Notes on cluster algebras and some all-loop Feynman integrals

Cluster algebras fomin2002cluster ; fomin2003cluster ; berenstein2005cluster ; fomin2007cluster are commutative algebras with a particular set of generators ${\cal A}_{i}$, known as the cluster ${\cal A}$-coordinates; they are grouped into clusters which are subsets of rank $d$. From an initial cluster, one can construct all the ${\cal A}$-coordinates by mutations acting on ${\cal A}$’s (the so-called frozen coordinates or coefficients can also be included, which do not mutate). Alternatively one can define cluster ${\cal X}$ coordinates which are given by monomials of ${\cal A}$’s.

There is a natural space of polylogarithm functions associated with a cluster algebra, given a set of cluster-${\cal A}$ or (${\cal X}$) coordinates. A cluster function $F^{(w)}$ Golden:2014xqa ; Parker:2015cia of transcendental weight $w$ is defined such that its differential has the form

$$dF^{(w)}=\sum_{i}F_{i}^{(w-1)}d\log{\cal A}_{i}$$

where $F^{(w-1)}_{i}$ are cluster functions of transcendental weight $w-1$ and ${\cal A}_{i}$ are cluster-${\cal A}$ coordinates. We see that if a multiple polylogarithm is a cluster function, then the alphabet can be identified with the corresponding cluster algebra.

For the purpose of this paper, it suffices to know that all finite-type cluster algebras, i.e. those with finite number of $A$-cluster coordinates (the dimension of the cluster algebra, denoted as $N$), has been classified in terms of Dynkin diagrams. There are series $A_{d},B_{d},C_{d},D_{d}$ and exceptional cases $E_{6},E_{7},E_{8},F_{4},G_{2}$. To identify an alphabet with certain finite-type cluster algebra, it is convenient to parametrize the coordinates of the latter in a nice way. For example, for type $A_{d}$ and $D_{d}$, we have the following $N=d(d{+}3)/2$ and $N=d^{2}$ letters respectively Chicherin:2020umh :

		$\displaystyle\Phi_{A_{d}}=\bigcup_{i=1}^{d}\{z_{i},1+z_{i}\}\cup\bigcup_{i<j}^{d}\{z_{i}-z_{j}\},$		(5)
		$\displaystyle\Phi_{D_{d}}=\bigcup_{i=1}^{d}\{z_{i},1{+}z_{i}\}\cup\bigcup_{i=1}^{d-2}\{z_{i}{+}z_{d-1}z_{d},z_{i}{-}z_{d-1},z_{i}{-}z_{d}\}\cup\bigcup_{i<j}^{d-2}\{z_{i}{-}z_{j},z_{i}{-}z_{j}{-}z_{i}z_{j}{+}z_{i}(z_{d-1}{+}z_{d}){-}z_{d-1}z_{d}\}$

In other words, once we find an alphabet which can be written as a collection of $N$ polynomials (of $d$ variables), the remaining task would be to look for some birational change of variables such that they become multiplicative combinations of letters in e.g. $\Phi_{A_{d}}$ or $\Phi_{D_{d}}$ (or a subset of them).

On the other hand, without any smart parametrization, there is a totally gauge-invariant way for describing any finite-type cluster algebra, known as the cluster configuration spaces Arkani-Hamed:2019plo ; Arkani-Hamed:2020tuz . One can simply represent $\Phi$ by $N$ variables called ${\bf u}$ variables, $\{{\bf u}_{\alpha}|\alpha=1,2,\cdots,N\}$ in bijection with ${\cal A}$-coordinates, which satisfy $N$ constraints known as the ${\bf u}$ equations (for $\alpha=1,2,\cdots,N$)

