On the Master Space for Brane Brick Models

We systematically study the master space of brane brick models that represent a large class of 2d (0,2) quiver gauge theories. These 2d (0,2) theories are worldvolume theories of D1-branes that probe singular toric Calabi-Yau 4-folds. The master space is the freely generated space of chiral fields subject to the J- and E-terms and the non-abelian part of the gauge symmetry. We investigate several properties of the master space for abelian brane brick models with U(1) gauge groups. For example, we calculate the Hilbert series, which allows us by using the plethystic programme to identify the generators and defining relations of the master space. By studying several explicit examples, we also show that the Hilbert series of the master space can be expressed in terms of characters of irreducible representations of the full global symmetry of the master space.


Introduction
In this work, we take a peek at the rich algebro-geometric structure of the master space of 2d (0, 2) supersymmetric gauge theories that arise as worldvolume theories of D1-branes probing toric Calabi-Yau 4-folds.These theories are realized by a Type IIA brane configuration that is connected to the D1-brane at the Calabi-Yau singularity via T-duality [1,2].These brane configurations and the corresponding 2d (0, 2) gauge theories are known as brane brick models and led to new interesting developments in various contexts [3][4][5][6][7].
In particular, these recent developments involve the study of the mesonic moduli space of brane brick models [1,2].The mesonic moduli space is defined as the space of gauge invariant operators under the J-and E-terms of the brane brick model.When the 2d theory has only U (1) gauge groups, the mesonic moduli space is exactly the toric Calabi-Yau 4-fold associated to the brane brick model.The gauge invariant operators carry charges under the mesonic flavor and U (1) R symmetries which combined become the isometry of the toric Calabi-Yau 4-fold.Brane brick models and their mesonic moduli spaces have been recently fully classified for large classes of toric Calabi-Yau 4-folds such as brane brick models corresponding to smooth Fano 3-folds [8] and cones over the Sasaki-Einstein 7-manifolds Y p.k (CP1 × CP 1 ) and Y p.k (CP 2 ) [9].
As introduced in [1], in analogy to brane tilings [10][11][12][13][14][15][16][17] realizing 4d N = 1 supersymmetric gauge theories associated to toric Calabi-Yau 3-folds, brane brick models have a master space that is larger than the mesonic moduli space.The master space of the brane brick model is defined as the space of chiral fields subject to the J-and E-terms constraints and the non-abelian part of the gauge symmetry of the brane brick model, while the mesonic moduli space requires gauge invariance under the full gauge symmetry.
In the case when the brane brick model has only U (1) gauge groups, the master space simply takes the form of the following algebraic variety, where n χ are the number of chiral fields X 1 , . . ., X nχ , and J a = 0 and E a = 0 are the Jand E-terms corresponding pairwise to a Fermi field Λ a of the brane brick model.We note that the master space is a toric variety [18][19][20] given that the J-and E-terms are all binomial relations.Furthermore, we note that the variety in (1.1) is generally reducible into irreducible components.We call the top-dimensional irreducible component as the coherent component Irr F .In the following work, when we refer to the master space of the brane brick model, we automatically refer to its irreducible coherent component Irr F .The U (1) symmetries that are not imposed on the master space become symmetries along the additional directions that are added to the mesonic moduli space in order to form the master space.We refer to these added directions as baryonic directions and the U (1) symmetries as the baryonic part of the global symmetry of the master space for brane brick models. 1 Out of the G U (1) symmetries in an abelian brane brick model, G − 1 are independent because of the bifundamental nature of the chiral fields.We have also the 3 global mesonic flavor symmetries and the U (1) R -symmetry of the mesonic moduli space, which together with the independent U (1) G−1 symmetries gives us the total rank of the global symmetry of the master space, G + 3.This also represents the dimension of the master space for abelian brane brick models with U (1) gauge groups.
In the following work, we systematically calculate the Hilbert series [21][22][23][24] for the master spaces of a selection of brane brick models corresponding to different toric [16,36] 7 SU (4) × SU (4) × U (1) R Q 1,1,1 [10,6] 7 SU (4) × SU (2) × SU (2) × U (1) R Y 2,4 (CP 2 ) [18,24] 9 Table 1: The mesonic moduli space M mes of the brane brick model, the coherent component of the master space Irr F , the dimension of Irr F and the full global symmetry of Irr F .The notation [n g , n r ] denotes the number of generators n g and the number of first order relations amongst the generators n r for Irr F if it is a non-complete intersection.
Calabi-Yau 4-folds.The Hilbert series for the master space is a generating function that counts operators invariant under only the non-abelian part of the gauge symmetry and are subject to the J-and E-terms of the brane brick model.In the following work, we will concentrate on master spaces of abelian brane brick models whose gauge groups are all U (1).Even with this restriction, we see that the master spaces exhibit extremely rich algebro-geometric properties.For example, by the use of the plethystic programme [21][22][23][24][25], we obtain expressions for the generators and defining first order relations amongst the generators of the master space.By identifying the full enhanced global symmetry of the master space, we show how the generators and defining relations of the master spaces transform under the full global symmetry.We also express the generators and relations of the master spaces in terms of GLSM fields that we obtain through the forward algorithm [1,26,27] for brane brick models.Table 1 summarizes the full global symmetry of the master spaces that we study in this work.By studying closely the geometric structure of the master spaces for brane brick models using the Hilbert series, we discover new phenomena specific to master spaces for brane brick models.These new discoveries involve the occurrence of extra GLSM fields that over-parameterize the master space for certain brane brick models, the enhancement of global symmetries of the master spaces, and the discovery that master spaces for brane brick models are toric but not necessarily Calabi-Yau.In the following work, we summarize these discoveries and illustrate their connection to the master spaces for brane brick models that correspond to SPP × C [3], C 4 /Z 4 with orbifold action (1, 1, 1, 1) [1,28,29], Q 1,1,1 [1,2,[30][31][32] and Y 2,4 (CP 2 ) [9,33,34].
Our work is organized as follows.In section §2.1, we give a brief introduction to brane brick models and discuss in section §2.2 the forward algorithm [1,26,27] that allows us to construct the mesonic moduli spaces for brane brick models.We introduce the master space for brane brick models in section §3.1 and summarize the computation for the Hilbert series [21][22][23][24] in section §3.2, using either directly the J-and E-terms of the brane brick model or the symplectic quotient description of the master space in terms of GLSM fields.Section §4 illustrates our findings in four explicit examples of master spaces for brane brick models.We conclude our work with a summary of our findings and a discussion for future research directions in section §5.

