Pith. Sign in

REVIEW

Brane Dimers and Quiver Gauge Theories

PaperPDF from arXiv

data files

ReviewThe objections and verdict, as JSON Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Sebastian Franco, Amihay Hanany, Kristian D. Kennaway, David Vegh, Brian Wecht

High Energy Physics - Theory · Submitted 13 Apr 2005 · 53 references

Pith's verdict: The brane tiling is now the standard tool for this job, and the paper earns that status — but Section 4's 'proof' is a proof sketch, and the authors say so themselves in the conclusions.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on October 29, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

  1. T0 AI review An AI read the paper and answered a fixed list of questions.
  2. T1 Audit Automatic checks that the references and files the paper relies on exist and match.
  3. T2 Compute The paper's code or calculations re-run with the same results.
  4. T3 Formal statement The claims written out in a formal math language a computer can read.
  5. T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 2 major objections · 5 minor · Ask about this paper · The compiled record · Read the full review →
Challenge this review Comment (0) Read on arXiv ↗
Cite ▾
arXiv IDhep-th/0504110 (version 2)
Pith IDP6AVWT5O, Pith's permanent record of this paper and its review
CitationAmihay Hanany, Brian Wecht, David Vegh, Kristian D. Kennaway, Sebastian Franco. "Brane Dimers and Quiver Gauge Theories." arXiv:hep-th/0504110 (2005). https://pith.science/paper/P6AVWT5O
Cite this reviewBibTeX for Pith's review of this paper
Claim paper

Why Pith says this

Highlight any text to comment on it.

Referee report

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

A single periodic tiling of a torus is claimed to encode both an $N=1$ quiver gauge theory and the toric Calabi-Yau geometry that theory probes.

The reading

The paper argues that one object — a periodic bipartite tiling of a torus, drawn physically as D5-branes ending on an NS5-brane that wraps a holomorphic curve — carries the complete data of an $N=1$ quiver gauge theory and, separately, the toric geometry of the singular Calabi-Yau the branes probe. Faces of the tiling are gauge groups, edges are bifundamental chiral multiplets, and vertices are superpotential terms, with the black/white coloring of the graph fixing the signs. The reason this matters is computational: the previous route from a quiver to its toric diagram, the Forward Algorithm, requires a dual-cone computation whose cost explodes with the number of nodes and becomes impractical past roughly ten. If the tiling works, one writes a small matrix, takes its determinant, and reads the toric diagram and the GLSM multiplicities straight off a Laurent polynomial. The authors check this against known results for del Pezzo surfaces, $F_0$ and the suspended pinch point, and then use it to derive the moduli space of the entire $Y^{p,q}$ family and to build tilings for the $X^{p,q}$ family. Two things are explicitly left open: why perfect matchings should correspond to GLSM fields in the brane picture, and how to go backwards from an arbitrary toric geometry to its tiling.

What carries the argument

The Kasteleyn matrix is the load-bearing device. It is a square matrix of size half the number of nodes in a unit cell, built by weighting each edge by $z^{n_z} w^{n_w}$ according to how the two flux lines winding the torus cycles cross it, and by a sign chosen so that the product of signs around any face is $+1$ for a face with $4k+2$ edges and $-1$ for one with $4k$. Its determinant is the characteristic polynomial, and the exponents of its monomials are the vertices of the toric diagram. The companion mechanism is the matching-to-field map: a bifundamental $X_i$ is written as the product of the perfect-matching variables that cover its edge, $X_i = prod_alpha p_alpha^{<e_i, p_alpha>}$, which makes the F-term relations hold identically because a matching covering one side of a superpotential vertex must cover the other. Together these turn a superpotential computation into a determinant.

