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Box Graphs and Singular Fibers

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Hirotaka Hayashi, Craig Lawrie, David R. Morrison, Sakura Schafer-Nameki

High Energy Physics - Theory · Submitted 11 Feb 2014 · 71 references

Pith's verdict: A genuinely useful reorganization of higher-codimension fibers and flop networks via box graphs; the combinatorial core is solid, and the new non-Kodaira fibers are plausible but rest on the reading rules rather than independent geometry.

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CitationCraig Lawrie, David R. Morrison, Hirotaka Hayashi, Sakura Schafer-Nameki. "Box Graphs and Singular Fibers." arXiv:1402.2653 (2014). https://pith.science/paper/KXGML6RD
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The paper

The paper claims

The paper claims that decorated sign diagrams on matter representation graphs classify every crepant resolution of a singular elliptic Calabi-Yau fourfold, including its flop network and higher-codimension fibers.

The reading

The paper tries to turn a hard classification problem in geometry into a finite combinatorial one. For an elliptically fibered Calabi-Yau fourfold, each crepant resolution is supposed to correspond to a phase of a three-dimensional supersymmetric gauge theory with matter. The authors argue that every such phase can be encoded by coloring the weights of the matter representation with plus and minus signs, subject to local flow rules; the boundary between the two signs is a path. If this is right, one can read off the effective curves, the flop transitions, and the fibers in codimensions two and three directly from the colored graph, including fiber types that do not appear in Kodaira's classification.

What carries the argument

The load-bearing device is the decorated box graph: draw the weight diagram of the matter representation, put a plus or minus sign in each box, and allow only sign patterns that obey the flow rules, which force signs to propagate in fixed directions across neighboring boxes. The interface between plus and minus boxes is a monotone path; for $mathrm{SU}(n)$ the tracelessness condition becomes the requirement that this path cross a diagonal, making it an anti-Dyck path. The phases are counted by the Weyl group quotient $|W_{tilde{mathfrak{g}}}|/|W_{mathfrak{g}}|$, and the same quotient explains why a phase network can look like a minuscule representation of $tilde{mathfrak{g}}$ or, when the commutant is $mathrm{SU}(2)$, like the Dynkin diagram of $tilde{mathfrak{g}}$.

Extended reading notes

Core claim

The central claim is that decorated box graphs—sign assignments on the representation graph of the matter, obeying flow rules—are in one-to-one correspondence with the Coulomb phases of the three-dimensional $mathcal{N}=2$ gauge theory and hence with crepant resolutions of the associated Weierstrass model. The interface between plus and minus signs is a monotone path; its corners give the extremal generators of the cone of effective curves, and flipping a corner is a flop. From the same data the authors reconstruct all fiber types in codimensions two and three, and report non-Kodaira fibers for $E_6$, $E_7$ and $E_8$. They also observe that the network of phases often has the shape of a minuscule or quasi-minuscule representation of a larger Lie algebra.

Load-bearing premise

The load-bearing premise is that every Coulomb phase of the three-dimensional gauge theory really is a crepant resolution of the Weierstrass model, so that the combinatorial phase diagram can be trusted as a complete geometric classification; that dictionary is motivated by M-theory/F-theory duality and used throughout, but it is not proved as a mathematical theorem here, and the paper also acknowledges that flatness of resolutions is not established in general for fourfolds.

Editorial extensions

If this is right

  • For any reductive gauge group and matter representation covered by the flow rules, the method gives a finite algorithm to enumerate crepant resolutions and their flop transitions without case-by-case algebraic construction.
  • The generators of the cone of effective curves are read from the corners of the anti-Dyck path together with the simple roots that do not cross it.
  • Codimension-two and codimension-three fibers are classified by the same graphs; for $E_6$, $E_7$ and $E_8$ this yields non-Kodaira types obtained by deleting one non-affine node from the Kodaira fiber.
  • When the commutant is $mathrm{SU}(2)$ the flop diagram is the Dynkin diagram of the enhanced algebra; when it is $mathrm{U}(1)$ the phase network is a minuscule representation.
  • For $mathrm{SU}(5)$ with $mathbf{5}$ and $mathbf{10}$ matter the box graphs reproduce the twelve known resolutions and distinguish the six known $E_6$ fibers from six new ones.