Background 2.1 Brane Brick Models
The worldvolume theories of D1-branes probing a toric Calabi-Yau 4-fold form a large class of 2d (0, 2) supersymmetric gauge theories, which can be realized by a Type IIA brane configuration known as a brane brick model [1,2].The brane brick model is connected to the D1-branes at the toric Calabi-Yau singularity by T-duality.The Type IIA brane configuration consists of D4-branes wrapping a 3-torus T 3 .They are also suspended from a NS5-brane that wraps a holomorphic surface Σ defined as, Σ : P (x, y, z) = 0 , ( where P (x, y, z) is the Newton polynomial in x, y, z ∈ C * of the toric diagram associated to the probed toric Calabi-Yau 4-fold.The D4-brane meets the NS5-brane precisely where the holomorphic surface Σ intersects the 3-torus T 3 .Table 2 shows the Type IIA brane configuration for brane brick models realizing 2d (0, 2) supersymmetric gauge theories corresponding to toric Calabi-Yau 4-folds.0 1 2 3 4 5 6 7 8 9 The Type IIA brane configuration corresponding to a brane brick model on T 3 .
The intersections between D4-brane and the NS5-brane form a tessellation of the 3-torus T 3 .This tessellation of the 3-torus is precisely what we call as the brane brick model.The following summarizes the dictionary between components of the brane brick model on T 3 and the corresponding 2d (0, 2) supersymmetric gauge theory [2]: • Bricks.These are 3-dimensional polytopes that form the fundamental building blocks of the tessellation of the 3-torus.Each brane brick corresponds to a U (N ) i gauge group of the 2d supersymmetric gauge theory.
• Faces.The boundary of a brane brick consists of even-sided 2-dimensional polygons.A subset of these polygonal faces are oriented along their boundary edges.Such oriented faces correspond to bifundamental chiral fields X ij in the corresponding 2d (0, 2) theory.All other polygonal faces that are unoriented along their boundary edges correspond to Fermi fields Λ ij (and their conjugate Λij ).These Fermi faces are always 4-sided in a brane brick model.The two brane bricks adjacent to a polygonal face correspond to the two gauge groups U (N ) i and U (N ) j under which X ij or Λ ij associated to the brick face is in the bifundamental representation.• Edges.In a brane brick model, edges are always connected to a single brick face corresponding to Λ ij and a collection of oriented brick faces corresponding to a set of chiral fields X ij .Accordingly, each brick edge is associated to monomial terms known as plaquettes in the brane brick model, which take one of the following forms where J ± ji and E ± ij are monomial products of chiral fields.Plaquettes corresponding to opposite edges of the 4-sided Fermi face are identified to either a binomial J-term or a binomial E-term of the 2d (0, 2) supersymmetric gauge theory, as illustrated in Figure 1.
In terms of the J-and E-terms of a brane brick model and the plaquettes associated to them, we are able to define special collections of chiral fields known as brick matchings [1,2].A brick matching p µ is a collection of chiral fields such that the chiral fields in the brick matching cover the plaquettes ( exactly once each.We can summarize the chiral fields X m contained in a brick matching p µ in terms of a brick matching matrix P , whose components take the following form, Additionally, the chiral fields X m of a brane brick model can be expressed in terms of products of brick matchings as follows, We note that brick matchings p µ correspond to GLSM fields [1,2,35] that describe the mesonic moduli spaces of brane brick models.GLSM fields play an important role in the symplectic quotient description [18,19] of the mesonic moduli space M mes as well as the master space F of brane brick models.The following sections are going to give a brief overview of GLSM fields and their role in describing master spaces of brane brick models.

The Forward Algorithm
The forward algorithm first introduced in [1] for brane brick models allows us to construct the mesonic moduli spaces of the corresponding 2d (0, 2) theories using GLSM fields.In the following section, we review the forward algorithm and illustrate the role played by GLSM fields to describe mesonic moduli spaces and master spaces of brane brick models.GLSM Fields.A key step in the forward algorithm is the construction of the Pmatrix in (2.4) encoding the GLSM fields p µ in terms of chiral fields X m in the brane brick model.We first observe that due to the binomial J-and E-terms of the 2d (0, 2) theory, where J ± ji and E ± ij are monomial products of chiral fields X m , not all chiral fields in the brane brick model are independent to each other.In fact, by relabelling the independent fields as v k , we can express all chiral fields X m as products of the following form, where m = 1, . . ., n χ labels the n χ chiral fields X m in the brane brick model, and k = 1, . . .G + 3 labels the independent fields v k with G being the number of gauge groups.Here, K is a n χ × (G + 3)-dimensional matrix which identifies the binomial Jand E-terms with the independent fields v k .
Because the J-and E-terms of the brane brick model are binomial, they form a binomial ideal which is related to toric geometry [20].In terms of toric geometry, the K-matrix in (2.7) defines a cone M + , which is generated by non-negative linear combinations of the vectors K m ∈ Z G+3 encoded in the K-matrix.The cone M + has a dual cone N + which is generated by non-negative linear combinations of another set of (G + 3)-dimensional vectors T µ where µ = 1, . . ., c.These vectors can be combined to form a (G + 3) × c-dimensional matrix known as the T -matrix.We note that the cones M + and N + are dual to each other such that which determines the number c of distinct vectors T µ .We can use now the T -matrix to identify the independent fields v k with a set of new fields p µ such that the chiral fields can be expressed in terms of products of p µ with strictly positive powers, (2.9) By combining the above expression with (2.7), we obtain an expression of the P -matrix originally defined in (2.5) as follows, where the new fields p µ with µ = 1, . . ., c are GLSM fields of the brane brick model.
The Mesonic Moduli Space and Toric Calabi-Yau 4-folds.Let us focus on abelian brane brick models with U (1) gauge groups.The mesonic moduli space M mes of the brane brick model [1,2] is the toric Calabi-Yau 4-fold geometry probed by a single D1-brane given that the worldvolume theory is a 2d (0, 2) supersymmetric gauge theory with U (1) gauge groups.The geometry of the toric Calabi-Yau 4-fold is encoded in the brane brick model.In order to obtain the geometry of the toric Calabi-Yau 4-fold from the brane brick model, we first express the J-and E-terms as well as the D-terms of the brane brick models as U (1) charges on the GLSM fields: • J-and E-terms.In terms of the GLSM fields summarized in the P -matrix, the U (1) charges under the J-and E-terms of the brane brick model are given by the kernel of the P -matrix, where the columns correspond to GLSM fields and the rows correspond to the U (1) c−G−3 charges carried by the GLSM fields due to the J-and E-terms.Here G refers to the number of gauge groups in the brane brick model.
• D-terms.Similarly, the D-terms of the brane brick model are captured in terms of U (1) charges carried by GLSM fields as follows, where ∆ is the reduced incidence matrix of the quiver for the brane brick model.The columns of the Q D -matrix correspond to the GLSM fields that carry the U (1) G−1 charges from the D-terms of the brane brick model.
Overall, the GLSM fields carry in total U (1) c−4 charges originating from both the J-and E-terms and the D-terms of the brane brick model.The total charge matrix is given by In terms of the total charge matrix Q t , we can define the mesonic moduli space of an abelian brane brick model as the following symplectic quotient, where C c is the freely generated space of GLSM fields, where c is the number of GLSM fields in the brane brick model.In terms of the total charge matrix Q t , we can also define the coordinate matrix for the toric diagram of the toric Calabi-Yau 4-fold, (2.15) The columns correspond to GLSM fields which map to vertices of the toric diagram.The coordinates of the vertices are given by 4-vertices in Z 4 .Under the Calabi-Yau condition, the end-points of the 4-vertices are all on a 3-dimensional hyperplane in Z 4 , allowing us to draw the toric diagram as a convex lattice polytope on Z 3 .
Extra GLSM Fields.It was noted in [1], that for certain brane brick models the toric diagram of the mesonic moduli space M mes obtained from (2.15) exhibits additional vertices that are not on the 3-dimensional hyperplane like the rest of the vertices of the toric diagram.That means, under a suitable GL(4, Z) transformation on the coordinates (x 1 , x 2 , x 3 , x 4 ) ∈ Z 4 of the vertices of the toric diagram, the set of vertices splits into two.The first set contains vertices that are on a 3-dimensional hyperplane GL(4, Z)-transformed such that x 4 = 1, and all other vertices are outside the hyperplane with x 4 = 1.These additional vertices with x 4 = 1 correspond to what we call as extra GLSM fields [1].
Although such extra GLSM fields manifest themselves as vertices in the toric diagram that seemingly break the Calabi-Yau condition on the mesonic moduli space M mes , it was shown in examples studied in previous work [1,2] that the extra GLSM fields act as an over-parameterization of M mes .Given that the mesonic moduli space is parameterized by mesonic gauge invariant operators that can be expressed products of GLSM fields, the presence or absence of extra GLSM fields does not affect the spectrum of operators.This means that the generators and the first order defining relations formed by the generators of the quotient in (2.14) remain unaffected by the presence or absence of extra GLSM fields.
In the following section, we describe the master space associated to a brane brick model.Like the mesonic moduli space for brane brick models, the master space can be parameterized in terms of GLSM fields of the brane brick model.For certain brane brick models, the master spaces exhibit extra GLSM fields when the mesonic moduli space also contains extra GLSM fields.The extra GLSM fields in the mesonic moduli spaces however are not the same as the ones for the master space.For some examples, as we are going to see in section §4, master spaces exhibit less extra GLSM fields than mesonic moduli spaces that contain extra GLSM fields.We are going to see in section §4 that extra GLSM fields of master spaces also have the features of an over-parameterization.
3 Master Spaces for Brane Brick Models

