Landau-Ginzburg skeletons

We study the class of indecomposable two-dimensional Landau-Ginzburg theories with (2,2) supersymmetry and central charge c<6 with the aim of classifying all such theories up to marginal deformations. Our results include cases overlooked in previous classifications. The results are rigorous for three or fewer fields and more generally are rigorous if we assume an extra bound. Numerics suggest that we have the complete set of indecomposable Landau-Ginzburg families with c<6. This set consists of 38 infinite families and a finite list of 418 sporadic cases. The basic tools are classic results of Kreuzer and Skarke on quasi-homogeneous isolated singularities and solutions to certain feasibility integer programming problems.


Introduction
Landau-Ginzburg (LG) theories have played an important role in the study of critical phenomena. This is especially true in the case of two dimensions, where they yield Lagrangian theories that flow to a wide class of interacting IR fixed points. There has been important work on LG theories with various amounts of supersymmetry. For instance, the seminal work of [1] pointed out the connection between minimal models and LG theories; theories with (1,1) supersymmetry were discussed in [2]; theories with (2,2) supersymmetry were introduced in [3,4]. The defining data of (2,2) LG theories is intimately related to the mathematics of isolated quasi-homogeneous hypersurface singularities, a subject studied for many years, going back to, for example, the classic work [5]. The graphical techniques of the sort we will be using go back to [6]. 1 The purpose of this work is to describe the combinatorics of (2,2) LG theories and classify indecomposable theories with central charge c < 6. Our interest in this question is threefold. First, a rich correspondence between these LG theories and N = 2, d = 4 superconformal field theories was proposed in [8]. It has been investigated in a geometric setting of rational Gorenstein singularities in [9][10][11] based on the earlier [12]. Our work generalizes those studies. We find several isolated rational Gorenstein hypersurfaces singularities in C 4 overlooked in [12], and, more generally, we also describe LG theories that do not correspond to hypersurface singularities in C 4 . Second, the c < 6 boundary is interesting since theories with c < 6 can arise in descriptions of various singular limits of (4, 4) superconformal theories with c = 6. We have in mind here the kinds of limits described in [13]. Finally, a good understanding of the (2,2) combinatorics should serve as a springboard for efforts to classify (0,2) LG theories. Although these were introduced and studied many years ago in [14,15], we have a very incomplete understanding and no classification even at fixed central charge!
The tools we use were developed for the purpose of classifying (2,2) LG theories at fixed central charge; c = 9 theories were classified in [16][17][18], and c = 12 theories in [19]. 2 We will review them, specifically the elegant approach developed in [17], in the next section and in section 3, and then we will turn to the c < 6 classification problem in sections 4 and 5. We end with a summary of our findings.

Quasi-homogeneous combinatorics
A (2,2) LG theory is a theory specified by a UV Lagrangian for n chiral superfields X i with canonical quadratic kinetic terms and polynomial superpotential interactions: Here (z, z) are coordinates on the Euclidean world-sheet, and θ ± , θ ± are the corresponding superspace coordinates. The θ ± (θ ± ) carry U(1) L (U(1) R ) charges ±1, and d 2 zdθ + dθ + is the chiral superspace measure. We demand that the interactions preserve a U(1) L × U(1) R Rsymmetry, which requires the superpotential W (X) to be a quasi-homogeneous polynomial.
That is there are non-negative rational charges q i ∈ Q such that for any t ∈ C * W (t q i i X i ) = tW (X) . (2. 2) The theory has a supersymmetric RG flow that preserves the U(1) L × U(1) R symmetry and flows to a strongly coupled theory in the IR. We will make the standard assumption that the U(1) L × U(1) R symmetries correspond to the left and right symmetries of a (2,2) SCFT. In particular, the chiral field X i will flow to a chiral primary (more precisely (c,c) ) operator with U(1) L × U(1) R charge (q i , q i ). Since the theory is left-right symmetric, in what follows we will simply refer to q i as the R-charge of X i . We will also require that the theory has a normalizable vacuum. We will assume this is equivalent to the scalar potential of the LG theory having an isolated vacuum, i.e. W (X) is compact.
We say that W (X) is compact if and only if X 1 = X 2 = · · · = X n = 0 is the unique solution to {W 1 = 0, W 2 = 0, . . . , W n = 0} ∈ C n ; in other words, W has a unique critical point.
The following are equivalent: 3. the Jacobian ring R W = C[X 1 ,...,Xn] W is finite dimensional as a vector space over C; 4. the determinant of the Hessian matrix of W (X) is non-zero in R W . These equivalences are well-known; see, for instance, [20].
With our assumptions the central charges of the (2,2) SCFT are determined by the anomalies of the UV U(1) L × U(1) R symmetries: they are equal and given by 3) The classification of LG theories at a fixed c amounts to finding all n and charges (q 1 , . . . , q n ) ∈ Q n that yield the desired c and admit a compact superpotential. While we will not use this result, we note [17] that for any fixed rational c there is a finite set of (q 1 , . . . , q n ) ∈ Q n that yield compact theories. It may be that a compact W is decomposable, i.e. it can be written as In that case W (1) and W (2) both define compact LG theories with, respectively, k and n − k variables. Since these are clearly non-interacting, the IR fixed point will be a product of the two SCFTs, and in our classification of LG families we may as well ignore such decomposable theories. 3 In the graphical notation that we will introduce shortly this will amount to ignoring disconnected graphs.

