# Partial Resolution of Complex Cones over Fanoℬ

Authors: Siddharth Dwivedi; P. Ramadevi
Journal: Advances in High Energy Physics
 (2013)

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## Abstract

In our recent paper, we systematized an inverse algorithm to obtain quiver gauge theory living on theM2-branes probing the singularities of a special kind of Calabi-Yau fourfold which were complex cones over toric Fano ℙ3, ℬ1, ℬ2, ℬ3. These quiver gauge theories cannot be given a dimer tiling presentation. We use the method of partial resolution to show that the toric data of ℂ4 and Fano ℙ3 can be embedded inside the toric data of Fano ℬ theories. This method indirectly justifies that the two-node quiver Chern-Simons theories corresponding to ℂ4, Fano ℙ3, and their orbifolds can be obtained by higgsing matter fields of the three-node parent quiver corresponding to Fano ℬ1, ℬ2, ℬ3, ℬ4 threefold.


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## Body

## 1. Introduction

Initial works of Bagger-Lambert [1–3] followed by Gustavsson [4, 5], Van Raamsdonk [6], and Aharony-Bergman-Jafferis-Maldacena (ABJM) [7] led to a flurry of interesting papers during the last four years between supersymmetric Chern-Simons gauge theory on coincident M2-branes at the tip of Calabi-Yau fourfold and their string duals. In a review article [8], these developments are discussed in detail.Martelli and Sparks [9] discussed the gauge-gravity correspondence (AdS4/CFT3) for some supersymmetric Chern-Simons theories with a quiver diagram description. Earlier works of Feng et al. in the context of Calabi-Yau threefold [10] called forward algorithm can be extended to obtain Calabi-Yau fourfold toric data from 2+1 dimensional quiver supersymmetric Chern-Simons theories.An elegant combinatorial approach called dimer tilings [11, 12] which gives both the toric data and the corresponding quiver gauge theories was generalised to study quiver Chern-Simons theories [13–20]. However, the dimer tiling approach is applicable for only a class of quiver gauge theories with m-matter fields, r gauge group nodes, and NW number of terms in the superpotential W satisfying r-m+NW=0. The Chern-Simons (CS) levels of the r-nodes can be denoted by the vector k→=(k1,k2,…,kr).One of the challenging problems was to determine quiver gauge theories corresponding to 18 toric Fano threefold. A Fano variety ind-complex dimension is characterized by positive curvature and one can construct the CY (d+1)-fold by taking a complex cone over it. If the Fano variety is toric, the Calabi-Yau constructed from it will also be toric, and one can attempt to find the dual quiver gauge theories. In 2 complex dimensions, there are 5 toric Fano 2-fold which are zeroth Hirzebruch surface 𝔽0 and the del Pezzo surfaces dP0, dP1, dP2, dP3. The quiver gauge theories living on D3 branes probing the toric CY 3-fold obtained from these 5 Fano 2-fold have been studied in the literature [10]. Moving on to 3 complex dimensions, there are 18 toric Fano 3-fold [21, 22] with nomenclature (as used in [20]) Fano ℙ3, ℬ1, ℬ2, ℬ3, ℬ4, 𝒞1, 𝒞2, 𝒞3, 𝒞4, 𝒞5, 𝒟1, 𝒟2, ℰ1, ℰ2, ℰ3, ℰ4, ℱ1, ℱ2. The next problem was to obtain the quiver CS theory living on the M2 branes probing the toric CY 4-fold obtained from these 18 Fano 3-fold.From the forward algorithm and dimer tiling method, quiver gauge theories corresponding to fourteen of the toric Fanoℬ4, 𝒞1, 𝒞2, 𝒞3, 𝒞4, 𝒞5, 𝒟1, 𝒟2, ℰ1, ℰ2, ℰ3, ℰ4, ℱ1, ℱ2 were determined [20]. In [23], we attempted the inverse algorithm of obtaining the quiver gauge theories for the remaining four Fano 3-fold ℙ3, ℬ1, ℬ2, ℬ3. As expected, these quiver gauge theories do not satisfy r-m+NW=0 confirming that they cannot be given dimer tiling presentation.The next immediate question is to understand the embeddings inside the toric Fanoℬ threefold by the method of partial resolution. In particular, we would like to obtain the quiver Chern-Simons theory corresponding to Fano ℙ3 by partial resolution of Fano ℬ threefold.Alternatively, we could determine embeddings from higgsing approach [19, 24, 25]: for the quivers with r nodes (usually called parent theories) which admit dimer tiling, we could higgsize some matter fields and obtain quivers with r-1 nodes (called daughter theories). Suppose we give a vacuum expectation value (VEV) to a bifundamental matter field Xab where the subscript denotes that the matter field is charged +1 with respect to node a with Chern-Simons level ka and charged -1 with respect to node b with Chern-Simons level kb. This results in coalescing of two nodes a and b into a node with Chern-Simons level ka+kb. Higgsing of the three node quiver corresponding to a toric Fano ℬ4 was discussed from dimer tilings in [19]. In fact, the daughter theories correspond to either phase II of ℂ4 or phase I of ℂ2/ℤ2×ℂ2 depending upon which bifundamental field was given VEV.Another approach of higgsing called the algebraic method [24] has been applied on parent quivers [25] corresponding to some toric Fano 3-fold. We will present the algebraic higgsing for Fano ℬ4 and show that the toric data corresponding to daughter theories has to be ℂ4 or orbifolds of ℂ4. We will also study the method of partial resolution [10] and obtain the ℂ4 toric data for the daughter theories.For the3-node quivers corresponding to Fano ℬ1,ℬ2, and ℬ3, which do not admit dimer tiling presentation, we have to obtain daughter theories by the algebraic higgsing method [25] or the partial resolution method [10]. We shall show that the algebraic higgsing on these 3-node quivers always gives ℂ4 or orbifolds of ℂ4. We also study the method of partial resolution to check whether the approach gives more information about the embedded theories. Interestingly, this method shows that the toric data of Fano ℙ3 or its orbifolds is embedded in the toric data of Fano ℬ2 and Fano ℬ3.The plan of the paper is as follows: in Section2, we briefly review the well-known 2-node quiver Chern-Simons theory with four matter fields and their toric data. In Section 3, we will first perform the higgsing of toric Fano ℬ4 using the algebraic method and determine the daughter quiver theories with 2 nodes. Finally, we show that the partial resolution method gives ℂ4 toric data. In Section 4, we will briefly present the necessary data of the quiver corresponding to Fano ℬ3. Then, we study the algebraic higgsing and the method of partial resolution for Fano ℬ3. Particularly, we show that the algebraic higgsing gives ℂ4 or orbifolds of ℂ4 whereas the method of partial resolution gives nontrivial embeddings inside Fano ℬ3. In Section 5, we study the method of partial resolution for Fano ℬ2. We present the results of partial resolution of Fano ℬ1 in Section 6. We summarize and discuss some open problems in Section 7.


