$\displaystyle{\bf g}_{C_{i}}={\bf e}_{i}.$ (78)

So the $g$-vectors of this triangulation $T_{\Gamma}$ span the positive
orthant of $V$.

###  4.1 Example: tree level at 5-points

Figure 18: The five curves on the $n=5$ tree fatgraph.

Take the comb graph $\Gamma$, with edges labelled by variables $x$ and $y$, as
in Figure 18. The five curves on $\Gamma$ are

$\displaystyle C_{13}=1LxR3,\qquad C_{14}=1LxLyR4,\qquad C_{24}=2RxLyR4,$ (79)
$\displaystyle C_{25}=2RxLyL5,\qquad C_{35}=3RyL5.$ (80)

Counting the peaks and valleys of these mountainscapes gives

$\displaystyle{\bf g}_{13}=\begin{bmatrix}1\\\
0\end{bmatrix},\leavevmode\nobreak\ {\bf g}_{14}=\begin{bmatrix}0\\\
1\end{bmatrix},\leavevmode\nobreak\ {\bf g}_{24}=\begin{bmatrix}-1\\\
1\end{bmatrix},\leavevmode\nobreak\ {\bf g}_{25}=\begin{bmatrix}-1\\\
0\end{bmatrix},\leavevmode\nobreak\ {\bf g}_{35}=\begin{bmatrix}0\\\
-1\end{bmatrix}.$ (81)

These $g$-vectors are shown in Figure 19. They define a _fan_ in the
2-dimensional vector space. The top-dimensional cones of this fan are spanned
by pairs of $g$-vectors, such as ${\bf g}_{14}$ and ${\bf g}_{24}$, whose
corresponding curves define triangulations.

Figure 19: The Feynman fan for $n=5$ tree level.

###  4.2 The Fan

The $g$-vectors of all the curves on $\Gamma$ together define an integer fan
$\mathfrak{F}\subset V$. To define a fan, we must specify all of its _cones_.
We adopt the rule that two or more $g$-vectors span a cone in $\mathfrak{F}$
if and only if their curves do not intersect. The main properties of
$\mathfrak{F}$ are:

  1. 1.

It is a polyhedral fan that is dense $V$.333A fan is _polyhedral_ if the
intersection of any two cones is itself, if nonempty, a cone in the fan, and
the faces of each cone are cones in the fan. A fan is _dense_ if any integer
vector is contained in some cone of the fan. In general, irrational vectors
are not always contained in our fans, but this will not play any role in this
paper.

  2. 2.

Its top dimensional cones are in 1:1 correspondence with triangulations.

  3. 3.

The $g$-vectors of each top-dimensional cone span a parallelepiped of unit
volume.

Since the top-dimensional cones of $\mathfrak{F}$ correspond to
triangulations, and hence to Feynman diagrams, we call $\mathfrak{F}$ the
_Feynman fan_ , or sometimes, the _$g$ -vector fan_.

The property that $\mathfrak{F}$ is _polyhedral and dense_ means that every
rational vector ${\bf g}\in V$ is contained in _some_ cone in the fan. This
implies that every such ${\bf g}$ can be _uniquely_ written as a positive
linear combination of $g$-vectors. In Section 5, we solve the problem of how
to do this expansion explicitly.

###  4.3 The Mapping Class Group

Figure 20: Triangulations (left) that are related to each other by the action
of of the MCG. These triangulations are all dual to the same Feynman diagram
(right).

The Feynman fan of a fat graph $\Gamma$ inherits from $\Gamma$ an action of a
discrete, finitely generated group called the _mapping class group_ , MCG. The
MCG of a fatgraph, $\Gamma$, is the group of homeomorphisms of $\Gamma$, up to
isotopy, that restrict to the identity on its boundaries. The action of MCG on
the fatgraph can be studied by considering its action on curves. Since we only
ever consider curves up to homotopy, a group element $\gamma\in\text{MCG}$
induces a map on curves

$\gamma:C\mapsto\gamma C.$ (82)

Since MCG acts via homeomorphisms, it does not affect curve intersections and
non-intersections. If $C$ and $C^{\prime}$ are two non-intersecting curves,
then $\gamma C$ and $\gamma C^{\prime}$ are likewise non-intersecting.
Similarly, if $C,C^{\prime}$ intersect, so do $\gamma C$ and $\gamma
C^{\prime}$. This means that if some curves, $C_{1},\ldots,C_{E}$, form a
triangulation, so do their images under MCG. Moreover, if the triangulation
$\\{C_{1},\ldots,C_{E}\\}$ is dual to a fatgraph $\Gamma^{\prime}$, then each
image $\\{\gamma C_{1},\ldots,\gamma C_{E}\\}$ is _also_ dual to the same
fatgraph, $\Gamma^{\prime}$.

For example, take the 2-point non-planar fatgraph $\Gamma$ in Figure 20. The
MCG acts on $\Gamma$ by _Dehn twists_ that increase the number of times a
curve winds around the fatgraph. All triangulations of $\Gamma$ are related to
each other by the MCG and they are all dual to the same fatgraph (right in
Figure 20).

In general, if $\Gamma$ has loop number $L$, then MCG has a presentation with
$L$ generators penner2012decorated . These can be identified with Dehn twists
around annuli in the fatgraph.

The MCG action on curves induces a piecewise linear action on the vector
space, $V$,

$\displaystyle\gamma:{\bf g}_{C}\mapsto{\bf g}_{\gamma C}.$ (83)

It follows from the above properties of the MCG action on curves that the
action of MCG on $V$ leaves the fan $\mathfrak{F}$ invariant (if we forget the
labels of the rays). Furthermore, two top-dimensional cones of the fan
correspond to the same Feynman diagram if and only if they are related by the
MCG action.

####  4.3.1 Aside on automorphisms

Figure 21: A fatgraph with $|\text{Aut}(\Gamma)|=3$. Cyclic permutations of
the edges leave it unchanged.

There is another discrete group that acts on the Feynman fan: the group of
graph automorphisms, $\text{Aut}(\Gamma)$. The elements of
$\text{Aut}(\Gamma)$ are permutations of the labels of the edges of $\Gamma$.
A permutation is an _automorphism_ if it leaves the list of fat vertices of
$\Gamma$ unchanged (including the vertex orientations). Each fat vertex can be
described by a triple of edge labels with a cyclic orientation, $(ijk)$.

$\text{Aut}(\Gamma)$ has a linear action on $V$ given by permuting the basis
vectors ${\bf e}_{1},\ldots,{\bf e}_{E}$. The action of $\text{Aut}(\Gamma)$
leaves the fan invariant (again if we forget the labels of the rays).

An example of a fatgraph with nontrivial automorphisms is Figure 21. In this
example, cyclic permutations of the 3 edges preserve the fatgraph. Most
fatgraphs that we will consider have trivial automorphism groups, and so the
action of $\text{Aut}(\Gamma)$ will not play a big role in this paper.

###  4.4 Example: the non-planar 1-loop propagator

Figure 22: The infinite family of curves, $C_{n}$, for the non-planar one loop
propagator.

Take the 1-loop fatgraph $\Gamma$ in Figure 22, with edges labeled by
variables $x$ and $y$. Some of the curves on $\Gamma$, $C_{n}$, are shown in
the Figure. These curves are related to each other by the action of MCG, which
is generated by a Dehn twist, $\gamma$. With the labelling in Figure 22, the
action of $\gamma$ is

$\gamma:C_{n}\mapsto C_{n+1}.$ (84)

There are infinitely many such curves on the fatgraph.

The paths of the curves on $\Gamma$ are

$\displaystyle C_{n}$
$\displaystyle=1L(xLyR)^{n}xR2\qquad\text{for}\leavevmode\nobreak\ n\geq 0,$
(85) $\displaystyle C_{n}$
$\displaystyle=1Ry(RxLy)^{1+n}L2\qquad\text{for}\leavevmode\nobreak\ n<0,$
(86) $\displaystyle\Delta$ $\displaystyle=xLyR,$ (87)

where $\Delta$ is the closed loop. Note that the curves $C_{n}$ differ from
one another by multiples of the closed path $\Delta$. In this way, we can see
the MCG directly in terms of the mountainscapes of the curves.

Counting peaks and valleys in the mountainscapes, the $g$-vectors for these
curves are:

$\displaystyle{\bf g}_{n}$ $\displaystyle=\begin{bmatrix}-n+1\\\
n\end{bmatrix}\qquad\text{for}\leavevmode\nobreak\ n\geq 0,$ (88)
$\displaystyle{\bf g}_{n}$ $\displaystyle=\begin{bmatrix}n+1\\\
-n-2\end{bmatrix}\qquad\text{for}\leavevmode\nobreak\ n<0,$ (89)
$\displaystyle{\bf g}_{\Delta}$ $\displaystyle=\begin{bmatrix}-1\\\
1\end{bmatrix}.$ (90)

These $g$-vectors define the fan in Figure 19. There are infinitely many rays
of this fan. The action of MCG on curves lifts to a piecewise linear action on
the fan, generated by the action of the Dehn twist $\gamma$. $\gamma$ acts on
the fan as

$\displaystyle{\bf g}_{{n+1}}$ $\displaystyle={\bf g}_{n}+{\bf
g}_{\Delta},\qquad\text{for}\leavevmode\nobreak\ n\geq 0,$ (91)
$\displaystyle{\bf g}_{0}$ $\displaystyle={\bf g}_{{-1}}+(1,1),$ (92)
$\displaystyle{\bf g}_{{n+1}}$ $\displaystyle={\bf g}_{{n}}-{\bf
g}_{\Delta},\qquad\text{for}\leavevmode\nobreak\ n<-1.$ (93)

This is (trivially) an isomorphism of the fan.

Figure 23: The Feynman fan for the non-planar 1-loop propagator.

###  4.5 The Delta plane

A _closed curve_ is a curve $\Gamma$ that forms a loop. For a closed curve
$\Delta$, consider the series of left and right turns that it makes. We can
record this series of turns as a _cyclic word_ , like $W_{\Delta}=(RRLRL)$.
Whenever $RL$ appears in $W_{\Delta}$ it corresponds to a _valley_ in the
mountainscape, which happens where the curve switches from turning right to
turning left. Likewise, $LR$ corresponds to a _peak_. If the cyclic word
$W_{C}$ has $n$ occurrences of ‘$RL$’, it must also have exactly $n$
occurrences of ‘$LR$’. For example, the cyclic word

$\displaystyle(RRLRLLLRRLL),$ (94)

switches from right-to-left 3 times, and from left-to-right 3 times.