$$1-{\bf u}_{\alpha}=\prod_{\beta=1}^{N}{\bf u}_{\beta}^{\beta|\alpha}\,,$$

where $\beta|\alpha$ are integers known as compatibility degrees Arkani-Hamed:2019plo . It is remarkable that the ${\bf u}$ equations are consistent and give a $d$-dimensional solution space we call cluster configuration space. For example, for type $A_{d}$ we have $N=d(d{+}3)/2$ ${\bf u}$ variables (one for each diagonal of $(d{+}3)$-gon):

$$\Phi_{A_{d}}=\{{\bf u}_{ij}|1\leq i<j\leq n,i\neq j-1\}$$

(6)

which satisfy $N$ equations of the form

$$1-{\bf u}_{i,j}=\prod_{(kl)\,{\rm cross}\,(ij)}{\bf u}_{k,l}$$

(7)

where on the RHS we have the product of all ${\bf u}_{k,l}$ with $(k,l)$ crossing $(i,j)$ (or ${\bf u}_{k,l}$ incompatible with ${\bf u}_{i,j}$, with compatibility $1$). Similarly for $D_{d}$ we have $N=d^{2}$ variables which we denote as

$$\Phi_{D_{d}}=\{{\bf u}_{i},\tilde{\bf u}_{i}|1\leq i\leq d\}\cup\{{\bf u}_{i,j}|1\leq i\neq j,j{+}1\leq d\}$$

(8)

which satisfy ${\bf u}$ equations as explicitly given in Arkani-Hamed:2019plo ; Arkani-Hamed:2020tuz .

Note that such a configuration space can be viewed as a “binary geometry” for the corresponding cluster polytope (or generalized associahedra). If we ask all ${\bf u}$ to be positive, we have $\{0<{\bf u}_{\alpha}<1|\alpha=1,\cdots,N\}$, which cuts out a “curvy” cluster polytope. Each of the $N$ boundaries of the space is reached by exactly one ${\bf u}_{\alpha}\to 0$, and the ${\bf u}$ equations force all incompatible (i.e. those with $\beta|\alpha>0$) ${\bf u}_{\beta}\to 1$, and we obtain the configuration space of the corresponding sub-algebra, which factorizes according to which node of the Dynkin diagram we remove. Note that though the complex configuration space is no longer a polytope, we still have such boundary structures exactly as any $u_{\alpha}\to 0$.

How do we find the ${\bf u}$ variables given an alphabet of $N$ polynomials? This has been proposed in Arkani-Hamed:2019mrd , and the basic idea is to use any positive parametrization such that the $N$ polynomials can be expressed as subtraction-free Laurant polynomials of positive coordinates $x_{1},\cdots,x_{d}$, which we denote as $p_{I}(\{x\})$ for $I=1,\cdots,N$; then we study the so-called stringy canonical forms Arkani-Hamed:2019mrd :

$$I_{\{p\}}(\{s\})=(\alpha^{\prime})^{d}\int_{\mathbb{R}_{>0}^{d}}\prod_{i=1}^{d}d\log x_{i}\prod_{I=1}^{N}p_{I}(\{x\})^{\alpha^{\prime}s_{I}}\,,$$

(9)

where we integrate $\prod_{i}{\rm d}\log x_{i}$ in the positive domain, and we take polynomials $p_{I}$ (to the power $\alpha^{\prime}s_{I}$ as “regulators” for potential divergences. The domain of the convergence for $I_{\{p\}}(\{s\})$ is given by a polytope in the exponent space, which is defined as the the Minkowski sum of Newton polytopes of $p_{I}$’s ²²2The $\alpha^{\prime}\to 0$ limit of the integral itself is given by the canonical form of the polytope (hence the name).. One can generally define ${\bf u}$ variables (and configuration spaces) for such an integral following the procedure in Arkani-Hamed:2019mrd , and quite beautifully all finite-type cluster algebras belong to a special case that the corresponding polytope has exactly $N$ facets and it is cut out by $X_{\alpha}(\{s\})\geq 0$ for $\alpha=1,\cdots,N$, known as the ABHY realization of cluster polytopes; for type $A,B,C,D$, the canonical form of these polytopes, or $\alpha^{\prime}\to 0$ limit of the integrals, have nice interpretation as planar $\phi^{3}$ amplitudes through one-loop Arkani-Hamed:2017mur ; Arkani-Hamed:2019vag . We can recombine the exponents into $X_{\alpha}$’s and the regulator becomes $\prod_{\alpha=1}^{N}{\bf u}_{\alpha}^{\alpha^{\prime}X_{\alpha}}$, where the polynomials combine into exactly the $N$ ${\bf u}_{\alpha}$ variables! Thus the ${\bf u}$ variables can be obtained by a Minkowski-sum calculation given any positive parametrization.