## 2. Two-Node Quiver Chern-Simons Theories

Our aim is to study partial resolution of toric data corresponding to 3-node parent quiver resulting in a toric data corresponding to a2-node daughter quiver. So, in this section we will briefly recapitulate the 2-node quivers with four matter fields corresponding to toric data of ℂ4, orbifolds of ℂ4 and Fano ℙ3. There can be three possible quivers as follows.(1) Theory with 4 bifundamental matter fieldsX12i and X21i (where i=1,2), with CS levels (k,  -k) and the superpotential W=Tr[ϵijX121X21iX122X21j]. For the abelian groups, W=0. The quiver diagram is shown in Figure 1. This theory admits tiling and the corresponding toric data is as follows [18]:
(1)𝒢a(k)=(p1p2p3p41111-10-100-1-1000k0).
The charge matrix Qa for this theory given by Qa·𝒢aT(k)=0 is trivial, Qa=0. This theory is ℤk orbifold of ℂ4, denoted by ℂ4/ℤk. For k=1, that is, when the CS-levels of the theory is (1, −1), there is no orbifolding action and this theory in the literature is known as Phase I of ℂ4 [18].Figure 1
Quiver diagram (a).(2) Theory with 2 adjointsϕ21,ϕ22 present at the same node (say node 2) and two bifundamentals X12,X21, with CS-levels (k,-k). Abelian W of this theory is again zero with nonabelian superpotential W=Tr[X12[ϕ21,ϕ22]X21]. The quiver diagram is shown in Figure 2. This theory also admits tiling and the toric data for this theory is as follows [18]:
(2)𝒢b(k)=(p1p2p3p4111110100-10000k0).
The charge matrix Qb of this theory given by Qb·𝒢bT(k)=0 is also trivial, Qb=0. This is (ℂ2/ℤk)×ℂ2 theory. For k=1, this theory is known as Phase II of ℂ4.Figure 2
Quiver diagram (b).It is pertinent to spell out the following obvious facts.(i)
We can deduce that the two quiver theories are distinct fork≠1 because the toric data of the two theories (𝒢a(k) and 𝒢b(k)) are not related by any GL(4,ℤ) transformation.(ii)
When there is no orbifolding—that is,k=1, both theories are the same up to some GL(4,ℤ) transformation. Hence, these theories are actually just the phases of ℂ4 theory, known as Phase I and Phase II of ℂ4, respectively.(3) Theory with 2 bifundamentalsX12,X21 and adjoints ϕ1,  ϕ2 at two different nodes with trivial W=0. The quiver of this theory is shown in Figure 3. This theory does not admit tiling and was first obtained using the inverse algorithm in [23], which was identified as the Fano ℙ3 theory.Figure 3
Quiver diagram (c).However, we cannot determine the CS levels from the inverse algorithm [23] for all the three 2-node quiver theories. From the tiling description, we can obtain the CS levels for the quiver diagrams (a) and (b). Comparing the inverse algorithm of the three theories, we had inferred that the theory corresponding to Fano ℙ3 could have CS levels (4,-4). As we do not know any method of determining the actual CS levels, the theory shown in quiver diagram (c) with k=1 could be Fano ℙ3 theory.A possible choice of toric data for the theory shown in Figure3 can be
(3)𝒢c(k)=(p1p2p3p4p5111111-100001-10000k-k0),
which is related to the toric data 𝒢c(k=1)≡𝒢ℙ3 by GL(4,ℤ) transformation 𝒯(k), such that the determinant of 𝒯 is k.The charge matrixQc(k) of this theory given by Qc(k)·𝒢c(k)T=0 consists only of QF and is given by [23]
(4)Qc(k)=QFℙ3=(1,1,1,1,-4),
for all k. This indicates that the charge QF will not be able to detect the orbifolding ℤk.We would like to see these Calabi-Yau 4-fold, particularly Fanoℙ3, as embeddings inside Calabi-Yau 4-fold toric data corresponding to some 3-node quivers. In the following section, we will extensively discuss the 3-node quiver corresponding to Fano ℬ4 which admits dimer tiling. Higgsing of the theory corresponding to Fano ℬ4 [19], from the tiling approach, has shown that the daughter theories are only quiver diagram (b). From the method of algebraic higgsing, we get the toric data to be ℂ4 and orbifolds of ℂ4 whereas partial resolution gives toric data of ℂ4. This will set the necessary tools and notations for studying the embeddings inside other three Fano ℬi’s in the later sections.