In other words, the mountainscape for a closed curve has exactly as many peaks
as valleys. It follows that the $g$-vector, ${\bf g}_{\Delta}$, for any closed
curve $\Delta$ is normal to the vector ${\bf n}=(1,1,1,...,1)^{T}$. We call
the plane normal to ${\bf n}$ the _$\Delta$ plane_: $V_{\Delta}\subset V$.

For example, in the previous subsection, the closed curve $\Delta$ had
$g$-vector ${\bf g}_{\Delta}=(-1,1)$, which is normal to the vector $(1,1)$.

Finally, note that a closed curve that makes _only_ right-turns (resp. left-
turns) corresponds to a path around a loop boundary of $\Gamma$. These curves
have no peaks and no valleys. So these loop boundaries are assigned zero
$g$-vector. They are also assigned zero momentum (by the reasoning in Section
3.4).

###  4.6 Example: the planar 1-loop propagator

Figure 24: Curves on the bubble fatgraph.

Take the 1-loop bubble diagram, $\Gamma$, with edges $x$ and $y$, and external
edges $1$ and $2$, as in Figure 24. Consider the four curves,
$C_{1},C_{2},S_{1},S_{2}$, shown in the Figure. These have paths

$\displaystyle C_{1}$ $\displaystyle=1RxLyR1$ (95) $\displaystyle C_{2}$
$\displaystyle=2RyLxR2$ (96) $\displaystyle S_{1}^{\prime}$
$\displaystyle=1RxLyLxLyL\cdots$ (97) $\displaystyle S_{2}^{\prime}$
$\displaystyle=2RyLxLyLxL\cdots.$ (98)

The curves $S_{1}^{\prime},S_{2}^{\prime}$ end in anticlockwise spirals around
the closed loop boundary. There are also two curves, $S_{1}$ and $S_{2}$,
which spiral _clockwise_ into the puncture:

$\displaystyle S_{1}$ $\displaystyle=1LyRxRyR\cdots$ (99) $\displaystyle
S_{2}$ $\displaystyle=2LxRyRxR\cdots.$ (100)

Counting peaks and valleys, the $g$-vectors of these curves are

$\displaystyle{\bf g}_{C_{1}}=\begin{bmatrix}-1\\\
1\end{bmatrix},\leavevmode\nobreak\ {\bf g}_{S_{1}}=\begin{bmatrix}1\\\
0\end{bmatrix},\leavevmode\nobreak\ {\bf g}_{S_{2}}=\begin{bmatrix}0\\\
1\end{bmatrix},\leavevmode\nobreak\ {\bf g}_{C_{2}}=\begin{bmatrix}1\\\
-1\end{bmatrix},\leavevmode\nobreak\ {\bf
g}_{S_{1}^{\prime}}=\begin{bmatrix}0\\\ -1\end{bmatrix},\leavevmode\nobreak\
{\bf g}_{S_{2}^{\prime}}=\begin{bmatrix}-1\\\ 0\end{bmatrix}.$ (101)

These $g$-vectors give the fan in Figure 25. Notice that the $g$-vectors of
the curves $C_{1},C_{2}$ lie on the Delta plane: $x+y=0$.

Including the anticlockwise spirals would lead to us counting every Feynman
diagram twice. This is because the triangulation with $C_{1},S_{1}$ is dual to
the same diagram as the triangulation by $C_{1},S_{1}^{\prime}$, and so on. To
prevent overcounting, it makes sense to restrict to the part of the fan that
involves only $C_{1},S_{1}^{\prime},S_{2}^{\prime},$ and $C_{2}$. This part of
the fan is precisely the half space, $x+y\leq 0$, cut out by the Delta plane.

Figure 25: The Feynman Fan for the 1-loop planar propagator.

##  5 A Counting Problem For Curves

There is a natural counting problem associated to mountainscapes, and this
counting problem plays the central role in our amplitude computations.

For a mountainscape, $C$, the idea is to form subsets of $C$ by _filling up_
the mountainscape from the bottom. A subset is valid if it includes everything
_downhill_ of itself in the mountainscape.

For example, consider the curve in Figure 26,

$\displaystyle C=1R2L3.$ (102)

The valid subsets of $C$, shown in the Figure, are $2,1R2,2L3$, and $1R2L3$.
In other words, if $3$ is in the subset, then $2$ must also be included,
because it is downhill of (left of) $3$. Likewise, if $1$ is in the subset,
then $2$ must also be included, because 2 is downhill of (right of) $3$.

This information can be summarised using a generating function or _$F$
-polynomial_. Introduce variables $y_{i}$, $i=1,\ldots,E$, labelled by the
edges of $\Gamma$. Then the $F$-polynomial of a curve $C$ is

$\displaystyle F_{C}=1+\sum_{C^{\prime}\subset C}\,\prod_{i\in
C^{\prime}}y_{i},$ (103)

where the sum is over all valid (non-empty) subsets of $C$, including $C$
itself.

In the example, (102), we have four valid subsets, and the $F$-polynomial is

$\displaystyle F_{C}=1+y_{2}+y_{1}y_{2}+y_{2}y_{3}+y_{1}y_{2}y_{3}.$ (104)

Figure 26: Two examples of mountainscapes and their sub-mountainscapes.

###  5.1 Curve Matrices

Consider a curve $C$ that starts at any edge $e_{i}$ and ends at any edge
$e_{j}$. It is natural to decompose its $F$-polynomial as a sum of four terms,

$F_{C}=F_{--}+F_{-+}+F_{+-}+F_{++},$ (105)

where: $F_{--}$ counts subsets that exclude the first and last edges; $F_{-+}$
counts subsets that exclude the first edge and include the last edge; and so
on.

Now consider what happens if we _extend_ $C$ along one extra edge. Let
$C^{\prime}$ extend $C$ by adding a left turn before $i$:

$C^{\prime}=e_{k}LC,$ (106)

for some edge $e_{k}$. The $F$-polynomial of $C^{\prime}$ can be deduced using
(105). Terms that involve $y_{i}$ _must_ contain $y_{k}$, since $e_{k}$ is
_downhill_ of $e_{i}$ in the curve. So

$F_{C^{\prime}}=(1+y_{k})F_{--}+(1+y_{k})F_{-+}+y_{k}F_{+-}+y_{k}F_{++}.$
(107)

Similarly, if $C^{\prime\prime}$ is obtained from $C$ by adding a right turn
before $e_{i}$, then $C^{\prime\prime}=e_{l}RC$, for some edge $e_{l}$, and we
find that the new $F$-polynomial is

$F_{C^{\prime\prime}}=F_{--}+F_{-+}+(1+y_{l})F_{+-}+(1+y_{l})F_{++}.$ (108)

This equation follows because any term not containing $y_{i}$ _cannot_ contain
$y_{l}$, since $e_{i}$ is _downhill_ of $e_{l}$ in the curve.

Equations (107) and (108) can be used to compute the $F$-polynomial for any
curve. It simple to do implement this is by defining a _curve matrix_ , whose
entries are given by the decomposition, (105):

$M_{C}=\begin{bmatrix}F_{--}&F_{-+}\\\ F_{+-}&F_{++}\end{bmatrix}.$ (109)

The curve matrix $M_{C^{\prime}}$ is obtained from the curve matrix $M_{C}$
via the matrix version of (107):

$M_{C^{\prime}}=\begin{bmatrix}1&0\\\ y_{k}&y_{k}\end{bmatrix}M_{C}.$ (110)

The matrix multiplying $M_{C}$ in this equation represents what happens when
$C$ is extended by adding a left turn at the start. Similarly, the matrix
version of (108) is

$M_{C^{\prime\prime}}=\begin{bmatrix}1&1\\\ 0&y_{l}\end{bmatrix}M_{C},$ (111)

which represents what happens when $C$ is adding a right turn at the start.

It can be convenient to decompose the new matrices appearing in (110) and
(111) as a product,

$\begin{bmatrix}1&0\\\ y_{k}&y_{k}\end{bmatrix}=\begin{bmatrix}1&0\\\
0&y_{k}\end{bmatrix}\begin{bmatrix}1&0\\\
1&1\end{bmatrix},\qquad\begin{bmatrix}1&1\\\
0&y_{l}\end{bmatrix}=\begin{bmatrix}1&0\\\
0&y_{l}\end{bmatrix}\begin{bmatrix}1&1\\\ 0&1\end{bmatrix}.$ (112)

Then, for any curve, $C$, we can compute its curve matrix, $M_{C}$, directly
from the word specifying the curve. To do this, we just replace each turn and
edge with the associated matrix:

$L\rightarrow\begin{bmatrix}1&0\\\ 1&1\end{bmatrix},\qquad
R\rightarrow\begin{bmatrix}1&1\\\ 0&1\end{bmatrix},\qquad
e_{i}\rightarrow\begin{bmatrix}1&0\\\ 0&y_{i}\end{bmatrix}.$ (113)

Every curve matrix $M_{C}$ is then a product of these simple matrices.

For example, for the curve $C=1R2L3$ considered above, its matrix is

$M_{C}=\begin{bmatrix}1&0\\\ 0&y_{1}\end{bmatrix}\begin{bmatrix}1&1\\\
0&1\end{bmatrix}\begin{bmatrix}1&0\\\
0&y_{2}\end{bmatrix}\begin{bmatrix}1&0\\\
1&1\end{bmatrix}\begin{bmatrix}1&0\\\
0&y_{3}\end{bmatrix}=\begin{bmatrix}1+y_{2}&y_{2}y_{3}\\\
y_{1}y_{2}&y_{1}y_{2}y_{3}\end{bmatrix}.$ (114)

The sum of the entries of this curve matrix recovers the curve’s
$F$-polynomial, (104).

Curve matrices neatly factorise. If several curves all begin with the same
word, $W$, their words can be written as $C_{i}=WC_{i}^{\prime}$. Their
matrices are then $M_{C_{i}}=M_{W}M_{C_{i}^{\prime}}$, so that we only have to
compute $M_{W}$ once to determine all the $M_{C_{i}}$. Moreover, if we add
extra legs to a fatgraph $\Gamma$, to form a larger fatgraph,
$\Gamma^{\prime}$, the matrices $M_{C}$ for the larger fatgraph can be
obtained directly from the matrices for the smaller fatgraph. In practice,
this is very useful, and allows us to exploit the methods in this paper to
compute all-$n$ formulas for amplitudes. us_alln

Figure 27: Getting a new fatgraph.