An Introduction to the Master Space
The master space F for brane brick models with U (1) gauge groups takes the form of an algebraic variety [12,14,15], where I JE = J a , E a is the quotienting ideal given by the relations J a = 0 and E a = 0 corresponding to all Fermi fields Λ a .2This algebraic variety is analogous to the master space of 4d N = 1 supersymmetric gauge theories given by brane tilings [10][11][12][13][14][15][16][17], where the master space here is the space of vanishing F -terms.We can summarize the properties of the master space F for a brane brick model as follows: • Because the J-and E-terms are binomial relations in chiral fields of the brane brick model, the quotienting ideal J a , E a is a binomial ideal and the resulting master space F is a toric variety [18][19][20].
• In general, the master space F is a reducible algebraic variety, which under primary decomposition [36,37] can be decomposed into irreducible components.Amongst these irreducible components, there is a top-dimensional irreducible component which is of the same dimension and degree as F .We call this the coherent component of the master space and denote it by Irr F .In the following discussion, whenever we refer to the master space of a brane brick model, we refer to the irreducible coherent component of the master space Irr F .
• The master space Irr F can be expressed as a symplectic quotient of the following form, where C c is parameterized by p 1 , . . ., p c , which are the GLSM fields of the brane brick model.The U (1) charges carried by the GLSM fields due to the J-and E-terms of the brane brick model are given by the Q JE -matrix, which we defined in (2.11).
• The master space Irr F is of dimension G + 3, where G refers to the number of U (1) gauge groups in the abelian brane brick model.
• For brane brick models with U (N ) gauge groups, the master space can be obtained by taking an additional quotient under the non-abelian SU (N ) part of the gauge symmetry, where G is the number of U (N ) gauge groups in the brane brick model.The resulting non-abelian master space Irr F N is expected to be non-toric.For the following work, we concentrate on master spaces Irr F for abelian brane brick models with U (1) gauge groups.
Global Symmetry of the Master Space.The master space Irr F of an abelian brane brick model exhibits the following global symmetries: • The mesonic symmetry of the master space is U (1) 4 or an enhancement with rank 4. The mesonic symmetry corresponds to the isometry of the mesonic moduli space, which is the toric Calabi-Yau 4-fold associated with the brane brick model.It contains the U (1) R symmetry and the mesonic flavor symmetries.
• The U (1) G gauge symmetry of the brane brick model acts as a symmetry of the master space.This part of the global symmetry is known as the baryonic part of the global symmetry for the master space Irr F .We take the name from the master spaces for brane tilings and 4d N = 1 supersymmetric gauge theories [12,[14][15][16][17].In total, we have G − 1 independent U (1) symmetries because all chiral fields transform in the bifundamental or adjoint representation of the U (1) in the quiver.
As a result, given the rank G − 1 baryonic part of the global symmetry and the rank 4 mesonic part of the global symmetry, the master space Irr F as expected has a global symmetry of rank G + 3.In terms of the brane brick model on the 3-torus, we can identify the U (1) 3 part of the global symmetry of the master space Irr F with the 3 S 1 -cycles of the 3-torus of the brane brick model.The remaining rank G symmetry corresponds to the G brane bricks of the brane brick model.

The Hilbert Series of the Master Space
Hilbert Series.A quintessential tool that is used to study the geometric structure of an algebraic variety is the Hilbert series [21][22][23][24].Given an affine variety Y in C k over which X is a cone, we define the Hilbert series to be the generating function for the dimension of the graded pieces of the coordinate ring of the form where f i is the quotienting ideal in terms of defining polynomials f i of Y .The dimension of the i-th graded piece Y i is the number of algebraically independent degree i polynomials on the variety Y .Accordingly, the Hilbert series takes the form, which always takes the form of a rational function.Here, the fugacity t keeps track of the degree i of the graded pieces Y i .In the case when the coordinate ring is multigraded with pieces Y i and grading i = (i 1 , . . ., i k ), the Hilbert series takes a refined form as follows, where the number of fugacities t i can be chosen to be as many as the dimension of the ambient space or as few as the dimension of X itself.
Hilbert Series of the Master Space.In our work, X is the master space F of a brane brick model with the corresponding coordinate ring given by, where X 1 , . . ., X nχ are the n χ bifundamental chiral fields of the brane brick model.I Irr JE is the ideal formed by the reduced J-and E-terms in the brane brick model which correspond to the coherent component of the master space Irr F .The Hilbert series of Irr F then can be obtained using the definition in (3.5).Given that the coordinate ring in (3.7) is in terms of chiral fields X 1 , . . ., X nχ in the brane brick model, we can introduce a grading corresponding to the global symmetry charges on the chiral fields.Taking m1 , m2 as the fugacities for the mesonic flavor part of the global symmetry, b1 , . . ., bG−1 as the baryonic part of the global symmetry, and t as the fugacity for the U (1) R symmetry, a chiral field X a carrying charges under the mesonic flavor part (q a 1 , q a 2 ), charges under the baryonic part (q a 3 , . . ., q a G+2 ), and a U (1) R charge q a G+3 can be associated with the following combination of fugacities, Accordingly, the general form of the refined Hilbert series of the master space Irr F based on (3.6) is as follows, The refined Hilbert series counts operators in terms of chiral fields that carry charges under the full global symmetry of the master space Irr F .n i ∈ Z + is the number of operators for a particular charge combination i.We can scale the fugacity t in such a way that it counts simply the overall degree of the operator.Given the coordinate ring in (3.7) and its grading under the global symmetry of the master space Irr F , the corresponding refined Hilbert series in (3.9) can be obtained using Macaulay2 [36].
Alternatively, the Hilbert series of the master space Irr F can be calculated using the symplectic quotient description of the master space in (3.3).Given the Q JE -matrix from (2.11), the refined Hilbert series of the master space Irr F is defined by the Molien integral formula [21,24] as follows, where c is the number of GLSM fields p α in the brane brick model and |Q JE | is the number of U (1) charges encoded in the Q JE -matrix.The fugacity y α identifies the global symmetry charges carried by the GLSM field p α as follows, where the GLSM field p α carries charges (q α 1 , q α 2 ) under the mesonic flavor part of the global symmetry and the charges (q α 3 , . . ., q α G+2 ) under the baryonic part of the global symmetry for the master space Irr F .We can choose the fugacity t to count the degree in GLSM fields p α instead of the U (1) R charge.
Note that the Hilbert series in (3.9) under fugacities for global symmetry charges carried by chiral fields X a is identical to the Hilbert series in (3.10) under the fugacities for global symmetry charges carried by GLSM fields since both Hilbert series describe the same master space Irr F .The two Hilbert series can be mapped to each other under a fugacity map between fugacities in (3.11) and fugacities in (3.8) using the expression of chiral fields in (2.5) in terms of products of GLSM fields.
Plethystics.The generators and the defining first order relations formed by the generators characterize the geometry of the master space Irr F .The plethystic logarithm [21][22][23][24][25] of the Hilbert series of the master space Irr F allows us to identify the generators and the first order relations formed by them.The plethystic logarithm is defined as where µ(k) is the Möbius function, and t α and y α are fugacities of the Hilbert series corresponding to the GLSM fields p α and the charges carried by them under the global symmetries of Irr F .If the expansion of the plethystic logarithm is finite, the master space Irr F is a complete intersection generated by a finite number of generators subject to a finite number of first order relations.In the case when the expansion is infinite, the master space Irr F is a non-complete intersection, where the first order relations amongst the generators form higher order relations known as syzygies [21][22][23][24].In the expansion of the plethystic logarithm, the first positive terms refer to the generators and the first negative terms in the expansion refer to the first order relations amongst the generators.

