Calabi-Yau Products: Graded Quivers for General Toric Calabi-Yaus

The open string sector of the topological B-model on CY $(m+2)$-folds is described by $m$-graded quivers with superpotentials. This correspondence generalizes the connection between CY $(m+2)$-folds and gauge theories on the worldvolume of D$(5-2m)$-branes for $m=0,\ldots,3$ to arbitrary $m$. In this paper we introduce the Calabi-Yau product, a new algorithm that starting from the known quiver theories for a pair of toric CY$_{m+2}$ and CY$_{n+2}$ produces the quiver theory for a related CY$_{m+n+3}$. This method significantly supersedes existing ones, enabling the simple determination of quiver theories for geometries that were previously out of practical reach.

This correspondence is particularly well understood in the case of toric CYs. For m = 1, brane tilings (a.k.a. dimer models), significantly simplify the map between CY 3-folds and 4d N = 1 gauge theories [17,19,25]. Progress in this area has considerably accelerated in recent years, initially fueled by a desire to develop brane constructions for lower dimensional gauge theories [26][27][28][29][30][31]. Lately, the scope of these investigations expanded to developing tools for toric CYs of arbitrary dimension. These efforts culminated in [32] with the introduction of m-dimers, which fully encode the m-graded quivers with superpotentials associated to toric CY (m + 2)-folds and streamline the connection between quivers and geometry.
The m-dimers associated to specific geometries can be determined via a variety of traditional approaches, such as partial resolution and mirror symmetry, which have been extended to general m [26]. Despite the considerable simplifications brought by m-dimers, their determination can sometimes become practically challenging and additional tools are desirable. Examples of such methods include orbifold reduction [33] and 3d printing [34] which were originally developed in the context of CY 4-folds but can be applied more broadly [24].
In this paper we introduce a substantially more powerful approach, which we denote Calabi-Yau product. This algorithm starts from the known quiver theories 1 for a pair of toric CY m+2 and CY n+2 and produces the quiver theory for a related CY m+n+3 . In doing so, it enables the computation of quiver theories that were previously out of practical reach. This paper is organized as follows. §2 presents a review of m-graded quivers. §3 introduces the basics of the CY product, in particular the input data for the construction and how the parent geometries give rise to the product geometry. §4 explains how to construct the periodic quiver for the product theory. §5 discusses the superpotential. The construction is illustrated in §6 with explicit examples. §7 considers the relation between the CY product and other constructions. We conclude and present ideas for future work in §8. Additional details are provided in two appendices.

A Brief Review of m-Graded Quiver Theories
In order to make our presentation self-contained, in this section we present a brief review m-graded quivers and their dualities. We refer the interested reader to [23,24,32] for further details.
Given an integer m ≥ 0, an m-graded quiver is a quiver with a grading for every arrow Φ ij by a quiver degree: |Φ ij | ∈ {0, 1, · · · , m} . (2.1) Every node i corresponds to a unitary "gauge group" U (N i ). Arrows connecting nodes correspond to bifundamental or adjoint "fields". The conjugate of every arrow Φ ij has the opposite orientation and degree m − |Φ ij |: where we use a superindex in parenthesis to explicitly indicate the degree of the corresponding arrow, i.e. |Φ (c) ij | = c. The integer m determines the possible degrees, i.e. the different types of fields, which can be restricted to the range: since other degrees can be obtained by conjugation. We refer to degree 0 fields as chiral fields.
Superpotential. Graded quivers admit superpotentials, which are linear combinations of gauge invariant terms of degree m − 1: Gauge invariant terms correspond to closed oriented cycles in the quiver, which may require conjugation of some of the fields.
Kontsevich bracket condition. The superpotential must also satisfy Here {f, g} denotes the Kontsevich bracket, which is defined as follows (2.6)

The Toric Case
The CY m+2 associated to an m-graded quiver arises as its classical moduli space which, generalizing the standard notion for m ≤ 3, is defined as the center of the Jacobian algebra with respect to fields of degree m − 1 [23]. Namely, it is obtained by imposing the relations: ∂W ∂Φ (m−1) = 0 , ∀ Φ (m−1) (2.7) plus gauge invariance. Since the superpotential has degree m − 1, the terms that contribute to the relations in (2.7) are of the general form Φ (m−1) J(Φ (0) ), with J(Φ (0) ) a holomorphic function of chiral fields. We will refer to such terms as J-terms. The relations (2.7) therefore comprise only chiral fields.
Toric superpotential. Every toric CY m+2 has at least one toric phase, which is a quiver theory satisfying the following properties. First, the ranks for all nodes can be equal. In addition, the superpotential of a toric phase has a special structure, which is referred to as the toric condition [32]. The toric condition implies that every field of degree m−1 appears in exactly two superpotential terms, with opposite signs. Namely, a J − a (Φ (0) ) + . . . , (2.8) where dots stand for terms that do not contain Φ (m−1) a . The relations (2.7) then take the form: J + a (Φ (0) ) = J − a (Φ (0) ) .
Generalized perfect matchings. We define a generalized perfect matching, or perfect matching for short, p as a collection of fields satisfying: 1) p contains precisely one field from each term in W .
2) For every field Φ in the quiver, either Φ orΦ is in p.
Perfect matchings provide variables that automatically satisfy the relations (2.9). Therefore, there is a one-to-one correspondence between them and GLSM fields in the toric description of the CY m+2 . Perfect matchings indeed substantially simplify the determination of the toric diagram (see [32] for details).
Since for every field a perfect matching contains either the field or its conjugate, a perfect matching determines a polarization of the quiver. We define polarization as a choice of orientation for every field in the quiver, i.e. a choice of what we regard as the original field and its conjugate. In what follows, we will adopt a convention for defining the polarization such that, given a perfect matching, we orient the fields in the quiver such that the fields in the perfect matching are the only ones that appear conjugated in the superpotential. 2 This choice of polarization implies that the corresponding perfect matching consists of the conjugates of all the fields in the quiver.

Generalized anomaly cancellation
Under a mutation at a node , its rank transform as: 10) where N 0 is the total number of incoming chiral fields. Invariance of the ranks under m+ 1 consecutive mutations of the same node leads to the generalized anomaly cancellation conditions. For odd m, these conditions are given by: ij ) denotes the number of arrows from i to j of degree c. For every i, the sum over j runs over all nodes in the quiver (including i), and n c is given by (2.3).
For even m, the conditions become For m = 0, 1, 2, 3, these conditions reproduce the cancellation of non-abelian anomalies in the corresponding d = 6, 4, 2, 0 gauge theories.

