Brane Tilings and Their Applications

TL;DR

Brane Tilings and Their Applications surveys how dimer models encode 4d N=1 quiver gauge theories arising from D3-branes at toric Calabi–Yau cones, recasting the data in a physical fivebrane system of D5 and NS5 branes. It develops foundations (quivers, anomalies, toric geometry) and builds to brane tilings as a unifying framework, including strong/weak coupling limits, untwisting, and the Kasteleyn/perfect-matching machinery. The second half explores AdS/CFT with Sasaki–Einstein manifolds, a-maximization, volume minimization, and the mirror D6 picture, linking field-theoretic central charges to geometric volumes and establishing toric dualities via Seiberg duality and marginal deformations. Overall, the work demonstrates how brane tilings provide a concrete, computable bridge between toric geometry, gauge theories, and holographic duals, with wide implications for both mathematics and string phenomenology.

Abstract

We review recent developments in the theory of brane tilings and four-dimensional N=1 supersymmetric quiver gauge theories. This review consists of two parts. In part I, we describe foundations of brane tilings, emphasizing the physical interpretation of brane tilings as fivebrane systems. In part II, we discuss application of brane tilings to AdS/CFT correspondence and homological mirror symmetry. More topics, such as orientifold of brane tilings, phenomenological model building, similarities with BPS solitons in supersymmetric gauge theories, are also briefly discussed. This paper is a revised version of the author's master's thesis submitted to Department of Physics, Faculty of Science, the University of Tokyo on January 2008, and is based on his several papers: math.AG/0605780, math.AG/0606548, hep-th/0702049, math.AG/0703267, arXiv:0801.3528 and some works in progress.

Figures (88)

• Figure 1: Example of a bipartite graph is shown in (a), which has three black/white vertices and seven edges. In this example, you can superimpose (a) with $2\times 3$ boxes (b), to obtain (c).
• Figure 2: Here we show an example of perfect matchings on the bipartite graph of Figure \ref{['bipartite']}. The problem is to count the number of perfect matchings, and you can easily verify that the answer is three, as shown in (b). Alternatively, you can see this problem as the tiling problem of the dual graph (in this case, $2\times 3$ boxes) by 'dominoes'. as shown in (b) and (c): we consider the tiling of $2\times 3$ regions ( Figure \ref{['bipartite']} (b)) using two types of dominoes (d) as basic constituents.
• Figure 3: Young diagram (a) is also represented as Maya diagram (b). Maya diagram, in our language, is in one-to-one correspondence with the choice of subset of edges of the bipartite graph shown in (c). In this sense enumeration of Young diagrams is equivalent to enumeration of such subsets of edges of bipartite graph (c), which is essentially one-dimensional version of the dimer model.
• Figure 4: This figure shows an example of three-dimensional version of Young diagram (a). If you rotate (a) by 180 degrees, we have (b), which looks like melting of a crystal. By projecting this figure onto two-dimensions, we have a perfect matching of a bipartite graph defined on honeycomb bipartite graph (c), or equivalently tiling of plane using three types of rhombi shown in (d) (this is an analogue of "domino tiling" in Figure \ref{['PM']}). This one-to-one correspondence between three-dimensional Young diagram and perfect matching in dimer model is a higher-dimensional generalization of more familiar correspondence shown in Figure \ref{['2dYoung']}. The interesting fact is that this type of three-dimensional Young diagram appears in string theory, in the "melting crystal" picture of Okounkov:2003sp.
• Figure 5: Example of brane tilings, or bipartite graphs on $\mathbb{T}^2$. The region represents fundamental region of torus. It suffices to write graphs only in the fundamental region, but sometimes it is convenient to write the graph as a periodic tiling of two-dimensional plane, which is the reason for the name "brane tiling". The left figure is the bipartite graph for $\mathbb{Z}_3$-orbifold of $\mathbb{C}^3$, and the right for canonical bundle over $\mathbb{P}^1\times \mathbb{P}^1$, which is denoted $K_{\mathbb{P}^1\times \mathbb{P}^1}$.

Theorems & Definitions (22)

• Theorem 3.1
• Theorem 4.1
• Theorem 4.2
• Example 4.1: SPP
• Example 4.2: Generalized Conifolds
• Example 4.3: conifold
• Example 4.4: SPP
• Example 4.5
• Example 4.6
• Example 5.1: $T^{1,1}$