Double Handled Brane Tilings
Abstract
We classify the first few brane tilings on a genus Riemann surface and identify their toric CalabiYau moduli spaces. These brane tilings are extensions of tilings on the 2torus, which represent one of the largest known classes of superconformal field theories for D3branes. The classification consists of distinct genus brane tilings with up to quiver fields and superpotential terms. The Higgs mechanism is used to relate the different theories.
Prince Consort Road, London SW7 2AZ, United Kingdom \preprint
Imperial/TP/13/AH/01
1 Introduction
Brane tilings Hanany:2005ve ; Franco:2005rj provide one of the largest known classes of supersymmetric gauge theories living on D3branes which probe CalabiYau 3fold singularities. As bipartite periodic graphs on the 2torus, which encode both field theory information and geometry, brane tilings represent an epitome of the rich interface between algebraic geometry and string theory. Our work attempts to upgrade this active relationship by introducing and classifying brane tilings not confined to the traditional 2torus.
Brane tilings have been used to classify toric quiver gauge theories with their mesonic and baryonic moduli spaces Benvenuti:2006qr ; Feng:2007ur ; Butti:2007jv ; Grant:2007ze ; Davey:2009bp ; 2007arXiv0710.1898I ; Forcella:2009vw ; Hanany:2007zz ; Hanany:2010zz ; Zaffaroni:2008zz ; Forcella:2009bv ; Forcella:2008eh ; Forcella:2008bb , dualities Feng:2002zw ; Feng:2001xr ; Feng:2001bn ; 2001JHEP…12..001B and symmetries Butti:2005vn ; Franco:2004rt . With the understanding of 3d ChernSimons theories as worldvolume theories of M2branes Aharony:2008ug ; Bagger:2006sk ; Bagger:2007jr ; Bagger:2007vi ; Gustavsson:2007vu ; Gustavsson:2008dy , this tour de force of research and discovery reached new heights and led to the introduction of ChernSimons levels on brane tilings Hanany:2008cd ; Hanany:2008fj ; Ueda:2008hx ; Davey:2009qx ; Davey:2011mz ; Benini:2009qs ; Benini:2011cma ; Closset:2012ep .
The work on brane boxes Hanany:1997tb described the construction of a prototypical brane tiling on a surface with boundaries such as a disc or cylinder. This idea recently reemerged as bipartite graphs on discs in relation to string scattering amplitudes Franco:2012mm ; ArkaniHamed:2012nw . The connection between supersymmetric gauge theories and brane tilings on surfaces with boundaries was further studied in Franco:2012wv .
In parallel, brane tilings associated to CalabiYau geometries whose toric diagrams are reflexive polygons Hanany:2012hi were found to have the same combined mesonic and baryonic moduli spaces under a map which is known as specular duality Hanany:2012vc . The fascinating properties of specular duality further motivates our work.
Specular duality makes use of the untwisting map Feng:2005gw ; Butti:2007jv which relates theories with the same master space Forcella:2008ng ; Hanany:2010zz ; Zaffaroni:2008zz ; Forcella:2009bv ; Forcella:2008eh ; Forcella:2008bb and generates new brane tilings that are not necessarily confined to the 2torus. The simplest example of this capability is the orbifold theory Hanany:2010cx ; Davey:2010px ; Davey:2011dd ; Hanany:2011iw ; Hanany:2010ne whose brane tiling can be untwisted to give a dual on a Riemann surface. This is an important example of a brane tiling beyond the 2torus and sheds light on a new infinite class of unexplored field theories.
This paper introduces a new procedure of classifying brane tilings on Riemann surfaces. We continue to call the new periodic bipartite graphs on Riemann surfaces as brane tilings since they are natural generalisations of the tilings on the 2torus. Although the brane construction for the generalisation is not yet fully understood, we believe that our classification is an important step towards a better understanding of the problem.
Despite the efficiency of generating brane tilings on or higher genus Riemann surfaces with specular duality, only a subset of these new brane tilings can be identified with this method. Most other brane tilings, often with much smaller number of fields and gauge groups, can only be obtained via a direct construction on the Riemann surface.^{1}^{1}1These are in fact under specular duality often related to inconsistent brane tilings on the 2torus. Consistency of brane tilings on the 2torus has been studied from many perspectives Franco:2005sm ; Hanany:2005ss ; Broomhead:2008an ; Gulotta:2008ef ; 2011arXiv1104.1592B , and the most important properties are reviewed in this work. The work will give the first classification of brane tilings on a Riemann surface with up to 8 quiver fields and 4 superpotential terms. Our classification identifies precisely 16 distinct brane tilings which can be related by a successive application of the Higgs mechanism.