The Master Space for SPP × C
The quiver diagram for the SPP × C model [3] is shown in Figure 2. The corresponding J-and E-terms take the following form, where we note that the SPP×C model can be obtained by dimensional reduction of the 4d N = 1 supersymmetric gauge theory corresponding to the suspended pinch point (SPP) [11,38].
< l a t e x i t s h a 1 _ b a s e 6 4 = " S C / + + x 0 x V v s v C e P 4 P 3 5 B 1 8 s q 4 A = < / l a t e x i t >

2
< l a t e x i t s h a 1 _ b a s e 6 4 = " V 7 b y X N P t m 9 u 5 l h I Q N G c z 6 p j 5 n P     We can rewrite the J-and E-terms in terms of G + 3 = 6 independent new variables, which are These independent fields are related to the rest of the chiral fields in the SPP × C model.This relationship is encoded in the following K-matrix, Using the forward algorithm, we obtain the P -matrix as follows, p 1 p 2 p 3 p 4 p 5 p 6 p 7 Φ 11 0 0 1 0 0 0 0 Φ 22 0 0 1 0 0 0 0 Φ 33 0 0 1 0 0 0 0 X 11 1 1 0 0 0 0 0 X 23 1 0 0 0 0 0 0 X 32 0 1 0 0 0 0 0 X 13 0 0 0 1 0 1 0 X 31 0 0 0 0 1 0 1 X 21 0 0 0 1 0 0 1 where p 1 , . . ., p 7 are the GLSM fields of the brane brick model.In order to obtain the toric diagram for the master space Irr F of the SPP × C model, we first summarize the U (1) charges on the GLSM fields due to the J-and E-terms of the SPP × C model.These charges are summarized in the Q JE -matrix as follows, where we note that all GLSM fields carry charges under the J-and E-terms.Additionally, the GLSM fields carry U (1) charges due to the D-terms of the SPP × C model.These charges are summarized in the Q D -matrix as follows, We observe that the Q JE -and Q D -matrices together indicate that the mesonic moduli space M mes has a completely broken symmetry of the form, In comparison, when we focus on the master space Irr F , the Q JE -matrix indicates that the global symmetry of Irr F is enhanced to where the total rank of the global symmetry of the master space Irr F is as expected G + 3 = 6.Here we note that the SU (3) enhancement in the global symmetry of Irr F is due to the fact that the GLSM fields (p 1 , p 2 , p 3 ) carry the same Q JE charges.3.
Given that the J-and E-terms in (4.1) are all binomial, the master space Irr F is toric and has a 5-dimensional toric diagram which is given by where the GLSM fields p 1 , . . ., p 7 correspond to extremal vertices of the toric diagram.
Using the symplectic quotient description of the master space Irr F and the corresponding formula in (3.10) for the Hilbert series, we obtain the Hilbert series of Irr F as follows, where the fugacities t α correspond to the GLSM fields p α in the brane brick model.By taking all fugacities to be t α = t, we can write down the unrefined Hilbert series of the master space Irr F , where the palindromic numerator indicates that the master space Irr F is Calabi-Yau.As a result, the master space Irr F is a 6-dimensional toric Calabi-Yau manifold.we further note that the master space Irr F is a complete intersection.
Using the P -matrix, we can express the chiral fields of the brane brick model as products of GLSM fields as follows, . (4.13) When we define a coordinate ring in terms of chiral fields, we can assign every chiral field a grading that corresponds to the degree of GLSM fields p α in the expressions in (4.13).Under primary decomposition of the J-and E-terms in (4.1), the coherent component takes the following form, The master space Irr F is then given by Using Macaulay2 [36], with the chiral fields graded in terms of the corresponding degree in GLSM fields p α , we obtain as expected exactly the same Hilbert series as in (4.11).
Using the following fugacity map, where [m 1 , m 2 ; n The corresponding highest weight generating function [39] is given by, where The plethystic logarithm of the refined Hilbert series takes the form, From the plethystic logarithm, we identify the generators of the master space Irr F as, The single relation at order −b −6 t 4 is given by, where we note that the subspace C 4 [B jk ]/ det B = 0 corresponds to the conifold C [40,41].As a result, we identify the master space Irr F of the SPP × C brane brick model as the following 6-dimensional product space, where C 3 is generated by A i and the conifold C is generated by B jk .Table 4 summarizes the generators of the master space Irr F with the corresponding charges under the global symmetry of Irr F .
generators The corresponding quiver diagram is shown in Figure 3.
The symplectic quotient description of the master space Irr F and the corresponding formula in (3.10) for the Hilbert series gives us the Hilbert series for Irr F as follows, where P (t α ) is the numerator of the Hilbert series and the fugacities t α count the degrees in GLSM fields p α .We can unrefine the Hilbert series by taking all fugacities to be t α = t.This gives us g(t; Irr F ) = 1 − 36t The master space Irr F is then given in terms of the ideal in (4.37) as follows, where the coordinate ring is in terms of the 16 chiral fields of the brane brick model.By grading the chiral fields in terms of their corresponding degree in GLSM fields p α , we can use Macaulay2 [36] in order to obtain the Hilbert series for the master space Irr F .As expected the Hilbert series takes exactly the same form as in (4.34).

3
< l a t e x i t s h a 1 _ b a s e 6 4 = " 8 4 a 4 7 w y / z M 5 P 6 e 7 t x b a q S 7 y O H 0 9 y e e A H R 3 7 1 v F p u + L P Y C m S X 7 J F 9 E p A 6 a Z A z 0 i Q t I g i Q e / J I n r w H 7 9 l 7 8 V 6 n r X P e b G a H / I D 3 9 g m 0 O 5 5 J < / l a t e x i t >