Extended reading notes

Core claim

The central claim is a dictionary. Take a bipartite graph embedded on $T^2$ with all faces of even length, and read it as a brane configuration. Then $2n$-sided faces are gauge groups with $n$ flavors, edges are bifundamental chiral multiplets oriented so the white node lies to the right, and $k$-valent vertices are order-$k$ superpotential terms whose sign is fixed by the vertex color. Euler's formula on the torus gives the identity $N_g + N_W - N_f = 0$ for these theories, and the same identity follows independently from requiring vanishing $\beta$ functions together with $R(W)=2$. The second half of the claim is that the determinant of the Kasteleyn matrix — a signed, weighted adjacency matrix with rows on white nodes and columns on black, with $z$ and $w$ inserted along two flux lines — is a Laurent polynomial $P(z,w)$ whose Newton polygon is the toric diagram and whose coefficients are the GLSM multiplicities. The authors argue this is not an accident: perfect matchings of the tiling solve the F-term equations automatically, so they can be identified with GLSM fields, which upgrades the earlier Newton-polygon conjecture to a claimed derivation. The paper also reports a counterexample to a working conjecture: two of the four toric phases of pseudo del Pezzo 5 have identical characteristic polynomials despite being different graphs, so GLSM multiplicities do not one-to-one label toric phases.

Load-bearing premise

The load-bearing premise is that perfect matchings of the tiling can be identified with the fields of the gauged linear sigma model, which rests on the assertion that a matching covering one side of a superpotential vertex must cover the other side; that assertion is argued for the generic case but not verified case by case for tilings containing adjoint fields or bidirectional arrows.

Editorial extensions

If this is right

  • Any quiver theory whose tiling can be drawn gets its toric diagram and GLSM multiplicities from the determinant of a matrix whose size is half the number of superpotential terms, replacing a dual-cone computation that scales badly with quiver size.
  • The moduli space of the whole $Y^{p,q}$ family follows from a Kasteleyn matrix written down by a codon rule, without drawing the tiling and without running the Forward Algorithm on each member.
  • Seiberg duality becomes a local move on the tiling and a corresponding operation on the Kasteleyn matrix, so the toric phases of a theory can be enumerated by computer starting from a single phase.
  • Partial resolution and Higgsing become edge deletion plus collapse of the resulting bivalent vertex, giving a direct tiling-level picture of how one toric singularity degenerates to another.
  • The McKay correspondence extends to non-compact toric Calabi-Yau threefolds, with the brane tiling playing the role the Dynkin diagram plays for ALE singularities.
  • Two distinct toric phases can share a toric diagram with identical multiplicities, so multiplicity data alone is not a complete classifier of toric phases.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the matching-to-GLSM-field dictionary is exact in general, the tiling should also give $R$-charges directly, since $R$-charges are the weights of the same perfect-matching variables under the residual abelian symmetry; that would make $a$-maximization a computation on the graph rather than on the Lagrangian.
  • The observed collision of characteristic polynomials implies that any automated enumeration of toric phases must eventually test graph isomorphism rather than compare polynomial invariants, which changes the complexity of the phase-classification problem from cheap to expensive in the worst case.
  • Dimer theory already has a variational principle — extremizing a surface-tension functional to get the limit curve — and the paper's own closing remarks note that a Monge-Ampere extremization also appears in $a$-maximization; if these are the same extremization, the Calabi-Yau cone's Reeb vector is readable off the tiling's asymptotic shape.
  • The deferral of the inverse problem suggests a concrete next target: a constructive procedure that takes a lattice polygon and outputs a bipartite torus tiling with that Newton polygon, which would close the loop between geometry and gauge theory.
Original abstract

