^{1,2}

^{3,4,5}

^{1}

^{2}

^{3}

^{4}

^{5}

We present the complete classification of smooth toric Fano threefolds, known to the algebraic geometry literature, and perform some preliminary analyses in the context of brane tilings and Chern-Simons theory on M2-branes probing Calabi-Yau fourfold singularities. We emphasise that these 18 spaces should be as intensely studied as their well-known counterparts: the del Pezzo surfaces.

A flurry of activity has, since the initial work of Bagger and Lambert [

Even though analogies with the case of D3-branes in Type IIB, whose world-volume theory is a

Brane probes and associated world-volume physics in various backgrounds.

Brane probe | Theory | Background | World-volume theory | Vacuum moduli space |
---|---|---|---|---|

D5 | Type IIB | (5+1)-d | CY2 | |

D3 | Type IIB | (3+1)-d | CY3 | |

M2 | M-theory | (2+1)-d | CY4 |

A crucial feature for all the brane embeddings in Table

It is perhaps naïvely natural to propose three-dimensional tilings for the case of M2-branes probing CY4, but in fact, it turns out not to be as useful as it may initially seem. These three-dimensional tilings have been nicely advocated in the crystal model [

We are thus led, for now, to keep on the path of two-dimensional tilings, while bearing in mind that the data needed to specify a QCS theory is given by gauge groups, matter fields, and interactions, as well as the additional data of the CS levels for the gauge groups. These nicely map, respectively, to tiles, edges, and nodes, while the corresponding CS levels are given by fluxes on the tiles. It would be interesting to check if this correspondence between tilings in one and two dimensions, that is, for toric Calabi-Yau

The cases for Calabi-Yau two- and threefolds are well established over the past decade. These are affine complex cones over base complex curves and surfaces, or real cones over real, compact, Sasaki-Einstein three and five manifolds. Perhaps the most extensively studied are, inspired by phenomenological concerns, D3-branes and Calabi-Yau threefolds and the widest class studied therein is

Another crucial family of Calabi-Yau threefold cones affords a clear construction, and the world-volume physics has been intensely investigated (cf., e.g., [

Indeed, all toric gauge theories in

It is therefore a natural and important question to ask what are the corresponding geometries for Calabi-Yau fourfolds and physically what are the associated

It is the purpose of the current short note, a prologue to [

Fano varieties are of obvious importance; these are varieties which admit an ample anticanonical sheaf; thus, whereas Calabi-Yau varieties are of zero curvature, they are of positive curvature. (Recently, lower bounds on the Ricci curvature of Fano manifolds have been found [

What are explicit examples of Fano varieties? In complex dimension one, there is only

We point out that, of course, the aforementioned are

Our chief interest lies in the situation of dimension three. These Fano threefolds can give rise to Calabi-Yau fourfolds which can then be probed by M2-branes in order to arrive at quiver Chern-Simons (QCS) theories on their world volume. A classification of the Fano vareities was achieved in the 80s [

With the rapid advance of computer algebra and algorithmic algebraic geometry, especially in applications to physics (cf. [

Given the enormity of the number, we were to allow singularities—against which, physically, there need be no prejudice—and being inspired by the 2-fold case of the del Pezzo surfaces all being smooth, we will henceforth restrict our attention to the

The 18 smooth toric Fano threefolds. For full explanation of notation, see the second paragraph of Section

Id of [ | Geometry | |||
---|---|---|---|---|

4 | ||||

35 | ||||

36 | ||||

37 | ||||

24 | ||||

105 | ||||

136 | ||||

62 | ||||

123 | ||||

68 | ||||

131 | ||||

139 | ||||

218 | ||||

275 | ||||

266 | ||||

271 | ||||

324 | ||||

369 |

Some detailed explanation of the nomenclature in Table

Indeed, our interest in (compact) Fano threefolds is that the complex cone thereupon is an (noncompact) affine Calabi-Yau fourfold which M2-branes may probe. Going form the data in the table to the fourfold is simple; we only need to add one more dimension, say, a row of 1s to each of the matrices. In such cases, the geometry will be cones over what is reported in the third column. In the physics literature, there have been several cases which have been studied in considerable depth and detail: the cone over

We, of course, recognise

Therefore, the cone in a sense undoes the said projectivisation, and the fourfold is simply the total space of the fibration. For example,

One piece of information, obviously of great importance, is the symmetry of the variety, which is encoded in the world-volume physics, either manifestly or as hidden global symmetries [

Note that the rank of the group of symmetries must total to 4 because we are dealing with a toric (affine) Calabi-Yau 4-fold. Indeed, one

We note that the three cases of there being only a

We have also listed, to the rightmost of the table, some geometrical data, such as topological invariants. In particular, we tabulate the second Betti number

However, in our present case of M2-branes probing the Calabi-Yau fourfold, the world-volume Chern-Simons theory in

On the other hand, a conserved baryonic charge corresponds to a gauge field in AdS. This is counted by the number of 2-cycles in the Sasaki-Einstein 7-fold (SE7), given by the 3-form on each 2-cycle. The number of 2-cycles in the SE7 is equal to the number of 5-cycles by Poincaré duality, which is in turn equal to the number

Next, let us discuss the genus

Now, it was first pointed out in [

The unrefined Hilbert series, computed for the canonical embedding stated above, is also presented in [

In the special cases where the Fano threefold

With a current want of an inverse algorithm, with or without the aid of dimer technology, it is difficult to systematically find the requisite quiver Chern-Simons theories whose moduli spaces are Calabi-Yau cones over the Fano threefolds listed above, a question certainly of considerable interest. Nevertheless, because the forward algorithm is now well established [

In accordance with the notation of [

Furthermore, as always, we let

The quiver and superpotential can be readily recalled from, for example, [

Now, take

Next, we recall the well-known two phases of the

From these progenitors, we can obtain quite a few Calabi-Yau fourfold cones with judicious choices of CS levels. We list these in Table

The two phases of the

In this table, we have used the notation

The theory for the cone over the

In this short note, a prelude to [

These 18 spaces are direct analogues of the toric del Pezzo surfaces, which have been the subject of much investigation in the past decade in association with the construction of

For some of these we have identified, using the forward algorithm, the quiver theories whose mesonic moduli spaces are precisely as desired. Such a

.

Scientiae et Technologiae Concilio Anglicae, et Ricardo Fitzjames, Episcopo Londiniensis, ceterisque omnibus benefactoribus Collegii Mertonensis Oxoniensis, sed super omnes, pro amore Catharinae Sanctae Alexandriae, lacrimarum Mariae semper Virginis, et ad Maiorem Dei Gloriam hoc opusculum Y.-H. He dedicat. The authors are indebted to John Davey, Kentaro Hori, Noppadol Mekareeya, Richard Thomas, Giuseppe Torri, and Alberto Zaffaroni for enlightening discussions. A. Hanany would like to thank the kind hospitality, during the initiation of this project, of IPMU in Tokyo and is further grateful to the University of Richmond, the Perimeter Institute as well as the KITP in Santa Barbara, during the completion. This research was supported in part by the National Science Foundation under Grant no. PHY05-51164.

_{4}/CFT

_{3}

_{4}/CFT

_{3}

^{1,1,1}

_{4}/CFT

_{3}duals and M-theory crystals

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