The mesonic moduli space of each brane tiling in the classification is computed by imposing Fand Dterm constraints. These moduli spaces are all toric CalabiYau 5folds. The moduli space dimension is in general where the number of homology 1cyles on the genus Riemann surface is . By computing the Hilbert series, we specify the explicit algebraic structure of the moduli space and relate new geometries to classical field theories.
For generic ranks of the gauge groups, it is not clear whether the beta functions of all couplings can be set to zero.^{2}^{2}2It is well known Hanany:2005ss that if the ranks of the gauge groups are all equal and none of the couplings vanish, the beta functions cannot all be zero. Accordingly, understanding the IR behaviour of the brane tilings may be challenging. For the moment, the classification of brane tilings should be considered as an important step towards a better understanding of recent lines of thought. We believe that such extensions to the field theories classified in this work along with a better understanding of the brane construction will lead to new exciting progress in the near future.
The structure of the paper is as follows. Section §2 gives a first glimpse of a brane tiling by untwisting the brane tiling for the theory and then proceeds to outline an algorithm for classifying all distinct brane tilings on a Riemann surface. The results are summarized in section §2.2. Section §2.3 continues with a discussion on consistency of brane tilings that plays an important role in the case of the 2torus. The section explains that restrictions are set on brane tilings to reduce the number of models in the classification even though the restrictions are not well motivated from a field theory perspective. Section §2.4 summarises the basic properties of the mesonic moduli spaces and continues with section §2.5 by explaining how the Higgs mechanism relates the theories in the classification and acts as a check of the classification. In the second part of the paper, section §3 summarises the full classification data for brane tilings, including the computation of the Hilbert series. Appendix §A includes a more concise summary of the classification. In addition, brane tilings with selfintersecting zigzag paths are presented in appendix §B. Appendix §C gives a short summary of the forward algorithm which is used to identify the mesonic moduli spaces.
2 Brane Tilings on Riemann Surfaces
In this section we present the classification scheme which we used for the brane tilings. A brief summary is given for what is meant by a brane tiling, with an overview of their field theoretic and geometric properties.
2.1 The Construction
As seen in Hanany:2012vc , specular duality and the untwisting map Butti:2007jv ; Feng:2005gw can be used to generate brane tilings on Riemann surfaces with genus . The simplest example is the brane tiling for with orbifold action , whose toric diagram is a lattice triangle with exactly two internal points. The toric diagram and the brane tiling are in Figure 2 with the quiver diagram in Figure 4. The superpotential has the form
Given that the superpotential has an overall trace, which is omitted for brevity, let us use the notation which replaces terms in the superpotential as a cyclic permutation of integers Jejjala:2010vb . The integers themselves label fields with the dictionary given in Figure 2,
(2.2)  
The specular dual tiling is on a Riemann surface and the corresponding supersymmetric field theory has a toric CalabiYau mesonic moduli space. The brane tiling is shown in Figure 3 with the quiver in Figure 4. The superpotential of the specular dual is easily obtained by reversing the permutations which correspond to the negative (or equivalently the positive) terms in the original superpotential in (2.2).
This brane tiling is the one that can be generated via specular duality with the least number of fields. In fact, there are brane tilings with much fewer fields that cannot be obtained via specular duality on 2torus tilings. In the following section, we illustrate a method of generating such tilings and give a full classification up to 8 quiver fields and 4 superpotential terms.
2.2 Classification of Brane Tilings
The brane tiling as a bipartite graph satisfies the Euler formula,
(2.3) 
where , and are respectively the number of edges, nodes and faces of the brane tiling and is the genus of the Riemann surface. The fundamental domain of the genus brane tiling is a sided polygon with our identification of sides being the one shown in Figure 5. Accordingly, there are fundamental cycles with every zigzag path^{3}^{3}3A zigzag path is a closed path along the edges on the brane tiling which alternates between white and black nodes. The path is such that it makes precisely one maximal clockwise turn around a white note and the a maximal anticlockwise turn around the next black node before reaching the next edge and node in the sequence. of the brane tiling having winding numbers. This leads to rank mesonic symmetry in the associated field theory Hanany:2005ve ; Kennaway:2007tq .
# Models  