## 3. Fanoℬ4

The quiver corresponding to the complex cone over Fanoℬ4 is a theory with 3 nodes and 9 bifundamental fields X12i, X23i, X31i, where i=1,2,3. This theory admits tiling and is known in the literature as Fano ℬ4 or M1,1,1 theory with CS levels k→=(k1,k2,k3)=(1,-2,1) [20]. The quiver diagram for this theory is shown in Figure 4. The superpotential of the theory is given by
(5)W=Tr[ϵijkX12iX23jX31k].
The incidence matrix of this theory is given by
(6)d=(X121X122X123X231X232X233X311X312X313G=1111000-1-1-1G=2-1-1-1111000G=3000-1-1-1111),
where the rows indicate the gauge groups or the nodes in the quiver, and columns indicate the matter fields.Figure 4
Quiver diagram for Fanoℬ4.The projected charge matrix (Δ) will consist of a single row whose elements will be given by
(7)Δi=k2d1i-k1d2i=-(2d1i+d2i).
Hence, Δ matrix will be given by
(8)Δ=(X121X122X123X231X232X233X311X312X313-1-1-1-1-1-1222).
From the superpotential W (5), one can find the F-term constraints given by the set of equations {∂W/∂Xi=0}, which means that the matter fields Xi’s can be written in terms of 5 independent v-fields, and the relation between them can be encoded in a matrix K. The matrix K and its dual matrix T are given as
(9)K=(v1v2v3v4v5X12(1)00100X12(2)00101X12(3)00110X23(1)01000X23(2)01001X23(3)01010X31(1)10000X31(2)10001X31(3)10010),T=(100001100010100010-101000-110000).
From K and T, we can write the matrix P=K·T. The entries of the matrix P are all nonnegative and they give the relation of the matter fields Xi’s with the GLSM fields pi’s:
(10)P=(p1p2p3p4p5p6X121100100X122010100X123001100X231100010X232010010X233001010X311100001X312010001X313001001).
The QF charge matrix is given by the null space of P:
(11)QF=(111-1-1-1).
The steps done so far in obtaining P-matrix and the charge QF are independent of the CS levels. The information about CS levels is contained only in the charge QD matrix which is a single row obeying the symmetry of the Calabi-Yau which is given as
(12)QD=P·Δt=[00011-2].
The total charge matrix can be obtained by concatenating (11) and (12) in a single matrix Q:
(13)Q=(QFQD)=(111-1-1-100011-2).The toric data of this theory is [20]:
(14)𝒢(1,-2,1)=(p1p2p3p4p5p61111111-1000001-10000001-10).Suppose that we rescale the levels of this theory as(k1,k2,k3)=(n,-2n,n), where n is some nonzero integer. Since this theory admits brane tiling description, the toric data for the scaled levels (n,-2n,n) can be readily obtained from the Kastelyn matrix method [20]. The toric data is dependent on the scale n and is given below:
(15)𝒢(n,-2n,n)=(11111110-1000-110000000n-n0).
Unfortunately, when we perform forward algorithm for the scaled CS levels (n,-2n,n), we see from (7) and (12) that QD gets scaled by factor n but QF remains unchanged. Clearly, the toric data 𝒢(1,-2,1) obtained as the null space of total charge matrix Q, that is, 𝒢·Qt=0, is still satisfied by Qt=(QFtnQDt).Equivalently, the toric data𝒢 obtained from forward algorithm has scaling ambiguity of any of the rows and hence not unique. These arguments on ambiguity of 𝒢, scaling of QD and Δ under k→→nk→ hold for inverse algorithm as well. Following the works on orbifolds of ℂ4, it is a well-known fact that 𝒢(n,-2n,n) obtained from tiling must be the ℤn orbifold of 𝒢(1,-2,1) theory. The two toric data are related by a GL(4,ℤ) transformation 𝒯 as
(16)𝒢(n,-2n,n)=𝒯·𝒢(1,-2,1),
where
(17)𝒯=(100001000010000n).
Here, |det(𝒯)|=n, which means that the volume of 𝒢(n,-2n,n) theory is n times that of 𝒢(1,-2,1) implying that the former theory is ℤn orbifold of latter theory. This orbifolding action, also scaling of CS levels, is explainable only for those theories which admit tiling. From our forward algorithm discussion, we have explicitly seen that the scaling of CS levels does not give the orbifolding information in the toric data. Reconciling with tiling results, we will hitherto use that fact that the scaling of CS levels represents ℤn orbifolding of all theories which may or may not admit tiling.From the tiling approach, higgsing of the quiver diagram in Figure4 has been presented in detail in [19]. In particular, we can obtain daughter quivers in Figure 2 corresponding to ℂ4 and orbifolds of ℂ4. We will now study the algebraic approach of higgsing which will be applicable for other quivers that do not admit dimer tiling presentation.


### 3.1. Algebraic Higgsing

We attempt the higgsing of Fanoℬ4 theory to obtain one of the 2 nodes theories discussed in Section 2. In algebraic higgsing, we choose some matter field, say Xi, and give a nonzero VEV to it. Giving a VEV makes the matter field massive and hence removed from the quiver. However, in the process of giving VEV, all those GLSM pj fields which contain the matter field Xi also become massive and hence must be removed. To do this, we delete the ith row from the P matrix which corresponds to the matter field Xi and also, we remove all those columns (p-fields) which have nonzero entry corresponding to ith row (and hence have become massive).As an example, let us take theX121 field of the ℬ4 theory and give a VEV to it. Thus, from the P, matrix (10), we must remove the first row and also the columns 1 and 4 which correspond to the GLSM fields p1,p4 which contain the X121 field. After giving VEV to the X121 field, the nodes 1 and 2 in the Figure 4 are collapsed giving a 2-node daughter theory with CS levels (1,-1). Removal of the corresponding row and columns in (10) will give the following reduced P-matrix:
(18)Pr=(p2p3p5p6X1221000X1230100X2310010X2321010X2330110X3110001X3121001X3130101).
The nullspace of this matrix gives the reduced charge matrix QFr=0. Thus, we see that the QFr of the daughter theory is trivial. The quivers (a) and (b) listed in Section 2 have QF=0. So, algebraic higgsing can give daughter quivers (a) as well as quiver (b) with CS levels (1,-1).By giving VEV to any other matter fields, we have checked that we get the same trivialQFr=0. This suggests that the algebraic higgsing of the parent quiver corresponding to Fano ℬ4 will give quivers corresponding to ℂ4 and orbifolds of ℂ4. It must be mentioned at this point that the algebraic higgsing of Fano ℬ1, ℬ2, ℬ3 also gives a daughter theory with trivial QF. So, the algebraic higgsing tells that only ℂ4 or the orbifolds of ℂ4 are embedded inside Fano ℬ1,ℬ2,ℬ3,ℬ4. It may be possible that we may get nontrivial QF from the method of partial resolution. So, we shall study the method of partial resolution for Fano ℬ4 toric data.