Reading between the lines

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

  • If the correspondence holds, the classification of higher-codimension fibers becomes a finite representation-theoretic computation that could be automated for any ADE enhancement, rather than an open-ended search through algebraic resolutions.
  • A natural extension is to test the same box-graph rules on elliptic Calabi-Yau threefolds, where flatness is better understood; a match there would strengthen the fourfold dictionary by analogy.
  • The minuscule-representation observation suggests an independent check: the flop graph might be recoverable from the crystal graph of the representation, giving a purely combinatorial prediction for the network.
  • For model building, the new $E_6$, $E_7$ and $E_8$ fibers imply that a codimension-three Yukawa point need not display an affine Dynkin diagram, so couplings read off a resolved geometry must account for the missing node.
Original abstract

We determine the higher codimension fibers of elliptically fibered Calabi-Yau fourfolds with section by studying the three-dimensional N=2 supersymmetric gauge theory with matter which describes the low energy effective theory of M-theory compactified on the associated Weierstrass model, a singular model of the fourfold. Each phase of the Coulomb branch of this theory corresponds to a particular resolution of the Weierstrass model, and we show that these have a concise description in terms of decorated box graphs based on the representation graph of the matter multiplets, or alternatively by a class of convex paths on said graph. Transitions between phases have a simple interpretation as `flopping' of the path, and in the geometry correspond to actual flop transitions. This description of the phases enables us to enumerate and determine the entire network between them, with various matter representations for all reductive Lie groups. Furthermore, we observe that each network of phases carries the structure of a (quasi-)minuscule representation of a specific Lie algebra. Interpreted from a geometric point of view, this analysis determines the generators of the cone of effective curves as well as the network of flop transitions between crepant resolutions of singular elliptic Calabi-Yau fourfolds. From the box graphs we determine all fiber types in codimensions two and three, and we find new, non-Kodaira, fiber types for E_6, E_7 and E_8.

The compiled record

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The abstract, sentence by sentence

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

We determine the higher codimension fibers of elliptically fibered Calabi-Yau fourfolds with section by studying the three-dimensional N=2 supersymmetric gauge theory with matter which describes the low energy effective theory of M-theory compactified on the associated Weierstrass model, a singular model of the fourfold. Each phase of the Coulomb branch of this theory corresponds to a particular resolution of the Weierstrass model, and we show that these have a concise description in terms of decorated box graphs based on the representation graph of the matter multiplets, or alternatively by a class of convex paths on said graph. Transitions between phases have a simple interpretation as `flopping' of the path, and in the geometry correspond to actual flop transitions. This description of the phases enables us to enumerate and determine the entire network between them, with various matter representations for all reductive Lie groups. Furthermore, we observe that each network of phases carries the structure of a (quasi-)minuscule representation of a specific Lie algebra. Interpreted from a geometric point of view, this analysis determines the generators of the cone of effective curves as well as the network of flop transitions between crepant resolutions of singular elliptic Calabi-Yau fourfolds. From the box graphs we determine all fiber types in codimensions two and three, and we find new, non-Kodaira, fiber types for E_6, E_7 and E_8.