3
< l a t e x i t s h a 1 _ b a s e 6 4 = " e Q 0 z x n h K 3 o 6 e 4 Z a c 0 Table 6: Generators of the master space Irr F for the C 4 /Z 4 ( with the global symmetry charges.
In order to obtain the toric diagram of the master space Irr F , we identify the U (1) charges due to the J-and E-terms on the GLSM fields of the brane brick model.These charges are summarized in the following Q JE -charge matrix, Additionally, the GLSM fields carry U (1) charges due to the D-terms of the brane brick model.These charges are summarized in the following Q D -matrix, where the total rank of the global symmetry of the master space Irr F is as expected G + 3 = 7.Table 7 summarizes how the GLSM fields of the Q 1,1,1 model are charged under the global symmetry of the master space Irr F .
The global symmetry charges of the master space Irr F on the GLSM fields p α for the Q 1,1,1 model.
The extra GLSM field o corresponds to a vertex which lies outside the 6-dimensional hyperplane in Z 7 .It is not part of the toric diagram of the master space Irr F of the Q 1,1,1 brane brick model and we can identify the extra GLSM field o as an overparameterization of the master space Irr F .In other words, when we identify the generators and defining relations of the master space Irr F in terms of GLSM fields, the presence or absence of the extra GLSM o does not affect the shape and number of generators and defining relations of the master space Irr F .This over-parameterization of the master space Irr F by the extra GLSM field o is best observed when we calculate the Hilbert series of Irr F in terms of fugacities that count degrees in GLSM fields.We can calculate the Hilbert series of the master space Irr F in terms of fugacities corresponding to GLSM fields by using the symplectic quotient description of Irr F and the corresponding Molien integral formula for the Hilbert series in (3.10).Accordingly, the Hilbert series for Irr F takes the form where the fugacities t α correspond to the GLSM fields p α and the fugacity u corresponds to the extra GLSM field o.Even if we set the fugacity u = 1, the Hilbert series in (4.54) describes the same master space Irr F for the Q The plethystic logarithm of the refined Hilbert series in terms of characters of irreducible representations of the master space global symmetry takes the following form, From the plethystic logarithm, we identify the generators of the master space Irr F as, The relation at order −[0, 1, 0; 0]b 2 1 b −2 2 t 3 is given by, where N lm = −N ml .As noted above, the presence or absence of the extra GLSM field o does not affect the algebraic description of the generators and first order relations of the master space Irr F of the Q 1,1,1 brane brick model.Table 8 summarizes the generators of the master space Irr F with their global symmetry charges.We have in this section identified the master space Irr F of the Q 1,1,1 brane brick model to be a 7-dimensional affine toric variety.However, although so far we have encountered master spaces Irr F for brane brick models which were toric and Calabi-Yau, the master space Irr F of the Q 1,1,1 brane brick model appears to be toric but not Calabi-Yau.This phenomenon can be seen when we unrefine the Hilbert series of the master space Irr F of the Q 1,1,1 brane brick model in (4.54) by setting the fugacities t α = t and u = 1.This results in the unrefined Hilbert series of the form where we discover that the numerator of the unrefined Hilbert series is not palindromic.By Stanley's theorem [42], the numerator of the Hilbert series in rational form is palindromic if the corresponding coordinate ring is Gorenstein and the variety is Calabi-Yau.
generators GLSM fields SU (4 Table 8: Generators of the master space Irr F for the Q 1,1,1 model with the global symmetry charges. While the coherent component of the J-and E-terms of the Q 1,1,1 brane brick model is binomial, implying that the master space Irr F is toric [18,19], the non-palindromic numerator of the unrefined Hilbert series in (4.67) indicates that the master space Irr F is indeed not Calabi-Yau.

4.4
The Master Space for Y 2,4 (CP 2 ) The quiver diagram for the Y 2,4 (CP 2 ) model [9,33,34] is shown in Figure 5.The corresponding J-and E-terms take the following form,

.73)
We observe that the Q JE -and Q D -matrices together indicate that the mesonic moduli space M mes has a symmetry of the form, The SU (3) enhancement is due to the fact that the GLSM fields (p 1 , p 2 , p 3 ) carry the same charges under the J-, E-and D-terms of the Y 2,4 (CP 2 ) model.In comparison, when we focus just on the master space Irr F , the Q JE -matrix indicates that the global symmetry of Irr F is where the total rank of the global symmetry of the master space Irr F is as expected G + 3 = 9.Here, we note that the additional SU (2) enhancements are due to the fact that the pairs of GLSM fields (p 6 , p 7 ) and (p 8 , p 9 ) carry the same charges in the Q JE -matrix.
The above features that we have observed for master spaces of brane brick models realizing 2d (0, 2) supersymmetric gauge theories have not been observed for master spaces of brane tilings realizing 4d N = 1 supersymmetric gauge theories [10][11][12][13][14][15][16][17].We believe that with the increase of dimensionality of the probed toric Calabi-Yau singularity, the master spaces of the worldvolume theories living on the probe branes not only increase in their dimension, but also exhibit much richer and surprising algebrogeometric features.We plan to explore and to identify the origin of these features in future work.