For example, by using any positive parametrization of polynomials in $\Phi_{A_{d}}$, we recognize $I_{\{p\}}$ as the usual $(d{+}3)$-point open-string integral, and the Minkowski sum gives exactly the ABHY associahedron in the kinematic (Mandelstam) space; the ${\bf u}_{ij}$ variables are then dihedral coordinates of ${\cal M}_{0,d{+}3}$ brown2009multiple , given by cross-ratios of the world-sheet coordinates $z_{i}$’s (with three additional fixed at $(0,-1,\infty)$).

Let us start with the well-known ladder integral, $I^{(L)}_{\rm b}(x_{1},x_{3},x_{5},x_{7})$ which depends on two cross-ratios and it is convenient to introduce $z$ and $\bar{z}$ variables defined by

$$\frac{z\bar{z}}{(1-z)(1-\bar{z})}=\frac{x_{13}^{2}x_{57}^{2}}{x_{15}^{2}x_{37}^{2}}\,,\quad\frac{1}{(1-z)(1-\bar{z})}=\frac{x_{17}^{2}x_{35}^{2}}{x_{15}^{2}x_{37}^{2}}\,.$$

(10)

The ladder integrals have been evaluated in ussyukina1993exact , and one way to do so is by solving differential equations they satisfy, which nicely relate $I^{(L)}_{\rm b}$ to $I^{(L{-}1)}_{\rm b}$. Recall that the integral has a natural overall normalization factor $I^{(L)}_{\rm b}:=f^{(L)}/(z-{\bar{z}})$ such that $f^{(L)}$ becomes pure function of weight $2L$. The functions $f^{(L)}$ satisfy second-order differential equations:

$$z\partial_{z}\bar{z}\partial_{\bar{z}}f^{(L)}(z,\bar{z})=f^{(L{-}1)}(z,\bar{z}),$$

(11)

with “tree-level source” $f^{(0)}=\frac{z\bar{z}}{(1-z)(1-\bar{z})}$. There is a closed-form expression for $f^{(L)}$ from solving the differential equations:

$$f^{(L)}=\sum_{m=L}^{2L}\frac{m!\ (\log(-z\bar{z}))^{2L-m}}{L!\ (m-L)!\ (2L-m)!}\left({\rm Li}_{m}(z)-{\rm Li}_{m}(\bar{z})\right)$$

(12)

which is a single-valued, analytic function of $z$ (In Euclidean signature, $z$ and $\bar{z}$ are complex conjugates to each other). From (12) it is obvious that the alphabet of the symbol consists of $4$ letters, $\{z,\bar{z},1-z,1-\bar{z}\}$, which we can immediately identify as that of $D_{2}\simeq A_{1}^{2}$.

We are mostly interested in positive external kinematics, where momentum twistors ${\bf Z}\in G_{+}(4,n)$, and it is easy to see that for such kinematics we have $z<0$ and $\bar{z}<0$. To relate this alphabet to the positive ${\bf u}$ variables of $D_{2}$, we make the change of variables ${\bf u}:=z/(z-1)$ and ${\bf\bar{u}}:=\bar{z}/(\bar{z}-1)$. The $D_{2}$ alphabet can be alternatively written in these variables ³³3Our convention follows that of Drummond:2010cz , and differs from He:2020lcu where the $z$ and $\bar{z}$ would be the variables ${\bf u}$ and ${\bf\bar{u}}$ here.:

$${\bf A}[I_{\rm b}^{(L)}]=\{{\bf u},1-{\bf u},{\bf\bar{u}},1-{\bf\bar{u}}\}$$

(13)

where all the letters are positive (between $0$ and $1$); this is the ${\bf u}$ space of $D_{2}$ which is literally a quadralateral.

