4d Crystal Melting, Toric Calabi-Yau 4-Folds and Brane Brick Models

Let us consider the gauge theory on D1-branes on $\mathbb{C}^{4}$, whose toric diagram is shown in Figure 2.

This theory can be obtained by dimensional reduction of $4d$ $\mathcal{N}$=4 SYM. The resulting $2d$ theory has $(8,8)$ SUSY. In $(0,2)$ language, the theory contains a vector multiplet associated with a single $U(N)$ gauge group, four chiral fields ($X$, $Y$, $Z$ and $W$) and three Fermi fields ($\Lambda_{i}$, $i=1,2,3$), all transforming in the adjoint representation of the gauge group. This information is summarized in the quiver shown in Figure 3.

The corresponding $J$- and $E$-terms are as follows

We can add flavor fields to the quiver by introducing higher dimensional branes. Let us consider a single D9-brane spanning the two dimensions of the gauge theory plus the entire $\mathbb{C}^{4}$. The gauge theory is the same one as before, with the addition of a single chiral arrow $q$, as shown in Figure 4. The flavor node, shown in blue, represents the D9-brane. The field $q$ does not participate in any $J$- or $E$-terms.

In what follows, we will interpret the class of theories described by brane brick models as the supersymmetric quantum mechanics on D0-branes probing the corresponding toric CY 4-folds. The extra flavor introduced in the previous section corresponds to a D8-brane with $B$-field spanning $\mathbb{C}^{4}$. In practice, we simply remove the two field theory dimensions from the previous discussion (or from the typical brane brick model literature, which is typically about theories on D1-branes).

In this context, the chiral and Fermi fields of the unflavored theories are in the D0-D0 sector. The flavor introduced in the previous section corresponds to a D8-D0 field. The flavored quiver theory in the previous section is indeed the one underlying the Magnificent Four model. More general flavor branes will be discussed in Section §11.

Figure 5 shows the periodic quiver and dual brane brick model for $\mathbb{C}^{4}$. Black and red faces in the brane brick model correspond to chiral and Fermi fields, respectively. To simplify the figure, the region shown in both cases is larger than a unit cell.

Throughout the paper, we will often consider the universal covers of the periodic quiver and the brane brick model. Figure 6 illustrates how the combination of several unit cells of each of them looks like.

Brick matchings are the brane brick model analogues of perfect matchings for dimer models. They play a central role in the connection between the quiver theories and the underlying CY 4-folds. Below we present three equivalent definitions for them.

The theory for $\mathbb{C}^{4}$ has four brick matchings, which are in one-to-one correspondence with the chiral fields associated to the four complex directions. They are presented in Figure 7. We only show the chiral fields in them since, as mentioned earlier, this information is sufficient for determining the Fermi content.

The $P$-matrix summarizing the perfect matchings is therefore

Figure 8 shows the correspondence between these brick matchings and points in the toric diagram of $\mathbb{C}^{4}$.

The difference between two brick matching $p_{1}-p_{2}$ results in the (disjoint union of) oriented surfaces on the brane brick model or its universal cover. This difference is defined as follows:

• •

  The orientation of every face in a perfect matching is given by the orientation of the corresponding bifundamental (or adjoint) fields.

• •

  The orientations of faces in $p_{2}$ are reversed.

• •

  Faces contained in both perfect matchings are combined with opposite orientations and hence disappear from the final result.

• •

  The orientation of the resulting surface is determined by the orientation of the chiral fields it contains.

Below we present two typical examples of surfaces obtained as differences between brick matchings.

Without loss of generality, let us assume that the flavor $q$ is connected to node $i_{0}$ in $\tilde{Q}$. We define the crystal such that every atom in it is in one-to-one correspondence with an oriented path of chiral fields in $\tilde{Q}$ starting from $q$ (equivalently starting from $i_{0}$), modulo $J$- and $E$-term relations.⁸⁸8$J$- and $E$-term relations lead to equivalences between paths in the periodic quiver with the same endpoints. In other words, atoms can generically be reached in multiple, equivalent ways. The crystal is built out of atoms stacked on top of the nodes of $\tilde{Q}$ on $\mathbb{R}^{3}$. Since the quiver for $\mathbb{C}^{4}$ has a single node, the crystal has a single type of atom. The crystal contains a fourth dimension, that we will denote the depth $d$.Chiral fields determine the relative depth of the atoms connected by them. If there is a chiral arrow from atom $i$ to atom $j$, atom $j$ is at a higher depth than $i$. We can think about two such atoms as partially overlapping.⁹⁹9We reserve the term overlapping for atoms that are directly on top of each other.