We describe a technique which enables one to quickly compute an infinite number of toric geometries and their dual quiver gauge theories. The central object in this construction is a ``brane tiling,'' which is a collection of D5-branes ending on an NS5-brane wrapping a holomorphic curve that can be represented as a periodic tiling of the plane. This construction solves the longstanding problem of computing superpotentials for D-branes probing a singular non-compact toric Calabi-Yau manifold, and overcomes many difficulties which were encountered in previous work. The brane tilings give the largest class of N=1 quiver gauge theories yet studied. A central feature of this work is the relation of these tilings to dimer constructions previously studied in a variety of contexts. We do many examples of computations with dimers, which give new results as well as confirm previous computations. Using our methods we explicitly derive the moduli space of the entire Y^{p,q} family of quiver theories, verifying that they correspond to the appropriate geometries. Our results may be interpreted as a generalization of the McKay correspondence to non-compact 3-dimensional toric Calabi-Yau manifolds.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,274 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.16 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.91 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We describe a technique which enables one to quickly compute an infinite number of toric geometries and their dual quiver gauge theories. The central object in this construction is a ``brane tiling,'' which is a collection of D5-branes ending on an NS5-brane wrapping a holomorphic curve that can be represented as a periodic tiling of the plane. This construction solves the longstanding problem of computing superpotentials for D-branes probing a singular non-compact toric Calabi-Yau manifold, and overcomes many difficulties which were encountered in previous work. The brane tilings give the largest class of N=1 quiver gauge theories yet studied. A central feature of this work is the relation of these tilings to dimer constructions previously studied in a variety of contexts. We do many examples of computations with dimers, which give new results as well as confirm previous computations. Using our methods we explicitly derive the moduli space of the entire Y^{p,q} family of quiver theories, verifying that they correspond to the appropriate geometries. Our results may be interpreted as a generalization of the McKay correspondence to non-compact 3-dimensional toric Calabi-Yau manifolds.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We describe a technique which enables one to quickly compute an infinite number of toric geometries and their dual quiver gauge theories. contribution 0.94 0.91 0.39 asserted 1.00
s1 The central object in this construction is a ``brane tiling,'' which is a collection of D5-branes ending on an NS5-brane wrapping a holomorphic curve that can be represented as a periodic tiling of the plane. method 0.72 0.48 0.08 asserted 1.00
s2 This construction solves the longstanding problem of computing superpotentials for D-branes probing a singular non-compact toric Calabi-Yau manifold, and overcomes many difficulties which were encountered in previous work. contribution 0.52 0.89 0.33 asserted 1.00
s3 The brane tilings give the largest class of N=1 quiver gauge theories yet studied. result 0.78 0.64 0.20 asserted 1.00
s4 A central feature of this work is the relation of these tilings to dimer constructions previously studied in a variety of contexts. contribution 0.61 0.43 0.07 asserted 1.00
s5 We do many examples of computations with dimers, which give new results as well as confirm previous computations. contribution 0.61 0.95 0.06 asserted 1.00
s6 Using our methods we explicitly derive the moduli space of the entire Y^{p,q} family of quiver theories, verifying that they correspond to the appropriate geometries. result 0.93 0.97 0.14 asserted 1.00