Product of Toric Calabi-Yaus: the Geometry
In this paper we will introduce the CY product. Before explaining the details of this novel algorithm, let us discuss its main ingredients and basics of the resulting geometry.
Initial data. The input for this procedure is given by: • An m-graded quiver theory P for a toric phase associated with a toric Calabi-Yau (m + 2)-fold CY m+2 . The toric diagram T CY m+2 is an (m + 1)-dimensional convex polytope consisting of points u i . We also pick a perfect matching p of P , which corresponds to the point u 0 of T CY m+2 .
• An n-graded quiver theory Q for a toric phase associated with a toric Calabi-Yau (n + 2)-fold CY n+2 . The toric diagram T CY n+2 is an (n + 1)-dimensional convex polytope consisting of points v i in it. We also pick a perfect matching q of Q, which corresponds to the point v 0 of T CY n+2 .
The product geometry. The output of this algorithm is an (m + n + 1)-graded quiver theory that we will call P p × Q q . This theory is a toric phase for the (m + n + 3)dimensional toric Calabi-Yau CY m+n+3 whose toric diagram T CY m+n+3 is the convex hull of points T CY m+n+3 is a lattice polytope in Z m+n+2 . In this lattice, the T CY m+2 gets embedded in a hyperplane spanned by the first m + 1 coordinates, while T CY n+2 gets embedded in a hyperplane spanned by the last n + 1 coordinates. These two hyperplanes are orthogonal and meet at a single point (u 0 , v 0 ). In other words, the final toric diagram T CY m+n+3 is the convex hull of the set of points obtained by "interlacing" T CY m+2 and T CY n+2 at the point (u 0 , v 0 ). Figure 1 shows two examples of this construction.
Higher dimensional examples are straightforward although, obviously, difficult to visualize. not the product of the two parent CYs. In particular, its dimension is not equal to the sum of the dimensions of the starting CYs. However, we feel that the term captures various aspects of the process and its su ciently simple to justify its adoption. It is clear that the product of CYs can very easily produce the quiver theories for extremely complicated geometries. Moreover, iterating the process, it becomes straightforward to deal with high dimensional geometries. We will present explicit examples in §6.
There is substantial freedom in this construction. Given a desired CY m+n+3 , it can generally be decomposed into other CY m+2 and CY n+2 geometries in multiple ways (even with di↵erent values of m and n), there is a choice of toric phase for each of the parent geometries and of perfect matchings for the points u 0 and v 0 . Therefore, generically, the product method can generate a large number of quiver theories for a given CY m+n+3 , reflecting the rich space of theories related by the corresponding order (m + n + 2) dualities.

Product of Toric Calabi-Yaus: the Periodic Quiver
Having discussed the connection between the parent and product geometries, we now explain how to construct the periodic quiver for the product. The periodic quiver contains all the information defining the quiver theory, namely not only the quiver but also the superpotential. Having said that, in §5 we will present explicit rules for constructing the superpotential directly, without having to read it from the periodic quiver.
The starting point of the construction is the initial data discussed in the previous section. As already mentioned, choosing di↵erent toric phases for the two parent geometries and/or using di↵erent perfect matchings for the u 0 and v 0 points can result in di↵erent phases for the same product geometry. Similar freedom has been observed -7 - not the product of the two parent CYs. In particular, its dimension is not equal to the sum of the dimensions of the starting CYs. However, we feel that the term captures various aspects of the process and its su ciently simple to justify its adoption.
It is clear that the product of CYs can very easily produce the quiver theories for extremely complicated geometries. Moreover, iterating the process, it becomes straightforward to deal with high dimensional geometries. We will present explicit examples in §6.
There is substantial freedom in this construction. Given a desired CY m+n+3 , it can generally be decomposed into other CY m+2 and CY n+2 geometries in multiple ways (even with di↵erent values of m and n), there is a choice of toric phase for each of the parent geometries and of perfect matchings for the points u 0 and v 0 . Therefore, generically, the product method can generate a large number of quiver theories for a given CY m+n+3 , reflecting the rich space of theories related by the corresponding order (m + n + 2) dualities.

Product of Toric Calabi-Yaus: the Periodic Quiver
Having discussed the connection between the parent and product geometries, we now explain how to construct the periodic quiver for the product. The periodic quiver contains all the information defining the quiver theory, namely not only the quiver but also the superpotential. Having said that, in §5 we will present explicit rules for constructing the superpotential directly, without having to read it from the periodic quiver.
The starting point of the construction is the initial data discussed in the previous section. As already mentioned, choosing di↵erent toric phases for the two parent geometries and/or using di↵erent perfect matchings for the u 0 and v 0 points can result not the product of the two parent CYs. In particular, its dimension is not equal to the sum of the dimensions of the starting CYs. However, we feel that the term captures various aspects of the process and its su ciently simple to justify its adoption.
It is clear that the product of CYs can very easily produce the quiver theories for extremely complicated geometries. Moreover, iterating the process, it becomes straightforward to deal with high dimensional geometries. We will present explicit examples in §6.
There is substantial freedom in this construction. Given a desired CY m+n+3 , it can generally be decomposed into other CY m+2 and CY n+2 geometries in multiple ways (even with di↵erent values of m and n), there is a choice of toric phase for each of the parent geometries and of perfect matchings for the points u 0 and v 0 . Therefore, generically, the product method can generate a large number of quiver theories for a given CY m+n+3 , reflecting the rich space of theories related by the corresponding order (m + n + 2) dualities.

Product of Toric Calabi-Yaus: the Periodic Quiver
Having discussed the connection between the parent and product geometries, we now explain how to construct the periodic quiver for the product. The periodic quiver contains all the information defining the quiver theory, namely not only the quiver but also the superpotential. Having said that, in §5 we will present explicit rules for constructing the superpotential directly, without having to read it from the periodic quiver.
The starting point of the construction is the initial data discussed in the previous section. As already mentioned, choosing di↵erent toric phases for the two parent ge- At first sight, the use of the term "product" to refer to the operation that acts on the geometry as described above, might be slightly confusing. The resulting geometry is not the product of the two parent CYs. In particular, its dimension is not equal to the sum of the dimensions of the starting CYs. However, we feel that the term captures various aspects of the process and its sufficiently simple to justify its adoption.
It is clear that the product of CYs can very easily produce quiver theories for extremely complicated geometries. Moreover, iterating the process, it becomes straightforward to deal with high dimensional geometries. We will present explicit examples in §6.
There is substantial freedom in this construction. Given a desired CY m+n+3 , it can generally be decomposed into other CY m+2 and CY n+2 geometries in multiple ways (even with different values of m and n), there is a choice of toric phase for each of the parent geometries and of perfect matchings for the points u 0 and v 0 . Therefore, generically, the CY product method can generate a large number of quiver theories -7 -for a given CY m+n+3 , reflecting the rich space of theories related by the corresponding order (m + n + 2) dualities.