To build the crystal, we first place an atom over node $i_{0}$, which will become the tip of the crystal. Then, we iteratively add new atoms according to the chiral fields that emanate from the corresponding nodes in the original quiver. The $(x,y,z)$ position of an atom is the one of the corresponding node in $\tilde{Q}$, while the depth is proportional to the $R$-charge (equivalently the conformal dimension) of the corresponding chiral operator. In the simple case of $\mathbb{C}^{4}$, in which the four chiral fields are equivalent, this is simply proportional to the length of the path (namely the number of chirals in it).¹⁰¹⁰10The explicit values of the depth are not necessary for determining melting configurations, as we discuss in the coming sections. This procedure results in an infinite crystal.

Modulo $J$- and $E$-term constraints, every oriented path $\gamma_{i_{0},j}$ defining atom $j$ of the crystal can be expressed as

where $v_{i_{0},j}$ is a shortest path connecting $i_{0}$ to $j$, $\omega$ is the closed loop associated to a chiral cycle, and $n\geq 0$.¹¹¹¹11As we explained in Section §4.1, all chiral cycles are equivalent up to $J$- and $E$-terms, and therefore can be identified with a single variable $\omega$. We can interpret $v_{i_{0},j}$ as defining an atom at the top layer of the crystal. An atom with an additional factor of $\omega^{n}$, is located directly below, $n$ levels down.

We now consider consider molten crystals, i.e. configurations that are obtained by removing atoms from the unmolten crystal. We will denote any crystal configuration (i.e. molten or not) as $\mathcal{I}_{\mu}$. We define the corresponding complement $\Omega_{\mu}$ as the difference between the unmolten crystal and $\mathcal{I}_{\mu}$, i.e. $\Omega_{\mu}$ is the set of removed atoms. For brevity, we will refer to the $\Omega_{\mu}$ as melting configurations.¹²¹²12Anticipating the connection between molten crystals and brick matchings that will be discussed in Section §10, we use the same type of subindex to label both.

Let us momentarily focus on the unflavored quiver $Q$. Let us denote $Q_{0}$ and $Q_{X}$ the sets of nodes and chiral arrows in $Q$, respectively. The set of all open oriented chiral paths in $Q$ gives rise to an algebra $\mathbb{C}[Q_{0},Q_{X}]$, that we will call the chiral path algebra. Given the ideal of relations coming from vanishing $J$- and $E$-terms

where $a$ runs over all Fermis, it is natural to define the factor algebra $A=\mathbb{C}[Q_{0},Q_{X}]/\mathcal{I}_{J,E}$. $A$ consists of the open chiral paths in the unflavored quiver modulo vanishing $J$- and $E$-terms.

Melting configurations are constructed according to the following melting rule.

It is straightforward to see that every molten crystal $\mathcal{I}_{\mu}$ defines an ideal of $A$. To show this, we consider the contraposition of the melting rule, which implies that for any $\gamma\in\mathcal{I}_{\mu}$ and any $\alpha\in A$, then $\gamma\alpha$ is also in $\mathcal{I}_{\mu}$. In simple words, starting from any atom in a molten crystal and moving from it along a path $\alpha\in A$, results in another atom in the molten crystal, i.e. an atom that has not been removed.

Explicit constructions of the unmolten crystal and melting configurations will be presented in Sections §6 and §8.

Visualizing a $4d$ crystal is of course challenging. In this paper we will present various ways to do so. The first approach consists of slicing the crystal at different depths. This approach can be applied to both unmolten and molten crystals. Let us illustrate this construction with the unmolten crystal. Figure 10 shows the atoms in it up to $d=4$.