5  2  1  1 
6  2  2  3 
7  2  3  1 
7  4  1  1 
8  2  4  2 
8  4  2  8 
For , the first few values of , and satisfying the Euler formula are given in Table 1. By setting for , we generate all possible permutations of integers. From this set of permutations, all possible pairings of permutations are taken. For each permutation pair one is marked as positive and the other one as negative. We associate a pairing to a brane tiling if it satisfies the following brane tiling conditions:

The number of cycles in the positive permutation is the same as the number of cycles in the negative permutation. This translates to the condition that there are the same number of positive and negative superpotential terms.

Every integer precisely appears once in a positive permutation cycle and a negative permutation cycle. This by construction satisfies the toric condition of the brane tiling.

The associated brane tiling has no selfintersecting zigzag paths and no multibonded edges Franco:2005sm ; Hanany:2005ss ; Broomhead:2008an as discussed in §2.3. We adopt these restrictions in the classification for brane tilings to reduce the number of identified models.
Two brane tilings on any genus Riemann surface are the same if they satisfy the following equivalence conditions:

The brane tilings are on the same Riemann surface with the same genus .

The quiver diagrams are equivalent graphs.

The superpotential as a permutation pairing is the same partition of integers.

The zigzag paths Gulotta:2008ef ; Butti:2005ps are the same partition of integers.

The mesonic moduli spaces Butti:2007jv ; Hanany:2012hi ; Feng:2000mi are the same.
Note that a subset of the conditions above may not be enough to identify brane tiling equivalence. An example is a pair of distinct toric dual brane tilings which are related by the urban renewal move. The dual brane tilings have the same mesonic moduli space Hanany:2005ve . In fact, for brane tilings, two distinct brane tilings which are not related by the urban renewal move can have the same mesonic moduli space.
Following the procedure which is outlined above, we classify all distinct brane tilings on a Riemann surface with up to edges and superpotential terms. We identify distinct brane tilings. They are summarized in Figure 6, and their mesonic moduli spaces are identified and discussed in Section §3. We emphasise that the 16 brane tilings are restricted, in other words they do not have selfintersecting zigzag paths and no multibonded edges. All other tilings are not discussed in detail in this paper and are subject for future studies.
2.3 Consistency of Brane Tilings on a 2torus
The notion of consistency of a brane tiling on the 2torus was first discussed in Hanany:2005ss . Consistent torus brane tilings are expected to flow in the IR to a superconformal fixed point with a preferred Rsymmetry which appears in the superconformal algebra and determines the scaling dimension of BPS operators. If the consistency conditions are not satisfied, one normally can expect zero superconformal Rcharges to be assigned to bifundamental fields under amaximisation Butti:2005ps ; Butti:2005vn ; Martelli:2005tp . In this case, some dibaryon operators would violate the unitarity bound on the scaling dimension.
In order to discuss brane tiling consistency from a geometric and combinatorial point of view, we recall that the classical vacuum moduli space of the 1brane theory^{4}^{4}4In the following sections, we call a quiver gauge theory with a superpotential associated to a brane tiling Abelian if it has only gauge factors. is a toric CalabiYau 3fold. It is represented by a convex lattice polygon known as the toric diagram. This mesonic moduli space can be expressed as a Kähler quotient in terms of a gauge linear sigma model (GLSM) Witten:1993yc description of the theory. Perfect matchings of the brane tiling are associated to GLSM fields and are identified as lattice points on the toric diagram. In summary, inconsistency can be observed when

Twice the area of the toric diagram is not the number of gauge groups in the brane tiling.