### 3.2. Partial Resolution

In partial resolution, we try to remove the points from the toric diagram. The resulting toric diagram corresponds to some daughter theory which is embedded in the parent theory. From the toric diagram ofℬ4, we will remove some points which amount to removing the corresponding columns (or the corresponding p-fields) from the toric data 𝒢. Next, we will check whether this reduced toric data (denoted by 𝒢r), which is obtained by removing the columns from original 𝒢, is related to any of the toric data of the 2-node theories.Take the toric data𝒢 of Fano ℬ4 in (14). We see that if we remove the points p1 and p4, we get the reduced toric data
(19)𝒢r=(1111-10001-10000-10).This reduced toric data (19) is equivalent to the toric data 𝒢a(k=1) (1) and 𝒢b(k=1) (2):
(20)𝒢a(k=1)=(100001010111000-1)·𝒢r,𝒢b(k=1)=(10000-10-10110000-1)·𝒢r.
Thus, partial resolution of ℬ4 toric data (14) only gives ℂ4 toric data. Explicit coalescing of nodes in tiling/algebraic higgsing allows both ℂ4 or ℤ2 orbifold of ℂ4 whereas we get only 𝒢a(k=1) or 𝒢b(k=1) in the partial resolution method.Alternatively, we can take the charge matrixQ of the ℬ4 theory given in (13) take a linear combination, and set the charge matrix elements corresponding to columns p1 and p4 to zero as described below: suppose that a row (r) of Q of the daughter theory is given as some linear combination of rows (Ri) of Q (13) of parent theory; that is,
(21)r=∑i=12‍(aiRi)=(a1,a1,a1,-a1+a2,-a1+a2,-a1-2a2).
Since we are removing p1 and p4 points, the corresponding columns in r are set to 0 and also removed. This will give the values of a1 and a2, and we find that a1=a2=0. Hence, the Q of the daughter theory is trivial.We have obtained𝒢r by removing all other possible set of points in the toric data of ℬ4—namely, {p1,p5}, {p2,p4}, {p2,p5}, {p3,p4}, and {p4,p5}, and checked that they are again related by GL(4,ℤ) to 𝒢a(k=1) and 𝒢b(k=1). For all these cases, the linear combination of the charge Q matrix (13) with the appropriate columns removed gives Q of the daughter theory to be trivial (Q=0). Thus, we see that ℂ4 theory is embedded in the ℬ4 theory.The quiver gauge theories corresponding to Fanoℬ3 and ℬ2 have nontrivial superpotential W [23] and hence can be described by both forward and inverse algorithms. We will briefly present in the next section the quiver and necessary data for Fano ℬ3 and study algebraic higgsing and partial resolution.


## 3.1. Algebraic Higgsing

We attempt the higgsing of Fanoℬ4 theory to obtain one of the 2 nodes theories discussed in Section 2. In algebraic higgsing, we choose some matter field, say Xi, and give a nonzero VEV to it. Giving a VEV makes the matter field massive and hence removed from the quiver. However, in the process of giving VEV, all those GLSM pj fields which contain the matter field Xi also become massive and hence must be removed. To do this, we delete the ith row from the P matrix which corresponds to the matter field Xi and also, we remove all those columns (p-fields) which have nonzero entry corresponding to ith row (and hence have become massive).As an example, let us take theX121 field of the ℬ4 theory and give a VEV to it. Thus, from the P, matrix (10), we must remove the first row and also the columns 1 and 4 which correspond to the GLSM fields p1,p4 which contain the X121 field. After giving VEV to the X121 field, the nodes 1 and 2 in the Figure 4 are collapsed giving a 2-node daughter theory with CS levels (1,-1). Removal of the corresponding row and columns in (10) will give the following reduced P-matrix:
(18)Pr=(p2p3p5p6X1221000X1230100X2310010X2321010X2330110X3110001X3121001X3130101).
The nullspace of this matrix gives the reduced charge matrix QFr=0. Thus, we see that the QFr of the daughter theory is trivial. The quivers (a) and (b) listed in Section 2 have QF=0. So, algebraic higgsing can give daughter quivers (a) as well as quiver (b) with CS levels (1,-1).By giving VEV to any other matter fields, we have checked that we get the same trivialQFr=0. This suggests that the algebraic higgsing of the parent quiver corresponding to Fano ℬ4 will give quivers corresponding to ℂ4 and orbifolds of ℂ4. It must be mentioned at this point that the algebraic higgsing of Fano ℬ1, ℬ2, ℬ3 also gives a daughter theory with trivial QF. So, the algebraic higgsing tells that only ℂ4 or the orbifolds of ℂ4 are embedded inside Fano ℬ1,ℬ2,ℬ3,ℬ4. It may be possible that we may get nontrivial QF from the method of partial resolution. So, we shall study the method of partial resolution for Fano ℬ4 toric data.


## 3.2. Partial Resolution

In partial resolution, we try to remove the points from the toric diagram. The resulting toric diagram corresponds to some daughter theory which is embedded in the parent theory. From the toric diagram ofℬ4, we will remove some points which amount to removing the corresponding columns (or the corresponding p-fields) from the toric data 𝒢. Next, we will check whether this reduced toric data (denoted by 𝒢r), which is obtained by removing the columns from original 𝒢, is related to any of the toric data of the 2-node theories.Take the toric data𝒢 of Fano ℬ4 in (14). We see that if we remove the points p1 and p4, we get the reduced toric data
(19)𝒢r=(1111-10001-10000-10).This reduced toric data (19) is equivalent to the toric data 𝒢a(k=1) (1) and 𝒢b(k=1) (2):
(20)𝒢a(k=1)=(100001010111000-1)·𝒢r,𝒢b(k=1)=(10000-10-10110000-1)·𝒢r.
Thus, partial resolution of ℬ4 toric data (14) only gives ℂ4 toric data. Explicit coalescing of nodes in tiling/algebraic higgsing allows both ℂ4 or ℤ2 orbifold of ℂ4 whereas we get only 𝒢a(k=1) or 𝒢b(k=1) in the partial resolution method.Alternatively, we can take the charge matrixQ of the ℬ4 theory given in (13) take a linear combination, and set the charge matrix elements corresponding to columns p1 and p4 to zero as described below: suppose that a row (r) of Q of the daughter theory is given as some linear combination of rows (Ri) of Q (13) of parent theory; that is,
(21)r=∑i=12‍(aiRi)=(a1,a1,a1,-a1+a2,-a1+a2,-a1-2a2).
Since we are removing p1 and p4 points, the corresponding columns in r are set to 0 and also removed. This will give the values of a1 and a2, and we find that a1=a2=0. Hence, the Q of the daughter theory is trivial.We have obtained𝒢r by removing all other possible set of points in the toric data of ℬ4—namely, {p1,p5}, {p2,p4}, {p2,p5}, {p3,p4}, and {p4,p5}, and checked that they are again related by GL(4,ℤ) to 𝒢a(k=1) and 𝒢b(k=1). For all these cases, the linear combination of the charge Q matrix (13) with the appropriate columns removed gives Q of the daughter theory to be trivial (Q=0). Thus, we see that ℂ4 theory is embedded in the ℬ4 theory.The quiver gauge theories corresponding to Fanoℬ3 and ℬ2 have nontrivial superpotential W [23] and hence can be described by both forward and inverse algorithms. We will briefly present in the next section the quiver and necessary data for Fano ℬ3 and study algebraic higgsing and partial resolution.