###  5.2 Headlight Functions

It follows from the definition of $M_{C}$, as a product of the matrices in
(114), that

$\det M_{C}=\prod_{e\in C}y_{e}.$ (115)

Expanding the determinant, this gives

$1=\frac{F_{-+}F_{+-}}{F_{--}F_{++}}+\frac{\prod y_{e}}{F_{--}F_{++}}.$ (116)

Motivated in part by this identity, define the _$u$ -variable_ of a curve $C$
as the ratio

$u_{C}=\frac{F_{-+}F_{+-}}{F_{--}F_{++}}.$ (117)

These $u$-variables vastly generalise those studied in brown2009 ;
Arkani_Hamed_2021 , and (116) is a generalisation of the _$u$ -equations_
studied there.

The _headlight function_ of a curve $C$ is the _tropicalization_ of the
$u$-variable,

$\alpha_{C}=-\text{Trop\ }u_{C}.$ (118)

For a polynomial $F(y)$, its tropicalization captures the behaviour of $F$ at
large values of $y_{i}$. Parametrise the $y_{i}$ as $y_{i}=\exp t_{i}$. Then,
in the large $t$ limit,

$\displaystyle F(y)\rightarrow\exp\text{Trop\ }F(t).$ (119)

For example, if $F(y)=1+y_{1}+y_{1}y_{2}$, then $\text{Trop\
}F(t)=\max(0,t_{1},t_{1}+t_{2})$. In practice, $\text{Trop\ }F$ is obtained
from $F$ by replacing multiplication with addition, and replacing sums with
taking the maximum.

In terms of the matrix $M_{C}$, the headlight function is

$\displaystyle\alpha_{C}=\mathrm{Trop\ }M_{C}^{1,1}+\mathrm{Trop\
}M_{C}^{2,2}-\mathrm{Trop\ }M_{C}^{1,2}-\mathrm{Trop\ }M_{C}^{2,1}.$ (120)

Headlight functions satisfy the following remarkable property:

$\displaystyle\alpha_{C}({\bf g}_{D})=\left\\{\begin{matrix}1&\qquad\text{if
$C=D$}\\\ 0&\qquad\text{otherwise.}\end{matrix}\right.$ (121)

This implies that headlight functions can be used to express any vector ${\bf
g}\in V$ as a positive linear combination of the generators of a cone of the
Feynman fan, by writing

$\displaystyle{\bf g}=\sum_{C}\alpha_{C}({\bf g})\,{\bf g}_{C}.$ (122)

This expansion has a geometrical interpretation. Any integer vector ${\bf
g}\in V$ corresponds to some curve (or set of curves), $L$, possibly with
self-intersections. Any intersections in $L$ can be uncrossed on $\Gamma$
using the _skein relations_. Repeatedly applying skein relations, $L$ can be
decomposed on the surface into a unique set of non-self-intersecting curves,
and $\alpha_{C}(g)$ is the number of times the curve $C$ appears in this
decomposition.

###  5.3 Example: tree level at 5-points

The curves for the 5-points tree level amplitude were given in Section 4.1.
Their curve matrices, using the replacements (114), are

$\displaystyle C_{13}$ $\displaystyle=LxR$
$\displaystyle\longrightarrow\qquad\qquad$ $\displaystyle
M_{13}=\begin{bmatrix}1&1\\\ 1&1+x\end{bmatrix},$ (123) $\displaystyle C_{14}$
$\displaystyle=LxLyR$ $\displaystyle\longrightarrow\qquad\qquad$
$\displaystyle M_{14}=\begin{bmatrix}1&1\\\ 1+x&1+x+xy\end{bmatrix},$ (124)
$\displaystyle C_{24}$ $\displaystyle=RxLyR$
$\displaystyle\longrightarrow\qquad\qquad$ $\displaystyle
M_{24}=\begin{bmatrix}1+x&1+x+xy\\\ x&x(1+y)\end{bmatrix},$ (125)
$\displaystyle C_{25}$ $\displaystyle=RxLyL$
$\displaystyle\longrightarrow\qquad\qquad$ $\displaystyle
M_{25}=\begin{bmatrix}1+x+xy&xy\\\ x+xy&xy\end{bmatrix},$ (126) $\displaystyle
C_{35}$ $\displaystyle=RyL$ $\displaystyle\longrightarrow\qquad\qquad$
$\displaystyle M_{35}=\begin{bmatrix}1+y&y\\\ y&y\end{bmatrix}.$ (127)

Given these matrices, the headlight functions are

$\displaystyle\alpha_{13}$ $\displaystyle=\max(0,x),$ (128)
$\displaystyle\alpha_{14}$ $\displaystyle=-\max(0,x)+\max(0,x,x+y),$ (129)
$\displaystyle\alpha_{24}$ $\displaystyle=-\max(0,x,x+y)+\max(0,x)+\max(0,y),$
(130) $\displaystyle\alpha_{25}$ $\displaystyle=-x-\max(0,y)+\max(0,x,x+y),$
(131) $\displaystyle\alpha_{35}$ $\displaystyle=-y+\max(0,y).$ (132)

It can be verified that $\alpha_{ij}({\bf g}_{C})=1$ if $C=C_{ij}$, and that
otherwise $\alpha_{ij}({\bf g}_{C})=0$. For example, the values taken by
$\alpha_{24}$ are shown in Figure 28.

Figure 28: The Schwinger parameter $\alpha_{24}$ on the Feynman fan.

###  5.4 Example: the non-planar 1-loop propagator

The mountainscapes for the non-planar 1-loop propagator are given in Section
4.4. Using these, we can compute the headlight functions, and find:

$\displaystyle\alpha_{n}$ $\displaystyle=f_{n}-2f_{n-1}+f_{n-2},\qquad n\geq
0,$ (133) $\displaystyle\alpha_{n}$
$\displaystyle=g_{n}-2g_{n+1}+g_{n+2},\qquad n<0.$ (134)

where the tropical functions $f_{n}$ and $g_{n}$ are given by

$\displaystyle f_{n}$ $\displaystyle=\max(0,(n+1)x,(n+1)x+ny),$ for
$\displaystyle n\geq 0,$ (135) $\displaystyle g_{n}$
$\displaystyle=\max(0,-(n+1)x,-(n+1)x-ny),$ for $\displaystyle n\leq-1,$ (136)

with the following special cases:

$\displaystyle f_{-2}=0,\leavevmode\nobreak\ \leavevmode\nobreak\
f_{-1}=0,\leavevmode\nobreak\ \leavevmode\nobreak\
g_{1}=-2x-y,\leavevmode\nobreak\ \leavevmode\nobreak\ g_{0}=-x.$ (137)

A full derivation of these functions using the matrix method is given in
Appendix F.

It is easy to verify that these $\alpha_{n}$ satisfy the key property:

$\displaystyle\alpha_{n}({\bf
g}_{m})=\left\\{\begin{matrix}1&\leavevmode\nobreak\ \leavevmode\nobreak\
\text{if $n=m$}\\\ 0&\leavevmode\nobreak\ \leavevmode\nobreak\
\text{otherwise.}\end{matrix}\right.$ (138)

For example, take $n,m\geq 0$. Then we find

$\displaystyle f_{n}({\bf g}_{m})=\max(0,1+n-m),$ (139)

so that

$\displaystyle\alpha_{n}({\bf g}_{m})$
$\displaystyle=\max(0,1+n-m)+\max(0,-1+n-m)-2\max(0,n-m).$ (140)

This agrees with (138).

###  5.5 Spirals

Suppose $C$ is a curve that ends in a spiral around a loop boundary of
$\Gamma$. If $1,2,...,m$ are the edges around that boundary, $C$ has the form

$\displaystyle C=W1L2L...LmL1L2L...,$ (141)

for some subpath $W$. We can compute the transfer matrix for the infinite tail
at the right end of $C$. The path for one loop around the boundary is

$C_{\Delta}:=1L2L...LmL,$ (142)

and the matrix for this path is

$\displaystyle M_{\Delta}=\begin{bmatrix}1&\leavevmode\nobreak\ 0\\\
F-1&\leavevmode\nobreak\ y_{*}\end{bmatrix},$ (143)

where

$y_{*}=\prod_{i=1}^{m}y_{i},\qquad\text{and}\qquad
F=1+y_{1}+y_{1}y_{2}+...+y_{1}y_{2}...y_{m}.$ (144)

Now consider the powers, $M_{\Delta}^{n}$. If $y^{*}<1$, the limit as
$n\rightarrow\infty$ converges to

$\displaystyle
M_{\Delta}^{\infty}\equiv\lim_{n\rightarrow\infty}M^{n}=\begin{bmatrix}1&\leavevmode\nobreak\
0\\\ F_{\infty}-1&\leavevmode\nobreak\ 0\end{bmatrix},$ (145)

where

$\displaystyle
F_{\infty}=1+\frac{y_{1}+y_{1}y_{2}+...+y_{1}y_{2}...y_{m}}{1-y_{*}}.$ (146)

The matrix for the curve $C$ is then

$M_{C}=M_{W}M_{\Delta}^{\infty}.$ (147)

We can use the formula (145) when computing the matrix for any curve that ends
in a spiral: the spiralling part can be replaced by $M_{\Delta}^{\infty}$
directly. If the curve also _begins_ with a spiral, this spiral contributes a
factor of $(M_{\Delta}^{\infty})^{T}$ to the beginning of the matrix product.