Constraints on the charges
To describe the classification further, we will need to introduce some more notation and results. Most of these are discussed in [17], and we include them here for completeness and reference. Throughout, W will be a polynomial quasi-homogeneous superpotential, i.e. it takes the form where the a p are complex coefficients, and ∆ is the Newton polytope for W . Because W is polynomial and quasi-homogeneous, ∆ lies in the positive orthant intersected with the hyperplane n i=1 q i p i = 1 . Thus, the charges (q 1 , . . . , q n ) specify a family of LG theories: the coefficients a p modulo the action of holomorphic field redefinitions correspond to marginal deformations of the theory. We will say that the family determined by (q 1 , . . . , q n ) is generically compact (or gc for short) if W is compact for a generic choice of the coefficients a p . For a gc family the basic properties of the SCFT, such as the Zamolodchikov metric and threepoint functions vary smoothly with these parameters away from a complex co-dimension one singular locus where W fails to be compact. Definition 2.2. For a fixed superpotential W a variable X i is a root if and only if W contains the monomial X m i i . 4 In this case we call the integer m i the exponent of the root. A variable that is not a root is a pointer if and only if W contains the monomial X m i i X j with j = i; in this case we say X i points at X j and has exponent m i . Lemma 2.3. If W is compact, then every variable is either a root or a pointer.
Proof. Suppose W is a superpotential with variable X 1 that is neither a root or a pointer. Every monomial in dW contains at least one X j for i > 1. Hence, dW | X i>1 =0 = 0 for all X 1 , and the potential is not compact. Proof. Let Y be any variable with q Y > 1/2 and denote the remaining fields by Z. With our assumptions on the charges Any quasi-homogeneous polynomial F (Z) with positive charge vanishes at Z = 0, so there are two possibilities: 1. q Y = 1. In this case F (Z) = F 0 , a constant, and W has no critical point.
Hence W is singular unless among the Z there is a field Z with q Z = 1 − q Y and F (Z) = Z + . . ., where ". . ." denote non-linear terms in the Z; these must be independent of Z. In other words, Hence, Y is a Lagrange mutliplier enforcing Z = F nlin . Since the right-hand-side is Z independent, we can solve the condition and obtain the equivalent superpotential This is IR-equivalent to W = G(Z, F ), a potential without either the Y or Z field.
Repeating this for every field with q Y > 1/2, we obtain an IR-equivalent potential where every field has q ≤ 1/2.
In fact, we have a stronger result.
Proposition 2.6. Up to IR equivalence we can assume 0 < q < 1/2 for every compact potential W ; in particular, every monomial in W is at least cubic in the fields.
Proof. By the previous lemma, we already have q i ≤ 1/2. Now we decompose the fields into Y a , a = 1, . . . , k, with q a = 1/2 and Z s with q s < 1/2. After an SO(k) linear transformation on the Y a , we find that the general form of the potential is (2.10) Since deg F a = 1/2 and q Z < 1/2, it follows that dW | Z=0 = a Y a m a dY a . Hence, W is non-compact if any m a = 0. On the other hand, if m a > 0 for all a, then we can use the equation of motion Y a = m −1 a F a to obtain the IR-equivalent potential (2.11)