The variables $z,{\bar{z}}$ (or equivalently ${\bf u},{\bf\bar{u}}$) are algebraic functions of momentum twistors, since they are two roots of the above quadratic equation. Without loss of generality for this specific problem, we can “rationalize” the square root explicitly by reducing the kinematics to two dimensions Caron-Huot:2013vda . Recall that when external kinematics lie in two-dimensional subspace, the polygon can only have even number of edges (which we denote as $2n$) and take a zigzag shape: edges with even and odd labels go along two light-like directions respectively. It is convenient to reduce momentum twistors as $Z_{2i-1}=(\lambda_{2i-1}^{1},0,\lambda_{2i-1}^{2},0)$ and $Z_{2i}=(0,\tilde{\lambda}_{2i}^{1},0,\tilde{\lambda}_{2i}^{2})$, which reduces the conformal group ${\rm SL}(4)$ to ${\rm SL}(2)\times{\rm SL}(2)$. The kinematics are encoded in even and odd ${\rm SL}(2)$ invariants $\langle i\,j\rangle$ (or $[i\,j]$) for odd (or even) $i,j$, which are in fact one-dimensional distances along odd (or even) direction.

Any cross-ratio factorizes into the product of an even and an odd cross-ratio, e.g. for $i,j,k,l$ all even, we have $\frac{x_{ij}^{2}x_{kl}^{2}}{x_{ik}^{2}x_{jl}^{2}}\to u^{\rm 2d}_{i-1,j-1,k-1,l-1}u^{\rm 2d}_{i,j,k,l}$ where $u^{\rm 2d}_{i,j,k,l}:=\frac{[i\,j][k,l]}{[i\,k][j\,l]}$ denote familiar cross-ratios of $A_{n{-}3}\sim G(2,n)/T$ in the even sector (and similarly an $A_{n{-}3}$ in the odd sector). Now for the box-ladder with $2n=8$, we see that the $2d$ kinematics naturally require two $A_{1}$’s (for even and odd sectors), and we the square root disappear to give Caron-Huot:2013vda

$${\bf u}=u^{\rm 2d}_{1,3,5,7}\,,\quad{\bf\bar{u}}=u^{\rm 2d}_{2,4,6,8}$$

and $1-{\bf u}=u^{\rm 2d}_{3,5,7,1}$ etc.. Thus we see that the ${\bf u}$ variables are literally the ${\bf u}$ variables for the two $A_{1}\sim G(2,4)/T$.

Moreover, it is trivial to see that we have $A_{1}$ sub-algebras of $D_{2}$ which can be reached when any of the ${\bf u}$ variables goes to zero. This is well known since e.g. at one-loop, as ${\bf u}\to 0$ the box function diverges, but we can look at the “finite part” ${\rm Li}_{2}(1-{\bf\bar{u}})$ which has the $A_{1}$ alphabet $\{{\bf\bar{u}},1-{\bf\bar{u}}\}$. Although this particular integral $I_{\rm b}^{(L)}$ diverges at any “boundary” of the $D_{2}$, more generic $D_{2}$ cluster functions can have such $A_{1}$ functions in these limits.