From Figure 10, we see that the $3d$ slice at depth $d$ is a tetrahedron of edge length $d+1$. The number of atoms at depth $d$ is therefore

which results in

Equation (7.18) can be generalized to $D$ dimensions, for which the number of atoms in the $(D-1)$-dimensional slices at depth $d$ is

Let us definite the “$3d$ surface” of a $4d$ crystal configuration $\mathcal{I}_{\mu}$ as its top layer, in analogy with the $2d$ surface of an ordinary $3d$ crystal.¹⁴¹⁴14We feel it is useful to stick to the term surface to denote the top layer of a crystal due to its intuitive interpretation, despite that, in this case, it is a 3-dimensional object. In other words, we define:

A corollary of this definition is that if we consider all chiral arrows connecting atoms in $S_{\mu}$, they are such that they do not form closed oriented loops. If there was such a loop, we would have two atoms at the same $(x,y,z)$ position but at different values of $d$. Therefore, one of them could not be in $S_{\mu}$. We will revisit this fact in Section §10 where we present a reformulation of the crystal in terms of brane brick models.

As an example, let us consider the surface of the unmolten crystal. Algorithmically, it can be constructed following the same procedure we used to build the full crystal, but keeping only the atoms of lowest depth for every $(x,y,z)$ coordinate. With the normalization we are using, the depths of overlapping atoms differ by multiples of 4, so at every depth $d$, we need to remove the atoms at every $d^{\prime}<d$ such that $d-d^{\prime}=0\mod 4$. Figure 11 shows the atoms on the surface up to $d=7$. Atoms on the surface and the interior of the crystal are shown in blue and orange, respectively. We observe that up to $d=3$ all atoms are on the surface.

The crystal for $\mathbb{C}^{4}$ is equivalent to a $4d$ corner, with melting configurations in one-to-one correspondence with solid partitions. The connection between the quiver theory for D0 and D8-branes on $\mathbb{C}^{4}$ and solid partitions was first explored in the context of the Magnificent Four model Nekrasov:2017cih ; Nekrasov:2018xsb ; Nekrasov:2023nai . The partition function counting solid partitions is

In the coming section, we will illustrate how this partition function arises from the crystal model.

Here we present explicit examples illustrating how the Hasse diagram can be used to classify melting configurations. According to the melting rule, whenever an atom is removed from the crystal, all atoms above it, i.e. all atoms contained in downward paths terminating in it, should also be removed.

The first step is to identify the perfect matching that describes the crystal before any melting. For $\mathbb{C}^{3}$, this configuration is often referred to as the empty room configuration for its similarity with the empty corner of a $3d$ room. The corresponding perfect matching, which we will call the canonical perfect matching $p_{0}$, is shown in Figure 20 Ooguri:2009ijd . The figure shows a finite region that should be extended to the infinite hexagonal lattice in the obvious way. To simplify the visualization, we have omitted the black and white nodes of the brane tiling. The canonical perfect matching consists of three regions. Inside each of them, $p_{0}$ is given by one of the perfect matchings of the original brane tiling associated to one of the corners of the $\mathbb{C}^{3}$ toric diagram ($p_{x}$, $p_{y}$ and $p_{z}$). Interestingly, the boundary between these regions, represented with dotted lines in Figure 20, agrees with the $(p,q)$ web in Figure 19 (up to an obvious transformation). Brane tilings give rise to a “discretized” version of the underlying CY₃ geometry Okounkov:2003sp ; Ooguri:2009ijd ; Chuang:2009crq ; Eager:2011ns . This fact holds for general toric CY₃’s. In Figure 20, we have indicated the hexagons on which two or three regions coincide in blue. On these hexagons, $p_{0}$ is given by the appropriate combination of the basic perfect matchings.

Let us now consider general melting configurations which, in this language, correspond to perfect matchings of the universal cover of the brane tiling.

Melting configurations are in one-to-one correspondence with brick matchings of the universal cover of the brane brick model. We will elaborate on the details of this correspondence in Sections §10.2 and §10.4. We now discuss how the fourth dimension of the crystal, i.e. the depth or the melting height, emerges from brick matchings.