Constraints on n from c
A gc family leads to a family of IR fixed points with Since every q i > 0 it follows that n > c/3. We also see that as q i → 0, we recover the free field central charge values 5 ; of course in the strict limit we would get a non-compact theory.
Here is a less trivial consequence, also proven in [17].
Proposition 2.7. A gc family has c/3 < n ≤ c.
Hence, to describe theories with c < 6 we consider 1 ≤ n ≤ 5 and In other words, we have n < 3 n = 3 n = 4 n = 5 (2.14) To prove the non-trivial part of the proposition, observe that every field with q i ≤ 1/3 contributes at least 1 to the central charge, and a pair of fields X, Y with 1/3 < q X < 1/2 and q Y = 1 − 2q X contributes at least 2 to the central charge. The claim then follows from Lemma 2.8. Fix a gc family. Suppose there are k fields Z a with q(Z a ) ∈ (1/3, 1/2). Then for every Z a there is a distinct field Y a with q(Y a ) = 1 − 2q(Z a ).
Proof. Z a must be a root or a pointer, but its charges only allow it to be a pointer of the form W ⊃ Z 2 a Y a , with q(Y a ) = 1 − 2q(Z a ). Suppose Z 1 and Z 2 point to the same field Y . Then Z 1,2 have the same charge, and the potential is of the form where U denote all the other fields, and Q 1,2 are quadratic polynomials. It is then easy to see that dW = 0 whenever Y = U = 0 and Q 1 (Z 1 , Z 2 ) = 0, and hence W is not compact.
To finish the proof of the proposition, we group our fields by the pairs (Z a , Y a ), as well as any remaining fields U with q(U ) ≤ 1/3.

A graphical notation and the Kreuzer-Skarke theorem
Every compact potential W with n variables yields a directed graph, known as the skeleton graph of W , obtained as follows. Every variable corresponds to a node indicated by an open circle •; there is an arrow from node i to node j if and only if X i points to X j . In general a W may yield several distinct graphs. For instance, we have the following potentials and corresponding graphs with n ≤ 2: Note that we decorate each node with an exponent label; this is a convenient way of keeping track of the combinatoric and integer data in the same graphical notation.
Every gc family with n = 2 has a potential of one of these forms. Note that a skeleton graph uniquely determines the charges in terms of the exponents.
A skeleton potential, even if taken with generic coefficients, need not be smooth. To see this, consider the n = 3 skeletons: Most of these are smooth and hence define gc families: just by including the indicated monomials with generic coefficients, we are guaranteed compactness. This is not the case for the second column with 1 or 0 roots: we need additional monomials. For instance, consider the example Without including another monomial, this will be non-compact. To see this, we compute 18) and this clearly has a zero with X 1,3 = 0. However, if we can add X p 1 1 X p 3 3 , then we find a gc family. This addition is indicated by drawing a "link" between the nodes. Definition 2.9. Given a potential and its skeleton, a link between two nodes, indicated by a dashed line, denotes a monomial that only depends on the connected fields.
Thus, in our refined example, we have We need two more graphical concepts before we present the theorem of [17] on compact gc families. We already defined a link between two nodes. Similarly, we can recursively define links between links.
Definition 2.10. A link L between two links L 1 and L 2 corresponds to a monomial constructed from fields participating in links L 1 and L 2 . In this case we join L 1 and L 2 by another dashed line. In our graphical notation we find it convenient to add a connecting node (indicated by •) for every link involved in links between links.
We also generalize the concept of links and pointers.
Definition 2.11. A pointing link L is a link between two nodes or two links that points to a node j corresponds to a monomial that is linear in the variable X j and otherwise depends only on the fields that are joined by the link L. Note that the variable X j is not counted as one of the variables of the pointing link. We will also indicate the pointing links with a •.
To illustrate these definitions consider two ways to complete the skeleton to a gc family: On the left we have a link between 3 and 4, which corresponds to the monomial X p 1 1 X p 3 3 ; on the right we have a pointing link, which corresponds to the monomial X p 1 1 X p 3 3 X 4 . We now state the main theorem of [17].
Theorem 2.12. Fix a family of potentials. A necessary and sufficient condition for this to be gc is that it contains a member W that can be associated a graph with the following properties: 1. each variable is a root or a pointer; 2. for any pair of variables or for a variable and link that point at some node v, there is a link L joining the two pointers, and L does not point at v or any of the nodes that are targets of the objects that are joined by L.
This theorem can be used to describe all possible gc families with fixed n. At each n we construct all the skeleton graphs. Some will already fulfill the conditions of the theorem and so correspond to gc families with c < 3n. For those skeletons that do not fulfill the conditions of the theorem, we must add links and pointer links in all possible ways so as to fulfill the second criterion of the theorem. This leads to a complete (in general redundant) list of gc families.
Before we leave the general discussion, we prove one more result that will be useful to us.
Theorem 2.13. Let m i denote the exponents for the roots and pointers of a LG skeleton. Then the total R-charge with equality if and only if every variable is a root.
Proof. Let X α , α = 1, . . . , r denote the roots of the skeleton, and let X I , I = 1, . . . , n − r denote the pointers, with X I pointing at X π(I) . In particular, it means that W ⊃ X mα α for every α and W ⊃ X m I I X π(I) . Thus, q α = 1 mα and q I = 1 m I (1 − q π(I) ), and Since 0 < q π(I) < 1/2 for every pointer variable, the result follows.
This has the following corollary that will be useful in characterizing infinite families of exponents.
1. If n = 3, and for some pair 2. If n = 4, and for some triplet i = j = k 1 m i + 1 m j + 1 m k < 1, then the remaining exponent is bounded. The only way this fails is if the triplet has the form 3. If n = 5, and for some quadruplet i = j = k = l 1 m i + 1 m j + 1 m k + 1 m l < 3/2, then the remaining exponent is bounded. The only way this fails is if the quadruplet has the form We also note that q tot is polynomial in µ i = 1 m i , and in fact is linear in each of the µ i . It is then easy to see that q tot is a monotonically decreasing function with respect to each of the exponents.