Next we consider a more non-trivial example, $I^{(L)}_{\rm pb}(x_{1},x_{2},x_{4},x_{5},x_{7})$, which depends on $3$ cross-ratios defined as

$$u=\frac{x_{17}^{2}x_{25}^{2}}{x_{15}^{2}x_{27}^{2}}\,,\quad v=\frac{x_{14}^{2}x_{57}^{2}}{x_{15}^{2}x_{47}^{2}}\,,\quad w=\frac{x_{15}^{2}x_{24}^{2}}{x_{14}^{2}x_{25}^{2}}\,.$$

(14)

As shown in He:2020uxy , we obtain a two-step recursion relation for $I_{\rm pl}(u,v,w)$:

$$\begin{split}I_{\rm pb}^{(L{+}\frac{1}{2})}(u,v,w)&=\int_{0}^{\infty}{\rm d}\log\frac{t+1}{t}\ I_{\rm pb}^{(L)}\left(\frac{u(t+w)}{t+uw},v,\frac{w(t+1)}{t+w}\right)\\ I_{\rm pb}^{(L{+}1)}(u,v,w)&=\int_{0}^{\infty}{\rm d}\log(s+1)\ I_{\rm pb}^{(L{+}\frac{1}{2})}\left(u,\frac{v(s+1)}{vs+1},\frac{s+w}{s+1}\right)\end{split}$$

(15)

We remark that the actual integrals with even weight are symmetric in exchange of $u$ and $v$ (the integral has a symmetry axis); we introduce odd-weight objects which break the symmetry, but we can alternatively write down recursion with $u,v$ swapped, which give different odd-weight functions but will not affect the even-weight integrals. Nicely, the recursion applies to $L=0$, where the tree case is defined to be $I_{\rm pb}^{(0)}=1-u-v+uvw$. By applying the first equation of  (15) to this tree result, we obtain a weight-$1$ function $I_{\rm dp}^{(1/2)}=\frac{1-u-v+uvw}{1-uw}\log(uw)$, and by applying the second equation, we arrive at the well-known one-loop chiral-pentagon $I_{\rm pb}^{(L=1)}:=I_{\rm p}(u,v,w)$:

$$I_{\rm p}(u,v,w)=\log u\log v+\operatorname{Li}_{2}(1{-}u)+\operatorname{Li}_{2}(1{-}v)+\operatorname{Li}_{2}(1{-}w)-\operatorname{Li}_{2}(1{-}uw)-\operatorname{Li}_{2}(1{-}vw).$$

The recursion makes it manifest that the result will always be pure functions starting $L=1$. We emphasize that by using the algorithm of CaronHuot:2011kk , it is straightforward to compute the symbol of $I_{\rm pb}^{(L)}$ to any loop order.

We are mainly interested in the alphabet of the resulting symbol. As we have seen at $L=1$ (weight $2$), and in fact also for $L=\frac{3}{2}$ (weight $3$) as obtained using the first line of (15), the alphabet consists of eight letters, $u,v,w,1-u,1-v,1-w,1-uw,1-vw$. However, these are just degenerate cases, and starting at $L=2$ we find $9$ letters where the additional one is nothing but the tree-level factor $1-u-v+uvw$:

$${\bf A}[I_{\rm pb}^{(L)}]=\{u,v,w,1-u,1-v,1-w,1-uw,1-vw,1-u-v+uvw\}.$$

(16)

We have checked up to $L=5$ (weight $10$), and in sec. 3 we will give an all-order proof using the recursion that this $9$-letter alphabet is true to all loops with $L\geq 2$.

Now we identify the alphabet ${\bf A}[I_{\rm pb}^{(L)}]$ with that of $D_{3}\simeq A_{3}$ cluster algebra, and we do so in two ways. First, similar to Chicherin:2020umh , we find the bi-rational change of variables

$$u=\frac{1}{1+z_{2}}\,,\quad v=\frac{1}{1+z_{3}}\,,\quad w=1+z_{1}$$

(17)

and, up to multiplicative redefinition, the alphabet (16) becomes

$${\bf A}[I_{\rm pb}^{(L)}]\simeq\{z_{1},z_{2},z_{3},1+z_{1},1+z_{2},1+z_{3},z_{1}-z_{2},z_{1}-z_{3},z_{1}+z_{2}z_{3}\}.$$

(18)

which we immediately recognize as that of $D_{3}$ (second line of (5) with $d=3$). A trivial change of variables turns it into that of $A_{3}$ (first line of (5) with $d=3$).