< l a t e x i t s h a 1 _
b a s e 6 4 = " W 2 e 0 d M n 1 n E a X x s l j I p l B u d m V U 9 8 = " > A A A C E H i c b V D L S g N B E J y N r x h f U Y 9 e B k P A U 9 g V U Y 8 B L x 4 T M A 9 I l j A 7 6 U 3 G z M 4 u M 7 2 B s O Q L v I n + i z f x 6 h / 4 K 5 6 c P A 6 a W N B Q V H X T 3 R U k U h h 0 3 S 8 n t 7 G5 t b 2 T 3 y 3 s 7 R 8 c H h W P T 5 o m T j W H B o 9 l r N s B M y C F g g H x d J M s 6 k N y 1 u N Z p 0 0 L y v e d e W q f l W q V p a x 5 c k Z O S c X x C M 3 p E r u S Y 0 0 C C d A n s g L e X W e n T f n 3 f l Y t O a c 5 c w p + Q P n 8 w e d C p 4 5 < / l a t e x i t > j < l a t e x i t s h a 1 _ b a s e 6 4 = " W 2 e 0 d M n 1 n E a X x s l j I p l B u d m V U 9 8 = " > A A A C E H i c b V D L S g N B E J y N r x h f U Y 9 e B k P A U 9 g V U Y 8 B L x 4 T M A 9 I l j A 7 6 U 3 G z M 4 u M 7 2 B s O Q L v I n + i z f x 6 h / 4 K 5 6 c P A 6 a W N B Q V H X T 3 R U k U h h 0 3 S 8 n t 7 G5 t b 2 T 3 y 3 s 7 R 8 c H h W P T 5 o m T j W H B o 9 l r N s B M y C F g g H x d J M s 6 k N y 1 u N Z p 0 0 L y v e d e W q f l W q V p a x 5 c k Z O S c X x C M 3 p E r u S Y 0 0 C C d A n s g L e X W e n T f n 3 f l Y t O a c 5 c w p + Q P n 8 w e d C p 4 5 < / l a t e x i t > j < l a t e x i t s h a 1 _ b a s e 6 4 = " W 2 e 0 d M n 1 n E a X x s l j I p l B u d m V U 9 8 = " > A A A C E H i c b V D L S g N B E J y N r x h f U Y 9 e B k P A U 9 g V U Y 8 B L x 4 T M A 9 I l j A 7 6 U 3 G z M 4 u M 7 2 B s O Q L v I n + i z f x 6 h / 4 K 5 6 c P A 6 a W N B Q V H X T 3 R U k U h h 0 3 S 8 n t 7 G5 t b 2 T 3 y 3 s 7 R 8 c H h W P T 5 o m T j W H B o 9 l r N s B M y C F g g e n T f n 3 f l Y t O a c 5 c w p + Q P n 8 w e d C p 4 5 < / l a t e x i t > j < l a t e x i t s h a 1 _ b a s e 6 4 = " Y / 5 l U d Z 4 d q X s l r P Y 8 u S C X J I r 4 p N b U i U P p E Y a h B M g T + S F v D r P z p v z 7 n y s W n N O N n N O / s D 5 / A G b X p 4 4 < / l a t e x i t > i < l a t e x i t s h a 1 _ b a s e 6 4 = " Y / 5 l U d Z 4 d q X s l r P Y 8 u S C X J I r 4 p N b U i U P p E Y a h B M g T + S F v D r P z p v z 7 n y s W n N O N n N O / s D 5 / A G b X p 4 4 < / l a t e x i t > i < l a t e x i t s h a 1 _ b a s e 6 4 = " Y / 5 l U d Z 4 d q X e n T f n 3 f l Y t O a c 5 c w p + Q P n 8 w e d C p 4 5 < / l a t e x i t > j < l a t e x i t s h a 1 _ b a s e 6 4 = " Y / 5 l U d Z 4 d q X k 3 g X 6 C Z 0 l U I d B + y p 0 c j X t m K N Z p D u B I y Y D 8 n X d F e a T 8 g / Z B e 6 6 a J U M 7 a h K e k B w e G c c 6 W r k + R S G A y C a 2 9 r + 8 H D R 4 9 3 n j S e P n v + 4 m X z 1 e t T k x W a Q 5 9 n M t N n C T M g h Y I + C p R w l m t g a S J h k r 2 I W Y 1 b 3 S a q F M m t J W Z W 0 a o f 8 n p 2 0 / 3 P e j z 1 H r y F / X t k P e k L f k H Q n J A T k i x + S E 9 A k n l + Q b + U G u v O / e j X f r / V x F t 7 z 1 z C 6 5 A + / 3 H 7 V B r 8 8 = < / l a t e x i t > J ji < l a t e x i t s h a 1 _ b a s e 6 4 = " i E d O j H l u P v E x g F 5 a y i w P w e u u t 7 E = " > A A A C R H i c d V D d i h M x G M 3 s + r P W v 7 p e e h M s C 4 I y Z K b T 3 e n d g g h e r m B 3 F 9 q x Z N K v 3 b i Z z J B 8 s 1 D C P I h P 4 5 3 o I / g O 3 o k 3 X o i Z a Q U V P R A 4 n P N 9 y c n J K y U t M v Y 5 2 N m 9 d v 3 G z b 1 b v d t 3 7 t 6 7 3 3 + w f 2 r L 2 g i Y i F K V 5 j z n F p T U M E G J C s 4 r A 7 z I F Z z l l 8 9 b / + w K j J W l f o 3 r C r K C r 7 R c S s H R S / P + 0 M 2 6 S 6 Z m l W e O h e M x S 5 L 0 G Q t H L I 6 T k S d s G K d p 1 L x 4 4 5 4 2 c y f f N s 2 8 P / D y a J w c x b Q d j M b J k L a r a X I 4 o l H I O g z I y H 2 9 r 2 y C P y m D w h E T k i x + Q l O S E T I s g 7 8 p 5 8 J J + C D 8 G X 4 G v w b T O 6 E 2 x 3 H p I / E P z 4 C Y 3 t r 7 g = < / l a t e x i t > E + ij < l a t e x i t s h a 1 _ b a s e 6 4 = " q j 8 3 o T r D d e U j + Q P D j J 5 F T r 7 o = < / l a t e x i t > E ij < l a t e x i t s h a 1 _ b a s e 6 4 = " f p 1X i / E j 4 u K l U W u o v t 5 E Z b t K W L I = " > A A A C T H i c d V B N a 9 t A E F 0 5 q Z M 6 / X D T Y y 9 L T K C H I m T F s V N y C f T S Q w 8 p 1 E n A E m a 1 H t t b r 1 Z i d 2 Q w i 3 5 M f k 1 u I T n 2 f / T U U u j K H 5 C U 5s H C 4 8 2 8 m d m X 5 F I Y D I I f X m 1 r + 1 l 9 Z / d 5 Y + / F y 1 e v m 2 / 2 L 0 x W a A 5 9 n s l M X y X M g B Q K + i h Q w l W u g a W J h M t k 9 q m q X 8 5 B G 5 G p b 7 j I I U 7 Z R I m x 4 A y d N G y e 2 m g 5 Z K A n S W w D / + N J N z w O P w R + E P T C o 2 5 F w l 4 n P C q j h G k b f X G T R 6 w c W v G 9 L I f N 1 s Z A N w a 6 M d C 2 U y q 0 y B r n w + a v a J T x I g W F X D J j B u 0 g x 9 g m 7 l 7 Q Z e M w K g z k j M / Y B A a O K i e b 2 C 7 T b X b / z t d M 6 8 9 e x 7 Z J 3 5 I C 8 J 2 3 S I 2 f k M z k n f c L J N b k h d + T e u / V + e r + 9 P 6 v W m r f 2 v C W P U K v / B b j I s k 0 = < / l a t e x i t > ⇤ij < l a t e x i t s h a 1 _ b a s e 6 4 = " f p 1
n r w H 7 9 l 7 8 V 6 n r X P e b G a H / I D 3 9 g m 0 O 5 5 J < / l a t e x i t > 3 < l a t e x i t s h a 1 _ b a s e 6 4 = " N q k L 5 w I Z W H 7 d h T / 7 W Z y l D S G s I U D I s g 1 u S F 3 5 N 6 7 9 R 6 8 R + / b e n T L 2 + y 8 I X / A + / 4 T W T C r f Q = = < / l a t e x i t > 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " V 7 b y X N P t m 9 u 5 l h I Q N G c z 6 p j 5 n P y 3 n U n B 0 0 r T 5 s Z j U l 4 z N I o 4 K 5 h 8 f 9 c J u + I n 5 j P X D w 1 5 F w n 4 n P C z D c t p s M b 9 7 7 N C n z O 8 x x o L u h r A u D X x W o 0 U 2 O J 0 2 f 0 x m q c g T 0 C g U t 3 Y c s A y j I n b x w J S N 9 i S 3 k H F x y R c w d l Q 7 2 U Z F H a e k b a f M 6 D w 1 7 m i k t d r 4 b a P g i b W r J H a T C c e l / d u r x H 9 5 4 x

2 <
U D I s g 1 u S F 3 5 N 6 7 9 R 6 8 R + / b e n T L 2 + y 8 I X / A + / 4 T W T C r f Q = = < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " V 7 b y X N P t m 9 u 5 l h I Q N G c z 6 p j 5 n P y 3 n U n B 0 0 r T 5 s Z j U l 4 z N I o 4 K 5 h 8 f 9 c J u + I n 5 j P X D w 1 5 F w n 4 n P C z D c t p s M b 9 7 7 N C n z O 8 x x o L u h r A u D X x W o 0 U 2 O J 0 2 f 0 x m q c g T 0 C g U t 3 Y c s A y j I n b x w J S N 9 i S 3 k H F x y R c w d l Q 7 2 U Z F H a e k b a f M 6 D w 1 7 m i k t d r 4 b a P g i b W r J H a T C c e l / d u r x H 9 5 4 x

2 Figure 2 :
Figure 2: The quiver diagram for the SPP × C model.

p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 0
t e x i t s h a 1 _ b a s e 6 4 = " S C / + + V v n x G y z S c u / K 3 Y 9 k 5 g B H j 0 = " > A A A C E H i c d V D L S g N B E J y N r x h f U Y 9 e B o P g a Z m V P P Q W 8 O I x g l E h C T I 7 6 c T B 2 d l l p j c Q l n y B N 9 F / 8 S Z e / Q N / x Z O T Z A U V L W g o q r r p 7 g o T J S 0 y 9 u 4

1 < l a t e x i t s h a 1 _
T a l r g W c P 4 A Q J Y o j Z J q P 5 H C W E Z 0 e F C s 7 y S Y u r K 9 E 6 P / k 8 s g P 6 n 7 1 v F p p + n l s R b J H 9 s k h C U i D N M k Z a Z E 2 E Q T I P X k k T 9 6 D 9 + y 9 e K / z 1 o K X z + y S H / D e P g G w 4 5 5 H < / l a t e x i t > b a s e 6 4 = " E Y a b k 2

3 < l a t e x i t s h a 1 _ b a s e 6 4 =
9 H 9 y e e A H R 3 7 1 v F p u + L P Y C m S X 7 J F 9 E p A 6 a Z A z 0 i Q t I g i Q e / J I n r w H 7 9 l 7 8 V 6 n r X P e b G a H / I D 3 9 g m 0 O 5 5 J < / l a t e x i t > " e Q 0 z x n h K 3 o 6 e 4 Z a c 0