The feasibility of links
Consider a connected graph G on n nodes for a superpotential W . The skeleton of the graph determines the charges q i in terms of the exponents. If a graph contains a link L linking, say, variables X 1 and X 2 , then W ⊃ X p 1 1 X p 2 2 , i.e. there exist non-negative integers p 1 , p 2 such that This is a non-trivial condition on the exponents, and for many exponents there will be no solution. We can cast this into a standard counting problem as follows. Let q 1 = r 1 /d and q 2 = r 2 /d, where r 1 , r 2 , d are integers and gcd(r 1 , r 2 , d) = 1. We then have to solve the equation for non-negative p 1 and p 2 . A necessary condition for a solution is that gcd(r 1 , r 2 )|d. When this holds, we can choose r 1 , r 2 , d so that gcd(r 1 , r 2 ) = 1, and we have the classic Frobenius 2-coin change problem and its solution: given two relatively prime positive integers r 1 and r 2 , the largest d which cannot be represented as If d < d max it may or may not be representable; indeed, we have the result that exactly half of the integers between 1 and (r 1 − 1)(r 2 − 1) are representable. 6 If G contains a pointing link with monomial W ⊃ X p 1 1 X p 2 2 X 3 , there is a similar problem to solve, with Hence, we have a collection of Frobenius coin change problems, one for every link. Each of these is a relatively simple 2-coin change problem. G may contain links between links (this is the case when two pointing links point to the same node). When this holds, we must consider a 4-coin change problem. In this case there is no known closed form expression for the largest non-representable integer. We make the following definition.

Examples of feasibility constraints
There are examples of skeletons and exponents for which there are no feasible graphs. Consider n = 4 and charges The most general superpotential is given by For any compact W the skeleton is constrained as follows: X 1 must be a root, X 2 and X 3 must be pointing at X 4 . Finally, X 4 is either a root or a pointer at 2. In either case there must be a link between X 2 and X 3 ; the link may point at X 1 . So, we have four possible graphs: However, with the charges as in (3.5) no graph is feasible. As we will see below, a graph G often allows for an infinite set of exponents consistent with c < 6; however, it may be that only a finite subset of those exponents is feasible. For instance, we give an example of an infinite family of exponents where only one member is feasible. We start with the skeleton superpotential To produce a gc family from this skeleton we need a link between X 1 and X 2 , which is possible if and only if is a positive integer. It is easy to see that k = 5 is the only feasible value, in which case this LG family is deformation-equivalent to the simpler and decomposable W = X 5 1 + X 5 2 + X 5 3 . As another example of linking subtleties, we can see that for a fixed skeleton and choice of exponents it may be that feasible graphs necessarily involve a pointing link. Consider the charges In this case, the most general superpotential is There are feasible graphs (the two bottom graphs in (3.7)), but they necessarily involve a pointing link. Note that the list of singularities in [12] does not involve any pointing links (no family contains a monomial in more than two variables) and is therefore incomplete.

Pointing links and disconnected skeletons
The combinatoric challenge of LG classification is in the analysis of links and pointing links.
There is one aspect of the problem that is simpler than it might first appear. While pointing links can connect distinct connected components of a skeleton graph, for n ≤ 5 the possibilities for this are limited. As we will see in the next section from the classification of connected skeletons, for n ≤ 2 there are no links, while for n ≤ 3 there is at most one link. An n = 4 connected skeleton has at most 3 links. Since we must have n ≤ 5 in total, there are just a few possibilities in which a pointing link can connect disconnected components of a skeleton: 1. a connected skeleton with n = 3 has a link that points at the n = 1 skeleton; 2. a connected skeleton with n = 3 has a link that points at a connected n = 2 skeleton; 3. a connected skeleton with n = 4 has k = 1, 2, 3 links that point at the n = 1 skeleton.