More than one GLSM field of the brane tiling is associated to a corner (extremal) point of the toric diagram.
From a purely graphical point of view, a brane tiling is consistent if it has the following properties:

No zigzag paths selfintersect.

No edges are ‘multibonded’ and hence no faces are 2sided.
The above consistency conditions are illustrated in Figure 7.
For the following classification of brane tilings on a Riemann surface, we restrict ourselves to brane tilings with no selfintersecting zigzag paths and no multibonded edges. We call these restricted brane tilings. We apply the restriction in order to reduce the number of brane tilings identified in the classification, even though we believe that it is of interest to study unrestricted brane tilings on Riemann surfaces. We leave the study of unrestricted brane tilings for future work.
2.4 Mesonic Moduli Spaces
#  Global Symmetry  

5.2  
6.2a  
6.2b  
6.2c  
7.2  
7.4  
8.2a  
8.2b  
8.4a  
8.4b  
8.4c  
8.4d  
8.4e  
8.4f  
8.4g  
8.4h 
The mesonic moduli space of a brane tiling is the vacuum moduli space of the corresponding supersymmetric gauge theory under both Fand Dterm constraints. The forward algorithm Feng:2000mi ; Feng:2001xr ; Feng:2002zw ; Feng:2002fv ; Hanany:2005ve ; Franco:2005rj ; Franco:2006gc has been used extensively in the case for brane tilings on to identify the mesonic moduli space of Abelian gauge theories with only gauge groups. It is summarized in appendix §C.
The forward algorithm can be used to identify for supersymmetric gauge theories represented by brane tilings on Riemann surfaces of arbitrary genus. The mesonic moduli spaces of the Abelian gauge theory is a dimensional toric CalabiYau variety.
In order to compute the structure of the mesonic moduli space, we evaluate the Hilbert series of . The Hilbert series is refined with fugacities which count charges under the global symmetries. The global symmetry group has total rank and can have for the case of brane tilings , , and enhancements. Table 2 summarises the global symmetries which are observed in the classification.
In field theory, the superpotential is conventionally assigned Rcharge , when the supercharges have unit Rcharge. For simplicity, we rescale the Rsymmetry generator: quiver fields are assigned Rcharges such that every perfect matching carries a Rcharge of 1. This is a notational simplification in the following sections. For the actual Rcharges the reader is reminded that the charges for perfect matchings should be rescaled such that the superpotential carries Rcharge 2 rather than equal to the number of perfect matchings.
5.2, 6.2a  
6.2b, 6.2c  
7.4, 8.4d  
8.2b, 8.4h  
8.4b, 8.4c 
By analysing the mesonic moduli spaces of the brane tilings in the classification shown in Figure 6, we observe interesting new phenomena. In the case of torus brane tilings, the mesonic moduli spaces of two brane tilings are the same if the brane tilings are related by an urban renewal move as depicted in Figure 8. Such a move seems to be still a sufficient condition for moduli space equivalence for brane tilings on higher genus Riemann surfaces. However, we observe examples of brane tilings which are not related by urban renewal, but have the same mesonic moduli space. The examples identified in the classification are shown in Table 3.
The above classification of the mesonic moduli spaces are based on the fact that we restrict to Abelian theories with only gauge groups. Whether as in the case of toric duality the supersymmetric theories share the same mesonic moduli spaces in the nonAbelian extension is unclear. It is of great interest to study this problem in future work.
2.5 Higgsing Brane Tilings
Section §2.2 explained the procedure which is followed in this work to identify brane tilings with up to fields and superpotential terms. We expect Higgsing Hanany:2005ve ; Feng:2002fv ; Hanany:2012hi to be an exploratory way to relate the discovered brane tilings and at the same time to check the classification for consistency. Higgsing is the procedure of giving VEVs to bifundamental fields in order to solve Dterm equations in the presence of FI parameters, and to integrate out mass terms in the resulting superpotential of the theory. It translates to removing edges in the brane tiling and reducing the graph such that there are no bivalent nodes. The procedure is illustrated in Figure 9.
Higgsing relates the restricted brane tilings in Figure 6 with each other. Intriguingly, Higgsing also relates restricted brane tilings with unrestricted ones which are not part of our classification. In fact, via Higgsing one identifies 10 unrestricted brane tilings with selfintersecting zigzag paths which are summarized with the corresponding superpotentials and quiver diagrams in appendix §B. A ‘Higgsing tree’, which illustrates brane tilings as nodes and VEVs as arrows, is shown in Figure 10. It is of great interest to understand the mechanism that relates restricted and unrestricted brane tilings in future studies.
3 A Classification of Brane Tilings
This section summarises the classification of brane tilings with up to fields and superpotential terms. The mesonic moduli spaces are studied by computing the Hilbert series of the corresponding algebraic variety. We discover several interesting geometries which are related to the new brane tilings.
3.1 5 Fields, 2 Superpotential Terms, 1 Gauge Group
3.1.1 Model 5.2:
1  2  3  4  5 