4 < l a t e x i t s h a 1 _ b a s e 6 4 = " 3 7
1 y S s 5 J m 3 S I I E A e y B N 5 9 h 6 9 F + 9 1 3 r j g l R O 7 5 A e 8 t 0 8 A 6 Z z 4 < / l a t e x i t >

6 < l a t e x i t s h a 1 _ b a s e 6 4 = " 3 7
p c 1 b o T + T y 6 9 c q V W r p 5 X S 8 d b o 9 r m y A b Z J D u k Q u r k m J y S M 9 I g n N y T B / J E n p 1 H 5 9 V 5 c 9 6 H o x P O a G e d / I D z 8 Q V a S K t + < / l a t e x i t >

6 < l a t e x i t s h a 1 _ b a s e 6 4 =
p c 1 b o T + T y 6 9 c q V W r p 5 X S 8 d b o 9 r m y A b Z J D u k Q u r k m J y S M 9 I g n N y T B / J E n p 1 H 5 9 V 5 c 9 6 H o x P O a G e d / I D z 8 Q V a S K t + < / l a t e x i t > " d j j v r I D H 7 P C 0 C M w q u m x H v 9 T v 3 e A = " > A A A C O 3 i c d V D P a 9 s w G J W z r u u 8 t f W 6 4 y 6 i W W G M Y u S S 9 M e t s M u O H S x t w T F B V r 6 k I r J s p M + B Y P w / 9 K / p r W z / x s 6 7 j V 1 3 2 W m y k 8 B W u g e C x 3 v f J z 2 9 t F D S I m P f v M 6 T j a e b z 7 a e + y 9 e b u / s

4 < l a t e x i t s h a 1 _ b a s e 6 4 =
" d j j v r I D H 7 P C 0 C M w q u m x H v 9 T v 3 e A = " > A A A C O 3 i c d V D P a 9 s w G J W z r u u 8 t f W 6 4 y 6 i W W G M Y u S S 9 M e t s M u O H S x t w T F B V r 6 k I r J s p M + B Y P w / 9 K / p r W z / x s 6 7 j V 1 3 2 W m y k 8 B W u g e C x 3 v f J z 2 9 t F D S I m P f v M 6 T j a e b z 7 a e + y 9 e b u / s

4 < l a t e x i t s h a 1 _ b a s e 6 4 =
" d j j v r I D H 7 P C 0 C M w q u m x H v 9 T v 3 e A = " > A A A C O 3 i c d V D P a 9 s w G J W z r u u 8 t f W 6 4 y 6 i W W G M Y u S S 9 M e t s M u O H S x t w T F B V r 6 k I r J s p M + B Y P w / 9 K / p r W z / x s 6 7 j V 1 3 2 W m y k 8 B W u g e C x 3 v f J z 2 9 t F D S I m P f v M 6 T j a e b z 7 a e + y 9 e b u / s

4 <
l a t e x i t s h a 1 _ b a s e 6 4 = " d j j v r I D H 7 P C 0 C M w q u m x H v 9 T v 3 e A = " > A A A C O 3 i c d V D P a 9 s w G J W z r u u 8 t f W 6 4 y 6 i W W G M Y u S S 9 M e t s M u O H S x t w T F B V r 6 k I r J s p M + B Y P w / 9 K / p r W z / x s 6 7 j V 1 3 2 W m y k 8 B W u g e C x 3 v f J z 2 9 t F D S I m P f v M 6 T j a e b z 7 a e + y 9 e b u / s

6 <
p c 1 b o T + T y 6 9 c q V W r p 5 X S 8 d b o 9 r m y A b Z J D u k Q u r k m J y S M 9 I g n N y T B / J E n p 1 H 5 9 V 5 c 9 6 H o x P O a G e d / I D z 8 Q V a S K t + < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " 5 5 i O 3 X + 7 t q 8 R 7 y q M T v E K 3 n B q X G s = " > A A A C O 3 i c d V B N a 9 t A F F w 5 a e u o H 3 G T Y y 9 L X E M p x c i y a y e 3 Q C 8 9 u h A 7 B l u Y 1 f r Z X r J a i d 0 n g x H 6 D / 0 1 u Y X 0 b / S c W 8 k 1 l 5 y 6 k m 1 o S j u w M M y 8 t z s 7 Y S K F Q c / 7 6 V T 2 9 p 8 9 f 1 E 9 c F + + e v 3 m s P b 2 a G

3 <
t W u E / p 8 M / W a r 2 + x 8 6 9 T P 3 2 9 r q 5 J 3 5 I R 8 I C 3 S I + f k K + m T A e H k O 7 k m t + S H c + P c O b + c + 8 1 o x d n u H J M n c B 5 + A 1 V B q 3 s = < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " 5 5 i O 3 X + 7 t q 8 R 7 y q M T v E K 3 n B q X G s = " > A A A C O 3 i c d V B N a 9 t A F F w 5 a e u o H 3 G T Y y 9 L X E M p x c i y a y e 3 Q C 8 9 u h A 7 B l u Y 1 f r Z X r J a i d 0 n g x H 6 D / 0 1 u Y X 0 b / S c W 8 k 1 l 5 y 6 k m 1 o S j u w M M y 8 t z s 7 Y S K F Q c / 7 6 V T 2 9 p 8 9 f 1 E 9 c F + + e v 3 m s P b 2 a G

3 < l a t e x i t s h a 1 _ b a s e 6 4 =
t W u E / p 8 M / W a r 2 + x 8 6 9 T P 3 2 9 r q 5 J 3 5 I R 8 I C 3 S I + f k K + m T A e H k O 7 k m t + S H c + P c O b + c + 8 1 o x d n u H J M n c B 5 + A 1 V B q 3 s = < / l a t e x i t > " 8 4 a 4 7 w y / z M 5 P 6 e 7 t x b a q S 7 y O H 0 8