Product of Toric Calabi-Yaus: the Periodic Quiver
Having discussed the connection between the parent and product geometries, we now explain how to construct the periodic quiver for the product. The periodic quiver contains all the information defining the quiver theory, namely not only the quiver but also the superpotential. Having said that, in §5 we will present explicit rules for constructing the superpotential directly, without having to read it from the periodic quiver.
The starting point of the construction is the initial data discussed in the previous section. As already mentioned, choosing different toric phases for the two parent geometries and/or using different perfect matchings for the u 0 and v 0 points can result in different phases for the same product geometry. Similar freedom has been observed in other constructions such as 3d printing [34] and it is natural to expect such different phases to be related by duality.
As discussed in §2.1, in order to simplify the product construction, given a perfect matching it is convenient to pick the polarization of the quiver in which the perfect matching turns out to simply consist of the conjugates of all the fields in the quiver. We will do so here. Using the polarization of P given by p and the polarization of Q given by q, we will define a polarization of the periodic quiver for P p × Q p . As we will see later, this polarization in fact corresponds to a perfect matching of the product theory and corresponds to the point (u 0 , v 0 ).
The periodic quiver of the product theory P p ×Q p can be elegantly defined in terms of the action of the product operation on the basic elements of the parent quivers: nodes and fields. Below, we will use the following convention to denote nodes and fields in the different quivers: i and X for P , j and Y for Q and (i, j) and Z for P q × Q q . We have three possible products: Node × node. The product of nodes i of P and j of Q gives rise to a node (i, j) of P p × Q q . This process is illustrated in Figure 2. Field × node. The product of a fieldX (c) i 1 ,i 2 of P which is in p with a node j of Q gives rise to a fieldZ (c+n+1) (i 1 ,j)(i 2 ,j) in P p × Q q . Similarly, the product of a node i of P and Figure 3 represents this operation. The horizontal and vertical directions encode the T m+1 and T n+1 tori, respectively. Figure 3: The four cases of elements of P and Q giving rise to elements of P p ⇥ Q q . In all cases we only consider the fields in P which are p and the fields in Q which are in q Figure 4: The four cases of elements of P and Q giving rise to elements of P p ⇥ Q q . In all cases we only consider the fields in P which are p and the fields in Q which are in q This process is depicted graphically in Figure 4. This process not only gives us the quiver for P p ⇥ Q q but also constructs the periodic quiver. This is because given an embedding of the periodic quiver P on T m+1 and an embedding of Q on T n+1 , these rules give us an embedding of P p ⇥ Q q on T n+m+2 .
For the sake of completeness we also describe the conjugates of the fields we have written above . This can be considered as arising from the product the gauge group i with the field Y (n d) j2j1 which is not in q.
• Similarly the conjugate ofZ . This can be considered as which is not in p and gauge group j.
• The conjugate ofZ and this should be regarded as Figure 3: The four cases of elements of P and Q giving rise to elements of P p ⇥ Q q . In all cases we only consider the fields in P which are p and the fields in Q which are in q j1 j2 d i Figure 4: The four cases of elements of P and Q giving rise to elements of P p ⇥ Q q . In all cases we only consider the fields in P which are p and the fields in Q which are in q This process is depicted graphically in Figure 4. This process not only gives us the quiver for P p ⇥ Q q but also constructs the periodic quiver. This is because given an embedding of the periodic quiver P on T m+1 and an embedding of Q on T n+1 , these rules give us an embedding of P p ⇥ Q q on T n+m+2 .
For the sake of completeness we also describe the conjugates of the fields we have written above . This can be considered as arising from the product the gauge group i with the field Y (n d) j2j1 which is not in q.
• Similarly the conjugate ofZ . This can be considered as arising from X (m c) i2i1 which is not in p and gauge group j.
• The conjugate ofZ and this should be regarded as Figure 4: The four cases of elements of P and Q giving rise to ele In all cases we only consider the fields in P which are p and the fiel in q This process is depicted graphically in Figure 4. This process no quiver for P p ⇥ Q q but also constructs the periodic quiver. This is embedding of the periodic quiver P on T m+1 and an embedding of rules give us an embedding of P p ⇥ Q q on T n+m+2 .
For the sake of completeness we also describe the conjugates of written above . This can be consider the product the gauge group i with the field Y which is not in p and gauge group j.
• The conjugate ofZ and this shoul Figure 4: The four cases of elements of P and Q giving rise to elements of P p ⇥ Q q . In all cases we only consider the fields in P which are p and the fields in Q which are in q This process is depicted graphically in Figure 4. This process not only gives us the quiver for P p ⇥ Q q but also constructs the periodic quiver. This is because given an embedding of the periodic quiver P on T m+1 and an embedding of Q on T n+1 , these rules give us an embedding of P p ⇥ Q q on T n+m+2 .
For the sake of completeness we also describe the conjugates of the fields we have written above . This can be considered as arising from the product the gauge group i with the field Y (n d) j2j1 which is not in q.
• Similarly the conjugate ofZ . This can be considered as which is not in p and gauge group j.
• The conjugate ofZ and this should be regarded as Figure 3: Field × node.
Field × field. The product of a fieldX (c) Figure 5: The four cases of elements of P and Q giving rise to elements of P p ⇥ Q q . In all cases we only consider the fields in P which are p and the fields in Q which are in q It is important to note that at the end of this process there is no field that comes from the product of a fieldX j2j1 which is not in q or vice versa. It is precisely this which makes the choice of p and q central to this construction.
Henceforth i will be used to denote a gauge group of P and j a gauge group of Q. Similarly we will always use X to refer to fields in P and Y to refer to fields in Q. Lastly we will use the pair (i, j) to denote a gauge group and Z to denote a field of P q ⇥ Q q . These will be implicitly assumed to arise from elements of P and Q as described above.

Anomaly Cancellation
We will now show that if P and Q satisfy the anomaly cancellation condition then so does P p ⇥ Q q . For this we start with enumerating all the fields that are charged under a given gauge group (i, j) of P p ⇥ Q q and their contributions to anomaly. These arise from 1. Product of incoming fields at i in P with gauge group j of Q.
(b) If X i 0 i is not in p then it gives rise to one field Z (c) (i 0 ,j)(i,j) incoming at (i, j) which contributes ( 1) c to anomaly.
2. Product of incoming field at j in Q with the gauge group i of P . Figure 3: The four cases of elements of P and Q giving rise to elements of P p ⇥ Q q . In all cases we only consider the fields in P which are p and the fields in Q which are in q j1 j2 d i Figure 4: The four cases of elements of P and Q giving rise to elements of P p ⇥ Q q . In all cases we only consider the fields in P which are p and the fields in Q which are in q This process is depicted graphically in Figure 4. This process not only gives us the quiver for P p ⇥ Q q but also constructs the periodic quiver. This is because given an embedding of the periodic quiver P on T m+1 and an embedding of Q on T n+1 , these rules give us an embedding of P p ⇥ Q q on T n+m+2 .
For the sake of completeness we also describe the conjugates of the fields we have written above . This can be considered as arising from the product the gauge group i with the field Y (n d) j2j1 which is not in q.
• Similarly the conjugate ofZ . This can be considered as arising from X (m c) i2i1 which is not in p and gauge group j.
• The conjugate ofZ and this should be regarded as Figure 3: The four cases of elements of P and Q giving rise to elements of P p ⇥ Q q . In all cases we only consider the fields in P which are p and the fields in Q which are in q j1 j2 d i Figure 4: The four cases of elements of P and Q giving rise to elements of P p ⇥ Q q . In all cases we only consider the fields in P which are p and the fields in Q which are in q This process is depicted graphically in Figure 4. This process not only gives us the quiver for P p ⇥ Q q but also constructs the periodic quiver. This is because given an embedding of the periodic quiver P on T m+1 and an embedding of Q on T n+1 , these rules give us an embedding of P p ⇥ Q q on T n+m+2 .
For the sake of completeness we also describe the conjugates of the fields we have written above . This can be considered as arising from the product the gauge group i with the field Y (n d) j2j1 which is not in q.
• Similarly the conjugate ofZ . This can be considered as arising from X (m c) i2i1 which is not in p and gauge group j.
• The conjugate ofZ and this should be regarded as arising from X  Table 1 summarizes the product construction. This procedure not only generates the quiver for P p × Q q but also constructs its periodic quiver. This is because given an embedding of the periodic quiver P in T m+1 and of Q in T n+1 , these rules result in an embedding of P p × Q q in T m+n+2 .
For the sake of completeness we also describe the conjugates of the fields we have written above. Their origin can be understood as follows: • The conjugate ofZ . It arises from the product between the node i and the field Y which is not in p and node j. 3 For clarity, we have emphasized that we go over the fieldsX (c) i1,i2 of P which are in p and the fields Y (d) j1j2 of Q in q. However, given our choice of polarization determined by p and q, these are simply the conjugates of all the fields in P and Q.
. It comes from the product between It is important to note that at the end of this process there is no field that comes from the product of anX (c) ∈ q or vice versa. This makes the choice of p and q central to this construction.