These changes of variables may seem a bit arbitrary, but as mentioned earlier we can reach at the conclusion in a totally invariant way. Pick any positive parametrization of the $9$ letters in (16); it does not matter what positive parametrization we choose as long as they give subtraction-free polynomials $p_{I}$ for $I=1,\dots,9$, and we write the stringy canonical form

$$I_{\{p\}}(\{s\})=(\alpha^{\prime})^{3}\int_{\mathbb{R}_{>0}^{3}}\prod_{i=1}^{3}d\log x_{i}\prod_{I=1}^{9}p_{I}(x_{1},x_{2},x_{3})^{\alpha^{\prime}s_{I}}\,.$$

(19)

Without being smart, we can simply compute the Minkowski sum of Newton polytopes of $p_{I}$’s which gives the convergence domain of $I_{\{p\}}$, and we find that the result is nothing but a $3$-dimensional associahedron! It is given by $9$ inequalities of the form $X_{a}(\{s\})\geq 0$, each of which can be written as a linear combination of the $s_{I}$’s. With these $9$ linear combinations, we can identify the $9$ ${\bf u}$ variables of $A_{3}$ by writing $p_{I}^{\alpha^{\prime}s_{I}}=\prod_{(i,j)}{\bf u}_{i,j}^{\alpha^{\prime}X_{i,j}}$. With a bit hindsight, we label the ${\bf u}$’s by diagonals of a hexagon as in (6), and they automatically satisfy the $9$ ${\bf u}$ equations in (7). This gives a description of the $A_{3}$ alphabet that is totally parametrization-independent, ${\bf A}[I_{\rm pb}^{(L)}]=\{{\bf u}_{i,j}|1\leq i<j-1<6,~{}(i,j)\neq(1,6)\}$, where the ${\bf u}$ variables are multiplicative combinations of the original letters:

		$\displaystyle{\bf u}_{1,3}=w,\,{\bf u}_{1,4}=\frac{1-v}{1-vw},\,{\bf u}_{1,5}=\frac{u(1-vw)}{1-v},\,{\bf u}_{2,4}=1-vw,\,{\bf u}_{2,5}=\frac{1-w}{(1-uw)(1-vw)},$
		$\displaystyle{\bf u}_{2,6}=1-uw,\,{\bf u}_{3,5}=\frac{v(1-uw)}{1-u},\,{\bf u}_{3,6}=\frac{1-u}{1-uw},\,{\bf u}_{4,6}=\frac{1-u-v+uvw}{(1-u)(1-v)}$		(20)

One can easily check that (2.2) satisfies ${\bf u}$ equations and with any positive parametrization all ${\bf u}$ variables are between $0$ and $1$.

As mentioned such a description using ${\bf u}$ variables is very useful, e.g. each of the $9$ boundaries can be obtained by sending exactly one ${\bf u}$ to zero. There are $6$ boundaries by ${\bf u}_{i,i{+}2}\to 0$ (for $i=1,\cdots,6$) which are $A_{2}$ (pentagon), and $3$ boundaries by ${\bf u}_{i,i{+}3}\to 0$ (for $i=1,2,3$) which are $D_{2}\simeq A_{1}^{2}$ (quadrilateral). However, for $I_{\rm pb}^{(L)}$ many boundaries are too degenerate as the symbol vanishes identically, only the following $4$ boundaries correspond to non-trivial limits: ${\bf u}_{1,3}\to 0$, ${\bf u}_{1,5}\to 0$, ${\bf u}_{3,5}\to 0$, which are $A_{2}$, and ${\bf u}_{2,5}\to 0$, which is $A_{1}$.