1 b 7 7 w j f 8 B 3 < 3 < 3 <
7 e A R g 2 a u B < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " 8 4 a 4 7 w y / z M 5 P 6 e 7 t x b a q S 7 y O H 0 8 = " > A A A C O 3 i c d V B N a x s x F N Q 6 / X C d t H H T Y y + i b i C U s m g T O x 8 3 Q y 8 9 O l A 7 A X s x W v n Z F t F q F + m t w S z 7 H / J r c g v p 3 + i 5 t 5 J r L j1 F u 7 Y h L e 2 A Y J h 5 T x p N l C p p k b E f X m 3 r 2 f M X L + u v G t s 7 r 9 / s N t / u D W y S G Q F 9 k a j E X E b c g p I a + i h R w W V q g M e R g o v o 6 k v p X y z A W J n o b 7 h M I Y z 5 T M u p F B y d N G 5 + y k f V J U M z i 8 K c + Z 0 z h 5 P P z D 9 m j A W d N W G d 4 q g Y N 1 s b n 2 5 8 u v F p 4 L M K L b J G b 9 z 8 P Z o k I o t B o 1 D c 2 m H A U g z z y M U D U z T 2 R 5 m F l I s r P o O h o 9 r J N s y r O A X d d 8 q E T h P j j k Z a q Y 0 n G z m P r V 3 G k Z u M O c 7 t 3 1 4 p / s s b Z j g 9 D X O p 0 w x B i 9 V D 0 0 x R T G h Z E J 1 I A w L V 0 h E u j E Q p q J h z w w W 6 G h s j C 7 j 6 A E K c K o 6 Q a 7 6 Q s 6 p O W g Z K l C3 y s q x N I / T / Z H D o B 8 d + + 7 z d 6 n 5 c 1 1 Y n 7 8 k H c k A C c k K 6 5 C v p k T 4 R 5 J r c k D v y 3 b v 1 f n q / v P v V a M 1 b 7 7 w j f 8 B 7 e A R g 2 a u B < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " 8 4 a 4 7 w y / z M 5 P 6 e 7 t x b a q S 7 y O H 0 8 = " > A A A C O 3 i c d V B N a x s x F N Q 6 / X C d t H H T Y y + i b i C U s m g T O x 8 3 Q y 8 9 O l A 7 A X s x W v n Z F t F q F + m t w S z 7 H / J r c g v p 3 + i 5 t 5 J r L j 1 F u 7 Y h L e 2 A Y J h 5 T x p N l C p p k b E f X m 3 r 2 f M X L + u v G t s 7 r 9 / s N t / u D W y S G Q F 9 k a j E X E b c g p I a + i h R w W V q g M e R g o v o 6 k v p X y z A W J n o b 7 h M I Y z 5 T M u p F B y d N G 5 + y k f V J U M z i 8 K c + Z 0 z h 5 P P z D 9 m j A W d N W G d 4 q g Y N 1 s b n 2 5 8 u v F p 4 L M K L b J G b 9 z 8 P Z o k I o t B o 1 D c 2 m H A U g z z y M U D U z T 2 R 5 m F l I s r P o O h o 9 r J N s y r O A X d d 8 q E T h P j j k Z a q Y 0 n G z m P r V 3 G k Z u M O c 7 t 3 1 4 p / s s b Z j g 9 D X O p 0 w x B i 9 V D 0 0 x R T G h Z E J 1 I A w L V 0 h E u j E Q p q J h z w w W 6 G h s j C 7 j 6 A E K c K o 6 Q a 7 6 Q s 6 p O W g Z K l C 3 y s q x N I / T / Z H D o B 8 d + + 7 z d 6 n 5 c 1 1 Y n 7 8 k H c k A C c k K 6 5 C v p k T 4 R 5 J r c k D v y 3 b v 1 f n q / v P v V a M 1 b 7 7 w j f 8 B 7 e A R g 2 a u B < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " S C / + + V v n x G y z S c u / K 3 Y 9 k 5 g B H j 0 = " > A A A C E H i c d V D L S g N B E J y N r x h f U Y 9 e B o P g a Z m V P P Q W 8 O I x g l E h C T I 7 6 c T B 2 d l l p j c Q l n y B N 9 F / 8 S Z e / Q N / x Z O T Z A U V L W g o q r r p 7 g o T J S 0 y 9 u 4

1 <
T a l r g W c P 4 A Q J Y o j Z J q P 5 H C W E Z 0 e F C s 7 y S Y u r K 9 E 6 P / k 8 s g P 6 n 7 1 v F p p + n l s R b J H 9 s k h C U i D N M k Z a Z E 2 E Q T I P X k k T 9 6 D 9 + y 9 e K / z 1 o K X z + y S H / D e P g G w 4 5 5 H < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " E Y a b k 2

4 <
1 y S s 5 J m 3 S I I E A e y B N 5 9 h 6 9 F + 9 1 3 r j g l R O 7 5 A e 8 t 0 8 A 6 Z z 4 < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " T M f k f R 2 R 3 9 3 Z g p n j + r P 9 I O t 7 Q + Y = " > A A A C E H i c d V D L S g N B E J y N r x h f U Y 9 e B o P g a Z m V b K K 3 g B e P E Y w K S Z D Z S S c O z s 4 u M 7 2 B s O Q L v I n + i z f x 6 h / 4 K 5 6 c P A Q V L W g o q r r p 7 o p S J S 0 y 9 u 4

5 < l a t e x i t s h a 1 _ b a s e 6 4 =
3 t h 2 w F L t 5 5 H a C G Z c O O p m F l I s 7 P o C 2 o 9 r J t p t P D x 3 T A 6 f 0 a D 8 x r j T S q V r 6 N p H z 2 N p R H L n O m O O t / e 1 N x L + 8 d o b 9 4 2 4 u d Z o h a D F b 1 M 8 U x Y R O v q Y 9 a U C g G j n C h Z E o B R W 3 3 H C B L p t S x w L O H k C I U 8 U R c s 2 H c j D N i E 4 O S p Q d 5 2 M X 1 l c i 9 H 9 y e e Q H N b 9 6 X q 0 0 / H l s R b J H 9 s k h C U i d N M g Z a Z I W E Q T I P X k k T 9 6 D 9 + y 9 e K + z 1 o I 3 n 9 k l P + C 9 f Q K 3 k 5 5 L < / l a t e x i t > " N H u + K Z s 0 p e b b q Y

1 b 7 7 w j f 8 B 3 < l a t e x i t s h a 1 _ b a s e 6 4 =
7 e A R g 2 a u B < / l a t e x i t > " 8 4 a 4 7 w y / z M 5 P 6 e 7 t x b a q S 7 y O H 0 8

3 <
1 b 7 7 w j f 8 B 7 e A R g 2 a u B < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " S C / + + V v n x G y z S c u / K 3 Y 9 k 5 g B H j 0 = " > A A A C E H i c d V D L S g N B E J y N r x h f U Y 9 e B o P g a Z m V P P Q W 8 O I x g l E h C T I 7 6 c T B 2 d l l p j c Q l n y B N 9 F / 8 S Z e / Q N / x Z O T Z A U V L W g o q r r p 7 g o T J S 0 y 9 u 4

1 <
s 7 y S Y u r K 9 E 6 P / k 8 s g P 6 n 7 1 v F p p + n l s R b J H 9 s k h C U i D N M k Z a Z E 2 E Q T I P X k k T 9 6 D 9 + y 9 e K / z 1 o K X z + y S H / D e P g G w 4 5 5 H < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " E Y a b k 2

3 < l a t e x i t s h a 1 _ b a s e 6 4 =
9 H 9 y e e A H R 3 7 1 v F p u + L P Y C m S X 7 J F 9 E p A 6 a Z A z 0 i Q t I g i Q e / J I n r w H 7 9 l 7 8 V 6 n r X P e b G a H / I D 3 9 g m 0 O 5 5 J < / l a t e x i t > " e Q 0 z x n h K 3 o 6 e 4 Z a c 0

4 < 5 < l a t e x i t s h a 1 _ 6 Figure 5 :
Figure 5: The quiver diagram for the Y 2,4 (CP 2 ) model.
.71) where the columns of the P -matrix correspond to the GLSM fields of the Y 2,4 (CP 2 ) model.p 1 , . . .p 11 correspond to regular GLSM fields, whereas o 1 , o 2 correspond to extra GLSM fields for the master space Irr F of the Y 2,4 (CP 2 ) model.We will see later on how we have identified the extra GLSM fields o 1 , o 2 from the toric diagram of the master space Irr F .

Table 3 :
The global symmetry charges of the master space Irr F on the GLSM fields p α for the SPP × C model.Furthermore, the SU (2) enhancements in the global symmetry are due to the fact that the GLSM fields (p 4 , p 5 ) and (p 6 , p 7 ) carry the same Q JE charges.The charges on the GLSM fields due to the global symmetry in (4.8) are summarized in Table When we calculate the plethystic logarithm, PL g(t α ; Irr F ) = t 1 + t 2 + t 3 + t 4 t 6 + t 5 t 6 + t 4 t 7 + t 5 t 7 − t 4 t 5 t 6 t 7 , (4.12) = t we can rewrite the refined Hilbert series in (4.10) in terms of characters of irreducible representations of the global symmetry of the master space Irr F .The refined Hilbert series in terms of characters of irreducible representations of the global symmetry takes the following form,
+ ...,(4.35)we obtain an infinite series which indicates that the master space Irr F is a non-complete intersection.

Table 9 :
The global symmetry charges of the master space Irr F on the GLSM fields p α for the Y 2,4 (CP 2 ).outside the 8-dimensional hyperplane and correspond to the extra GLSM fields o 1 and o 2 .