## 4. Fanoℬ3

It is a theory with 3 nodes, 2 adjoint fieldsX1,X2 on node-1 and 6 bifundamental fields X3,X4,X5,X6,X7,X8. The quiver diagram for this theory is shown in Figure 5. This theory does not admit tiling and was first studied in [23], where it was identified as the quiver gauge theory for Fano ℬ3 with CS levels (6,-6,0). The superpotential of the theory is given by
(22)W=Tr[(X1X4-X2X3)(X5X8-X6X7)].
From W given in (22), we construct K, which gives T and hence P:
(23)P=(p1p2p3p4p5p6p7X13010001X20310001X33001001X40301001X50040601X60004601X70040061X80004061).
Total charge matrix Q is given by
(24)Q=(QFQD)=(1133-1-1-6111100-40022-20-2).
The toric data of the theory is given as [23]:
(25)𝒢(6,-6,0)=(p1p2p3p4p5p6p711111111-100000001-1000010-1-1-10).
Similar to Fano ℬ4 theory, if we rescale the levels here to (k1,k2,k3)=(6n,-6n,0), where n is some non-zero integer, we can write the toric data which has information about this scaling. Note that this theory does not admit tiling and the forward/inverse algorithm does not give an explicit dependence of 𝒢(6n,-6n,0) on n, but there is a scaling ambiguity of any row of toric data. A choice of the toric data will be
(26)𝒢(6n,-6n,0)=(11111111-100000001-10000n0-n-n-n0).
This toric data is related to that of 𝒢(6,-6,0) by a volume factor n and hence represents the ℤn orbifolding:
(27)𝒢(6n,-6n,0)=(100001000010000n)·𝒢(6,-6,0).
We will now study algebraic higgsing and obtain two-node daughter quivers.Figure 5
Quiver diagram for Fanoℬ3.


### 4.1. Algebraic Higgsing

If we give a VEV to any of theXi fields, thereby removing the corresponding row and columns from the P-matrix (23), we see that we get a reduced matrix (Pr), whose null space (QFr) is always trivial. Starting from the parent quiver as shown in Figure 5, we see that the nontrivial CS levels of the 2-node daughter quiver to be (6,-6). Thus, we can only say that the higgsing of the ℤn orbifolds of Fano ℬ3 theory will give ℂ4/ℤ6n as the daughter theory. Moreover, we see that the last column (p7) of P-matrix (23) which corresponds to the internal point in the toric diagram of Fano ℬ3 will always be removed because it contains all the matter fields. So, this method cannot give a daughter theory corresponding to Fano ℙ3 which has an internal point in the toric diagram. Hence, we are forced to study the method of partial resolution to check whether toric Fano ℙ3 is embedded inside ℬ3 Fano.


### 4.2. Partial Resolution

Here, we do the partial resolution of Fanoℬ3 theory and also check whether we get nontrivial charge Q for the daughter theory.


#### 4.2.1. Embedding of Fanoℙ3 inside Fano ℬ3

It is interesting to see that the method of partial resolution does embed the toric Fanoℙ3 inside Fano ℬ3 giving the correct QF (4). Hence, we can claim that this method gives more information than algebraic higgsing for quiver which do not admit dimer tiling presentation. Suppose we remove the points {p5,p6} from the 𝒢 (25), we get a reduced toric data
(28)𝒢r=(p1p2p3p4p7111111-1000001-10010-10),
and the toric data 𝒢ℙ3≡𝒢c(k=1) (3) is related to 𝒢r by a GL(4,ℤ) transformation:
(29)𝒢ℙ3=(1000010000-110010)·𝒢r.
A row (r) of Q of the daughter theory will be given as linear combination of rows (Ri) of Q (24); that is,
(30)r=∑i=13‍(aiRi)=(a1+a2,a1+a2,3a1+a2+2a3,3a1+a2---+2a3,-a1-2a3,-a1,-6a1-4a2-2a3).
Setting the columns 5, 6 in r to 0 gives a1=a3=0. Removing these columns gives the reduced charge matrix as follows:
(31)r=(a2,a2,a2,a2,-4a2)=a2(1,1,1,1,-4).
Thus, Q will have only one row generated by (1,1,1,1,-4). Hence, the charge matrix of daughter theory is the same as charge matrix of the Fano ℙ3 (4). Thus, we see that Fano ℙ3 is embedded in the Fano ℬ3 theory. Similar to the partial resolution of ℬ4 theory, we only get toric data of Fano ℙ3 but not the ℤ6 orbifold of ℙ3, which is expected from the coalescing of the nodes in Figure 5.