The first brane tiling of our classification and the corresponding quiver diagram are shown in Figure 11 and Figure 12 respectively. The brane tiling is made of a single decagonal face which is the single gauge group with 5 adjoints in the quiver diagram. The superpotential is
(3.4) 
A single adjoint on its own forms a perfect matching of the brane tiling. Accordingly, the perfect matching matrix is the identity matrix
(3.5) 
The perfect matching matrix is always the identity matrix for models with just 2 superpotential terms. The zigzag paths of the brane tiling are
(3.6) 
There are only trivial F and Dterms. The mesonic moduli space is a toric CalabiYau 5fold. More specifically, Model 5.2’s mesonic moduli space is with the refined Hilbert series being
(3.7) 
where the fugacities count the perfect matchings respectively.
Given that the mesonic moduli space is , the global symmetry group is found as , where the is the Rsymmetry. The global symmetry charges assigned to perfect matchings are shown below.
fugacity  

1  
1  
1  
1  
1 
Under the above global symmetry charge assignment, the Hilbert series can be expressed in terms of characters of irreducible representations of ,
(3.8) 
The toric diagram of the mesonic moduli space is a 4 dimensional lattice polytope. The coordinates of the toric points are encoded in the matrix
(3.9) 
The projected toric diagram is a unit lattice 4simplex.
3.2 6 Fields, 2 Superpotential Terms, 2 Gauge Groups
3.2.1 Model 6.2a:
1  2  3  4  5  6 

The brane tiling on a Riemann surface and the corresponding quiver diagram are shown in Figure 13 and Figure 14 respectively. The superpotential is
(3.10) 
The quiver incidence matrix is
(3.11) 
The brane tiling has perfect matchings. Since there are only 2 superpotential terms, every field on its own represents a perfect matching. The perfect matching matrix is therefore the identity matrix,
(3.12) 
The zigzag paths in the brane tiling of Model 6.2a are
(3.13) 
The superpotential for a theory with only gauge groups vanishes , and therefore the kernel of the perfect matching matrix is empty. There are no Fterms, and there are no Fterm charges
(3.14) 
The Dterm charges are encoded in the quiver incidence matrix and are summarized in the following charge matrix,
(3.15) 
Accordingly, the total charge matrix , and the mesonic moduli space is given by the symplectic quotient of the form
(3.16) 
By associating the fugacities to the perfect matchings respectively, the fully refined Hilbert series of is given by the following Molien integral
(3.17)  
Accordingly, the mesonic moduli space is a freely generated space, .
The charge matrix in (3.15) indicates a symmetry of .
fugacity  