Connected skeletons in n ≤ 5 variables
In this section we classify all connected LG skeletons in n ≤ 5 variables. These results are surely known to many; see, for instance [7], however, we have not been able to find the n = 5 case in the literature.
A LG skeleton is a directed graph with n nodes and no edge from a node to itself. For each node let ℓ in and ℓ out denote, respectively, the number of incoming and outgoing edges. In a LG skeleton each node has at most one outgoing edge: ℓ out (v) = 0 when v is a root, and ℓ out (v) = 1 when v is a pointer. If the number of roots r ≥ 2 then the skeleton graph is necessarily disconnected. For a LG skeleton with p pointers we split the nodes v into the r = n − p roots {u 1 , . . . , u r } and the p pointers {w 1 , . . . , w r }. (4.1) Thus, every LG skeleton with p pointers yields a partition of p into n non-negative integers [ℓ in (u 1 ), . . . , ℓ in (u r ); ℓ in (w 1 ), . . . , ℓ in (w p )]. Without loss of generality we may assume ℓ in (u 1 ) ≥ ℓ in (u 2 ) ≥ · · · ≥ ℓ in (u r ) and similarly for the pointers. Suppose G is a graph for a gc LG family. When G is disconnected, there is a locus in the parameter space where the theory is compact and can be written as a direct product of SCFTs, one for each connected component. Hence, to classify the LG theories up to marginal deformations we can restrict attention to connected graphs.
Every graph G has a skeleton subgraph S obtained by dropping all of the links in G. The resulting skeleton may either be connected or disconnected. Turning this around, given a disconnected skeleton with components S 1 and S 2 , the only element in the construction that might yield a connected graph G involves a pointing link from either S 1 to S 2 or from S 2 to S 1 . So, to classify the indecomposable LG SCFTs with c < 6 we can proceed as follows: 1. construct all connected skeletons with n ≤ 5 nodes and 1 root; 2. construct all connected skeletons with n ≤ 5 nodes and 0 roots; 3. for every singular skeleton (i.e. those with ℓ in (v) > 1) add the required links; 4. find restrictions on exponents for n = 3, 4, 5 such that c < 6 and every link is feasible; 5. for every singular skeleton allow for any possible pointing links from the skeleton to itself (this will in general require links between links) and check for feasibility; 6. construct all connected graphs where a disconnected skeleton with c < 6 is made into a connected graph by pointing links.
In what follows we will carry out these steps with one important simplification: we will only be interested in indecomposable theories. An indecomposable gc-family has charges that do not allow W to be written as a sum of two non-interacting terms for any values of the complex coefficients away from the singular locus. We will work with a slightly more general notion.
Definition 4.1. We will say that a graph is reducible if one of the following holds: 1. the graph has more than 1 root; 2. the graph has 1 root, but the exponents are such that some pointer variable can appear in the superpotential as a root ; 3. the graph has 0 roots, but the exponents are such that some pointer variable can appear in the superpotential as a root.
In the first and second case the gc-family is decomposable; in the third case if precisely one pointer can appear as a root, the gc-family may not be decomposable, but it is then necessarily equivalent to a gc-family obtained from a graph with 1 root. So, to classify indecomposable theories by our procedure it is sufficient to describe irreducible theories.
In the remainder of this section we will construct all the connected skeletons, and we will indicate all the required links, but we will not illustrate the possible pointing links in the diagrams. In the following sections we will discuss the constraints on exponents, combinatorics of links and pointing links.

The connected skeletons with n ≤ 5
We will denote the connected skeletons in k nodes by S k,α , where α is a label for the graph type at fixed k. For each skeleton we will include the links necessary to yield a gc family, but at this point we will not distinguish the pointing links.

n ≤ 2 skeletons
These are straightforward. There are no links, and the skeletons are These are easy enough to construct directly, but with the n = 4, 5 cases in view, let us describe a procedure for obtaining all of the connected skeletons. We first note that a connected skeleton has either 1 root or 0 roots. Second, each node v has ℓ in (v) arrows pointing to it, which requires ℓ in (v) 2 links. So, we can organize the skeletons by the number of roots and the partition of p among the ℓ in (v). Applying this to n = 3, we first consider the case of 1 root, which can have at most 1 link: Next, we consider the cases without roots.
At this point we hope that the reader sees the translation from the graphic notation to the superpotential. It is then easy to find the charges in terms of the exponents. So, for the remaining cases we will simply present the graphs.
The skeletons with 1 root, organized by the number of links, are The skeletons without roots are as well as The skeletons with 1 root and 1 link are There is one skeleton with two links and one with three links: Finally, we have a skeleton with six links: We now move on to skeletons with 0 roots. Those with one link or less are Next, the skeletons with two links are The skeletons with three links are Finally, we have the skeletons with 4 and 6 links: • m 4 (4.14)