background contribution method result interpretation main claim

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We determine the higher codimension fibers of elliptically fibered Calabi-Yau fourfolds with section by studying the three-dimensional N=2 supersymmetric gauge theory with matter which describes the low energy effective theory of M-theory compactified on the associated Weierstrass model, a singular model of the fourfold. contribution 0.84 0.93 0.16 asserted 1.00
s1 Each phase of the Coulomb branch of this theory corresponds to a particular resolution of the Weierstrass model, and we show that these have a concise description in terms of decorated box graphs based on the representation graph of the matter multiplets, or alternatively by a class of convex paths on said graph. contribution 0.53 0.93 0.22 asserted 1.00
s2 Transitions between phases have a simple interpretation as `flopping' of the path, and in the geometry correspond to actual flop transitions. interpretation 0.61 0.82 0.07 asserted 1.00
s3 This description of the phases enables us to enumerate and determine the entire network between them, with various matter representations for all reductive Lie groups. result 0.57 0.94 0.16 asserted 1.00
s4 Furthermore, we observe that each network of phases carries the structure of a (quasi-)minuscule representation of a specific Lie algebra. result 0.85 0.91 0.15 asserted 0.98
s5 Interpreted from a geometric point of view, this analysis determines the generators of the cone of effective curves as well as the network of flop transitions between crepant resolutions of singular elliptic Calabi-Yau fourfolds. interpretation 0.86 0.92 0.38 asserted 1.00
s6 From the box graphs we determine all fiber types in codimensions two and three, and we find new, non-Kodaira, fiber types for E_6, E_7 and E_8. result 0.99 0.97 0.34 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.79 · evidence from proof or derivation 0.99.

