Cluster Alphabets from Generalized Worldsheets: A Geometric Approach to Finite Types

The search for a geometric description and a simple set of variables has guided the study of scattering amplitudes in quantum field theory and string theory. Historically, the Veneziano amplitude consistent with the Regge poles and crossing symmetry was written down first, then extended to $n$-point amplitudes, and the notion of a worldsheet swept out by the motion of strings emerged only later.

The generalization of the Veneziano amplitude to $n$ mesons is expressed as [1, 2, 3]

Here the integral is over the $n-3$ compatible resonances, $\alpha^{\prime}$ is the Regge slope, and $X_{i,j}$ are functions of the Mandelstam variables $s_{i,j}$. It was soon realized that the $u$ variables may be written as cross ratios of the Koba-Nielsen $z$ variables [4].

This leads to the expression of the integration measure in the Parke-Taylor form $\prod_{i=1}^{n}dz_{i}/{(z_{i}-z_{i+1})}$ and the Koba-Nielsen factor $|z_{i}-z_{j}|^{\alpha^{\prime}s_{i,j}}$ familiar in modern textbooks, with the residual gauge symmetry used to fix the positions of three points, e.g., $z_{1}\to-1,z_{2}\to 0,z_{n}\to\infty$. It is now recognized that this integral describes the tree-level amplitude of open strings, whose worldsheet is a disc with $n$ marked points at the boundary. The string amplitude enjoys properties such as crossing symmetry, factorization, and Regge behavior.

More recently, the factorization property of the string integral was put on the center stage to define a class of generalized string integrals associated with Dynkin diagrams [5]. The so-called cluster string integrals factorize at the poles that correspond to the boundaries of the configuration space of $u$ variables [6, 7]. For example, the $A_{n-3}$ integral (1) factorizes into an $A_{n-k-2}$ integral and an $A_{k-2}$ integral at its poles. The factorization property is reflected in the geometry of the generalized associahedra [8] and the integrals are interpreted as volume forms. However, like the multimeson amplitudes, the integrals are written in terms of the $u$ variables as a generalization of (1). It was not clear what the underlying worldsheet picture is.

A second motivation for this work comes from the structure of field-theory amplitudes. The amplitudes are expressed in terms of generalized polylogarithms. The cluster bootstrap program attempts to constrain the form of the amplitude using a set of symbol alphabets [9, 10, 11]. In a related development [12], a class of alphabets based on cluster algebras of finite type was proposed using birational maps from the kinematic variables:

It was shown that the Feynman integral for the one-loop Bhabha scattering correspond to the $A_{3}$ cluster alphabet, a certain six-dimensional hexagon integral to the $D_{4}$ cluster alphabet, etc. However, the cluster alphabets were found by a clever choice of birational maps, and it was not clear how to derive them for other finite-type cluster algebras. The $A$-type alphabet is the set of gauge-fixed factors in the $u$ variables (2), as string amplitudes reduce to field-theory amplitudes in the $\alpha^{\prime}\to 0$ limit. It was also not clear whether the other alphabets have any geometric origin or if there is an underlying worldsheet at all.

In [13], a systematic derivation of such variables was proposed based on $Y$ systems, and the results for $D$ types were presented in detail. The strategy is to construct the generalized worldsheets through a “gluing” construction. We solve the $u$ variables in terms of the worldsheet coordinates. Like in the $A$ type, the elements of an alphabet, called letters, are the factors that appear in the $u$ variables. As it stands, there are more ungauged letters than the number of cluster variables. Upon a choice of gauge, the alphabets are then read off from the factors. The number of letters in a cluster alphabet is shown in Table 1.

This paper aims to derive the cluster alphabets for all the finite types. Our main results are as follows:

• 1.

  Systematic construction of the generalized worldsheet for all finite types.

  We provide a systematic derivation of the gluing construction of the exceptional types. We begin by reviewing the gluing construction of the $D_{n}$ worldsheet. We show how the construction extends to the exceptional types, and derive an explicit cross-ratio representation of all the $E_{6}$ $u$ coordinates. For the nonsimply laced types, we present the folding map that identifies the worldsheet coordinates. Our results may also be seen as an explicit verification of Zamolodchikov’s periodicity conjecture for $Y$ systems [14].

• 2.

  New cluster alphabets.

  In the standard gauge choice, one may obtain the cluster alphabet for $BCD,E_{6},F_{4},G_{2}$ types as polynomials of degrees at most 2, 4, 5, 4, respectively. By choosing a different gauge, we produce a new, simpler set of alphabets. We obtain a linear alphabet for $B$ type, quadratic alphabets for $CD$ types, and for $E_{6},E_{7},E_{8},F_{4},G_{2}$ types, polynomials of degrees at most 4, 5, 7, 4, 2, respectively.

Recall that one of the main features of the $A_{n-3}$ worldsheet is that each diagonal divides an $n$-gon into an $(n-k+1)$-gon and a $(k+1)$-gon. The string amplitude factorizes at each pole (boundary of the $u$ space) as

We can make similar statements for the other types with the picture of glued polygons. Each diagonal in the initial triangulation corresponds to a node on the Dynkin diagram and slices the polygon into two parts. The $D_{5}$ example is shown in Fig. 3.

The multiplicity $n$ is determined by the periodicity $n$ of the $D_{n}$ $Y$-system.

The boundaries of the $E_{n}$ $u$ space can be obtained similarly, as shown in Fig. 4 for $E_{6}$.

There is an overall factor because the $E_{6}$ system has period 7. Note that for $E_{n}$ types, there are more boundaries than diagonals available. We will only identify the possible types of boundaries using the initial cluster, and the remaining boundaries will be obtained by evolving the $Y$-system equations. All the $u_{i}(t)$ at the same $i$th node correspond to the same type of boundary. For example, the 128 boundaries of the $E_{8}$ worldsheet are

The worldsheet parametrization for the nonsimply laced types can be achieved by a process known as folding. The folding map on the $z$ parameters is derived from the standard folding of the root systems combined with the birational map in the $ADE$ types. See Figs. 5 and 6.

In the standard gauge choice, which is usually taken to be $z_{1}\to-1,z_{2}\to 0,z_{n}\to\infty$, we recover the known cluster alphabets of types $A,C,D$ and obtain new ones for type $E$ and the nonsimply laced ones. However, it is reasonable to suspect that we have not found the simplest possible choice. Unlike the $A_{n}$ case, not all worldsheet variables are on an equal footing. Different gauge choices will lead to different alphabets. While there is no canonical choice of the alphabet, choosing a gauge that yields letters that are polynomials of the lowest order is preferable. We say two alphabets are equivalent if seen as a collection of hypersurfaces, they have the same topological property. That is,

1. 1.

   The number of letters equals the dimension of the cluster algebra.

2. 2.

   They give the same point count in the hypersurface complement.

A simpler $D_{n}$ alphabet is obtained by the gauge choice $z_{n+3}\to-1,z_{1}\to 0,z_{n+1}\to\infty$. Let

be the gauge-fixed version of $w_{i,j}$. The $D_{n}$ alphabet is

A similar gauge choice ($z_{9}\to-1,z_{1}\to 0,z_{6}\to\infty$) removes the terms containing $z_{i}^{2}$ in the $E_{6}$ alphabet (18). Let