A priori bounds on exponents
To produce a LG theory each of the skeletons above is assigned a set of exponents, i.e. the integers m i for each root or pointer variable. Since we are not interested in variables with q i = 1/2, m i ≥ 3 when X i is a root, and m i ≥ 2 when X i is a pointer. For n < 3 these are the only constraints on the m i . Hence, each superpotential in (4.2) yields a gc family with c < 6. For n ≥ 3 there are non-trivial bounds on q tot given in (2.14), and these translate into bounds on the exponents m i . In order to avoid drowning in a case-by-case analysis, we would like to start with some a priori bounds on the m i that do not depend on the details of the skeleton. Such a bound follows from theorem 2.13. The details are relegated to the Appendix, and here we just state the results.
Let M = (m 1 , m 2 , . . . , m n ) be a set of exponents ordered according to A necessary condition for (2.14) is that where Σ ∞ n is composed is composed of infinite families of ordered n-tuples of integers, while Σ spor n is a finite list of ordered n-tuples of integers. To list these we will use a short-hand: e.g. (6..7, 6, 5) will stand for (6, 6, 5), (7,6,5) . For n = 3 we find We are finally ready to tackle our graphs. In the remainder of this section we will describe the feasible irreducible graphs. Since n ≤ 2 is trivial, we begin with the n = 3 skeletons.

Exponents for n = 3 graphs
Here there are no links, and we just need to describe the bounds on the exponents that lead to c < 6, i.e. q tot > 1/2. Our first step is to use (2.22) to limit the possible infinite families of exponents. Running through the finite list of cases in Σ ∞ 3 , we find the following infinite families: (2, * , * ) , ( * , 2, * ) , There are many similar decompositions, but there are also irreducible families; as an example, consider M = (m 1 , 3, 3), with m 1 not divisible by 7. The most general superpotential is 20) and this is irreducible.
This diagram is symmetric in the 1, 3 indices, so we will assume m 3 ≥ m 1 . In fact, we can assume m 3 > m 1 because if m 3 = m 1 and the link is feasible, then both X 3 and X 1 can be roots, i.e. the graph will be reducible. Since X 2 is a root we also have m 2 > 2. With these restrictions we find from (2.22) that the infinite families satisfying q tot > 1/2 are The last three cases are not feasible if m 2 is larger than 16. To see this, consider the exponent M = (3, m 2 , 5). The charges in this case are ( m 2 −1 3m 2 , 1 m 2 , m 2 −1 5m 2 ). Thus, to have the link between 1 and 3 we need a solution to So, m 2 ≤ 16. Proceeding in the same fashion, we find stronger bounds on m 2 for the exponents M = (3, * , 4) and M = (3, * , 6). Next, we consider the cases (4, 3, * ), (3, 4, * ), and (3, 3, * ).
1. M = (4, 3, m 3 ). In this case the charges are ( 1 6 , 1 3 , 2 3m 3 ). Since X 1 and X 2 can both be roots, this is feasible; on the other hand, it is manifestly a reducible LG family, so we need not consider it further.  If m 2 is even, then the first equation implies ℓ = 2r, but then q 3 = 1 2kr . So, this possibility leads to a reducible family.
If, on the other hand, m 2 = 2k + 1, then m 3 = kℓ, and the second equation becomes If p 1 = 0 or p 3 = 0, then the family will be reducible. So, we need only consider p 1 = 1 or p 1 = 2. If p 1 = 2, then ℓ = 2p 3 , and thus m 3 = 2kp 3 . This implies q 3 = 1 p 3 (2k+1) , so we again find a reducible family. The last interesting possibility is p 1 = 1. If ℓ is even, we again find a reducible family, but if ℓ = (2s + 1), we find an interesting possibility. In that case a solution for p 3 is possible if and only if k = 2t + 1, so that we find a family M = (2, 4t + 3, (2s + 1)(2t + 1)) . This is the only infinite family of exponents that yields irreducible feasible graphs for S 3,2 . Eliminating this infinite family from Σ 3 , we find just three remaining irreducible exponents: (3,7,4) , (3,13,4) , (3,16,5) . (4.28) The graph has cyclic symmetry. Using (2.22) we find that up to cyclic symmetry the infinite families are In addition to these Σ 3 contains, up to cyclic permutations, 48 sets of exponents with maximum exponent of 18.  This completes the analysis of exponents for n = 3 graphs without pointing links.