Anomaly Cancellation
Let us begin checking the consistency of the CY product construction we have just introduced. In this section we will show that if P and Q satisfy the corresponding anomaly cancellation conditions, then so does P p × Q q . We assume that the ranks of all nodes are equal to N and normalize the anomaly by this number. We first enumerate all the fields that are charged under a given node (i, j) of P p × Q q and consider their contributions to the anomaly. These fields are given by: incoming at (i, j) which contributes (−1) d to the anomaly.

Product of a fieldX
j j that is in q. This gives rise to the incoming fieldZ (c+d) (i ,j )(i,j) which contributes (−1) c+d to the anomaly. This is just the product of the the contribution to anomaly at i of the incoming field X ii at i that is in p with an outgoing fieldȲ at j that is in q. This gives rise to the outgoing fieldZ Adding all these contributions, the anomaly at node (i, j) becomes where a p is the contribution to the anomaly by incoming fields at i which are in p and a ¡ p is the contribution to the anomaly by incoming fields that are not in p. Similarly, b p is the contribution to the anomaly at node j by incoming fields that are in q, while b ¡ q is the contribution from the fields that are not in q.
At this point we distinguish three cases depending on the parity on m and n.
Odd m and n. In this case A becomes For odd m and n, the anomaly cancellation conditions for i in P and j in Q respectively are Plugging these back into the expression for A results in A = 0, which is the anomaly cancellation condition, since m + n + 1 is odd.
Even m and even n. In this case A becomes The anomaly cancellation conditions for i and j respectively are Plugging these back also results in A = 0, which is again the anomaly cancellation condition since m + n + 1 is odd in this case, too.
Odd m and even n. Lastly, in this case The anomaly cancellation conditions at i and j are which gives A = 2, i.e. the anomaly cancellation condition is satisfied since m + n + 1 is even for this case.

Superpotential
The construction introduced in §4, produces the periodic quiver for P p ×Q q from which, in principle, its superpotential can be read off. In general, this can be rather challenging. Therefore, in this section we introduce explicit rules for the direct construction of the superpotential. The superpotential of the product theory takes the general form W P and W Q descend from the superpotentials of P and Q, respectively. W C consists of new cubic interactions. Finally, W P Q depends on superpotentials of both P and Q.
We now describe each of them in detail.
W P : terms descending from the superpotential of P . Let us consider a single term T P in the superpotential W P of the parent theory P . It has the general form where n c n = m − 1 due to degree constraint. Our convention for the polarization makes the perfect matching p manifest. The fields in p appear as a single conjugated -12 -field per term in W P . Furthermore, we will order the fields in every term such that the fields in p occur last. Every term T P gives rise to various terms in W P , as we now discuss. First, some of these terms correspond to the product between the fields in this term and a node j of Q. They take the form where the sum is over the set J of nodes j of Q. After this operation, the degree of the superpotential changes by n + 1 and becomes m + n, as required for the superpotential of an (m + n + 1)-graded quiver.
The additional terms descending from T P are constructed as follows. We first pick a field X (c) i i from those in T P . Since this field does not appear conjugated, it is obviously not contained in p. We also pick a field Y (d) j j that is not in q. We then replace X . This operation increases the degree by d + 1. We also replaceX j ) . This changes the degree by n − d. Finally, we simply replace the remaining fields in T P by their product with appropriate node in Q, which does not change the degrees since these fields are not in p. When combined, all these replacements change the degree of the superpotential term by n + 1, as desired. Explicitly these terms are (5.4) To obtain W P , we repeat this process for all the terms in W P . In addition to the signs written above, we must include the signs with which the parent superpotential terms enter W P .
W Q : terms descending from the superpotential in Q. These terms are determined by the same procedure, after the exchange (P, p) ↔ (Q, q). Let us present the final result. Every term T Q in the superpotential W Q of Q is of the form: j k j 1 .
(5.5) -13 -As before, T Q gives rise to superpotential terms of two types, analogous to (5.3) and (5.4). The first set of terms is with I the set of nodes of P . The second set of terms is Repeating this process for all the terms in W P , we obtain W P . Once again, we need to include the signs of the parent terms in W P .
j 1 j 2 ∈ q we have a pair of cubic terms where the fields involved are descendants ofX jj via the rules in Table 1, or their conjugates. Namely, (5.9) W C is the sum of (5.8) over all the pairs ofX (c) W P Q : mixed terms. The last part of the superpotential involves contributions coming from P and Q. A term T P in the superpotential of P and a term T Q in the superpotential of Q give rise to a number of terms in the superpotential of the product theory. W P Q is the sum of all such terms. To describe them, let us first consider the special case in which both T P and T Q are cubic terms, i.e.
In this case, they give rise to a single term that involves the pairwise product of fields, 4 i.e. (−1) m+n+c 2 +d 2 Z (c 1 +d 1 +1) .

(5.11)
If T P and/or T Q are of order greater than 3, no such simple terms can be written. The reason is that the pairwise product of fields is only possible if they have the same order and the resulting terms will have correct degree, i.e. m + n, if and only if T P and T Q are cubic. 5 One way of addressing this issue is to turn T P and T Q into a sum of cubic terms and mass terms, by integrating in auxiliary massive fields. Then we can construct W P Q as described above, consisting exclusively of terms descending from the cubic terms. The final quiver and superpotential can then be obtained by integrating out the massive fields.
Naively, it might seem that this procedure dramatically changes our construction. A massive field in P gives rise to one descendant for every field or node of Q and vice versa. Nevertheless, it can be verified that all these descendants are massive, resulting in the same quiver we would have obtained without integrating in massive fields. Therefore, we can use the rule for cubic terms above as the starting point to efficiently compute the rules for higher order terms. The result is that there are terms in W P Q descending from terms T P of order k and T Q of order l. All these terms are of order k + l − 3. We provide a thorough discussion of these terms and the first few steps of this iteration in Appendix A.
The geometry of the product theory. It is relatively straight forward, yet quite laborious, to show that the desired geometry (3.1) arises as the classical moduli space of the P p × Q q theory we have constructed. 6 We present the proof in Appendix B. 4 It is useful to reflect on why we obtain a single term. First of all, we defined the polarizations of the parent theories such that every term in their superpotentials contains a single conjugated field. In addition, following the rules introduced in §4, we cannot multiply unbarred and barred fields. As a result, there are not multiple possibilities associated to cyclic permutations of the fields in (5.10). 5 It is interesting to compare this to the B-model computation of the superpotential: cubic terms are special in that they correspond to m 2 of the A ∞ algebra, which is composition of maps, while higher order terms correspond to higher m k , which are more involved. 6 The notion of moduli space has been extended to general m in [23].