#### 4.2.2. Embedding ofℂ4 inside Fano ℬ3

If we remove points{p1,p3,p5} from the toric data (25), we will get a reduced toric data given by
(32)𝒢r=(p2p4p6p71111-10000-1001-1-10).
Similar to the case of ℬ4 theory, 𝒢a(k=1) and 𝒢b(k=1) are related to (32):
(33)𝒢a(k=1)=(100002-1101010-11-1)·𝒢r,𝒢b(k=1)=(10000-21-100100-11-1)·𝒢r.
Also, we can find the reduced charge matrix Qr in a similar way as was done for other cases, and we find that it is trivial. Similarly, if we remove the other set of points {p1,p4,p5}, {p2,p3,p5} and {p2,p4,p5}, we will get trivial Qr and the reduced toric data in each case is only related to 𝒢a and 𝒢b toric datas for k=1 which implies ℂ4 toric data.In the following section, we will briefly present the quiver corresponding to Fanoℬ2 and study the partial resolution.


## 4.1. Algebraic Higgsing

If we give a VEV to any of theXi fields, thereby removing the corresponding row and columns from the P-matrix (23), we see that we get a reduced matrix (Pr), whose null space (QFr) is always trivial. Starting from the parent quiver as shown in Figure 5, we see that the nontrivial CS levels of the 2-node daughter quiver to be (6,-6). Thus, we can only say that the higgsing of the ℤn orbifolds of Fano ℬ3 theory will give ℂ4/ℤ6n as the daughter theory. Moreover, we see that the last column (p7) of P-matrix (23) which corresponds to the internal point in the toric diagram of Fano ℬ3 will always be removed because it contains all the matter fields. So, this method cannot give a daughter theory corresponding to Fano ℙ3 which has an internal point in the toric diagram. Hence, we are forced to study the method of partial resolution to check whether toric Fano ℙ3 is embedded inside ℬ3 Fano.


## 4.2. Partial Resolution

Here, we do the partial resolution of Fanoℬ3 theory and also check whether we get nontrivial charge Q for the daughter theory.


### 4.2.1. Embedding of Fanoℙ3 inside Fano ℬ3

It is interesting to see that the method of partial resolution does embed the toric Fanoℙ3 inside Fano ℬ3 giving the correct QF (4). Hence, we can claim that this method gives more information than algebraic higgsing for quiver which do not admit dimer tiling presentation. Suppose we remove the points {p5,p6} from the 𝒢 (25), we get a reduced toric data
(28)𝒢r=(p1p2p3p4p7111111-1000001-10010-10),
and the toric data 𝒢ℙ3≡𝒢c(k=1) (3) is related to 𝒢r by a GL(4,ℤ) transformation:
(29)𝒢ℙ3=(1000010000-110010)·𝒢r.
A row (r) of Q of the daughter theory will be given as linear combination of rows (Ri) of Q (24); that is,
(30)r=∑i=13‍(aiRi)=(a1+a2,a1+a2,3a1+a2+2a3,3a1+a2---+2a3,-a1-2a3,-a1,-6a1-4a2-2a3).
Setting the columns 5, 6 in r to 0 gives a1=a3=0. Removing these columns gives the reduced charge matrix as follows:
(31)r=(a2,a2,a2,a2,-4a2)=a2(1,1,1,1,-4).
Thus, Q will have only one row generated by (1,1,1,1,-4). Hence, the charge matrix of daughter theory is the same as charge matrix of the Fano ℙ3 (4). Thus, we see that Fano ℙ3 is embedded in the Fano ℬ3 theory. Similar to the partial resolution of ℬ4 theory, we only get toric data of Fano ℙ3 but not the ℤ6 orbifold of ℙ3, which is expected from the coalescing of the nodes in Figure 5.


### 4.2.2. Embedding ofℂ4 inside Fano ℬ3

If we remove points{p1,p3,p5} from the toric data (25), we will get a reduced toric data given by
(32)𝒢r=(p2p4p6p71111-10000-1001-1-10).
Similar to the case of ℬ4 theory, 𝒢a(k=1) and 𝒢b(k=1) are related to (32):
(33)𝒢a(k=1)=(100002-1101010-11-1)·𝒢r,𝒢b(k=1)=(10000-21-100100-11-1)·𝒢r.
Also, we can find the reduced charge matrix Qr in a similar way as was done for other cases, and we find that it is trivial. Similarly, if we remove the other set of points {p1,p4,p5}, {p2,p3,p5} and {p2,p4,p5}, we will get trivial Qr and the reduced toric data in each case is only related to 𝒢a and 𝒢b toric datas for k=1 which implies ℂ4 toric data.In the following section, we will briefly present the quiver corresponding to Fanoℬ2 and study the partial resolution.


## 4.2.1. Embedding of Fanoℙ3 inside Fano ℬ3

It is interesting to see that the method of partial resolution does embed the toric Fanoℙ3 inside Fano ℬ3 giving the correct QF (4). Hence, we can claim that this method gives more information than algebraic higgsing for quiver which do not admit dimer tiling presentation. Suppose we remove the points {p5,p6} from the 𝒢 (25), we get a reduced toric data
(28)𝒢r=(p1p2p3p4p7111111-1000001-10010-10),
and the toric data 𝒢ℙ3≡𝒢c(k=1) (3) is related to 𝒢r by a GL(4,ℤ) transformation:
(29)𝒢ℙ3=(1000010000-110010)·𝒢r.
A row (r) of Q of the daughter theory will be given as linear combination of rows (Ri) of Q (24); that is,
(30)r=∑i=13‍(aiRi)=(a1+a2,a1+a2,3a1+a2+2a3,3a1+a2---+2a3,-a1-2a3,-a1,-6a1-4a2-2a3).
Setting the columns 5, 6 in r to 0 gives a1=a3=0. Removing these columns gives the reduced charge matrix as follows:
(31)r=(a2,a2,a2,a2,-4a2)=a2(1,1,1,1,-4).
Thus, Q will have only one row generated by (1,1,1,1,-4). Hence, the charge matrix of daughter theory is the same as charge matrix of the Fano ℙ3 (4). Thus, we see that Fano ℙ3 is embedded in the Fano ℬ3 theory. Similar to the partial resolution of ℬ4 theory, we only get toric data of Fano ℙ3 but not the ℤ6 orbifold of ℙ3, which is expected from the coalescing of the nodes in Figure 5.