Exponents for the n = 4 graphs
We now turn to n = 4. Although the number of skeletons is large, we can reuse the n = 3 results because most of the n = 4 graphs arise as decorations of n = 3 graphs. In other words, they have one of two forms: Thus, we have 8 (4.34) 8 Recall that π(i) denotes the node pointed to by node i.
Since the pointing exponent m 4 ≥ 2, we conclude that If G 4 is a decoration of G 3 , then to find the exponents on G 4 consistent with (2.14), i.e. those consistent with c < 6 we can restrict attention to G 3 that satisfy the bound for n = 3. Moreover, whenever G 3 is reducible, then G 4 is necessarily reducible as well: this means that we need only worry about decorating the irreducible n = 3 graphs. Finally, suppose that Then for every irreducible G 4 that is a decoration of G 3 the m 4 exponent is bounded. We will now apply these ideas to the n = 4 graphs.
S 4,1 S 4,1 is a decoration of S 3,1 with X 4 pointing at X 1 , and therefore Since q tot is monotonically decreasing in each exponent, it is easy to see that q tot (S 3,1 ) ≤ 1 for all exponents, with equality precisely for M = (2, 2, 3); however, this has q 1 = q 2 = q 3 = 1/3 and obviously decomposes. Hence, for irreducible S 3,1 we have a better bound We also have q tot (S 3,2 ) ≤ 6/7 for irreducible theories, with equality realized by M = (2,7,3) or M = (3,7,2). This immediately leads to a bound m 4 ≤ 4. Taking a look at q tot (S 4,2 ), we then obtain just one infinite family with irreducible theories: M = (2, 4t + 3, (2s + 1)(2t + 1), 2) . Eliminating this from Σ 4 , we find that there are 5 additional irreducible theories with exponents: (2, 7, 3, 3) , (2,7,3,4) , (3,7,2,2) , (3,7,2,3) , (3,7,4,2) . Since the left-hand-side must be an integer, m 2 is bounded. 9 After checking through every infinite family of exponents in Σ ∞ 3 , we find that every exponent yields a graph that is reducible whenever it is feasible. We then also check that the same holds for every M ∈ Σ spor 3 . Thus, S 4,3 contributes no new irreducible theories. This has a nice consequence for the n = 5 analysis: every n = 5 graph that is a decoration of S 4,3 can now be ignored. All of these graphs behave as S 4,3 : there are no irreducible theories with c < 6. 9 In fact, here there are no m2 ≥ 2 for which the right-hand-side is an integer, but the point is that even if there were non-trivial solutions, they would be bounded.

S 4,5
This graph is not a decoration of an n = 3 graph. We use (2.23) to restrict the possible unbounded exponents, and we find that up to cyclic symmetry there are just three unbounded families:

Exponents for the n = 5 graphs
There is nothing conceptually different about the analysis of the n = 5 cases, but it does become a bit more tedious because there are more possibilities for the potentially infinite families of exponents. To handle these possibilities, we made a scan through all exponents with max M ≤ 100 for irreducible feasible graphs with q tot > 3/2. We found no evidence of infinite families and in fact just a short list of possible exponents with max M = 10. We believe we have the full set of exponents, and we will discuss this further below. For now we will simply summarize the results of our scan through the exponents. Eleven of the twenty n = 5 graphs are decorations of n = 4 graphs that are reducible for every exponent. For instance, S 5,5 is a decoration of S 4,3 , with X 5 pointing at X 1 . Following the general discussion of decorations, we need not consider such n = 5 graphs. This leaves us to consider nine graphs: S 5,α with α ∈ {1, 2, 3, 4, 10, 11, 12, 15, 18}. Of these only S 5,1 , S 5,2 and S 5,10 admit irreducible theories, and, furthermore, the single irreducible exponent for S 5,2 produces a set of charges already contained in the exponents of S 5,1 . The full list of exponents is then as follows: S 5,1 has 19 exponents and S 5,10 has 7 exponents (up to cyclic symmetry): (2, 2, 2, 2, 3) , (2, 2, 2, 2, 4) , (2, 2, 2, 2, 5) , (2, 2, 2, 2, 6) , (2, 2, 2, 2, 7) , (2, 2, 2, 3, 3) , (2, 2, 3, 2, 3) .