0  1  
0  1  
0  1  
0  1  
1  1  
1  1 
Under the above charge assignment, the Hilbert series of can be expressed as
(3.18) 
Since the moduli space space is , we expect a symmetry. The fully enhanced global symmetry is therefore . This can be observed by modifying the global charges on the perfect matchings and . A possible choice can be:
fugacity  

1  
1  
1  
1  
1/2  
1/2 
Under the above charge assignment, the mesonic Hilbert series can be expressed as expected in terms of characters of irreducible representations,
(3.19) 
The toric diagram of the mesonic moduli space is a 4 dimensional lattice polytope. The coordinates of the toric points are encoded in the matrix
(3.20) 
Recall that perfect matchings correspond to toric points. We observe that the perfect matchings and correspond to the same toric point.
3.2.2 Model 6.2b:
1  2  3  4  5  6 

The second brane tiling on a Riemann surface with 2 superpotential terms with 6 fields is shown with the corresponding quiver diagram in Figure 15 and Figure 16 respectively. The superpotential is
(3.21) 
The quiver incidence matrix is
(3.22) 
The brane tiling has perfect matchings, each of them given by a bifundamental field. The perfect matching matrix is the identity matrix,
(3.23) 
The zigzag paths of the brane tiling are
(3.24) 
The Abelian superpotential vanishes , and the kernel of the perfect matching matrix is empty. There are no Fterms, therefore no Fterm charges. The Dterm charges are encoded in the quiver incidence matrix :
(3.25) 
The total charge matrix , and the mesonic moduli space is the symplectic quotient
(3.26) 
By associating the fugacities and to the perfect matchings and respectively, the fully refined Hilbert series of is given by the Molien integral
where
(3.28) 
Accordingly, the mesonic moduli space is a noncomplete intersection of dimension 5. By setting the fugacities , the unrefined Hilbert series is
(3.29) 
The palindromic numerator of the Hilbert series indicates that is a CalabiYau 5fold. The plethystic logarithm of the refined Hilbert series of is of the form
(3.30) 
The generators of the mesonic moduli space in terms of perfect matching variables are
generator  perfect matchings 

which are subject to the first order relations
(3.31) 
One can assign the following enhanced global charges to the perfect matching variables
fugacity  

0  1  
0  1  
0  1  
0  1  
0  1  
0  1 
Under the above charge assignment, the Hilbert series of can be expressed as
(3.32) 
where . The generators and the first order relations formed by them are encoded in the plethystics logarithm, which now takes the form
(3.33) 
The toric diagram of the mesonic moduli space is a 4 dimensional lattice polytope. The coordinates of the toric points are encoded in the matrix
(3.34) 
Note that the mesonic moduli space here is the same as the master space of Butti:2007jv .
3.2.3 Model 6.2c:
1  2  3  4  5  6 

The brane tiling and quiver for Model 6.2c are shown in Figure 17 and Figure 18 respectively. The superpotential is
(3.35) 
In the Abelian gauge theory the superpotential vanishes, giving the same model as in the previous section. (The nonAbelian gauge theories differ by superpotential interactions.) There is a difference in the zigzag paths, which now are
(3.36) 
3.3 7 Fields, 2 Superpotential Terms, 3 Gauge Groups
3.3.1 Model 7.2:
1  2  3  4  5  6  7 

The brane tiling and corresponding quiver for Model 7.2 is shown in Figure 19 and Figure 20 respectively. The superpotential is
The quiver incidence matrix is
(3.38) 
Model 7.2 has perfect matchings, each made out of a single field in the quiver. The perfect matching matrix is therefore the identity matrix,
(3.39) 
The brane tiling has the following zigzag paths,