###  5.6 Example: the planar 1-loop propagator

We can put these formulas to work for the planar 1-loop propagator. The curves
for this amplitude are given in Section 4.6. Evaluating the curve matrices
gives:

$\displaystyle M_{C_{1}}$
$\displaystyle=\begin{bmatrix}1+x&\leavevmode\nobreak\ 1+x+xy\\\
x&\leavevmode\nobreak\ x+xy\end{bmatrix},\leavevmode\nobreak\ $ $\displaystyle
M_{C_{2}}$ $\displaystyle=\begin{bmatrix}1+y&\leavevmode\nobreak\ 1+y+xy\\\
y&\leavevmode\nobreak\ y+xy\end{bmatrix},$ (148) $\displaystyle
M_{S_{1}^{\prime}}$
$\displaystyle=\begin{bmatrix}\frac{1+x}{1-xy}&\leavevmode\nobreak\
\leavevmode\nobreak\ 0\\\ \frac{x(1+y)}{1-xy}&\leavevmode\nobreak\
\leavevmode\nobreak\ 0\end{bmatrix},\leavevmode\nobreak\ $ $\displaystyle
M_{S_{2}^{\prime}}$
$\displaystyle=\begin{bmatrix}\frac{1+y}{1-xy}&\leavevmode\nobreak\
\leavevmode\nobreak\ 0\\\ \frac{y(1+x)}{1-xy}&\leavevmode\nobreak\
\leavevmode\nobreak\ 0\end{bmatrix}.$ (149)

The headlight functions are

$\displaystyle\alpha_{C_{1}}$
$\displaystyle=\max(0,x)+\max(0,y)-\max(0,x,x+y),$ (150)
$\displaystyle\alpha_{C_{2}}$
$\displaystyle=\max(0,x)+\max(0,y)-\max(0,y,x+y),$ (151)
$\displaystyle\alpha_{S_{1}^{\prime}}$ $\displaystyle=-x-\max(0,y)+\max(0,x),$
(152) $\displaystyle\alpha_{S_{2}^{\prime}}$
$\displaystyle=-y-\max(0,x)+\max(0,y).$ (153)

Once again, using the $g$-vectors from Section 4.6, we verify that these
functions satisfy

$\displaystyle\alpha_{C}({\bf
g}_{D})=\left\\{\begin{matrix}1&\leavevmode\nobreak\ \leavevmode\nobreak\
\text{if $C=D$}\\\ 0&\leavevmode\nobreak\ \leavevmode\nobreak\
\text{otherwise.}\end{matrix}\right.$ (154)

###  5.7 Example: the genus one 2-loop vacuum

Figure 29: The 2-loop vacuum graph with genus one.

We now introduce a more complicated example: the 2-loop vacuum amplitude at
genus one. A fatgraph for this amplitude, $\Gamma$, is given in Figure 29. The
colour factor of this graph has only one factor, $\text{tr}(1)$, because
$\Gamma$ only has one boundary. In fact, the curves on $\Gamma$ must all begin
and end in spirals around this one boundary. Using Figure 29 we can identify
the curves which have precisely _one valley_ in their mountainscape: i.e.
which only have one switch from turning right to turning left. These three
curves are

$\displaystyle C_{1/0}$ $\displaystyle=(wRzRxR)^{\infty}w(LxLzLw)^{\infty},$
(155) $\displaystyle C_{0/1}$
$\displaystyle=(xRwRzR)^{\infty}x(LzLwLx)^{\infty},$ (156) $\displaystyle
C_{1/1}$ $\displaystyle=(zRxRwR)^{\infty}z(LwLxLz)^{\infty}.$ (157)

These curves are non-intersecting and form a triangulation. The surface
associated to $\Gamma$ is the torus with one puncture, and the labels we
assign to these curves are inspired by drawing the curves on the torus,
pictured as a quotient of a $\mathbb{Z}^{2}$ lattice.

Besides $C_{1/1}$, we find that the only other curve compatible with both
$C_{1/0}$ and $C_{0/1}$ is

$\displaystyle C_{-1/1}=(xRwRzR)^{\infty}xLzRx(LzLwLx)^{\infty}.$ (158)

This curve has a peak at $z$, but no peaks at either $x$ or $w$ (which is what
would result in an intersection with $C_{1/0}$ or $C_{0/1}$).

As we will see later, the four curves $C_{1/0},C_{0/1},C_{1/1},C_{-1/1}$ are
all we need to compute the 2-loop vacuum genus one amplitude. Evaluating these
curves’ matrices gives

$\displaystyle M_{1/0}$
$\displaystyle=\begin{bmatrix}\frac{1+x+xz}{1-xzw}&0\\\ 0&0\end{bmatrix},$
$\displaystyle M_{0/1}$
$\displaystyle=\begin{bmatrix}\frac{1+z+zw}{1-xzw}&0\\\ 0&0\end{bmatrix},$
(159) $\displaystyle M_{1/1}$
$\displaystyle=\begin{bmatrix}\frac{1+w+wx}{1-xzw}&0\\\ 0&0\end{bmatrix},$
$\displaystyle M_{-1/1}$
$\displaystyle=\begin{bmatrix}\frac{1+2x(1+z)+x^{2}(1+3z+(3+2w)z^{2}+(1+w)^{2}z^{3})}{(1-wxz)^{2}}&0\\\
0&0\end{bmatrix}.$ (160)

The headlight functions for these curves are

$\displaystyle\alpha_{1/1}$ $\displaystyle=\max(0,w,w+x)-\max(0,w+z+x),$ (161)
$\displaystyle\alpha_{1/0}$ $\displaystyle=\max(0,x,x+z)-\max(0,w+z+x),$ (162)
$\displaystyle\alpha_{0/1}$ $\displaystyle=\max(0,z,z+w)-\max(0,w+z+x),$ (163)
$\displaystyle\alpha_{-1/1}$
$\displaystyle=\max(0,2x,2x+3z,2x+3z+2w)-2\max(0,w+z+x).$ (164)

##  6 Integrand Curve Integrals

We want to compute the partial amplitudes of our theory. For some fatgraph
$\Gamma$, let ${\cal A}$ be the amplitude that multiplies the colour factor
$c_{\Gamma}$.

The momentum assignment rule in Section 3.3 defines one set of loop momentum
variables for all propagators contributing to the amplitude, even beyond
planar diagrams. This means that ${\cal A}$ can be obtained as the integral of
a single _loop integrand_ ${\cal I}$:

$\displaystyle{\cal A}=\int\left(\prod_{i=1}^{L}d^{D}\ell_{i}\right){\cal I}.$
(165)

However, beyond planar diagrams, there is a price to pay for introducing our
momentum assignment. For any triangulation by curves, $C_{1},C_{2},...,C_{E}$,
we associate the product of propagators

$\displaystyle\frac{1}{X_{C_{1}}X_{C_{2}}\ldots X_{C_{E}}},$ (166)

where $X_{C}$ is given by the momentum assignment rule. If we sum over every
such term, (166), for all triangulations of $\Gamma$, we obtain some rational
function ${\cal I}_{\infty}$. But the loop integral of ${\cal I}_{\infty}$ is
not well defined if $\Gamma$ has a nontrivial mapping class group, MCG. This
is because two triangulations related by the MCG action integrate to the
_same_ Feynman diagram. So the loop integral of ${\cal I}_{\infty}$ contains,
in general, infinitely many copies of each Feynman integral.

Fortunately, we can compute integrands ${\cal I}$ for the amplitude by
‘dividing by the volume of MCG’. As a function, ${\cal I}$ is not uniquely
defined. But all choices for ${\cal I}$ integrate to the same amplitude.

We will compute integrands ${\cal I}$ using the headlight functions,
$\alpha_{C}$. The formula takes the form of a _curve integral_ ,

$\displaystyle{\cal I}=\int\frac{d^{E}{t}}{\text{MCG}}\,e^{-S({\bf t})}.$
(167)

Here, $E$ is the number of edges of the fatgraph $\Gamma$. We call it a _curve
integral_ because the integral is over the $E$-dimensional vector space, $V$,
whose integral points correspond to curves (or collections of curves) on
$\Gamma$. As discussed in Section 4.2, the mapping class group MCG has a
piecewise linear action on $V$, and we mod out by this action in the integral.
We call $S(t)$ the _curve action_. It is given by a sum

$S({\bf t})=\sum_{C}\alpha_{C}(\mathbf{t})X_{C},$ (168)

where we sum over all curves, $C$, on the fatgraph.444We exclude _closed
curves_ from this sum. Including the closed curves corresponds to coupling our
colored field to an uncolored scalar particle. For simplicity, we delay the
discussion of uncolored amplitudes For a general derivation of this curve
integral formula, see Appendix A. In this section, we show how to practically
use (167) to compute some simple amplitudes.

In fact, (167) also makes the loop integrals easy to do. This leads to a
direct curve integral formula for the amplitude ${\cal A}$, which we study in
Section 7.

Later, in Section 10, we also show that the integrands ${\cal I}$ can be
computed recursively, starting from the curve integral formula, (167). This
result generalises the standard _forward limit_ method for 1-loop amplitudes
to _all_ orders in the perturbation series.

###  6.1 Example: the tree level 5-point amplitude

Curve integrals give new and simple amplitude formulas, even at tree level.
Take the same fatgraph studied in Sections 4.1, 5.3 and 6.1. The kinematic
variables for the curves on this graph are ($i<j-1$)

$X_{ij}=(k_{i}+...+k_{j-1})^{2}+m^{2}.$ (169)

Then the amplitude, given by (165), is

$\displaystyle{\cal A}(12345)=\int dy_{1}dy_{2}\,Z,$ (170)

where

$\displaystyle-\log
Z=\alpha_{13}X_{13}+\alpha_{14}X_{14}+\alpha_{24}X_{24}+\alpha_{25}X_{25}+\alpha_{35}X_{35}.$
(171)

Using the formulas for $\alpha_{ij}$ from Section 5.3, $Z$ can be further
simplified to

$\displaystyle\log
Z=X_{25}\,x+X_{35}\,y+s_{13}f_{13}+s_{14}f_{14}+s_{24}f_{24},$ (172)

where $s_{ij}=2k_{i}\cdot k_{j}$ and the $f_{ij}$ are the simple functions

$f_{13}=\max(0,x),\qquad f_{14}=\max(0,x,x+y),\qquad f_{24}=\max(0,y).$ (173)

The 5-point amplitude is then

$\displaystyle{\cal A}(12345)=\int
dy_{1}dy_{2}\,\exp\left(X_{25}\,x+X_{35}\,y+s_{13}f_{13}+s_{14}f_{14}+s_{24}f_{24}\right).$
(174)

It is already interesting to note that the formula for the amplitude has been
written in terms of the simple functions $f_{13},f_{14},f_{24},y_{1},y_{2}$,
and the Mandelstam invariants $s_{ij}$. These $s_{ij}$ are automatically
summed together by the formula to form the appropriate poles of the tree level
amplitude.