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Figures
Figure 1. Example for Bruhat ordering of the phases or equivalently Weyl group quotients,
Figure 2. The representation n for su(n). Each box represents a weight, Li and the walls separating the boxes represent the action of the negative (positive) simple roots on the weights. Based on the Bruhat order for WJ we define a diagram. The nodes correspond to the elements in WJ . The lines between the nodes means that the elements are ordered by Bruhat order and the difference between their lengths is one. From the properties of the Bruhat order and the quotient WJ , this diagram matches with the phase diagram of Φρ 1···r of the theory with g⊕u(1)
Figure 3. The representation Λ2n for su(n). Each box represents a weight, Li + Lj of the representations labeled by (i, j), the walls separating the boxes represent the action of the negative (positive) simple roots on the weights. We can then ask which such decorated box graphs correspond to a phase, Φρ 1···r , where the i are the signs decorating the box corresponding to the ith weight of the representation. We show the existence of the following flow rules governing the placement of signs which are a necessary condition for any decorated box graph to correspond to a consistent phase (or
Figure 4. Phases of the su(5) theory with fundamental representation 5 matter. The blue/yellow boxes correspond to decorating with ±. The green lines and red dots will play a role later on in understanding the flops between these phases: red dots correspond to extremal points that can be flopped, whereas white dots correspond to flops that would map out of the su(5) phases to the u(5) phases. Recall that a sign i = + means that Li > 0 is positive, and i = − means that Li < 0. There are precisely n+ 1 phases for the u(n) theory, which can be counted by the Weyl group quotient |Wsu(n+1)| |Wsu(
Figure 5. Weight diagrams for n even (LHS) and odd (RHS). The weights marked in red are the ones appearing in the sign constraints (3.9) and (3.11), which determine the su(n) phases. The examples drawn here are su(14) and su(13). The nodes represent the weights as explained in figure 3. These conditions are depicted in terms of red boxes in figure 5. Note that the signs “flow” as explained in (2.18), i.e. a consistent phase sign assignment will always respect the following sign implications (flow rules), (3.13) A phase for su(n) with the antisymmetric representation is characterized by a…
Figure 6. Example phase diagram, where blue are + and yellow
Figure 7. Phases of SU(5) with anti-symmetric representation, with ± corresponding to blue/yellow boxes, including the corresponding anti-Dyck paths (green lines) and extremal points (red nodes), which will be discussed in the section on flops. All phases satisfy the anti-Dyck property with respect to the diagonal defined by E5 in (3.11). The numbering is as in figure 1. 27
Figure 8. Setup for phases for antisymmetric and fundamental representation. The black
Figure 9. Phases of SU(5) with anti-symmetric (dark blue and yellow boxes) as in figure 7 and fundamental representation (light blue and yellow boxes) as in figure 4, including the corresponding anti-Dyck paths (green lines) and extremal points (red nodes). Note that the flops need to retain the anti-Dyck property with respect to the diagonal defined by E6 in (3.9), which would be violated if the white nodes would be flopped. The numbering is as in figures 4 and 7, and following the flops along the red nodes of the anti-Dyck path, we reproduce the phase diagram 11. 30
Figure 10. A flop is the transition between these two decorated box graphs (which are details
Figure 11. The phase diagram for SU(5) with fundamental and anti-symmetric matter, as derived from flops among the decorated box graphs/anti-Dyck paths in figure 9. Identification along the dark blue lines yields the phase diagram of the anti-symmetric representation, identification of nodes along the light blue lines yields the phase diagram, which is a line, for the fundamental representation. The curious observation which is drawn in terms of the green dotted lines is, that this diagram can be cut up into pairs of Dynkin diagrams of E6, D6 and A6, respectively. The case of the E6 flop…
Figure 12. SU(5) with fundamental and anti-symmetric representation, by attaching a (+ + + − −) fundamental phase (which is phase II) to phase 8 for the anti-symmetric repre￾sentation. In each diagram the red nodes corresponds to the extremal points, and we labeled the corresponding extremal weights by (i, j) = Li +Lj and (i) = Li . By joining the diagrams, to obtain the phase of the anti-symmetric and fundamental theory, L4 and L1 + L5 cease to be extremal and split. as follows −(L1 + L5) → −(L2 + L4) − L3 −L4 → −(L3 + L4) + L3 . (4.1) Let us remark in view of later discussions of the geometr
Figure 13. Induction step for su(n) with n = 2k even. Each box in this graph represents one of the weights in the lower diagonal of the weight diagram of the anti-symmetric representation. 5.1 SU(n) with antisymmetric representation We can determine the number of su(n) phases with antisymmetric representation by counting the complementary phases which violate the sign conditions (3.9, 3.11). Note that the number of phases with En = (+ · · · +) is the same as for En = (− · · · −). By following the flow rules (3.13) the number of phases for En = (+ · · · +), for instance, can be determined by a…
Figure 14. Example of two Dyck paths for k = 4. These are staircase paths, starting at (0, 0) and ending at (k, k) which do not cross, but can touch, the diagonal. To prove this, again consider the non-su(n) phases, which violate (3.11). First consider again the case with E2k+1 all +. The sign assignment is given in figure 15. Again we count the paths from s to pi , i = 1, · · · , n. In this case, however, there is a subtlety: starting with n and passing to n+ 2, we obtain the extension of the diagram as shown in figure 15. Most paths in the diagram for su(n) will again give rise to paths for…