Kontsevich Bracket
As another consistency check of our construction, let us verify that the superpotential we have written satisfies {W, W } = 0, where .

(5.12)
To do this, we divide {W, W } into eight pieces, {W, W } = 2(KB P + KB Q + KB P C + KB QC + KB P Q + KB P QP + KB P QQ + KB P QC ) , (5.13) each of which vanishes individually.
KB P = 1 2 {W P , W P } is the contribution that arises exclusively due to W P . Explicitly, its nontrivial terms are . (5.14) It is straightforward to show that KB P vanishes if the superpotential W P of P satisfies {W P , W P } = 0. The reason is that the terms in KB P descend from the terms of {W P , W P } in a manner that is analogous to how terms in W P descend from terms in W P and the signs in (5.4) are such that the required cancellations still occur. Similarly, , (5.15) and it vanishes if the superpotential W Q of Q satisfies {W Q , W Q } = 0.
KB P C and KB QC involve the Kontsevich bracket between W P and W Q with W C . Explicitly, . They reduce to Both KB P C and KB QC vanish independently of any conditions on W P and W Q . This can be verified directly using the explicit form of W C . Let us now consider KB P Q = 1 2 {W P Q , W P Q }. Its non-trivial part is .

(5.17)
First, let us consider the case in which W P and W Q are cubic, since in this case W P Q comes just from the pairwise product of fields, as explained earlier. In this case, both {W P , W P } and {W Q , W Q } are entirely quartic and a term in KB P Q comes from the pairwise product of fields from a term in {W P , W P } and a term in {W Q , W Q }, As result, KB P Q vanishes.
To show that KB P Q vanishes even when W P and W Q are not cubic, we can rewrite W P and W Q as sums of cubic terms and mass terms by appropriately integrating in massive fields and using the argument above. There is an added subtlety: after integrating in these massive fields, {W P , W P } and {W Q , W Q } vanish only after using the equations of motion for massive fields. This is enough for our purposes, and it can be shown that KB P Q vanishes once we integrate out massive fields from the product theory.
All the remaining contributions, KB P QP , KB P QQ and KB P QC , involve W P Q and therefore it is convenient to express W P and W Q as a sum of cubic terms and mass terms. Explicitly, they are A lengthy but straightforward bookkeeping calculation shows that all of these contributions vanish up to the equations of motion for massive fields. KB P QP vanishes as a result of {W P , W P } = 0, while vanishing of KB P QQ follows from {W Q , W Q } = 0. Lastly, KB P QC vanishes independently of any restriction on W P and W Q .

Toric Condition
To conclude our discussion of the superpotential, we now show that our construction is such that if P and Q satisfy the toric condition then P q × Q q also does so. We do so by considering the different ways a field of degree m + n can arise in the superpotential of P p × Q q . It is useful to note that all such terms must come from W P , W Q and W C , but not from W P Q . As explained in Appendix A, every term in W P Q contains two fields coming from the product of a field not in p and a field not in q. The degrees of such fields are greater or equal to 1, so none of these terms can contain a degree m + n field. The different scenarios are: Its product with a node j of Q gives rise to a fieldZ (m+n) (i 1 ,j)(i 2 ,j) of degree m + n. This field only appears in W P , in the form shown in (5.3). Therefore, ifX (m−1) i 1 i 2 participates in two terms with opposite signs, then so doesZ (m+n) (i 1 ,j)(i 2 ,j) . Similarly, if there is a fieldȲ (n−1) j 1 j 2 ∈ q, its product with a node i of P gives rise toZ (m+n) (i,j 1 )(i,j 2 ) . It only participates in W P , as shown in (5.6), namely in two terms with opposite sign.
• The product of a conjugate chiralX (m) i 1 i 2 ∈ p and a conjugate chiral fieldȲ (n) j 1 j 2 ∈ q gives rise to a fieldZ (m+n) (i 1 ,j 1 )(i 2 ,j 2 ) of degree m + n. Since conjugate chiral fields do not appear in the superpotential,Z (m+n) (i 1 ,j 1 )(i 2 ,j 2 ) does not appear in W P or W Q . It only appears in two terms of W C with opposite sign as shown in (5.8).
• The product of a field X (m−1) i 1 i 2 / ∈ p and a conjugate chiral field Y (n) appears in two terms with opposite sings in W P , Z (i 1 ,j 1 )(i 2 ,j 2 ) appears in two terms of the final superpotential with opposite signs. These terms arise as described by (5.4). Since Y (n) j 1 j 2 is a conjugate chiral, it does not appear in W Q , which implies that Z (m+n) (i 1 ,j 1 )(i 2 ,j 2 ) does not appear in W Q . It does not appear in W C , either.
Similarly, the product of a conjugate chiral field X (m) , which only appears in two terms with opposite signs. These terms are in W Q , specifically among those described in (5.7).
-18 - The discussion above covers all the fields of degree m + n. We conclude that the product between an m-graded toric phase P and an n-graded toric phase Q using arbitrary perfect matchings is an (m + n + 1)-graded toric phase.

Examples
In this section we illustrate the product construction with two explicit examples. The first theory we will construct is the well-known phase 2 of F 0 [14]. 7 The second example is a product of the conifold quiver theory with itself, which results in a 0d N = 1 matrix model. While, to our knowledge, this the first time the second theory appears in the literature, our primary goal is to demonstrate the simplicity of this procedure.

F 0
Let us consider the complex cone over F 0 CY 3-fold, or F 0 for short. The m = 1, i.e. 4d N = 1, quiver theories for this geometry have been extensively studied in the literature (see e.g. [14]). The toric diagram for F 0 can be constructed as the product of two copies of C 2 /Z 2 using one of the two perfect matchings for the central point in each case, as illustrated in Figure 6.
The m = 0, i.e. 6d N = (1, 0), quiver theory of the parent C 2 /Z 2 geometry consists of two U (N ) gauge groups with two hypermultiplets stretching between them, as shown in Figure 5. This theory has 4 perfect matchings, which translate into the 4 ways in which we can orient the 2 hypermultiplets. Two of them correspond to the two endpoints of the toric diagram (shown on the left of Figure 6) while the other 2 correspond to the central point. As a result, we have 2 perfect matching choices for the central point of each of the C 2 /Z 2 factors. But the 2 central perfect matchings are conjugates of each other and as a result any choice of perfect matchings gives the same theory up to chiral conjugation. 8 7 By phase 2, we mean the phase whose quiver is shown in (8). Various papers label the two phases of F 0 in different ways. 8 We note that m = 0 is the only case for which the conjugates of the field in a perfect matching also form a perfect matching. This is only possible because there is no superpotential.
-19 - Figure 6: The toric diagram of F 0 can be obtained as the product of two copies of the toric diagram of C 2 /Z 2 . In both cases we use the central point of the toric diagram to take the product.
The product of the periodic quivers is presented in Figure 7. The first step shows the two parent 6d N = (1, 0) quivers. The arrows are oriented to indicate the choice of perfect matchings. The second step shows the nodes of F 0 that arise from the product of nodes in the parent theories. In the third step, we add vertical fields (which come from the product of a node in the first parent and a field in the second one) and horizontal fields (which come from the product of a field in the first parent and a node in the second one). The last step adds the diagonal fields that arise from the product of a field in the first parent with a field in the second one. -20 - The result is the phase 2 of F 0 [14]. Since in this the parent theories do not have superpotentials, the final superpotential only consists of the new cubic terms that arise in the product. These terms can be straightforwardly read from the minimal plaquettes of the quiver.
For completeness, in Figure 8 we show the standard quiver for this theory. Its superpotential is This family was first introduced in [24], where the corresponding quiver theories were also constructed. Roughly speaking the periodic quiver for F for m > 1 and it is natural to expect that different choices of perfect matching lead to different phases related by the dualities discussed in §2.2. 9 The quiver theory Q (m) of one particular phase of F (m) 0 can be constructed inductively as follows where we use the product perfect matching p × q of P p × Q q as defined in Appendix B. This phase of F (m) 0 was discussed at length in [24,32], to which we refer for details.