s7 Our results may be interpreted as a generalization of the McKay correspondence to non-compact 3-dimensional toric Calabi-Yau manifolds. interpretation 1.00 0.71 0.16 hedged possibility 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from proof or derivation 0.91.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. A finite region in the infinite brane tiling and quiver diagram for Model I of dP3. We indicate the correspondence between: gauge groups ↔ faces, bifundamental fields ↔ edges and superpotential terms ↔ nodes. 2.1 Unification of quiver and superpotential data An N = 1 quiver gauge theory is described by the following data: a directed graph represent￾ing the gauge groups and matter content, and a set of closed paths on the graph representing the gauge invariant interactions in the superpotential. An equivalent way to characterise this data is to view it as defining a CW-complex; in…
Figure 2. The logical flowchart. We will show that we can think of the superpotential and quiver together as a tiling of a two-dimensional surface, where bifundamentals are edges, superpotential terms are faces, and gauge groups are nodes. We refer to this as the “periodic quiver” representation. The toric condition, which states that each matter field appears in precisely two superpotential terms of opposite sign, means that the faces all glue together in pairs along the common edges. Since every field is represented exactly twice in the superpotential, this tiling has no boundaries. Thus,…
Figure 3. The quiver gauge theory associated to one of the toric phases of the cone over F0. In the upper right the quiver and superpotential (2.6) are combined into the periodic quiver defined on T 2 . The terms in the superpotential bound the faces of the periodic quiver, and the signs are indicated and have the dual-bipartite property that all adjacent faces have opposite sign. To get the bottom picture, we take the planar dual graph and indicate the bipartite property of this graph by coloring the vertices alternately. The dashed lines indicate edges of the graph that are duplicated by…
Figure 4. a) Brane tiling for Model I of dP3 with flux lines indicated in red. b) Unit cell for Model I of dP3. We show the edges connecting to images of the fundamental nodes in green. We also indicate the signs associated to each edge as well as the powers of w and z corresponding to crossing flux lines. From the unit cell, we derive the following Kasteleyn matrix K =   2 4 6 1 1 + w 1 − zw 1 + z 3 1 −1 −w −1 5 −z −1 −1 1   (3.4) We observe that is has twelve monomials, associated to the twelve bifundamental fields. This matrix leads to the characteristic polynomial P(z, w) = w…
Figure 5. Toric diagram for Model I of dP3 derived from the characteristic polynomial in (3.5). is remarkable, since this size can be very modest even for very complicated gauge theories. The simplicity of computing the toric data using this procedure should be contrasted with the difficulty of the Forward Algorithm. This procedure has a profound impact on the study of quiver theories for arbitrary toric singularities. Given a candidate quiver theory for D3-branes over some geometry, instead of running the lengthy Forward Algorithm, one simply constructs the associated brane tiling using the…
Figure 6. P (X ) 1 i X P (X ) 2 i =
Figure 7. Quiver diagram for the SPP. It is interesting to see how our methods apply to this example, since it has both adjoint fields and bidirectional arrows
Figure 8. Brane tiling for the SPP. K =   2 4 1 1 + w −1 z + w −1 z 3 1 1 + w −1   (4.9) from which we determine the characteristic polynomial 22
Figure 9. Toric diagram for the SPP. We indicate the perfect matchings corresponding to each node in the toric diagram. There are six perfect matchings of the SPP tiling. We show them in
Figure 10. Perfect matchings for the SPP. We indicate the slopes (hw, hz), which allow the identification of the corresponding node in the toric diagram as shown in