###  6.2 Example: the planar 1-loop propagator

Consider again the 1-loop planar propagator (Sections 4.6 and 5.6). The
amplitude is

$\displaystyle{\cal A}=\int d^{D}\ell\int\limits_{x+y\leq 0}dxdy{Z},$ (175)

where

$\displaystyle-\log{Z}=\alpha_{C_{1}}X_{C_{1}}+\alpha_{C_{2}}X_{C_{2}}+\alpha_{S_{1}^{\prime}}X_{S_{1}^{\prime}}+\alpha_{S_{2}^{\prime}}X_{S_{2}^{\prime}}.$
(176)

We can assign the momenta of the curves to be

$\displaystyle P_{C_{1}}=0,\leavevmode\nobreak\ \leavevmode\nobreak\
P_{S_{1}^{\prime}}=\ell,\leavevmode\nobreak\ \leavevmode\nobreak\
P_{S_{2}^{\prime}}=\ell+k,\leavevmode\nobreak\ \leavevmode\nobreak\
P_{C_{2}}=0.$ (177)

Substituting these momenta (with $k^{2}+m^{2}=0$) into the integrand gives

$\displaystyle-\log{Z}=\ell^{2}(\alpha_{S_{1}^{\prime}}+\alpha_{S_{2}^{\prime}})+2\ell\cdot
k\alpha_{S_{2}^{\prime}}+m^{2}(\alpha_{C_{1}}+\alpha_{C_{2}}+\alpha_{S_{1}^{\prime}}).$
(178)

At this point, we can either integrate over $x+y\leq 0$, or do the loop
integral. Doing the loop integral first is a Gaussian integral, which gives

$\displaystyle{\cal A}=\int\limits_{x+y\leq
0}dxdy\left(\frac{\pi}{\alpha_{S_{1}^{\prime}}+\alpha_{S_{2}^{\prime}}}\right)^{\frac{D}{2}}\exp\left(k^{2}\frac{\alpha_{S_{2}^{\prime}}^{2}}{\alpha_{S_{1}^{\prime}}+\alpha_{S_{2}^{\prime}}}-m^{2}(\alpha_{C_{1}}+\alpha_{C_{2}}+\alpha_{S_{1}^{\prime}})\right).$
(179)

This resembles the Symanzik formula for a single Feynman integral, but instead
includes contributions from all three Feynman diagrams for this amplitude.
Finally, substituting the headlight functions gives

$\displaystyle{\cal A}=\int\limits_{x+y\leq
0}dxdy\left(\frac{-\pi}{x+y}\right)^{\frac{D}{2}}\exp\left[m^{2}\frac{(\max(0,y)-y-\max(0,x))^{2}}{x+y}+m^{2}(2\max(0,y)+x)\right].$
(180)

It is not immediately obvious that this reproduces the Feynman integrals for
this amplitude. But note that, for example, restricting the domain of the
integral to the negative orthant gives

$\displaystyle\int\limits_{x,y\leq
0}dxdy\left(\frac{-\pi}{x+y}\right)^{\frac{D}{2}}\exp\left(m^{2}\left(\frac{y^{2}}{x+y}+x\right)\right).$
(181)

After writing

$\frac{y^{2}}{x+y}+x=-\frac{xy}{x+y}+(x+y),$ (182)

this recovers the Feynman integral for the bubble graph. By extending the
integral to the full region, $x+y\leq 0$, we recover not just this bubble
integral, but the full amplitude!

###  6.3 Example: the planar 1-loop 3-point amplitude

For a more complicated planar example, consider the 1-loop planar 3-point
amplitude, with the fatgraph $\Gamma$, in Figure 30. There are nine curves on
this graph: three curves $C_{i,i+2}$, connecting external lines $i,i+2$; three
curves $C_{i,i}$, which loop around and come back to external line $i$; and
three curves $C_{i,0}$ that start from the external line $i$ and end in a
spiral around the closed loop.

In the planar sector, a convenient way to assign momenta is to use _dual
variables_. Let $z_{i}^{\mu}$ ($i=1,2,3$) be dual variables for the external
lines, and $z_{0}$ be the dual variable for the closed loop. Then curves from
external lines $i$ to $j$ have

$X_{i,j}=(z_{j}-z_{i})^{2}+m^{2},$ (183)

whereas a curve from $i$ that ends in a spiral around the loop has

$X_{i,0}=(z_{i}-z_{0})^{2}+m^{2}.$ (184)

If the external momenta are $p_{1},p_{2},p_{3}$, then we can take
$z_{1}=0,z_{2}=p_{1},z_{3}=p_{1}+p_{2}$. The closed loop variable, $z_{0}$,
can be used as a loop momentum variable.

Figure 30: A fatgraph for the 3-point 1-loop planar amplitude.

The 3-point one-loop planar amplitude is then

$\displaystyle\mathcal{A}=\int d^{D}z_{0}\int\limits_{\sum t_{i}\geq 0}d{\bf
t}\,{Z},$ (185)

where (taking cyclic indices mod $3$)

$\displaystyle-\log
Z=\sum_{i=1}^{3}\alpha_{i,i+2}X_{i,i+2}+\sum_{i=1}^{3}\alpha_{i,i}X_{i,i}+\sum_{i=1}^{3}\alpha_{i,0}X_{i,0}.$
(186)

The headlight functions for these curves are

$\displaystyle\alpha_{i,0}=t_{i}+g_{i+1}-g_{i},$ (187)
$\displaystyle\alpha_{i,i+2}=g_{i}-f_{i}-f_{i+1},$ (188)
$\displaystyle\alpha_{i,i}=f_{i+1}+h_{i}-g_{i}-g_{i+1},$ (189)

where

$\displaystyle f_{i}$ $\displaystyle=\max(0,t_{i}),$ (190) $\displaystyle
g_{i}$ $\displaystyle=\max(0,t_{i},t_{i}+t_{i+1}),$ (191) $\displaystyle
h_{i}$ $\displaystyle=\max(0,t_{i},t_{i}+t_{i+1},t_{i}+t_{i+1}+t_{i+2}).$
(192)

###  6.4 Note on factorization

The integrands defined by curve integrals factorise in the correct way. Take
again the curve integral

$\displaystyle{\cal I}=\int\frac{d^{E}t}{\text{MCG}}\,Z.$ (193)

In Appendix B, we show that the residue at $X_{C}=0$ is given by

$\displaystyle\text{Res}_{X_{C}=0}\,{\cal
I}=\int\frac{d^{E-1}t}{\text{MCG}^{\prime}}Z^{\prime},$ (194)

which is now the curve integral for the fatgraph $\Gamma_{C}$, obtained by
cutting $\Gamma$ along $C$. In this formula, $\text{MCG}^{\prime}$ is the MCG
of $\Gamma_{C}$, and the momentum $P_{C}^{\mu}$ of the curve $C$ is put on
shell. In the fatgraph $\Gamma_{C}$, the curve $C$ gives two new boundaries,
which are assigned momenta $\pm P_{C}^{\mu}$.

For example, before loop integration, the non-planar 1-loop fatgraph $\Gamma$
has loop integrand

$\displaystyle{\cal I}=\int
dxdy\,\exp\left(-\sum_{n=-\infty}^{\infty}\alpha_{n}X_{n}\right).$ (195)

Here, the momenta of the curves are $P_{n}^{\mu}=\ell^{\mu}+nk^{\mu}$.
Consider the $X_{0}=0$ pole. The parameter $\alpha_{0}$ vanishes outside
$x\geq 0$. In this region, the only non-vanishing parameters are $\alpha_{1}$
and $\alpha_{-1}$. The residue at $X_{0}=0$ is then

$\displaystyle\text{Res}_{X_{0}=0}{\cal I}=\int
dy\,\exp\left(-\alpha_{1}^{\prime}X_{1}-\alpha_{-1}^{\prime}X_{-1}\right),$
(196)

where the restriction to $x=0$ gives $\alpha_{1}^{\prime}=\max(0,y)$ and
$\alpha_{-1}^{\prime}=y-\max(0,y)$. This is the $n=4$ tree level amplitude,
with external momenta are $k,\ell,-k,-\ell$, and $\ell^{\mu}$. The two
propagators are $X_{1}=(k+\ell)^{2}+m^{2}$ and $X_{-1}=(k-\ell)^{2}+m^{2}$.

##  7 Amplitude Curve Integrals

Following the previous section, the curve integral formula for the full
amplitude is

${\cal A}=\int\frac{d^{E}\mathbf{t}}{\text{MCG}}\int\left(\prod
d^{D}\ell_{a}\right)\exp(-S({\bf t})).$ (197)

The loop integration variables, $\ell_{a}$, appear quadratically in the curve
action $S({\bf t})$. So, if we perform the loop integral _before_ performing
the curve integral over the $t_{i}$, it is a Gaussian integral. The result is
a curve integral

$\displaystyle{\cal
A}=\int\frac{d^{E}\mathbf{t}}{\text{MCG}}\,\left(\frac{\pi^{L}}{\mathcal{U}}\right)^{\frac{D}{2}}\exp\left(\frac{\mathcal{F}_{0}}{\mathcal{U}}-\mathcal{Z}\right),$
(198)

where $\mathcal{U},\mathcal{F}_{0}$ and $\mathcal{Z}$ are homogeneous
polynomials in the $\alpha_{C}$’s that we call _surface Symanzik polynomials_.

The curve integral (198) resembles the Schwinger form of a single Feynman
integral, but it integrates to the full amplitude. Once again, it is important
to mod out by the action of the mapping class group, to ensure that the
integral does not overcount Feynman diagrams.

We now summarise how to compute the surface Symanzik polynomials, ${\cal
U},{\cal F}_{0},{\cal Z}$. Suppose that a choice of loop momentum variables,
$\ell_{a}^{\mu}$, has been fixed. The momentum assigned to a curve $C$ is of
the form

$P_{C}^{\mu}=K_{C}^{\mu}+\sum h_{C}^{a}\ell_{a}^{\mu},$ (199)

for some integers $h_{C}^{a}$. These $h_{C}^{a}$ geometrically can be
understood in terms of intersections between $C$ and a basis of $L$ closed
curves on the fatgraph. Using the $h_{C}^{a}$ intersection numbers, define an
$L\times L$ matrix

$A^{ab}=\sum_{C}h_{C}^{a}h_{C}^{b}\alpha_{C},$ (200)

and a $L$-dimensional vector (with momentum index $\mu$)

$B^{a,\mu}=\sum_{C}h_{C}^{a}\alpha_{C}K_{C}^{\mu}.$ (201)

The then surface Symanzik polynomials are

$\mathcal{U}=\det A,\qquad\frac{\mathcal{F}_{0}}{\cal
U}=B^{a}_{\mu}\left(A^{-1}\right)_{ab}B^{b,\mu},\qquad\mathcal{Z}=\sum_{C}\alpha_{C}\left(K_{C}^{2}+m^{2}\right).$
(202)

These arise in the usual way by performing the Gaussian integral, as discussed
in detail in Appendix C.