Figure 15. Induction for n = 2k + 1. The condition that E2k+1 = (+ · · · +) is given by the black + and the blue +. In the induction step, indicated by the additional green boxes, the blue + is not required by the condition E2k+3 = (+ · · · +). n Phases of u(n) with Λ2n Phases of su(n) with Λ2n 5 16 8 6 32 12 7 64 34 8 128 58 9 256 144 10 512 260 11 1024 604 12 2048 1124 13 4096 2512 14 8192 4760 15 16384 10378
Figure 16. Counting of phases for the combined antisymmetric and fundamental representa
Figure 17. Representation graph for Λ36 of SU(6). The weights in each box are (ijk) = Li + Lj + Lk, and they are aligned so that the action of the simple roots corresponds to the edges of the diagram. The dotted lines indicate the action of the simple root L4 − L3, and the representation graph is really three-dimensional, with the two red boxes being on top of each other. The right hand side shows the flow rules, with + corresponding to blue and − to yellow. root space of e6 can be identified with the orthogonal complement of k = −3L0+L1+· · ·+L6. We can choose six independent bases in the…
Figure 18. Phases of the SU(6) × SU(2) theory with Λ36 of SU(6) and singlets ± P6 i=1 Li . The boxed phases are the SU(6) phases for which P6 i=1 Li is not fixed from the other weights, and thus can be either positive or negative, i.e. these diagrams appear twice in the phases. There are 72 box graphs, of which 60 are not invariant under the Z2 and are mapped into each other. This correspond to reflection along the central axis in the above diagram. 44
Figure 19. Phases of the SU(6) theory with Λ36 of SU(6), i.e., these satisfy in addition to the flow rules that the sum of all Li vanishes. In particular these are also the diagrams which are invariant under the Z2 automorphism. The green lines show the corresponding anti-Dyck paths. The 72 phases in terms of the box graphs are depicted in figure 18. The right half of figure 18 corresponds to phases where P6 i=1 Li > 0 and the left half to P6 i=1 Li < 0. One can clearly see a symmetry associated with the Weyl reflection from figure 18. The Weyl reflection Z2 of the SU(2) changes the weight Li…
Figure 20. Flop diagram of the SU(6) × SU(2) theory with Λ36 of SU(6). The top part depicts the flop diagram for the right half of figure 18 as well as the 6 phases which are labeled by 31-36, which correspond to the invariant diagrams at the bottom of figure 18. The mirrored numbers correspond to the left hand 36 box graphs. The Z2 automorphism acts by reflection along the horizontal axis. Nodes which overlap are connected, and each color indicates one layer. This flop graph is exactly the quasi-minuscule representation of E6 with the zero weights removed. 47
Figure 21. The fundamental representation of the Sp(n) theory. The only consistent phase of the Sp(n) theory is given by this coloring. Blue indicates that the decoration of the weight is plus, and yellow negative
Figure 22. The box graph of the Sp(3) theory with Λ26. Blue boxes indicate the sign of the corresponding weight is plus, yellow indicates negative, and orange that there is no way to associate a definite sign. Finally we can ask if there exists some consistent phase structure when considering sp(n) with both the V and Λ2V representations. This can be seen by the embedding of su(2n + 1) into so(4n + 4) with an intermediate embedding by su(2n + 2) like so(4n + 4) ⊃ su(2n + 2) ⊃ su(2n + 1). (6.9) The outer automorphism of su(2n + 2), which is again an element of the Weyl group of so(4n + 4)…
Figure 23. The phases of the Sp(3) theory with Λ36. Blue indicates the box is decorated with a plus sign and yellow with a negative. representation. The Γ0,0,1 representation is, up to multiplicity of the weights, identical to the reducible Λ36 representation, which arises under the quotient by the outer automorphism of the Λ36 representation of su(6). It is here where we first observe a non-trivial phase structure for the Sp series. The two phases of the SU(6) theory (figure 19) which consistently descend to the Sp(3) theory are depicted in figure 23. 7 Box Graphs and Elliptic Fibrations 7.1…
Figure 24. Surfaces for SU(5). The fiber above the blue locus on the base expresses the irreducible effective curves when including classes of curves in the fundamental representation. The red locus indicates the irreducible effective curves when the antisymmetric representation is included, with effective curves specified by phase 9, figure 26. for k > `. We can thus write the effective curves as a sequence C + ` + F1 + · · · + F`−1 C + ` + F2 + · · · + F`−1 . . . C + ` + F`−1 C + ` C − `+1 C − `+1 + F`+1 C − `+1 + F`+1 + F`+2 . . . C − `+1 + F`+1 + · · · + Fn−1, (7.7) and note that the…
Figure 25. Box graph for U(n) with fundamental representation. The arrows indicate the direction of the action of the simple roots, as they generate the phase. The green line is the “anti-Dyck path” in this case, which separates the blue/yellow, i.e., positive and negative, weights. This is the phase diagram corresponding to the effective curves in (7.7). 9 (2,3) (1,5) (2,4) L2-L3 L3-L4 L4-L5 L3-L4 L2-L3
Figure 26. Again, this is phase 9 of SU(5), with entries (i, j) in the boxes denoting the weights Li + Lj . The double-headed arrows indicate where crossing from + to − (blue to yellow), the corresponding roots, namely α1 = L1 − L2 and α3 = L3 − L4, split. The single arrows correspond to the action of the simple roots, that remain irreducible, as they generate the representation from the extremal weights, along the anti-Dyck path. curves are KU(n),n,phase l = {F1, · · · , Fl−1, C+ l , C− l+1, Fl+1, · · · , Fn−1} . (7.9) The splitting occurs at Fl = C + l + C − l+1. We can read off the Dyn