## 4.2.2. Embedding ofℂ4 inside Fano ℬ3

If we remove points{p1,p3,p5} from the toric data (25), we will get a reduced toric data given by
(32)𝒢r=(p2p4p6p71111-10000-1001-1-10).
Similar to the case of ℬ4 theory, 𝒢a(k=1) and 𝒢b(k=1) are related to (32):
(33)𝒢a(k=1)=(100002-1101010-11-1)·𝒢r,𝒢b(k=1)=(10000-21-100100-11-1)·𝒢r.
Also, we can find the reduced charge matrix Qr in a similar way as was done for other cases, and we find that it is trivial. Similarly, if we remove the other set of points {p1,p4,p5}, {p2,p3,p5} and {p2,p4,p5}, we will get trivial Qr and the reduced toric data in each case is only related to 𝒢a and 𝒢b toric datas for k=1 which implies ℂ4 toric data.In the following section, we will briefly present the quiver corresponding to Fanoℬ2 and study the partial resolution.


## 5. Fanoℬ2

The quiver Chern-Simons theory corresponding to Fanoℬ2 [23] has 3 nodes, 12 matter fields with 2 possible quiver diagrams as shown in Figures 6 and 7 with CS levels (2,-2,0). The superpotential of the theory is given by
(34)W=Tr(X1X4X8X12-X1X4X9X11-X2X5X7X12=Tr-+X2X5X9X10+X3X6X7X11-X3X6X8X10).
This theory does not admit dimer tiling presentation but can be studied using forward algorithm. Using W given in (34), one can construct K which gives T whence P:
(35)P=(p1p2p3p4p5p6p7p8X120022001X202022001X300222001X420002201X502002201X600202201X740020021X804020021X900420021X1040000221X1104000221X1200400221).
A possible choice of the total charge matrix Q is given by
(36)Q=(QFQD)=(111-1-1-1-220001-11-100000101-20001120-4).
The toric data of the theory is given as [23]:
(37)𝒢(2,-2,0)=(p1p2p3p4p5p6p7p8111111111-100000001-100000001-1-1110).
Taking the P-matrix (35), giving VEV to any of the matter fields gives QFr=0. Also from coalescing of nodes in the quiver diagrams given in Figures 6 and 7, we know that the nontrivial CS level of the 2-node daughter theory will be (2,-2). Thus, the daughter theory will be ℤ2 orbifolds of ℂ4. From (35), we see that giving a VEV to any of the matter fields will always remove the last column (p8) of P which corresponds to an internal point in the toric diagram of ℬ2. So if we do the algebraic higgsing of ℬ2, we are never going to get the embedding as Fano ℙ3 which has an internal point in its toric diagram. We will now work out the partial resolution to see if we can get more information.Figure 6
Cyclic quiver for Fanoℬ2.Figure 7
Linear quiver for Fanoℬ2.


### 5.1. Partial Resolution

In this case, we found that then if we remove the set of points{p1,p4,p5,p7}, {p1,p4,p6,p7}, {p2,p4,p5,p7}, {p2,p4,p6,p7}, {p3,p4,p5,p7}, {p3,p4,p6,p7} or {p4,p5,p6,p7}, we will get a reduced toric data which is related to 𝒢a(k=1) or 𝒢b(k=1) by a GL(4,ℤ) transformation. Further, the null space, that is, the reduced charge matrix Qr, is trivial. This implies that the toric data of the daughter theory is ℂ4.


#### 5.1.1. Embedding of Fanoℙ3 inside Fano ℬ2

If we remove the points{p5,p6,p7} from the toric data 𝒢 given in (37), we will get the reduced toric data:
(38)𝒢r=(p1p2p3p4p8111111-100001-100001-10),
which is exactly same as the toric data of ℙ3. Thus, we see that ℙ3 is embedded inside ℬ2.Taking a row (r) of total charge Q of the daughter theory as a linear combination of rows (Ri) of total charge matrix of ℬ2 given in (36),
(39)r=(a1,a1,a1,-a1+a2+a4,-a1-a2+a3+a4,-a1+a2--+2a4,-2a1-a2+a3,2a1-2a3-4a4).
Setting the columns 5, 6, and 7 in r to 0, we get a1=a2/3=a3/5=-a4. The reduced charge matrix after removal of the columns 5, 6, and 7 gives
(40)r=(a1,a1,a1,a1,-4a1)=a1(1,1,1,1,-4).
Thus, Q will have only 1 row generated by (1,1,1,1,-4). Hence, the charge matrix of daughter theory is Q=(QF)=(1,1,1,1,-4) which is the charge matrix of ℙ3. Thus, partial resolution only gives ℙ3. However, from coalescing of nodes in Figures 6 and 7, we expect the daughter theory to be ℤ2 orbifold of ℙ3.


## 5.1. Partial Resolution

In this case, we found that then if we remove the set of points{p1,p4,p5,p7}, {p1,p4,p6,p7}, {p2,p4,p5,p7}, {p2,p4,p6,p7}, {p3,p4,p5,p7}, {p3,p4,p6,p7} or {p4,p5,p6,p7}, we will get a reduced toric data which is related to 𝒢a(k=1) or 𝒢b(k=1) by a GL(4,ℤ) transformation. Further, the null space, that is, the reduced charge matrix Qr, is trivial. This implies that the toric data of the daughter theory is ℂ4.


### 5.1.1. Embedding of Fanoℙ3 inside Fano ℬ2

If we remove the points{p5,p6,p7} from the toric data 𝒢 given in (37), we will get the reduced toric data:
(38)𝒢r=(p1p2p3p4p8111111-100001-100001-10),
which is exactly same as the toric data of ℙ3. Thus, we see that ℙ3 is embedded inside ℬ2.Taking a row (r) of total charge Q of the daughter theory as a linear combination of rows (Ri) of total charge matrix of ℬ2 given in (36),
(39)r=(a1,a1,a1,-a1+a2+a4,-a1-a2+a3+a4,-a1+a2--+2a4,-2a1-a2+a3,2a1-2a3-4a4).
Setting the columns 5, 6, and 7 in r to 0, we get a1=a2/3=a3/5=-a4. The reduced charge matrix after removal of the columns 5, 6, and 7 gives
(40)r=(a1,a1,a1,a1,-4a1)=a1(1,1,1,1,-4).
Thus, Q will have only 1 row generated by (1,1,1,1,-4). Hence, the charge matrix of daughter theory is Q=(QF)=(1,1,1,1,-4) which is the charge matrix of ℙ3. Thus, partial resolution only gives ℙ3. However, from coalescing of nodes in Figures 6 and 7, we expect the daughter theory to be ℤ2 orbifold of ℙ3.