In fact, the surface Symanzik polynomials have simple expressions when
expanded as a sum of monomials. For a set of curves,
$\mathcal{S}=\\{C_{1},...,C_{L}\\}$, write $\alpha_{\mathcal{S}}$ for the
corresponding monomial

$\displaystyle\alpha_{\mathcal{S}}=\prod_{i=1}^{L}\alpha_{C_{i}}.$ (203)

The determinant, $\det A$, can be expanded to give

$\displaystyle\mathcal{U}=\sum_{\begin{subarray}{c}\mathcal{S}\leavevmode\nobreak\
\text{cuts}\,\Sigma\\\ \text{to disk}\end{subarray}}\alpha_{\mathcal{S}}\,,$
(204)

where we sum over all sets $\mathcal{S}$ whose curves cut $\Gamma$ down to a
tree fatgraph. In other words, ${\cal U}$ is the sum over all _maximal cuts_
of the graph $\Gamma$. Moreover, using the Laplace expansion of the matrix
inverse, $\mathcal{F}_{0}$ can be expanded to find

$\displaystyle\mathcal{F}_{0}=\sum_{\begin{subarray}{c}\mathcal{S}^{\prime}\leavevmode\nobreak\
\text{cuts}\leavevmode\nobreak\ \Sigma\\\ \text{to 2
disks}\end{subarray}}\alpha_{\mathcal{S}^{\prime}}\left(\sum_{C\in\mathcal{S}^{\prime}}K_{C}^{\mu}\right)^{2},$
(205)

where the sum in this formula is now over sets $\mathcal{S}^{\prime}$ of $L+1$
curves that factorise $\Gamma$ into two disjoint tree graphs. Each monomial in
the sum is multiplied by the total momentum flowing through the factorisation
channel.

A complete derivation of (204) and (205) is given in Appendix C.

###  7.1 Example: the planar 1-loop propagator

We return to the planar 1-loop propagator (Sections 4.6, 5.6, 6.2). Of the
four curves $C_{1},C_{2},S_{1},S_{2}$, only $S_{1}$ and $S_{2}$ carry loop
momentum and cut $\Gamma$ open to a tree. The first surface Symanzik
polynomial is therefore

$\displaystyle\mathcal{U}=\alpha_{S_{1}}+\alpha_{S_{2}}.$ (206)

The $B$-vector is

$B^{\mu}=\alpha_{S_{2}^{\prime}}k^{\mu},$ (207)

so that the second surface Symanzik polynomial is

$\displaystyle\mathcal{F}_{0}=\alpha_{S_{2}^{\prime}}^{2}k^{2}.$ (208)

Finally,

${\cal Z}=m^{2}(\alpha_{S_{1}^{\prime}}+\alpha_{C_{1}}+\alpha_{C_{2}}).$ (209)

The amplitude is then given by the curve integral

$\displaystyle{\cal A}=\int\limits_{x+y\geq
0}dxdy\left(\frac{\pi}{\alpha_{S_{1}}+\alpha_{S_{2}}}\right)^{\frac{D}{2}}\exp\left(\frac{\alpha_{S_{2}}p^{2}}{\alpha_{S_{1}}+\alpha_{S_{2}}}-m^{2}\left(\alpha_{S_{1}}+\alpha_{C_{1}}+\alpha_{C_{2}}\right)\right).$
(210)

This again recovers the formula (179), which we obtained by direct integration
in the previous section.

###  7.2 Example: the non-planar 1-loop propagator

We return to the non-planar 1-loop propagator (Sections 4.4 and 5.4). The
momentum of the curve $C_{n}$ is

$\displaystyle P_{n}^{\mu}=\ell^{\mu}+np^{\mu}.$ (211)

Every curve $C_{n}$ cuts $\Gamma$ to a tree graph with 4 external legs. So the
first Symanzik polynomials is

$\displaystyle\mathcal{U}=\sum_{n=-\infty}^{\infty}\alpha_{n},$ (212)

where $\alpha_{n}$ is the headlight function for $C_{n}$. Every pair of
distinct curves $C_{n},C_{m}$ cuts $\Gamma$ into two trees, and so

$\displaystyle\mathcal{F}_{0}=\sum_{n,m=-\infty}^{\infty}nm\alpha_{n}\alpha_{m}p^{2}.$
(213)

Finally,

$\displaystyle\mathcal{Z}=\sum_{n=-\infty}^{\infty}\alpha_{n}(m^{2}+n^{2}p^{2}).$
(214)

The amplitude is then

$\displaystyle{\cal
A}=\int\frac{dxdy}{\text{MCG}}\left(\frac{\pi}{\mathcal{U}}\right)^{\frac{D}{2}}\exp\left(\frac{\mathcal{F}_{0}}{\mathcal{U}}-\mathcal{Z}\right).$
(215)

The MCG acts on the fan in this case as ${\bf g}_{n}\mapsto{\bf g}_{n+1}$. A
fundamental domain for this action is clearly the positive orthant, spanned by
${\bf g}_{0},{\bf g}_{1}$. In this orthant, the surface Symanzik polynomials
are

$\displaystyle\mathcal{U}$ $\displaystyle=x+y,$ (216)
$\displaystyle\mathcal{F}_{0}$ $\displaystyle=y^{2}p^{2},$ (217)
$\displaystyle\mathcal{Z}$ $\displaystyle=xm^{2}.$ (218)

So we find

$\displaystyle{\cal A}=\int\limits_{x,y\geq
0}dxdy\left(\frac{\pi}{x+y}\right)^{D/2}\exp\left(m^{2}\left(-\frac{y^{2}}{x+y}-x\right)\right),$
(219)

where we have put $p^{\mu}$ on shell, $p^{2}+m^{2}=0$. Or, equivalently,

$\displaystyle{\cal A}=\int\limits_{x,y\geq
0}dxdy\left(\frac{\pi}{x+y}\right)^{D/2}\exp\left(-p^{2}\frac{xy}{x+y}-m^{2}(x+y)\right).$
(220)

###  7.3 Example: The non-planar 3-point amplitude

Even at 1-loop, it is not always easy to identify the fundamental domain of
the MCG. To see the problem, consider the non-planar one-loop 3-point
amplitude. Let the first trace factor have external particle $p_{1}^{\mu}$,
and the second trace factor have $p_{2}^{\mu}$ and $p_{3}^{\mu}$. The curves,
$C_{ij}^{n}$, connecting a pair of distinct start and end points, $i,j$, are
labelled by the number of times, $n$, they loop around the graph. The curves
$C_{22}$ and $C_{33}$ begin and end at the same edge, and are invariant under
the MCG. Then, for a specific choice of loop momentum variable, we find the
momentum assignments

$P_{12}^{n}=np_{1}^{\mu},\qquad P_{13}^{n}=np_{1}^{\mu}-p_{2}^{\mu},\qquad
P_{22}=0,\qquad P_{33}=0.$ (221)

We can readily give the curve integral formula for the amplitude,

$\displaystyle{\cal
A}=\int\frac{dxdydz}{\text{MCG}}\left(\frac{\pi}{\mathcal{U}}\right)^{\frac{D}{2}}\exp\left(\frac{\mathcal{F}_{0}}{\mathcal{U}}-\mathcal{Z}\right),$
(222)

where the surface Symanzik polynomials are

$\mathcal{U}=\sum_{n=-\infty}^{\infty}\alpha_{13}^{n}+\alpha_{12}^{n},\qquad\mathcal{F}_{0}=B^{\mu}B^{\mu},\qquad{\cal
Z}=m^{2}\left(\alpha_{22}+\alpha_{33}+\sum_{n=-\infty}^{\infty}\alpha_{12}^{n}\right).$
(223)

In the formula for ${\cal F}_{0}$, the $B$-vector is

$B^{\mu}=\sum_{n=-\infty}^{\infty}np_{1}^{\mu}\alpha_{12}^{n}+(np_{1}^{\mu}-p_{2}^{\mu})\alpha_{13}^{n}.$
(224)

However, at this point we confront the problem of quotienting by MCG. The MCG
is generated by

$\displaystyle{\bf g}_{12}^{n}\mapsto{\bf g}_{12}^{n+1},\leavevmode\nobreak\
{\bf g}_{13}^{n}\mapsto{\bf g}_{13}^{n+1},$ (225)

and it leaves ${\bf g}_{22}$ and ${\bf g}_{33}$ invariant. Naively, we might
want to quotient by the MCG by restricting the integral to the region spanned
by: ${\bf g}_{12}^{0},{\bf g}_{13}^{0},{\bf g}_{22},{\bf g}_{33}$. However,
this region is too small. It does not include any full cones of the Feynman
fan. We could also try restricting the integral to the region spanned by:
${\bf g}_{12}^{0},{\bf g}_{13}^{0},{\bf g}_{12}^{1},{\bf g}_{13}^{1},{\bf
g}_{22},{\bf g}_{33}$. But this region is too large! The amplitude has _three_
Feynman diagrams, but this region contains _four_ cones, so it counts one of
the diagrams twice.

As this example shows, it is already a delicate problem to explicitly specify
a fundamental domain for the MCG action.

###  7.4 Example: genus-one 2-loop amplitudes

The problem of modding by MCG becomes even more acute for non-planar
amplitudes. The genus one 2-loop vacuum amplitude, considered in Section 5.7,
is computed by a 3-dimensional curve integral. But the MCG action in this case
is an action of ${\rm SL}_{2}\mathbb{Z}$. The action on $g$-vectors is of the
form

${\bf g}_{p/q}\mapsto{\bf
g}_{(ap+bq)/(cp+dq)},\qquad\text{for}\leavevmode\nobreak\
\begin{bmatrix}a\leavevmode\nobreak\ &b\\\ c\leavevmode\nobreak\
&d\end{bmatrix}\in{\rm SL}_{2}\mathbb{Z}.$ (226)

For the vacuum amplitude, a simple example of a fundamental region is the
region spanned by ${\bf g}_{1/0},{\bf g}_{0/1},$ and ${\bf g}_{1/1}$. However,
for the $n$-point genus one 2-loop amplitude, identifying a fundamental region
of this ${\rm SL}_{2}\mathbb{Z}$-action becomes very difficult.