Figure 11. Integrating out a massive field corresponds to collapsing the two vertices adjacent to a bivalent vertex into a single vertex of higher valence. both neighboring vertices to the bivalent vertex are identified in the graph). In the example of figure 11, if the bivalent white node has label 1 and the adjacent black nodes are 1′ and 2 ′ (this can always be arranged by a reordering of rows or columns, with the corresponding action of (−1) to preserve the determinant), the Kasteleyn matrix (or its transpose) has the following structure: K =   v (1) 1 v (2) 1 0 . . . 0 v (
Figure 12. The action of Seiberg duality on a periodic quiver to produce another toric phase of the quiver. Also marked are the signs of superpotential terms, showing that the new terms (faces) are consistent with the pre-existing 2-coloring of the global graph. The neighboring vertices to V are not necessarily all distinct (they may be identified by the periodicity of the torus on which the quiver lives). However by the periodic quiver construction, if there are multiple fields in the quiver connecting the same two vertices, these appear as distinct edges in the periodic quiver. Note that…
Figure 13. Seiberg duality acting on a brane tiling to produce another toric phase. This is the planar dual to the operation depicted in
Figure 14. The operation of Seiberg duality on a phase of F0. It is useful to see how this action of Seiberg duality can be understood from the brane perspective. Since the area of each cell (volume of the D-brane) is related to the gauge coupling of the corresponding group [47], one would expect that Seiberg duality could be viewed as a cell shrinking and then growing with the opposite orientation, e.g. as branes move through one another. It is possible to see this from
Figure 15. Removing the edge from between faces 5 and 6 Higgses Model I of dP3 (top) to one of the two toric phases of dP2 (bottom). 4 
Figure 16. The dP2 tiling (top) can be taken to either dP1 (bottom left) or F0 (bottom right), depending on which edge gets removed. In the F0 tiling, one should collapse the edge between regions 2 and 4 to a point; this corresponds to the bifundamentals on the diagonal of the quiver. so one expects that we can reach any tiling by removing edges from hexagons. This is indeed the case, as noted by [48] and used extensively in [33]. 34
Figure 17. Quiver diagram admitting two toric superpotentials. fields. From (2.1), we see that the number of superpotential terms is NW = 14 − 6 = 8. There are two possible toric superpotentials consistent with the node symmetry group of the quiver. They have been considered in [20] and [49] and are WA = [X12X26X61 − X12X25X51 + X36X64X43 − X35X54X43] +[−X61X13X36 + X51Y13X35] + [−X26X64X41Y13X32 + X25X54X41X13X32] (8.1) WB = Y13X36X61 + X13X35X51 − X61X12X26 − X43X35X54 +X12X25X54X41 + X26X64X43X32 − X25X51Y13X32 − X64X41X13X36 (8.2) WA corresponds to a brane tiling with six valence-3 and…
Figure 18. Brane tiling corresponding to the quiver diagram in
Figure 19. Brane tiling corresponding to the quiver diagram in
Figure 20. Toric diagram for the quiver in
Figure 21. Brane tiling for Model I of dP2. 37
Figure 22. Brane tiling for Model II of dP2. K =   1 − z −1 w 1 1 1 z −1 + w −1 z −1 1 1   (9.3) which leads to the following characteristic polynomial P(z, w) = w −1 z −1 − z −1 + 5 − w − z − wz (9.4) From (9.2) and (9.4) we can determine the toric diagrams along with the GLSM multiplicities, which are in agreement with the results in [20]. 38
Figure 23. Brane tiling for Model III of dP3. K =   2 4 6 8 1 1 w −1 w −1 z −1 1 3 1 1 −z −1 0 5 wz 1 −1 −w 7 z 0 1 1   (9.5) from which we compute the determinant P(z, w) = w −1 z −1 + z −1 − w −1 − 8 + w + z + wz (9.6) This corresponds to the toric diagram of dP3 with multiplicity 8 for the central point. This result agrees with the Forward Algorithm computations in [20]. 39
Figure 24. Brane tiling for Model IV of dP3. Model IV
Figure 25. Brane tilings for the four toric phases of P dP5. 41