Conifold × Conifold
The conifold is one of the most thoroughly studied toric CY 3-folds. Its toric diagram is shown in Figure 9. The corresponding gauge theory was constructed in the seminal work [37]. It consists of two U (N ) gauge groups and four bifundamental chiral fields X 01 ,X 01 , X 10 , andX 10 , as shown in Figure 9. The superpotential is W con = X 01 X 10X01X10 −X 01 X 10 X 01X10 (6.5)

Conifold ⇥ Conifold
The conifold is one of the most thoroughly studied toric CY 3-folds. Its toric diagram is shown in Figure 9. The corresponding gauge theory was constructed in the seminal work [37]. It consists of two U (N ) gauge groups and four bifundamental chiral fields X 01 ,X 01 , X 10 , andX 10 , as shown in Figure 9. The superpotential is W con = X 01 X 10X01X10 X 01 X 10 X 01X10 (6.5)

Conifold ⇥ Conifold
The conifold is one of the most thoroughly studied toric CY 3-folds. Its toric diagram is shown in Figure 9. The corresponding gauge theory was constructed in the seminal work [37]. It consists of two U (N ) gauge groups and four bifundamental chiral fields X 01 ,X 01 , X 10 , andX 10 , as shown in Figure 9. The superpotential is This theory has 4 perfect matchings, each of them consists of one of the chiral fields and corresponds to a corner in the toric diagram. Given the symmetry between the perfect matchings, the result is independent of which perfect matching we use for the product, up to relabeling. We will therefore drop the reference to the perfect matching and refer to this theory as conifold⇥conifold. Without loss of generality, we choose the toric diagrams of the two conifolds to coincide at the origin. The conifold⇥conifold is therefore a toric CY 5-fold with toric diagram (0, 0, 0, 0) (1, 0, 0, 0) (0, 1, 0, 0) (1, 1, 0, 0) (0, 0, 0, 1)

Conifold ⇥ Conifold
The conifold is one of the most thoroughly studied toric CY 3-folds. Its toric diagram is shown in Figure 9. The corresponding gauge theory was constructed in the seminal work [37]. It consists of two U (N ) gauge groups and four bifundamental chiral fields X 01 ,X 01 , X 10 , andX 10 , as shown in Figure 9. The superpotential is This theory has 4 perfect matchings, each of them consists of one of the chiral fields and corresponds to a corner in the toric diagram. Given the symmetry between the perfect matchings, the result is independent of which perfect matching we use for the product, up to relabeling. We will therefore drop the reference to the perfect matching and refer to this theory as conifold⇥conifold. Without loss of generality, we choose the toric diagrams of the two conifolds to coincide at the origin. The conifold⇥conifold is therefore a toric CY 5-fold with toric diagram

Conifold ⇥ Conifold
The conifold is one of the most thoroughly studied toric CY 3-folds. Its toric diagram is shown in Figure 9. The corresponding gauge theory was constructed in the seminal work [37]. It consists of two U (N ) gauge groups and four bifundamental chiral fields X 01 ,X 01 , X 10 , andX 10 , as shown in Figure 9. The superpotential is This theory has 4 perfect matchings, each of them consists of one of the chiral fields and corresponds to a corner in the toric diagram. Given the symmetry between the perfect matchings, the result is independent of which perfect matching we use for the product, up to relabeling. We will therefore drop the reference to the perfect matching and refer to this theory as conifold⇥conifold. Without loss of generality, we choose the toric diagrams of the two conifolds to coincide at the origin. The conifold⇥conifold is therefore a toric CY 5-fold with toric diagram  This theory has 4 perfect matchings, each of them consists of one of the chiral fields and corresponds to a corner in the toric diagram. Given the symmetry between the perfect matchings, the result is independent of which perfect matching we use for the product, up to relabeling. We will therefore drop the reference to the perfect matching and refer to this theory as conifold⇥conifold. Without loss of generality, we choose the toric diagrams of the two conifolds to coincide at the origin. The conifold⇥conifold is therefore a toric CY 5-fold with toric diagram (0, 0, 0, 0) (1, 0, 0, 0) (0, 1, 0, 0) (1, 1, 0, 0) (0, 0, 0, 1)

Conifold ⇥ Conifold
The conifold is one of the most thoroughly studied toric CY 3-folds. Its toric diagram is shown in Figure 9. The corresponding gauge theory was constructed in the seminal work [37]. It consists of two U (N ) gauge groups and four bifundamental chiral fields X 01 ,X 01 , X 10 , andX 10 , as shown in Figure 9. The superpotential is W con = X 01 X 10X01X10 X 01 X 10 X 01X10 (6.5)

Conifold ⇥ Conifold
The conifold is one of the most thoroughly studied toric CY 3-folds. Its toric diagram is shown in Figure 9. The corresponding gauge theory was constructed in the seminal work [37]. It consists of two U (N ) gauge groups and four bifundamental chiral fields X 01 ,X 01 , X 10 , andX 10 , as shown in Figure 9. The superpotential is This theory has 4 perfect matchings, each of them consists of one of the chiral fields and corresponds to a corner in the toric diagram. Given the symmetry between the perfect matchings, the result is independent of which perfect matching we use for the product, up to relabeling. We will therefore drop the reference to the perfect matching and refer to this theory as conifold⇥conifold. Without loss of generality, we choose the toric diagrams of the two conifolds to coincide at the origin. The conifold⇥conifold is therefore a toric CY 5-fold with toric diagram (0, 0, 0, 0) (1, 0, 0, 0) (0, 1, 0, 0) (1, 1, 0, 0) (0, 0, 0, 1)