Figure 27. Discriminant in higher codimension. The weights associated with the extremal points correspond to the flopped curves. As we demonstrated the phases of the SU(5) gauge theory with the anti-symmetric representation and the fundamental representation in figure 11, each box graph corresponds to a resolution, and a single-box sign change among the box graphs or equivalently a flop of a corner of the anti-Dyck path corresponds to a flop transition between distinct resolutions. 8 Elliptic fibrations in codimensions two and three 8.1 Local models for fibers in codimension two Consider an…
Figure 28. Phases and Fiber types of the SU(6) theory with Λ36 of SU(6). In each box the left most diagram is the intersection graph of the fiber with multiplicities and the white node denoting the zero-section. The middle diagram is the fiber graph with the extremal rays, and on the left we show the decorated box graph, where the weights (ijk) that are explicitly labeled in the blue/yellow boxes correspond curves C ± ijk in the fibers, which together with the irreducible Cartans Fi are the extremal rays. Note that the Z2 quotient identifies these weights pairwise, which is shown also in the…
Figure 29. Example phase and fiber type of the U(6) theory with Λ36, corresponding to trivial monodromy action, with standard Kodaira IV ∗ fiber. Description of the various diagrams is as in figure 28. which means that F0 (which corresponds to L6 − L1) does not intersect this extremal ray transversally, but splits F0 → C − 145 + C + 456 . (8.17) The resulting additional curve, C + 456 is the component of the fiber that remains large in the singular limit, and thus corresponds to the zero section. In summary the new fiber types can be characterized by deleting one (non-extended) node in th
Figure 30. On the left hand side the fiber type and box graph of the
Figure 31. Codimension-three fiber degenerations along
Figure 32. Representation graph for the 32 spin representation of SO(12). The red boxes are layered on top of each other. The ith entry in boxes correspond to the signs of 1 2 Li . The representation graph is shown in figure 32. The Weyl group of su(2) acts as a sign change in the weights. Again, we can determine all the possible so(12) ⊕ su(2) fibers from the box graphs, and the number of phases is
Figure 33. Box graphs and fiber types for Z2 monodromy-reduced E7, obtained from SO(12) with 32 representation. The fibers are obtained by deleting a single node from the Kodaira type III∗ fiber, and are connected by the flops indicated by the lines connecting the different phases. The phase diagram is the Dynkin diagram of E7. 73
Figure 34. The intersection diagram of the fiber above a codimension-three locus for the
Figure 35. Dynkin diagram for En, establishing our notation, with numbers i labeling the simple roots αi . (A.16). Then U(1)r ⊂ Sp(r) is a maximal torus. The roots are L` with weight space spanned by z` , and −L` with weight space spanned by w` . The complex dimension of the fundamental representation is 2r. The adjoint representation of Sp(r) is the second symmetric power of the fundamental representation, so the roots are given by the nonzero sums of pairs of weights of the fun￾damental representation, not necessarily distinct. We obtain zero as an adjoint weight via L` + (−L`) = 0, ` = 1,…
Figure 36. The standard representation of SO(2n). ...
Figure 38. The phases of the SO(2n) theory. Blue indicates the box is decorated with a plus sign and yellow with a negative. In fact the weights of the representation of SO(2n) × U(1) are w i , i = 1, · · · , 2n, where the u(1) generator is given by identifying w i with −w n+i U(1) : X 2n i=1 w i − w n+i . (C.6) One can present the representation V as in figure 36. Similarly to section 2.4 there exist flow rules which determine whether the decorated box graph gives a consistent phase. For SO(2n) × U(1) these are + ← + ↑ + − → − ↓ − (C.7) One can do some combinatorics with the fl
Figure 39. The undecorated box graph of E6 with the 27 representation. (ij) denotes the weight L0 − Li − Lj , and (i) the weight 2L0 − P5 k=1 Ljk where jk 6= i are distinct. flop diagrams. The examples we consider are e7 → e6 ⊕ u(1) 133 → 780 ⊕ 10 ⊕ 272 ⊕ 27−2 e8 → e7 ⊕ su(2) 248 → (133, 1) ⊕ (3, 1) ⊕ (56, 2) (D.1) D.1 Phases of E6 with 27 matter The phases of the E6 theory with 27 has the standard flow rules given in (3.13) acting on the representation graph in figure 39. The weights of the representation shown there are in the standard representation in terms of Li is given in appendix A.4.…
Figure 40. Phases in terms of decorated box graphs of the
Figure 41. Phase diagram of the e6 ⊕ u(1) theory with matter in the 27. The flop diagram agrees with the representation graph of the 56 of E7. The nodes colored in blue are the phases of the e6 theory. The numbering corresponds to the decorated box graphs in figure 40, by considering each column in that diagram, read from left to right, top to bottom. 95
Figure 42. The undecorated box graph for E7 with 56. (ij) denotes weight L0 − Li − Lj , (ijklm) denotes weight L0−Li−Lj−Lk−Ll−Lm, and (i) denotes weight 3L0− P5 k=1 Ljk−2Li , where the jk 6= i are distinct. The flow rules are as in (3.13) and between layers +, resp. −, flows in the direction of the blue, resp. yellow, arrows. with corresponding multiplicities {2, 3, 4, 5, 6, 3, 4}. The box graph tells us the following in￾tersections of the curve C + 12345 with the roots C + 12345 · F1 = 0, C+ 12345 · F2 = 0, C+ 12345 · F3 = 0, C + 12345 · F4 = 0, C+ 12345 · F5 = +1, C+ 12345 · F6 = 0
Figure 43. Phases in terms of decorated box graphs of the
Figure 44. Monodromy-reduced fibers for E8, we shown the intersection graph including multiplicities for each of the decorated box graphs. Note the the fibers correspond to type II∗ Kodaira fibers, with one non-affine node deleted. The lines connecting black boxes correspond to flop transitions, and form a (non-affine) E8 Dynkin diagram. 101