Figure 26. Toric diagrams with multiplicities for the four toric phases of P dP5. We observe that the GLSM multiplicities are the same for Models I and III. that the application of the Forward Algorithm to large quivers becomes computationally prohibitive. In addition, it is obviously impossible to apply the Forward Algorithm to an infinite number of theories. Hence, the determination of gauge theories dual to an infinite number of geometries usually involve indirect evidence such as: (un)higgsing, global sym￾metries, computation of R-charges and central charges and comparison to volumes in…
Figure 27. Brane tiling for Y 3,3 43
Figure 28. Brane tiling for Y 3,2 . The impurity is the blue area. an identical shaded region into the tiling, separated from the first one by some hexagonal faces. For Y 3,1 this is not possible because the fundamental cell consists of only six hexagons, whereas two separated single impurities would cover eight of them. Instead, we can consider the case in which the two impurities are adjacent. This corresponds to a similar impurity graph, which is shown in
Figure 29. Brane tiling for Y 3,1 . One can continue adding impurities and discover the simplicity of the Kasteleyn matrix 44
Figure 30. Toric diagram of a phase of Y 5,3 We note that the above rules for constructing the Kasteleyn matrix produce the toric diagrams for all Y p,q with p > q > 0. To check this, we can see that the correct monomials appear in the determinant. First, the only powers of z that appear in det K are -1,0, and 1. Terms of the form z 0w k appear for all k = 0, . . . , p; these come from the diagonal. Second, there is a term z −1w 0 that comes from the lower off-diagonal. Finally, the term zwn , where n = p − q is the number of single impurities, comes from the upper off-diagonal and gets c
Figure 31. Toric diagram of a phase of Y 3,0 with three single impurities. These Kasteleyn matrices give the toric diagrams of a certain phase of the theories (the one with only single impurities, all of them together). Other phases can be obtained by performing Seiberg duality transformations. As discussed in section 6 this may be efficiently implemented on a computer and used to enumerate the toric phases of the theory, together with the duality graph showing the interconnections between phases. 9.3.2 Y 3,1 with double impurity In [50], it was shown that all toric phases of Y p,q theories…
Figure 32. Dualizing face 3 in Y 3,1 with two single impurities. In resulting tiling, we indicate the double impurity in pink. 47
Figure 33. The double impurity in Y 3,1 K =   1 1 0 0 0 z −1 w −1 1 w −1 0 0 0 0 1 1 1 0 0 0 0 w −1 1 1 0 0 0 0 1 −1 −1 z −1 0 0 0 w 1   (9.22) P(w, z) = −7 − w −2 + 9w −1 + w + w −1 z −1 + zw−1 (9.23) 7 9 9 7 =
Figure 34. Toric diagram for Y 3,1 in the double impurity phase All the multiplicity results in this and previous section agree with the ones derived using the Forward Algorithm in [51]. 9.3.3 Xp,q tilings We now describe the brane tilings for the Xp,q spaces constructed in [41]. Recall that these spaces are defined by the property that an Xp,q theory can be Higgsed to both Y p,q and 48
Figure 35. Brane tilings for Y 3,q . Constructing the Xp,q brane tilings is now straightforward. We give the example of X3,1 below; the other Xp,q tilings work similarly. To build the tilings, simply insert diagonal lines in hexagons such that removing the line from the Xp,q tiling gives the Y p,q to which it descends. This diagonal line should always share a node with one of the horizontal lines subdividing a hexagon in two; this is what allows one to blow down to Y p,q−1 as well as Y p,q . In
Figure 36. A brane tiling for X3,1 . The Kasteleyn matrix for this tiling is K =   2 4 6 8 1 1 + w −1 1 0 z 3 1 −1 − w 1 0 5 0 1 w −1 1 7 z −1 1 1 −1   (9.24) which has determinant det K = 7w −1 +w −2 + 8 +w −z −1 −z +w −1 z. This yields the proper toric diagram and multiplicities for this phase of X3,1 [51]. 10. Conclusions In this paper, we have presented a piece of technology which not only simplifies previously difficult calculations, but also draws new connections to other interesting areas of study. In this section, we present some conlcuding remarks abo