In the next section, we present a simple method to compute the integrals in
our formulas, for any MCG action.

##  8 Modding Out by the Mapping Class Group

Our formulas for amplitudes and integrands take the form of integrals over
$\mathbb{R}^{E}$ modulo the action of the Mapping Class Group, MCG,

${\cal A}=\int\frac{d^{E}t}{\text{MCG}}\,f(t),$ (227)

for some MCG-invariant function, $f(t)$. One way to evaluate this integral is
to find a fundamental domain for the MCG action. But it is tricky to identify
such a region in general. Instead, it is convenient to mod out by the MCG
action by defining a kernel, $\mathcal{K}$, such that

${\cal A}=\int d^{E}t\,\mathcal{K}(t)f(t).$ (228)

In this section, we find kernels, $\mathcal{K}$, that can be used at all
orders in perturbation theory, for all Mapping Class Groups.

###  8.1 Warm up

Consider the problem of evaluating an integral modulo a group action on its
domain. For example, suppose $f(x)$ is invariant under the group of
translations, $T$, generated by $x\mapsto x+a$, for some constant, $a$. We
want to evaluate an integral

$I=\int\limits_{\mathbb{R}/T}dxf(x).$ (229)

One way to do this is to restrict to a fundamental domain of $T$:

$I=\int\limits_{0}^{a}dxf(x).$ (230)

But we can alternatively find a kernel $\mathcal{K}(x)$ such that

$I=\int\limits_{-\infty}^{\infty}dx\,\mathcal{K}(x)f(x).$ (231)

One way to find such a kernel is to take a function $g(x)$ with finite support
around $0$, say. Then we can write

$1=\frac{\sum_{n=-\infty}^{\infty}g(x-na)}{\sum_{n=-\infty}^{\infty}g(x-na)},$
(232)

provided that $\sum_{n=-\infty}^{\infty}g(x-na)$ is nowhere vanishing.
Inserting this into (229),

$I=\int\limits_{\mathbb{R}/T}dx\,\frac{\sum_{n=-\infty}^{\infty}g(x-na)}{\sum_{n=-\infty}^{\infty}g(x-na)}f(x)=\int\limits_{-\infty}^{\infty}dx\,\frac{g(x)}{\sum_{n=-\infty}^{\infty}g(x-na)}f(x).$
(233)

So that we can use

$\mathcal{K}(x)=\frac{g(x)}{\sum_{n=-\infty}^{\infty}g(x-na)}$ (234)

as a kernel to quotient out by the translation group. For example, suppose
that we take $g(x)=\Theta(x+a)\Theta(-x+a)$, where $\Theta(x)$ is the
Heaviside function. Inserting this into (233) gives

$I=\int\limits_{-a}^{a}dx\,\frac{1}{2}f(x).$ (235)

The domain of this integral contains two copies of a fundamental domain for
$T$, but this is compensated for by the $1/2$ coming from $\mathcal{K}(x)$ to
give the correct answer.

###  8.2 A Tropical Mirzakhani kernel

The headlight functions, $\alpha_{C}$, give a very natural solution to the
problem of defining an integration kernel, $\mathcal{K}$.

Consider the case when MCG has _one generator_. Let $\mathcal{S}$ be the set
of curves which are _not_ invariant under MCG. The sum of their headlight
functions,

$\displaystyle\rho=\sum_{C\in\mathcal{S}}\alpha_{C},$ (236)

is itself a MCG-invariant function. Moreover, $\rho$ does not vanish on any
top-dimensional cone (because no diagram can be formed without using at least
one propagator from $\mathcal{S}$). So we can consider inserting the function

$\displaystyle 1=\frac{\rho}{\rho}$ (237)

into our integrals.

The set $\mathcal{S}$ is the disjoint union of cosets under the MCG action, by
the Orbit-Stabilizer theorem. When MCG has a single generator, these cosets
are easy to describe. MCG does not alter the endpoints of curves. So if
$C_{ij}\in\mathcal{S}$ is a curve connecting external lines $i$ and $j$, the
orbit of $C_{ij}$ is a coset of $\mathcal{S}$. By the Orbit-Stabalizer
theorem, these cosets are disjoint. So $\rho$ can be resumed as

$\displaystyle\rho=\sum_{i,j}\sum_{\gamma\in\text{MCG}}\alpha_{\gamma
C_{ij}}.$ (238)

Given this, we can mod out by the MCG action by defining

$\displaystyle\mathcal{K}=\sum_{i,j}\frac{\alpha_{C_{ij}}}{\rho},$ (239)

where we choose a distinguished representative, $C_{ij}$, for each coset. We
call (239) a _tropical Mirzakhani kernel_ , because it is a tropical version
of the kernel introduced by Mirzakhani to compute Weil-Petersson volumes
mirzakhani2007 . Each headlight function, $\alpha_{C_{ij}}$, is non-vanishing
in a convex region $V_{C_{ij}}$ that is spanned by all the cones in the fan
that contain ${\bf g}_{C_{ij}}$. These regions _over-count_ the diagrams, but
this over-counting is corrected by the kernel, $\mathcal{K}$.

###  8.3 Example: the non-planar 1-loop propagator

As a sanity check, let us repeat the calculation of the non-planar 1-loop
propagator from Section 7.2, but now using the tropical Mirzakhani kernel. The
MCG has one generator, and no curves are MCG-invariant. So take the set
$\mathcal{S}$ to be the set of all curves, $C_{n}$, and write

$\displaystyle\rho=\sum_{n=-\infty}^{\infty}\alpha_{n}.$ (240)

Choose $C_{0}$, say, as the coset representative (all other curves are in the
orbit of $C_{0}$). Then the tropical Mirzakhani kernel, (239), is

$\mathcal{K}=\frac{\alpha_{0}}{\rho}.$ (241)

Using this kernel, we find a pre-loop-integration integrand,

$\displaystyle\mathcal{I}=\int
dxdy\,\mathcal{K}(x,y)\,\mathrm{exp}\left(-\sum_{i=-\infty}^{\infty}\alpha_{i}X_{i}\right).$
(242)

The headlight functions for this example were given in (134). In particular,
$\alpha_{0}=\max(0,x)$, which is vanishing outside of the region $x\geq 0$. In
this region, the only other non-vanishing headlight functions are

$\displaystyle\alpha_{-1}=\mathrm{max}(0,y)\qquad\text{and}\qquad\alpha_{1}=-y+\mathrm{max}(0,y).$
(243)

The formula is therefore

$\displaystyle\mathcal{I}=\int\limits_{x\geq
0}dxdy\,\frac{x}{x+|y|}\mathrm{exp}\left(-\alpha_{-1}X_{-1}-\alpha_{0}X_{0}-\alpha_{1}X_{1}\right).$
(244)

We can now perform the loop integral. Recall that $X_{n}=(\ell+nk)^{2}+m^{2}$.
Using this, the exponent, $Z$, in (244) is

$\displaystyle-\log Z=\rho\,\ell^{2}+2\ell\cdot
k(\alpha_{1}-\alpha_{-1})+m^{2}\alpha_{0}.$ (245)

The Gaussian integral gives

$\displaystyle{\cal A}=\int\limits_{x\geq
0}dxdy\frac{x}{x+|y|}\left(\frac{\pi}{x+|y|}\right)^{\frac{D}{2}}\mathrm{exp}\left(k^{2}\frac{|y|^{2}}{x+|y|}-m^{2}x\right).$
(246)

This doesn’t immediately look like the Feynman integral for the 1-loop bubble.
However, writing

$\displaystyle\frac{2x}{x+y}=1+\frac{x-y}{x+y},$ (247)

we find

$\displaystyle{\cal A}=\int\limits_{x,y\geq
0}dxdy\left(\frac{\pi}{x+y}\right)^{\frac{D}{2}}\exp\left(k^{2}\frac{y^{2}}{x+y}-m^{2}x\right).$
(248)

since the integrand over $x,y\geq 0$ is even under $x\leftrightarrow y$,
whereas $x-y$ is odd. This is still not exactly the same as the conventional
integral. To recover the conventional form, note that the exponent can be
rewritten as

$-\frac{y^{2}}{x+y}-x=\frac{xy}{x+y}-(x+y).$ (249)

###  8.4 General Tropical Mirzakhani Kernels

Tropical Mirzakhani kernels can be defined to _any_ mapping class group, with
more than one generator. Fix some fatgraph $\Gamma$, with mapping class group
MCG.

A conceptually simple way to define a kernel is to consider the set of
$L$-tuples of curves that cut $\Gamma$ to a tree graph. These define the
_first Symanzik polynomial_ ,

$\mathcal{U}=\sum_{\begin{subarray}{c}S\\\ \text{cuts to
tree}\end{subarray}}\alpha_{S},$ (250)

which can also be computed as a determinant of a matrix (Section 7). This
function does not vanish on top-dimensional cones of the Feynman fan, since
every diagram contains a subset of propagators that cut $\Gamma$ to a tree. We
can therefore insert

$1=\frac{\mathcal{U}}{\mathcal{U}}$ (251)

into our integrals. Under the MCG action, the set of $L$-tuples appearing in
$\mathcal{U}$ is partitioned into cosets. Each coset represents an MCG-
inequivalent way of cutting $\Gamma$ down to a tree. By choosing a
representative $L$-tuple for each such loop cut, we arrive at a kernel

$\mathcal{K}=\sum_{\text{distinct loop cuts}}\frac{\alpha_{S}}{\mathcal{U}}.$
(252)

Our integrals can then be computed as a sum over maximal cuts:

$\mathcal{A}=\int\frac{d^{E}y}{\text{MCG}}{\cal I}=\sum_{\text{distinct loop
cuts}}\int d^{E}y\,\frac{\alpha_{S}}{\mathcal{U}}\,{\cal I}.$ (253)

The disadvantage of this formula is that it can be difficult to systematically
identify a set of MCG-inequivalent maximal cuts.

###  8.5 The General Iterative Method

A more systematic way to quotient out by MCG is to break the MCG-action one
generator at a time. This iterative method has the advantage of being
completely algorithmic.