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  • Said that its own intersection diagrams for incomplete resolutions in its Figures 5, 7 and 8 are the ones at the bottom, third from the bottom and second from the top of the left column of Figure 33 of this paper.

    Half-hypermultiplets and incomplete/complete resolutions in F-theory (2020) · 1.00 · paraphrased: the licence does not allow quoting

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69 arXiv papers cite this in 129 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

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  • We can see that the intersection diagrams for incomplete resolutions shown in Figures 5, 7 and 8 of the present paper are the ones at the bottom, the third from the bottom and the second from the top of the left column of Figure 33 of [ref] .

    Half-hypermultiplets and incomplete/complete resolutions in F-theory · Comparison with the results of the M-theory Coulomb branch analysis · 1.00

  • For these, we showed that the “usual” fiber analysis in spirit of [ref] matches previous toric constructions of 5d SCFTs [ref] .

    Phases of 5d SCFTs from M-/F-theory on Non-Flat Fibrations · Summary and Outlook · 0.89

  • The enumeration of chambers of $$ announced in thm:chamberStructureassert:chamberBiject recovers the count of phases of $U(n)$ with fundamental and antisymmetric matter from the physics literature [ref] .

    Incidence Geometry in a Weyl Chamber I: $GL_n$ · Structure of chambers, faces, flats, and extreme rays of · 0.92

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  • If the Weierstrass coefficients $f$ and $g$ have multiplicities at $P_μ$ which exceed $4$ for $f$ and $6$ for $g$, then there is no desingularization of the Weierstrass model which is flat [ref] , which means that there would be tensionless strings in the low-energy theory.

    Codimension-Three Bundle Singularities in F-Theory · Local models for fibers in codimension three · 0.96

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