## 5.1.1. Embedding of Fanoℙ3 inside Fano ℬ2

If we remove the points{p5,p6,p7} from the toric data 𝒢 given in (37), we will get the reduced toric data:
(38)𝒢r=(p1p2p3p4p8111111-100001-100001-10),
which is exactly same as the toric data of ℙ3. Thus, we see that ℙ3 is embedded inside ℬ2.Taking a row (r) of total charge Q of the daughter theory as a linear combination of rows (Ri) of total charge matrix of ℬ2 given in (36),
(39)r=(a1,a1,a1,-a1+a2+a4,-a1-a2+a3+a4,-a1+a2--+2a4,-2a1-a2+a3,2a1-2a3-4a4).
Setting the columns 5, 6, and 7 in r to 0, we get a1=a2/3=a3/5=-a4. The reduced charge matrix after removal of the columns 5, 6, and 7 gives
(40)r=(a1,a1,a1,a1,-4a1)=a1(1,1,1,1,-4).
Thus, Q will have only 1 row generated by (1,1,1,1,-4). Hence, the charge matrix of daughter theory is Q=(QF)=(1,1,1,1,-4) which is the charge matrix of ℙ3. Thus, partial resolution only gives ℙ3. However, from coalescing of nodes in Figures 6 and 7, we expect the daughter theory to be ℤ2 orbifold of ℙ3.


## 6. Fanoℬ1

This theory was studied in [23] where the inverse algorithm was used to find a quiver gauge theory shown in Figure 8 with CS levels (2,0,-2). The superpotential of this theory was obtained as
(41)W=Tr[X2X5X8(X1X4X9X3X6X7-X1X6X7X3X4X9)].
Note that the abelian W=0 and the quiver does not admit tiling. So, there is no way to obtain the toric data of this theory using forward algorithm or dimer tiling approach. The toric data for ℬ1 theory with multiplicity is given as [23]:
(42)𝒢=(p1p2p3p4p5p6p711111111-10000001-10000002-1110).
From the ansatz given in [23], we can take a choice of QF and QD obeying the symmetry of Calabi-Yau over Fano ℬ1, and the total charge matrix will be given by
(43)Q=(QFQD)=(111-2-2-23000211-4000101-2).Figure 8
Quiver diagram for Fanoℬ1.The algebraic higgsing in this case givesℤ2 orbifolds of ℂ4 and we also observe that the internal point in the toric Fano ℬ1 gets removed. The partial resolution by removing the points {p1,p4,p5}, {p2,p4,p5}, or {p3,p4,p5} gives toric data of ℂ4. We could not obtain ℙ3 as embedding inside ℬ1 by this method.


## 7. Conclusions

Our main motivation was to determine the2-node daughter quivers by the method of higgsing the 3-node parent quivers. Particularly, we wanted to higgsize matter fields of the 3-node quivers corresponding to Fano ℬ1,ℬ2,ℬ3,ℬ4 3-fold and obtain the daughter quivers. This procedure will determine the CS levels of the daughter quiver theories. Unfortunately, the algebraic method of higgsing does not give any nontrivial charge matrix Q.For the quiver corresponding to Fanoℬ4, which has dimer tiling presentation, higgsing could be studied by tiling approach as well as by algebraic method. By the algebraic method of higgsing, we get the reduced charge matrix Qr=0 which suggests that the two-node daughter theories can be ℂ4 and orbifolds of ℂ4. From the method of partial resolution of the toric data corresponding to Fano ℬ4, we obtained only ℂ4. We have shown that the scaling of CS levels (1,-2,1)→(n,-2n,n) results in ℤn orbifolding of Fano ℬ4 theory. Hence, partial resolution of the orbifolded Fano ℬ4 will give ℂ4/ℤn toric data with trivial QF.The3-node quivers for other Fano ℬ which were determined from the inverse algorithm do not admit dimer tiling presentation. So, we cannot study higgsing of these theories from tiling approach. The algebraic higgsing of any matter field removes the information of the internal point in the toric data and always gives trivial reduced charge matrix Qr=0. Unlike ℂ4 and its orbifolds, the toric data of Fano ℙ3 has an internal point. So, we studied the method of partial resolution for toric Fano ℬ1,ℬ2,ℬ3 3-fold. We found that the Fano ℙ3 can be embedded inside Fano ℬ3 and ℬ2 theories.Algebraic higgsing and unhiggsing of quiver theories corresponding to some Fano 3-fold have been studied recently [25]. Higgsing certain matter fields in the 4-node quivers corresponding to toric Fano 𝒞4 gives the 3-node quiver corresponding to Fano ℬ4. Also higgsing of the quiver corresponding to Fano 𝒟2 gives daughter quiver corresponding to Fano ℬ4. These results can also be reproduced using the method of partial resolution.It is not obvious whether we can obtain other Fanoℬ’s toric data as embeddings inside Fano 𝒞4 and Fano 𝒟2  3-fold. One of the issues is about our choice of multiplicity of certain points in the toric data for Fano ℬ1,ℬ2,ℬ3.We had chosen a charge matrixQF,QD with respect to the ansatz [23] which determined the multiplicity of certain points in the toric data of Fano ℬ’s. The toric data with the specific multiplicity of certain points was important to obtain sensible quivers. In principle, we would like to apply the method of partial resolution for toric data corresponding to some 4-node quivers and reproduce the toric data of Fano ℬ1,ℬ2,ℬ3 with the correct multiplicity of some of the points. We hope to study in the future the embeddings of the toric Fano ℬ’s inside toric fourfold corresponding to 4-node quiver Chern-Simons theories.



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*Source: 295842-2013-05-22.xml*