Figures reproduced from arXiv:hep-th/0504110 by the authors.

References
How later papers used it

214 later papers cite it

Evidence: substantial. 156 of the citing papers share no author with it, and 30 of those carry a confident label from Pith's citation classifier.

Pith read 11 of the 537 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 2 of 35 papers with this label
How the literature uses it

210 arXiv papers cite this in 532 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 291 sentences without a confident label

Builds on it 5 papers
  • Observe that the rule of how to perform Seiberg duality (for $SU(N)$ groups) in brane tilings [ref] can be extended to the orientifoded case as well.

    Brane Tilings for Orientifolds · Orientifolds acting as line reflection · 0.96

  • This generalizes the definition of perfect matchings for brane tilings [ref] and of brick matchings for brane brick models [ref] .

    Graded quivers and B-branes at Calabi-Yau singularities · Toric graded quivers and toric singularities · 0.98

  • We consider BFTs as natural generalizations of brane tilings [ref] defined on a torus $T^2$ with $B=0$.

    Hilbert Series of Bipartite Field Theories · Bipartite Field Theories · 0.89

Uses its method or tool 36 papers
  • Our analysis, which is done exploiting dimers techniques [ref] , relies also on results obtained in [ref] , where a thorough investigation of consistent, anomaly free, D-brane models at orientifold singularities has been performed.

    Dimers, Orientifolds and Stability of Supersymmetry Breaking Vacua · Introduction · 0.87

  • In this paper we apply the powerful description of D3-branes at toric CY threefold singularities in terms of dimer diagrams [ref] to unravel the underlying structure of discrete symmetries in general orbifolds of general toric singularities.

    Discrete Symmetries in Dimer Diagrams · Introduction and conclusions · 0.98

  • Alternatively, the perfect matching technology gives the toric diagram directly via the fast-forward algorithm [ref] (which improves the forward algorithm of [ref] ): * To each perfect matching $p_α$ we associate a field in an abelian gauged linear sigma model (GLSM) [ref] , which we call a perfect matching variable and also denote by $p_α$, with a slight abuse of notation.

    Zig-zag deformations of toric quiver gauge theories. Part I: reflexive polytopes · Fast-forward algorithm and toric diagrams · 0.97

Compares with it 5 papers
  • Comparing with model in model $9a$ in [ref] , with [ref] , phase III (see also [ref] , model B), we see that the relevant quiver has one additional bidirectional arrow corresponding to the one of the vanishing entries $S^∨_14,S^∨_25, S^∨_36$ in the adjacency matrix [ref].

    Vafa-Witten invariants from exceptional collections · $PdP_3b$ · 1.00

  • The identification above has been observed already in [ref] and our composite move II corresponds a special case of their figure 13 (i.e. they are not working within the class of smooth quadrilateral tilings).

    On the structure of quadrilateral brane tilings · Smooth quadrilateral tilings · 0.92

  • 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 [ref] .

    On the Master Space for Brane Brick Models · Observations and Discussions · 0.98

Cites it as support 2 papers
  • (However, it is worth mentioning that the effect of this duplication can be seen in the multiplicities of fields in the gauged linear sigma model description of $_τ_G⃗$ [ref] .) Despite seeming rather innocuous, notice that any such duplication must trigger an inconsistency in the associated superconformal field theory since [ref] implies $R_b + R_c =0$ for the new face bounded by $b$ and $c$.

    On the structure of quadrilateral brane tilings · Duplication and lagging · 0.96

  • As a further check, the non-anomalous orthogonal Abelian symmetries become baryonic symmetries in the IR fixed point, and it is known (e.g., [ref] ) that the number of such symmetries matches that expected from the gravity dual, hence supporting the assumption that the IR $R$-symmetry is already visible in the UV Lagrangian.

    Invariants of Toric Seiberg Duality · Dimer models and Seiberg duality · 0.97

Cites it as background 122 papers
  • In the last year there has been significant progress in the computation of BPS spectra of 4d $=2$ supersymmetric gauge theories using the Representation Theory of quivers with superpotentials [ref] (for previous work see [ref] ; for other approches to the $=2$ BPS spectra see [ref] ).

    4d N=2 Gauge Theories and Quivers: the Non-Simply Laced Case · Introduction · 0.93

  • A prominent example is the correspondence [ref] between 4d $= 1$ quiver gauge theories realized by certain brane configurations in string theory, known as brane tilings [ref] , and the 2d lattice models that Bazhanov and Sergeev [ref] constructed using elliptic hypergeometric integrals discovered by Spiridonov [ref] .

    Surface defects as transfer matrices · Introduction · 0.91

  • In order to extract the data of the effective quiver gauge theory from the geometry of $Z$, several procedures have been developed in [ref] , most notably the fast inverse algorithm and the brane tiling method.

    Remarks on quiver gauge theories from open topological string theory · Introduction and summary · 0.85

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.