Conifold ⇥ Conifold
The conifold is one of the most thoroughly studied toric CY 3-folds. Its toric diagram is shown in Figure 9. The corresponding gauge theory was constructed in the seminal work [37]. It consists of two U (N ) gauge groups and four bifundamental chiral fields X 01 ,X 01 , X 10 , andX 10 , as shown in Figure 9. The superpotential is This theory has 4 perfect matchings, each of them consists of one of the chiral fields and corresponds to a corner in the toric diagram. Given the symmetry between the perfect matchings, the result is independent of which perfect matching we use for the product, up to relabeling. We will therefore drop the reference to the perfect matching and refer to this theory as conifold⇥conifold. Without loss of generality, we choose the toric diagrams of the two conifolds to coincide at the origin. The conifold⇥conifold is therefore a toric CY 5-fold with toric diagram

Conifold ⇥ Conifold
The conifold is one of the most thoroughly studied toric CY 3-folds. Its toric diagram is shown in Figure 9. The corresponding gauge theory was constructed in the seminal work [37]. It consists of two U (N ) gauge groups and four bifundamental chiral fields X 01 ,X 01 , X 10 , andX 10 , as shown in Figure 9. The superpotential is This theory has 4 perfect matchings, each of them consists of one of the chiral fields and corresponds to a corner in the toric diagram. Given the symmetry between the perfect matchings, the result is independent of which perfect matching we use for the product, up to relabeling. We will therefore drop the reference to the perfect matching and refer to this theory as conifold⇥conifold. Without loss of generality, we choose the toric diagrams of the two conifolds to coincide at the origin. The conifold⇥conifold is therefore a toric CY 5-fold with toric diagram  This theory has 4 perfect matchings, each of them consists of one of the chiral fields and corresponds to a corner in the toric diagram. Given the symmetry between the perfect matchings, the result is independent of which perfect matching we use for the product, up to relabeling. We will therefore drop the reference to the perfect matching and refer to this theory as conifold⇥conifold. Without loss of generality, we choose the toric diagrams of the two conifolds to coincide at the origin. The conifold⇥conifold is therefore a toric CY 5-fold with toric diagram (0, 0, 0, 0) (1, 0, 0, 0) (0, 1, 0, 0) (1, 1, 0, 0) (0, 0, 0, 1) This theory has 4 perfect matchings, each of them consists of one of the chiral fields and corresponds to a corner of the toric diagram. Given the symmetry between the perfect matchings, the result is independent of which perfect matching we use for the product, up to relabeling. We will therefore drop the reference to the perfect matching 9 For example, F (2) 0 is also known as Q 1,1,1 /Z 2 . This theory has 14 toric phases, which were classified in [34]. where we have indicated the two conifold factors as the row and column. Table 2 summarizes the nodes and fields in the product 0d N = 1 matrix model. 10 The corresponding quiver is shown in Figure 10.  Table 2: Summary of how the nodes and fields in the conifold×conifold theory descend from the two parents. For simplicity, we converted the pairs of indices labeling nodes in the product to single indices. We also indicate the degree of the fields as a superindex. We use Latin and Greek letters to indicate chiral and Fermi fields, respectively.
-23 - Superpotential. Since the periodic quiver in this case lives on T 4 we cannot display it diagrammatically. Instead, we can construct the superpotential explicitly using prescription given in §5. We divide the total superpotential into four parts where W 1 and W 2 come from the first and second conifold factors respectively, W C contains the new cubic terms and W 12 contains the mixed terms. Recall that in Table  2 we used Latin and Greek letters to indicate chiral and Fermi fields, respectively. With this in mind the various parts of superpotential are: Since there are only two terms in the superpotential of the conifold we write the descendants of each of them separately. We thus write W 1 = W 1+ − W 1− , with W 1+ and W 1− the descendants of the positive and negative terms, respectively. We get W C : As explained in §5 there are two cubic terms in the superpotential of P p × Q q for every pair of fieldsX (c) In the present case, these terms are: W C is the sum of all these terms.
W 12 : As explained in §5 and Appendix A, for every pair of terms T P and T Q , there are terms in the product superpotential that combine them. For every pair of quartic terms, there are 9 quintic terms. As in the case of W 1 and W 2 , we write the corresponding terms separately. So we decompose W 12 as - 25 -where the signs correspond to the signs of the parent terms in the two conifolds. The individual contributions are: This completes our description of the superpotential. All in all, it consists of 124 terms. Of these, 38 are J-terms, i.e. they contain precisely one degree m − 1 field (namely degree 2 in this case) and the rest are chiral fields. Each one of the 19 degree m − 1 fields (see the quiver in Figure 10) appear in two of these terms with opposite sign, so the superpotential satisfies the toric condition. Finally, with some effort we can verify that the Kontsevich bracket {W, W } vanishes.

Relation to Other Constructions
We now briefly discuss how the product construction relates to other known methods for determining the quiver theories corresponding to a given geometry.
Algebraic dimensional reduction is indeed a specific instance of products and corresponds to the product of the quiver theory for CY m+2 with the simplest m = 0 quiver theory, the one for C 2 . This theory is shown in Figure 11 and has two perfect matchings. We can use any of them and get the same result. Similarly any perfect matching used for the CY m+2 theory gives the same quiver theory for CY m+2 × C up to a relabeling of fields.  Figure 11: The periodic quiver for C 2 and its perfect matchings, represented here as orientations of the quiver.

Orbifold Reduction
Orbifold reduction is a generalization of dimensional reduction that constructs a quiver theory for a toric CY 4 from a that of a toric CY 3 [33]. 12 It adds a third dimension to the toric diagram T CY 3 by adding images of one of its points up to some height k + above the central plane containing the T CY 3 and some depth k − below it (see Figure  12). Figure 12: Toric diagrams for: a) the dimensional reduction of dP 3 to dP 3 × C, b) a (dP 3 × C)/Z k orbifold with k = 2 and c) an orbifold reduction of dP 3 with k + = 2 and k − = 1.
This process again corresponds to a specific case of a product. The orbifold reduction of a 4d N = 1 quiver theory with periodic quiver P using a perfect matching q corresponds to the product P p × A (k + +k − ) q . Here A (k) is the 6d N = (1, 0) quiver theory for C 2 /Z k , i.e. the affine necklace quiver of type A with k nodes. A perfect matching of an m = 0 quiver is just a choice of orientation of its edges, so the perfect matching p is such that k + arrows point up while k − arrows point down. There are k + +k − k − such perfect matchings. They all realize theories corresponding to the same geometry and are related by a sequence of trialities.