To apply the method, pick a trace-factor of $\Gamma$, $\beta$, which has some
external particles, $1,...,m$. Let $\mathcal{S}_{\beta}$ be the set of curves
that have at least one endpoint in $\beta$, excluding any curves that are MCG-
invariant, and write

$\displaystyle\rho_{\beta}=\sum_{C\in\mathcal{S}_{\beta}}\alpha_{C}.$ (254)

$\rho_{\beta}$ is MCG-invariant. This is because the MCG action does not alter
the endpoints of a curve. The set $\mathcal{S}_{\beta}$ therefore has a coset
decomposition. For each MCG orbit in $\mathcal{S}_{\beta}$, pick a
representative curve, so that

$\displaystyle\rho_{\beta}=\sum_{i=1}^{k}\sum_{\gamma\in\mathrm{MCG}(\Sigma)}\alpha_{\gamma
C_{i}},$ (255)

for some $k=|\mathcal{S}_{\beta}/\text{MCG}(\Sigma)|$ coset representatives
$C_{1},...,C_{k}$. We give more details about how to pick a set of coset
representatives below.

Every top-dimensional cone is generated by at least _one_ curve from the set
$\mathcal{S}_{\beta}$, because otherwise that cone would not correspond to a
complete triangulation of $\Gamma$. This means that $\rho_{\beta}$ is non-
vanishing everywhere, except on some lower-dimensional cones. Away from this
vanishing locus, we can write

$\displaystyle 1=\frac{\rho_{\beta}}{\rho_{\beta}}.$ (256)

Given this, we define a tropical Mirzakhani kernel

$\displaystyle\mathcal{K}_{\beta}=\sum_{i=1}^{k}\frac{\alpha_{C_{i}}}{\rho_{\beta}}.$
(257)

This has the effect of breaking the MCG symmetry of the integrand, and
reducing us to evaluating simpler integrals. In particular, we have

${\cal
A}=\int\frac{d^{E}t}{\text{MCG}}\,\mathcal{I}=\sum_{i=1}^{k}\,\int\frac{d^{E}t}{\mathrm{Stab}(C_{i})}\,\frac{\alpha_{C_{i}}}{\rho_{\beta}}\,\mathcal{I},$
(258)

where $\mathrm{Stab}(C_{i})\leq\text{MCG}$ is the _stablizer subgroup_ for
$C_{i}$. The factor

$\frac{\alpha_{C_{i}}}{\rho_{\beta}}$ (259)

is _itself_ invariant under $\mathrm{Stab}(C_{i})$. So the integrals,

$\int\frac{d^{E}t}{\mathrm{Stab}(C_{i})}\,\frac{\alpha_{C_{i}}}{\rho}\,\mathcal{I},$
(260)

can themselves be evaluated by finding a Mirzkhani kernel for the new group,
$\mathrm{Stab}(C_{i})$. This iterative method ultimately yields an integral
with no group action,

${\cal A}=\int\frac{d^{E}y}{\text{MCG}}\,\mathcal{I}=\int
d^{n}y\,\mathcal{K}\,\mathcal{I},$ (261)

where $\mathcal{K}$ is a sum of products of kernels of the form (259).

To complete the description of the iterative method, we describe how to choose
coset representatives from the set $\mathcal{S}_{\beta}$. The curves in this
set break into two subsets, as in Figure 31:

  1. 1.

Curves $C$ whose endpoints lie in two distinct trace factors. These curves cut
$\Gamma$ to a fatgraph $\Gamma_{C}$ which has one fewer trace factors.

  2. 2.

Curves $C$ with both endpoints in the same trace factor. These curves cut
$\Gamma$ to a fatgraph $\Gamma_{C}$ with one lower genus.

Both of these subsets have decompositions into cosets specified by the
endpoints of the curves. So, for every pair of particles, $i,j$ (with $i$ in
trace factor $\beta$), pick _any_ curve $C_{ij}^{0}$ connecting them. These
can be taken as coset representatives. The caveat is that, if $i,j$ are both
in trace factor $\beta$, we must choose a curve $C_{ij}^{0}$ which is not MCG-
invariant. An MCG-invariant curve generates a trivial coset. The first step to
break the MCG is then to insert the kernel

$\sum_{i\in\beta}\sum_{j}\frac{\alpha_{ij}^{0}}{\sum_{{\cal
S}_{\beta}}\alpha_{C}}.$ (262)

Figure 31: The two types of curves that are not invariant under the MCG, drawn
here on the surface $S(\Gamma)$ associated to a fatgraph: curves connecting
distinct trace factors (right), and topologically nontrivial curves that begin
and end on the same trace factor (left).

For amplitudes involving a large number of external particles, this iterative
method naively requires a lot of work (growing like $n^{L}$ with the number of
particles, $n$). However, this apparent complexity goes away completely if we
choose an appropriate fatgraph, $\Gamma$, for our calculation. We use this to
obtain simple formulas for amplitudes at all-$n$ in a separate paper, us_alln
. But for now we will focus on low-point amplitudes, to illustrate the method
in its simplest form.

###  8.6 Example: the genus one 2-loop vacuum amplitude

As an example, we briefly describe what happens for the genus one 2-loop
vacuum amplitude (Sections 5.7 and 7.4). The MCG is now
$\mathrm{SL}_{2}\mathbb{Z}$. In this case, there is only _one_ coset to
consider, since every curve is related to every other by

${\bf g}_{p/q}\mapsto{\bf
g}_{(ap+bq)/(cp+dq)},\qquad\text{for}\leavevmode\nobreak\
\begin{bmatrix}a\leavevmode\nobreak\ &b\\\ c\leavevmode\nobreak\
&d\end{bmatrix}\in{\rm SL}_{2}\mathbb{Z}.$ (263)

For the first step of the iteration, we can take any curve, say $C_{1/0}$, as
a coset representative. The kernel for the first step is

${\cal K}_{1/0}=\frac{\alpha_{1/0}}{\sum_{C}\alpha_{C}}.$ (264)

The subgroup that leaves $C_{1/0}$ invariant is

$\text{Stab}\,C_{1/0}=\left\\{\begin{bmatrix}1&n\\\
0&1\end{bmatrix}\leavevmode\nobreak\ :\leavevmode\nobreak\
n\in\mathbb{Z}\right\\}<{\rm SL}_{2}\mathbb{Z}.$ (265)

The curves compatible with $C_{1/0}$ form a single coset for the action of
this subgroup. So, for the second step, we can choose just one of them,
$C_{0/1}$, say, as a coset representative. The kernel for the second step is

${\cal K}_{0/1}=\frac{\alpha_{0/1}}{\sum_{C^{\prime}}\alpha_{C^{\prime}}},$
(266)

where we sum only over curves, $C^{\prime}$, that are non-intersecting with
$C_{1/0}$. The final kernel is simply

${\cal
K}=\frac{\alpha_{1/0}}{\alpha_{1/0}+\alpha_{0/1}+\alpha_{1/1}+\alpha_{-1/1}}\,\frac{\alpha_{0/1}}{\alpha_{0/1}+\alpha_{1/1}+\alpha_{-1/1}},$
(267)

where the simplification arises because $C_{1/1}$ and $C_{-1/1}$ are the only
curves compatible with both $C_{1/0}$ and $C_{0/1}$.

##  9 Examplitudes

We now show how to use the tropical Mirzakhani kernels to evaluate curve
integrals. We give detailed low-dimensional examples of amplitudes up to 3
loops.

###  9.1 The non-planar 1-loop 3-point amplitude

The formula for the 1-loop non-planar 3-point amplitude was given in Section
7.3. However, we did not show how to quotient by the MCG. Using the tropical
Mirzakhani kernel, we now find the formula

$\displaystyle{\cal A}=\int d^{3}t\,{\cal K}\,\left(\frac{\pi}{\cal
U}\right)^{\frac{D}{2}}\,\exp\left(\frac{\mathcal{F}_{0}}{\mathcal{U}}-\mathcal{Z}\right),$
(268)

where the Mirzakhani kernel is

${\cal K}=\frac{\alpha_{12}^{0}+\alpha_{13}^{0}}{\rho},$ (269)

with $\rho$ the sum over all $\alpha_{C}$ (except for those curves which are
invariant under the MCG, namely $C_{22}$, $C_{33}$). The surface Symanzik
polynomials are, as before,

$\mathcal{U}=\sum_{n=-\infty}^{\infty}\alpha_{13}^{n}+\alpha_{12}^{n},\qquad\mathcal{F}_{0}=B_{\mu}B^{\mu},\qquad{\cal
Z}=m^{2}\left(\alpha_{22}+\alpha_{33}+\sum_{n=-\infty}^{\infty}\alpha_{12}^{n}\right).$
(270)

In the formula for ${\cal F}_{0}$, the $B$-vector is

$B^{\mu}=\sum_{n=-\infty}^{\infty}np_{1}^{\mu}\alpha_{12}^{n}+(np_{1}^{\mu}-p_{2}^{\mu})\alpha_{13}^{n}.$
(271)

Let us first see why (269) is a Mirzakhani kernel. The MCG has one generator.
It leaves $C_{22}$ and $C_{33}$ invariant, but acts non-trivially on the set
$\\{C_{12}^{n},C_{13}^{n}\\}$ of all curves that connect the first trace
factor to the second trace factor. $\rho$ is the sum of $\alpha_{C}$ for all
these curves,

$\displaystyle\rho=\sum_{n=-\infty}^{\infty}\left(\alpha_{12}^{n}+\alpha_{13}^{n}\right).$
(272)

This set has two MCG cosets, labelled by the start and end points of the
curves. We can take $C_{12}^{0}$ and $C_{13}^{0}$ as the two coset
representatives. $C_{12}^{0}$, for instance, represents the coset of all
curves that begin at $1$ and end at $2$. (Recall Section 8.)

Naively, it looks as if (268) involves infinitely many $\alpha_{C}$, which it
would be laborious to compute. However, the Mirzakhani kernel ensures that
only a few $\alpha_{C}$ are needed. To see how this works, consider, say, the
first term in the kernel,

${\cal K}_{12}=\frac{\alpha_{12}^{0}}{\rho}.$ (273)

In the region where $\alpha_{12}^{0}\neq 0$, all other $\alpha_{C}$ are
vanishing, except for:

$\displaystyle\alpha_{12}^{-1},\leavevmode\nobreak\
\alpha_{12}^{1},\leavevmode\nobreak\ \alpha_{13}^{0},\leavevmode\nobreak\
\alpha_{13}^{1},\leavevmode\nobreak\ \alpha_{22}.$ (274)

So in this region, ${\cal U}$ and $B^{\mu}$ simplify to

$\displaystyle\mathcal{U}$