Figure 27 shows the canonical brick matching $p_{0}$ of the universal cover of the brane brick model for $\mathbb{C}^{4}$ that describes the unmolten crystal. Generalizing the discussion in Section §9.1, $p_{0}$ consists of four regions, inside each of which it is given by one of the brick matchings of the brane brick model associated to a corner of the $\mathbb{C}^{4}$ toric diagram. The boundaries between every pair of these regions are six 2-dimensional surfaces that are in one-to-one them correspondence with the six edges connecting corners in the toric diagram. In other words, we can regard these surfaces as a “$(p,q,r)$-web” dual to the toric diagram of a CY₄. We have indicated the bricks on which two or three regions coincide in green and blue, respectively. There is also a brick at the center of the configuration, and hence not visible in the figure, on which the four regions come together. On all these special loci, brick matchings are given by the appropriate combinations of the basic brick matchings.¹⁷¹⁷17The region shown in Figure 27 should be regarded as a finite subset of the infinite universal cover of the brane brick model. Its beautiful rhombic dodecahedron shape is an artifact of how it was generated, which is analogous to how the hexagonal region for $\mathbb{C}^{3}$ in the figures of Section §9 was created. In the normalization used in this figure, the universal cover of the brane brick model consists of bricks on a lattice generated by the following vectors: $v_{x}=(0,-2,\sqrt{2})$, $v_{y}=(0,2,\sqrt{2})$, $v_{z}=(2,0,-\sqrt{2})$ and $v_{W}=(-2,0,-\sqrt{2})$. Each of the regions covered by one of the basic brick matchings $p_{i}$, with $i=x,y,z,d$, are given by bricks located at $n_{j}v_{j}+n_{k}v_{k}+n_{l}v_{l}$, with $j,k,l\neq i$ and $n_{j},n_{k},n_{l}=1\ldots,n_{max}$, for some maximum size $n_{max}$. Also, the reason why the facets of the region in Figure 27 are not triangles as the ones in Figure 10 is simply that they do not correspond to surfaces of equal depth. This perfect matching represents the $4d$ empty room configuration.

Let us explore this configuration in further detail. Figure 28.a shows a view of $p_{0}$ along one of the blue rays, e.g. the vector $v_{W}$ in our construction. Remarkably, from this viewpoint, the configuration reduces to the canonical perfect matching for $\mathbb{C}^{3}$, which we presented in Figure 21. The truncated octahedra bricks and their brick matchings get projected onto the hexagonal lattice and its perfect matchings! Figure 28.b shows a view from the antipode, i,e. from $-v_{W}$, from where we observe the entire plane covered by the remaining perfect matching $p_{D}$.

As for $\mathbb{C}^{3}$, the reference brick matching $p_{r}$ for $\mathbb{C}^{4}$ is basically given by the “complement” of $p_{0}$ in each region, surface, etc. Since we will be primarily interested in the melting height of configurations, instead of the depth, and visualizing these perfect matchings is challenging, we will not present a figure with $p_{r}$.

The canonical perfect matching we introduced above was determined such that, in combination with the reference brick matching, it gives rise to the depth determined by the quiver construction and illustrated in Figure 10. This procedure translates into an algorithmic prescription for determining $p_{0}$ from the quiver, which we now review. The analogous construction for CY 3-folds was discussed in Ooguri:2009ijd .

Let us focus on projection of the surface of the unmolten crystal onto quiver space and consider the chiral arrows connecting these atoms. Since we restrict to the surface of the crystal, these arrows do not form closed loops. If closed loops were present, they would give rise to atoms with the same coordinates on quiver space but different depths, therefore not on the surface. Equivalently, the arrows in this construction are those in the Hasse diagram of Figure 12 when restricted to atoms on the surface. The resulting quiver is shown in Figure 29, where red arrows indicate the four primary directions of this crystal, $v_{i}$, $i=X,Y,Z,W$.