3d Printing
Another algorithm for efficiently constructing quiver theories for toric CYs starting from simpler parent geometries is 3d printing. 3d printing allows one to add images of multiple points in the toric diagram (we refer to [34] for details). 3d printing is indeed more general than the CY product in two senses: • All the geometries that can be addressed with CY products can also be reached by a sequence of 3d printings that increase m by one at a time. The converse is not true; there are geometries that can be realized by 3d printing but not as CY products. The simplest such example is the conifold. As it is evident from its toric diagram, shown in Figure 9, it can be constructed by lifting both the points in the toric diagram of C 2 . On the other hand, it is clear that it is not possible to produce it by a product.
• Even if the same geometry can be realized by both processes, there might be phases of the quiver theories that can be obtained via 3d printing but not via a product. A simple example of this phenomenon is F 0 . Phase 2 of F 0 can be obtained using either construction but only 3d printing is able to construct phase 1.
Despite these relative disadvantages, the CY product is a superior method for geometries that can be reached via both methods for several reasons: • The CY product is much more efficient. This is true even for simple geometries. As an example, let us consider the construction of a quiver theory for the conifold×conifold. In order to 3d print this theory starting from the conifold, we first need to produce an intermediate CY 4-fold that is the dimensional reduction of the conifold, i.e. conifold×C. Then two points of its toric diagram must be lifted to produce the conifold×conifold. To carry out this process we will have to compute the perfect matchings not only for the conifold but also for the intermediate conifold×C theory. The difficulties of constructing the necessary quiver blocks and computing perfect matchings at every intermediate step makes 3d printing impractical if the difference between the dimensions of the input and target geometries is large.
• The CY product always produces reduced theories, which is not the case with 3d printing which often results in reducible, also known as inconsistent, theories which need to be reduced [34].
• Unlike 3d printing the CY product does not generate mass terms in the superpotential. This not only reduces the computational burden but it also means that CY product provides a more direct way of arriving to the final quiver theory, without the need to integrate out massive fields at the end.
• More importantly, in addition to these computational advantages, the CY product provides us with a concise and much clearer relationship between the input and target geometries. This becomes more striking as the difference between the dimensions of the input and target geometries increases.
Having considered the relative merits of the two constructions we turn to some speculation about their relation. While we have restricted ourselves to the case in which the periodic quivers for both theories are embedded in tori, more generally we can regard the product construction as a method for producing a quiver embedded in S × T given two quivers embedded in manifolds S and T . We can also consider cases where the manifolds have a boundary. Imagine T has a boundary ∂T . In that case the resulting quiver will be embedded in a manifold S × T with boundary S × ∂T . Arguably the simplest case of this situation is when T is the line segment I. The basic building block of 3d printing, a quiver block Q (m+1) p , is a graph embedded in T m × I and indeed can be regarded as a product of an m-graded periodic quiver Q (m) using a perfect matching p with a simple quiver embedded in a line segment as follows 13 As usual, we have indicated the perfect matching of the m = 0 quiver by specifying an orientation of its fields. This construction realizes both the field content and the superpotential of the quiver block.
It is therefore natural to expect that 3d printing and product are two instances of a single overarching construction. Such procedure would include both the products of m-graded quivers embedded in manifolds, possibly with boundaries, and an operation to glue two such manifolds along their boundaries under suitable conditions. We leave the task of understanding this construction in complete generality and its physical realization to a future work.

Conclusions
Over the years, there has been tremendous progress in the map between the geometry of singularities and the corresponding quiver theories on branes. This started with a few isolated examples of CY 3-folds and evolved into the development of brane tilings, tools that vastly simplify that study of infinite classes of geometries. Similar tools were later developed for higher dimensional CYs. We regard the CY product as a significant development in the arsenal of tools to connect geometry and quiver theories. It allows us to straightforwardly compute quiver theories in cases that were previously out of practical reach.
We envision multiple directions for future research. To name a few: • The CY product will help investigating the order (m + 1) dualities of the mgraded quiver theories associated to CY (m + 2)-folds. There is a large amount of freedom in this construction: choice of phases for the quiver theories of the parent geometries and choice of perfect matchings for the interlacing points. 14 Therefore, given a target CY, there are multiple possible decompositions into CY factors. In fact, different decompositions can even differ in the dimension of the components. It is therefore worthwhile to study the interplay between this vast landscape of possibilities and the intricate space of dual theories.
• The CY product is particularly amenable to automatic computer implementation. It is therefore ideally suited for generating large datasets of CYs/quiver theories. Such datasets would provide valuable insights into the structure of these theories. Moreover, they can be used to test the applicability of modern ideas such as machine learning to problems involving quiver theories, such as the classification of duals for general m. Initial explorations of these ideas have been undertaken in [38].
• As mentioned in §6.1 in the case of F (m) 0 , the CY product can be applied iteratively, equivalently using multiple factors. In this way, it is possible to build 14 Moreover, the perfect matchings are phase dependent.
-30 -quiver theories for complicated, higher dimensional geometries using very simple, low dimensional building blocks. A similar approach has been exploited to build some of the infinite classes of theories in [24].
• From a first principle perspective, we can calculate the quivers associated with a CY m+2 via the topological B-model [21][22][23][24]. However, this approach requires knowledge of the fractional branes as a starting point, which is often challenging. It would be interesting to investigate the correspondence between the B-model and CY product approaches.  Figure 14: The 9 terms in W P Q coming from both T P and T Q quartic. Red arrows represent the products of a field in p and a field in q. Black arrows descend from fields that are not in p or q.
to expect that these two theories are related by duality. Knowing the terms arising from an order k − 1 term and an order l term, we can recursively derive the terms arising from an order k term and an order l term. To do so, we can simply split the order k term into an order k − 1 term, a cubic term and a mass term. Continuing this iterative process for a few more steps we can infer the structure of the general case, which is depicted graphically in Figure 15. Every term in W P Q contains one field that is the product of a field in p and a field in q, and two fields that are the product of a field not in p and a field not in q. They correspond the red and two black diagonal arrows. There is exactly one term for every choice of two diagonal black arrows. Every one of the three blue boxes contains a path between two of these fields composed exclusively of horizontal and vertical arrows, i.e. of fields that are the product of a field and a node. The precise path depends on the breaking of the order k term into an order k − 1 term, a cubic term and a mass term.
classes needed for computations of the products m k with k > 2. ip i p+1 iq i q+1 i k i 1 Figure 15: The general structure of a term in W P Q descending from an order k and an order l terms. The blue boxes contain paths involving horizontal and vertical fields, i.e. products of a field and a node. The multiplicity of terms corresponds to the different ways of choosing the two black diagonal fields.

B Products and Geometry
Here we explain how the product theory gives rise to the desired geometry, which arises as its classical moduli space. To do so, we show how the perfect matchings of P p × Q q result in the toric diagram described by (3.1). First, we note that the collection of all the conjugated fields forms a perfect matching. 18 This is the perfect matching that corresponds to the "central point" (u 0 , v 0 ) of T CY m+n+3 .
Given a perfect matchingp of P we can construct a perfect matching that we will callp × q of P p × Q q . Ifp corresponds to the point u i in T CY m+2 thenp × q corresponds to the point (u i , v 0 ) of T CY m+n+3 . In order to constructp × q, we divide the fields iñ p into two sets. The first setp 0 contains the fields inp that are also in p, while the second setp * contains the fields inp that are not in p, namelỹ It is clear that with these definitions bothp × q and p ×q contain either the field or its conjugate for every field in P p × Q q . We will now show that the fields in them also cover every term in the superpotential exactly once.
We begin withp × q and consider W P , W Q and W C and W P Q separately. Starting with W P let us consider a term T P in the superpotential of P . This term gives rise to a number of terms in W P as shown in (5.3) and (5.4). Sincep is a perfect matching of P , then T P contains exactly one field fromp. There are three possibilities for how such field appears in a term of W P descending from T P : • It gets replaced by its product with a node of Q. The resulting field is inp × J so this term is covered exactly once byp × q.
• This field is common top and p and gets replaced by its product with a field in q. The result is a field inp 0 × q.
• This field is inp but not in p and gets replaced by its product with a field not in q. The result is a field inp * ×