Dimer models from mirror symmetry and quivering amoebae

By Bo Feng, Yang-Hui He, Kristian D. Kennaway, C. Vafa. 2005

319 citations recorded by the source corpus.

Dimer models are 2-dimensional combinatorial systems that have been shown to encode the gauge groups, matter content and tree-level superpotential of the world-volume quiver gauge theories obtained by placing D3-branes at the tip of a singular toric Calabi-Yau cone. In particular the dimer graph is dual to the quiver graph. However, the string theoretic explanation of this was unclear. In this paper we use mirror symmetry to shed light on this: the dimer models live on a T^2 subspace of the T^3 fiber that is involved in mirror symmetry and is wrapped by D6-branes. These D6-branes are mirror to the D3-branes at the singular point, and geometrically encode the same quiver theory on their world-volume.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ac38f5996b9a18b3cf2a3fbb8846a2d341778b08

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Astronomy and Astrophysics

In Nuclear and High Energy Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Brane tilings and their applications (2008, methodology): “As explained in §4.8, this corresponds to the 3-cycles of D6-branes projected onto the Riemann surface W (x, y) = 0, or in fivebrane setup, the boundary Ca of D5-brane discs Da in §3.3.2.”
Chiral flavors and M2-branes at toric CY4 singularities (2009, methodology): “In case the CS-matter quiver theory is a brane tiling [53, 54, 55, 49], this branch of the moduli space is a toric CY4.”
A simple class of N = 3 gauge/gravity duals (2008, background): “The field theories can be engineered from a brane configuration; their geometry is summarized by a “hyperKähler toric fan” that can be read off easily from the relative angles of the branes.”
N=1 theories of class S_k (2015, background): “A classical example is the class of theories associated to dimer models [1, 2, 3, 4], labelled by bi-partite graphs drawn on a torus and a choice of rank N .”
Non-commutative Donaldson-Thomas theory and the conifold (2007, background): “hem to enumerative invariants of non-commutative Calabi–Yau geometries, seems to be new. It would be interesting to relate this idea to the other direct derivation of dimer models from toric geometry [8], based on mirror symmetry and special Lagrangian geometry. Section 1 studies quiver algebras deﬁ”
Counting BPS operators in gauge theories: quivers, syzygies and plethystics (2006, background): “The 3 here, of course, correspond to the (5)Since we are studying BPS mesons, we may emphasize the relationship between the concept of the “extremal BPS meson” (introduced in [28]) and that of the “zig-zag path” in [21, 22, 24].”
Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space (2011, unknown): “It appeared in [28] (with the name 'alga'), where it was used as motivation to construct Landau-Ginzburg mirrors to some toric surfaces.”
Phases Of N=2 Theories In 1+1 Dimensions With Boundary (2008, background): “The image of S under the map Arg is called the co-Amoeba or the Alga of S [60, 61].”
On the homological mirror symmetry conjecture for pairs of pants (2010, methodology): “It appeared in [16] (with the name ‘alga’), where it was used as motivation to construct LandauGinzburg mirrors to some toric surfaces.”
$\mathcal{N} = 2$ quantum field theories and their BPS quivers (2011, methodology): “• The toric methods of [46, 47] and the relation to dimer models [48] were used in [49] to construct a large class of quivers, their construction using mirror symmetry was studied in [50].”