From Figure 11, we know that the slices of the surface at each depth have a tetrahedral shape. When projected onto quiver space, the entire surface becomes an onion-like collection of tetrahedral shells of increasing size and depth. Since the arrows in Figure 29 point in the direction of increasing depth, they provide a quiver representation for the gradient of the depth. The canonical brick matching, once combined with the reference brick matching, gives rise to surfaces of increasing height (i.e. of decreasing depth). Therefore, we conclude that the arrows in the quiver in Figure 29 corresponds to $p_{r}$, while its complement correspond to $p_{0}$. The fact that the resulting objects are brick matchings follow from the absence of closed loops on this quiver and the definition of brick matchings in terms of chiral cycles given in Section §4.1.

In this section, we discuss how general melting configurations are described in terms of brane brick models and brick matchings. For concreteness, we illustrate our ideas using Example 3 of Section §8.1.

To start, let us visualize how the atoms in this partition translate into brane brick models. Figure 30 shows the constant depth slices of this melting configuration, originally shown in Figure 16, in terms of the brane brick model.¹⁹¹⁹19Figures 30 and 16 differ by a $45^{\circ}$ rotation around the vertical axis. This follows from the different generators for the lattice chosen in each case. They are given in (7.15) and footnote 17. It is a straightforward exercise to connect the two constructions. Figure 31 shows the projection of these atoms onto quiver space.

The brick matching $p$ corresponding to the melting configuration under consideration is shown in blue in Figure 32, where we also show the canonical brick matching $p_{0}$ in red. We only give the chiral field content of both brick matchings, since Fermis can be reconstructed from this information. To simplify the figure, we restrict both brick matchings to the relevant region in Figure 31. $p$ and $p_{0}$ coincide outside of this region and, therefore, they cancel out when subtracted. To be able to peek inside the configuration, we have split it open through the middle. The other halves of these brick matchings are identical the ones shown.

The difference $p-p_{0}$ results into two nested oriented surfaces, as shown in Figure 32. Figure 33 shows the resulting melting height. The blue region at the origin has height 2, i.e. it corresponds to two overlapping atoms in the $4d$ crystal as expected. This is in perfect agreement with Figure 17. The configuration has 14 bricks with melting height 1 and 1 brick with melting height 2, corresponding to 16 atoms, as expected.

It is worth noting that, from a graphing point of view, for general toric singularities it is often simpler to consider the periodic quiver version of crystal melting than its brane brick model counterpart, since representing the latter might be harder. Having said that, the brane brick model realization of the crystal melting model is conceptually important since, among other things, it maps the problem to the combinatorics of brick matchings.

To illustrate the construction of crystals for more general flavor configurations, let us consider the case of D0-branes and a single D6-brane on $\mathbb{C}^{4}$. Figure 34 shows the corresponding quiver. In the notation introduced above, this example has $N_{q}=N_{\Psi}=1$ and $N_{\tilde{q}}=0$.

Without loss of generality, let us assume that the D6-brane spans the $X$, $Y$ and $Z$ directions, and it is located at $W=0$. Then, in addition to the $J$- and $E$-terms of the D0-brane theory, which were given in (3.1), we have a $J$-term involving the flavors, given by the gauge invariant coupling $q\,W\Psi$. This coupling is easy to understand, since a (classical) expectation value for $W$ would result on a mass term for the flavor fields $q$ and $\Psi$, which agrees with the fact that separating the D6 and D0-branes along the $W$ direction makes the strings stretched between them massive.

The atoms in the unmolten crystal correspond to open oriented paths of chiral fields starting from $q$, modulo $J$- and $E$-term relations. If we considered only the $J$- and $E$-terms in (3.1), we would generate the same crystal we previously construceted in Section §7. However, the vanishing of the $J$-term for $\Psi$ gives rise to the additional relation

Since every atom corresponds to a path containing $q$, (11.25) implies that any path containing a $W$ field vanishes and the corresponding atom is not present in the crystal. Not surprisingly, we recover the well-known crystal melting model for D0-branes and a D6-brane on $\mathbb{C}^{3}$, which we also discussed in Section §9. The crystal turns out to be infinite but 3-dimensional, with melting configurations in one-to-one correspondence with plane partitions. Figure 35 shows constant depth slices of the unmolten crystal up to $d=4$. In contrast with Figure 10, we observe that for this model the slices are 2-dimensional, in agreement with the fact that in this case the full crystal is 3-dimensional.