Pointing links
We now turn to the final aspect of the classification: the analysis of pointing links. Again, in principle the problem is straightforward: for every skeleton in 5 variables or less, we are to consider every possible way of making every link point at some node. The only requirement modulo feasibility of the link is that a link between variables X i , X j cannot point at X i , X j or any X k that is a target of a pointer. The last point may be a little bit surprising, so we illustrate it with a simple example based on S 5,2 . Suppose the link between X 1 and X 3 points at the variable X 4 , so that a representative superpotential is In this case dW = 0 if X 2 = X 4 = 0 and and this has solutions with X 1 , X 3 , X 5 non-zero.

Pointing links for disconnected skeletons
The most interesting feature of a pointing link is that it can connect graphs with disconnected skeletons; we already saw examples in section 3.1.
Let us investigate this more systematically by examining a graph with disconnected skeleton S 3,2 + S 1,1 . The only way this can be irreducible is if the link of S 3,2 points at the node of S 1,1 : We examine the exponents in Σ ∞ 4 and find a large number of infinite families consistent with q tot > 1. However, in all but one case, a solution to the pointing link only has a finite number of solutions. For instance, we have the family M = (2, * , 3, 5), for which the link requires 4) and this has a unique solution for k = 25. For all but one other case, the feasibility condition always takes this form of for a, b, c, d some positive integers: in every one of these situations, the possible values of k are therefore bounded. The exception is the exponent M = (2, k, 2, l), for which the charges are This is a reducible family: since q 1 = q 3 , if a link is feasible, then X 1 can point at X 4 , so that we simply obtain the disconnected skeleton S 2,1 + S 2,1 . It follows that only a finite number of exponents yields feasible and irreducible theories.

General analysis via numeric study
While the analysis of the general case is in principle straight-forward, the number of cases to check grows rather large as we increase n and the number of pointing links. A link typically has more than one possible target, and we must consider all possibilities. It is easy to scan through Σ spor n for all of these possibilities, but the existence of the potentially infinite families in Σ ∞ n presents a challenge. In every graph that we checked manually, we were able to rule out irreducible theories for all but a finite subset of Σ ∞ n . As we illustrated in the examples above, this sort of result typically relies on three factors: the bound on q tot , the necessary conditions for irreducibility, and the feasibility of links. Typically these factors are intertwined, and we have not been able to distill the constraints into a rigorous finiteness result.
Instead, we made a numeric investigation of all pointing links for exponents bounded by max M ≤ 100. 10 We found no evidence of infinite families of exponents that contain irreducible theories and in fact a rather small list of distinct charges. The connected n = 4 graphs with at least one pointing link yielded 20 distinct charges from the following skeletons: S 3,4 + S 1,1 , S 3,2 + S 1,1 , S 4,2 , and S 4,4 . These realized a maximum exponent of 31, while the n = 5 graphs with at least one pointing link produced 2 distinct charges from the skeleton S 4,2 + S 1,1 .
These results, together with investigations of infinite families for specific graphs, amount to substantial evidence that we do indeed have a complete list of irreducible theories, but it would certainly be useful to have a rigorous proof of the assertion -such a proof would likely yield insights into LG combinatorics. We leave this for future work.

Summary of Results
We studied the combinatorics of LG theories. Our work of course has a close relation to the classification of quasi-homogeneous singularities that has been of substantial interest to mathematicians, and to our surprise we found some singularities that were overlooked in previous work. The physics perspective offers some useful simplifications, most of which were indeed anticipated in [4]: it is natural to organize the theories by n, and to restrict attention to indecomposable theories; moreover, instead of organizing theories at fixed Milnor number (in the SCFT this is the number of relevant and marginal deformations of the theory), it is natural to focus on the central charge. We applied this philosophy to the problem of finding all indecomposable theories with c < 6.
Let us now summarize our results. We will describe the set of irreducible theories for various 1 ≤ n ≤ 5. We remind the reader that the results for n = 4, 5 do rely on some numeric studies. Two more points should be borne in mind: first, while we claim to have every indecomposable theory, our lists do contain some decomposable theories; second, because of the infinite families present some indecomposable theories may occur more than once. We claim, however, that every indecomposable theory is included.
In what follows we present the infinite families of charges. The sporadic cases are provided in the supplementary files included with the ArXiv submission.  With n = 4 variables the bound of interest is We proceed in the same fashion as in the previous section and establish that integer solutions m 1 ≥ m 2 ≥ m 3 ≥ m 4 ≥ 2 to this inequality fall into one of the following classes: A bound for n = 5 Our last bound is (A.14) The integer solutions with m 1 ≥ m 2 ≥ m 3 ≥ m 4 ≥ m 5 ≥ 2 are There are no additional infinite families and just three more sporadic cases.